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[
"POSITIVE GREEN'S FUNCTIONS FOR SOME FRACTIONAL-ORDER BOUNDARY VALUE PROBLEMS",
"POSITIVE GREEN'S FUNCTIONS FOR SOME FRACTIONAL-ORDER BOUNDARY VALUE PROBLEMS"
]
| [
"Douglas R Anderson "
]
| []
| []
| We use the newly introduced conformable fractional derivative, which is different from the Caputo and Riemann-Liouville fractional derivatives, to reformulate several common boundary value problems, including those with conjugate, right-focal, and Lidstone conditions. With the fractional differential equation and fractional boundary conditions established, we find the corresponding Green's functions and prove their positivity under appropriate assumptions.2010 Mathematics Subject Classification. 26A33. | null | [
"https://arxiv.org/pdf/1411.5616v1.pdf"
]
| 117,052,300 | 1411.5616 | c55ce49fcc81be16a6c27214afeb52bd37eb933a |
POSITIVE GREEN'S FUNCTIONS FOR SOME FRACTIONAL-ORDER BOUNDARY VALUE PROBLEMS
20 Nov 2014
Douglas R Anderson
POSITIVE GREEN'S FUNCTIONS FOR SOME FRACTIONAL-ORDER BOUNDARY VALUE PROBLEMS
20 Nov 2014
We use the newly introduced conformable fractional derivative, which is different from the Caputo and Riemann-Liouville fractional derivatives, to reformulate several common boundary value problems, including those with conjugate, right-focal, and Lidstone conditions. With the fractional differential equation and fractional boundary conditions established, we find the corresponding Green's functions and prove their positivity under appropriate assumptions.2010 Mathematics Subject Classification. 26A33.
Introduction
The search for the existence of positive solutions and multiple positive solutions to nonlinear fractional boundary value problems has expanded greatly over the past decade; for some recent examples please see [1][2][3][4][5][6][7]9,[12][13][14][15][16]. In all of these works and the references cited therein, however, the definition of the fractional derivative used is either the Caputo or the Riemann-Liouville fractional derivative, involving an integral expression and the gamma function. Recently [8,9,12] a new definition has been formulated and dubbed the conformable fractional derivative. In this paper, we use this fractional derivative of order α, given by
D α f (t) := lim ε→0 f (te εt −α ) − f (t)
ε , D α f (0) = lim t→0 + D α f (t); (1.1) note that if f is differentiable, then
D α f (t) = t 1−α f ′ (t),(1.2)
where f ′ (t) = lim ε→0 [f (t + ε) − f (t)]/ε. Using this new definition of the fractional derivative, we reformulate several common boundary value problems, including those with conjugate, right-focal, and Lidstone conditions. With the fractional differential equation and fractional boundary conditions established, we find the corresponding Green's functions and prove their positivity under appropriate assumptions. This work thus sets the stage for using fixed point theorems to prove the existence of positive and multiple positive solutions to nonlinear fractional problems based on the conformable fractional derivative and these boundary value problems, as the kernel of the integral operator is often Green's function.
Two Iterated Fractional Derivatives
We begin by considering two iterated fractional derivatives in the differential operator, together with two-point boundary conditions, as illustrated in the nonlinear boundary value problem
−D β D α x(t) = f (t, x(t)), 0 ≤ t ≤ 1, (2.1) γx(0) − δD α x(0) = 0 = ηx(1) + ζD α x(1), (2.2)
where α, β ∈ (0, 1] and the derivatives are conformable fractional derivatives (1.1), with γ, δ, η, ζ ≥ 0 and d := ηδ + γζ + γη/α > 0. To verify the existence of positive solutions to this problem, one often finds the corresponding Green's function, shows that it is non-negative, and then uses a fixed point theorem on an integral operator whose kernel is Green's function. Throughout this and subsequent sections, we will state the fractional boundary value problem, find Green's function for it, and prove that it is positive except at various boundary points.
G(t, s) = 1 d δ + γ α t α ζ + η α (1 − s α ) : t ≤ s 1 d δ + γ α s α ζ + η α (1 − t α ) : s ≤ t (2.3)
where we assume with γ, δ, η, ζ ≥ 0 and d = ηδ + γζ + γη/α > 0.
Proof. Let h be any continuous function. We will show that
x(t) = 1 0 G(t, s)h(s)s β−1 ds,
for G given by (2.3), is a solution to the linear boundary value problem
−D β D α x(t) = h(t)
with boundary conditions (2.2).
For any t ∈ [0, 1], using the branches of (2.3) we have
x(t) = 1 d ζ + η α (1 − t α ) t 0 δ + γ α s α h(s)s β−1 ds + 1 d δ + γ α t α 1 t ζ + η α (1 − s α ) h(s)s β−1 ds.
Taking the α-fractional derivative yields
D α x(t) = − η d t 0 δ + γ α s α h(s)s β−1 ds + γ d 1 t ζ + η α (1 − s α ) h(s)s β−1 ds.
Checking the first boundary condition, we see that
γx(0) − δD α x(0) = 0.
Moreover, in checking the second boundary condition we get
ηx(1) + ζD α x(1) = 0.
Taking the β-fractional derivative of the α-fractional derivative yields
D β D α x(t) = − η d δ + γ α t α h(t)t β−1 t 1−β − γ d ζ + η α (1 − t α ) h(t)t β−1 t 1−β = − 1 d h(t) ηδ + γζ + γη α = −h(t),
which is what we set out to prove.
G(t, s) = 1 α t α (1 − s α ) : t ≤ s, 1 α s α (1 − t α ) : s ≤ t,(2.
4)
and the corresponding Green's function for the homogeneous problem
−D β D α x(t) = 0
satisfying the right-focal-type boundary conditions x(0) = D α x(1) = 0 is given by
G(t, s) = 1 α t α : t ≤ s, 1 α s α : s ≤ t.
(2.5)
Remark 2.1. Note that the fractional Green's function given above for the conjugate boundary conditions in (2.4) differs from that found for example in Bai and Lü [1].
Theorem 2.2. For G(t, s) given in (2.3), we have that g(t)G(s, s) < G(t, s) ≤ G(s, s) (2.6) for t, s ∈ [0, 1], where g(t) := min αδ + γt α αδ + γ , αζ + η (1 − t α ) αζ + η . (2.7)
Proof. It is straightforward to see that
G(t, s) G(s, s) = δ + γ α t α δ + γ α s α , t ≤ s, ζ + η α (1 − t α ) ζ + η α (1 − s α ) , s ≤ t;
this expression yields both inequalities in (2.6) for g as in (2.7).
Remark 2.2.
It is also common to skip the Green's function representation and to express solutions directly. For instance, let α, β ∈ (0, 1]. The motivated reader can verify that the fractional boundary value problem
−D β D α x(t) = h(t), t ∈ (0, 1),
satisfying the three-point boundary conditions
x(0) = 0, δx(η) = x(1),
where h is a continuous function, η ∈ (0, 1) and 0 < δη α < 1, is given by
x(t) = −1 α t 0 (t α − s α ) h(s)s β−1 ds − δt α α (1 − δη α ) η 0 (η α − s α ) h(s)s β−1 ds + t α α (1 − δη α ) 1 0 (1 − s α ) h(s)s β−1 ds.
Three Iterated Fractional Derivatives
Next we consider the three iterated fractional derivative nonlinear right-focal problem
D γ D β D α x(t) = f (t, x(t)), 0 ≤ t ≤ 1, (3.1) x(0) = D α x(τ ) = D β D α x(1) = 0, (3.2)
where α, β, γ ∈ (0, 1] with 0 < τ < 1. One could impose further the conditions α + β ∈ (1, 2] and α + β + γ ∈ (2, 3] if one wishes to explore a fractional problem of order (2,3]. Our approach to the existence of positive solutions would again involve Green's function for this problem. Once the following three theorems are established, an interested reader could then apply a fixed point theorem to get positive solutions to (3.1), (3.2), although the details are omitted here.
Theorem 3.1 (Fractional Right-Focal Problem). Let α, β, γ ∈ (0, 1] and 0 < τ < 1.
The corresponding Green's function for the homogeneous problem
D γ D β D α x(t) = 0
satisfying boundary conditions (3.2) is given by
G(t, s) = s ∈ [0, τ ] : u(t, s) : t ≤ s x(0, s) : s ≤ t s ∈ [τ, 1] : u(t, τ ) : t ≤ s u(t, τ ) + x(t, s) : s ≤ t (3.3) where u(t, s) = (α + β)t α s β − αt α+β αβ(α + β) (3.4)
and x(·, ·) is the Cauchy function given
x(t, s) = αt α t β − s β + βs β (s α − t α ) αβ(α + β) . (3.5)
Proof. Let h be any continuous function. We will show that
x(t) = 1 0 G(t, s)h(s)s γ−1 ds,
for G given by (3.3), is a solution to the linear boundary value problem
D γ D β D α x(t) = h(t)
with boundary conditions (3.2).
First let t ∈ [0, τ ]. Then
D α x(t) = x(0, t)h(t)t γ−1 t 1−α + τ t s β − t β β h(s)s γ−1 ds −u(t, t)h(t)t γ−1 t 1−α + 1 τ τ β − t β β h(s)s γ−1 ds = 1 β τ t s β − t β h(s)s γ−1 ds + 1 β 1 τ τ β − t β h(s)s γ−1 ds.
Clearly the second boundary condition D α x(τ ) = 0 is met. Differentiating again, we have
D β D α x(t) = 1 β τ t −βt β−1 t 1−β h(s)s γ−1 ds + 1 β 1 τ −βt β−1 t 1−β h(s)s γ−1 ds = t 1 h(s)s γ−1 ds. It follows that D β D α x(1) = 0 and D γ D β D α x(t) = h(t)
, proving the claim in this case.
Next let t ∈ [τ, 1]. Then
x(t) = τ 0 x(0, s)h(s)s γ−1 ds + t τ x(t, s)h(s)s γ−1 ds + u(t, τ ) 1 τ h(s)s γ−1 ds; (3.7) again x(0) = 0 by (3.4). Differentiating (3.7), we have D α x(t) = 1 β t τ t β − s β h(s)s γ−1 ds + τ β − t β β 1 τ h(s)s γ−1 ds.
The second boundary condition D α x(τ ) = 0 is clearly met. Differentiating again, we have
D β D α x(t) =d dt u(t, s) = t α−1 (s β − t β ) β ≥ 0, s ≥ t.
Moreover, from (3.5) we have x(s, s) = 0 and
d dt x(t, s) = t α−1 (t β − s β ) β ≥ 0, t ≥ s,
so that G(t, s) is non-decreasing on [0, τ ] and non-increasing on [τ, 1]. Thus (3.8) will hold if G(1, τ ) > 0 holds, which occurs if u(1, τ ) > 0. This is equivalent to (3.9), completing the proof.
g(t)G(τ, s) ≤ G(t, s) ≤ G(τ, s) (3.10) where g(t) := min t α βτ α+β (α + β)τ β − αt β , 1 − t 1 − τ . (3.11)
Proof. From the preceeding theorem, we have G(t, s) ≤ G(τ, s) for all t, s ∈ [0, 1]. For the lower bound, we proceed by cases on the branches of the Green's function (3.3), that is we use (3.4) and (3.5).
(i) 0 ≤ t ≤ s ≤ τ : Here G(t, s) = u(t, s), G(τ, s) = x(0, s) = 1 α(α+β) s α+β . For these t, s we have u(t, s) x(0, s) ≥ u(t, τ ) x(0, τ ) ≥ t α βτ α+β (α + β)τ β − αt β which implies G(t, s) ≥ t α βτ α+β (α + β)τ β − αt β G(τ, s).
(ii) 0 ≤ t ≤ τ ≤ s ≤ 1: In this case G(t, s) = u(t, τ ) and G(τ, s) = u(τ, τ ), so again we have
G(t, s) ≥ t α βτ α+β (α + β)τ β − αt β G(τ, s). (iii) 0 ≤ s ≤ t ≤ τ or 0 ≤ s ≤ τ ≤ t ≤ 1: Since G(t, s) = G(τ, s) = 1 α(α+β) s α+β , it follows that G(t, s) = G(τ, s).
(iv) τ ≤ t ≤ s ≤ 1: As in (ii), G(t, s) = u(t, τ ) and G(τ, s) = u(τ, τ ). Define
w(t) := u(t, τ ) − 1 − t 1 − τ u(τ, τ ) (3.12) = G(t, s) − 1 − t 1 − τ G(τ, s).
Now w(τ ) = 0, w ′ (τ ) > 0, and w(1) = G(1, s) > 0 by (3.9). Since w is concave down, w(t) ≥ 0 on [τ, 1], hence
G(t, s) ≥ 1 − t 1 − τ G(τ, s).
(v) τ ≤ s ≤ t ≤ 1: Note that G(τ, s) = u(τ, τ ), while G(t, s) = u(t, τ ) + x(t, s) ≥ u(t, τ ); consequently, the employment of w as in (3.12) yields
G(t, s) ≥ 1 − t 1 − τ G(τ, s).
Four Iterated Fractional Derivatives
In this final section we consider four iterated fractional derivatives in the differential operator, with two types of boundary conditions. First, consider the nonlinear two-point cantilever beam eigenvalue problem
D δ D γ D β D α x(t) = λa(t)f (x), 0 ≤ t ≤ 1, (4.1) x(0) = D α x(0) = D β D α x(1) = D γ D β D α x(1) = 0, (4.2)
where α, β, γ, δ ∈ (0, 1], and with α + β ∈ (1, 2], α + β + γ ∈ (2, 3], and α + β + γ + δ ∈ (3,4] if one wishes to explore this problem as a fractional order between 3 and 4. The theme throughout this work has been to approach the existence of positive solutions to (4.1), (4.2) by involving Green's function for this problem. To obtain symmetry in Green's function below we must take γ = α in the fractional differential equation, but we will maintain the more general form (γ not necessarily equal to α) in the proofs to follow.
D δ D γ D β D α x(t) = 0 (4.3)
satisfying boundary conditions (4.2) is given by
G(t, s) = t α+β γ s γ β(α + β) − t γ (β + γ)(α + β + γ) : t ≤ s, s β+γ α t α β(β + γ) − s α (α + β)(α + β + γ)
: s ≤ t. Proof. Let h be any continuous function. We will show that
x(t) = 1 0 G(t, s)h(s)s δ−1 ds,
for G given by (4.4), is a solution to the linear boundary value problem
D δ D γ D β D α x(t) = h(t)
with boundary conditions (4.2).
For any t ∈ [0, 1], using the branches of (4.4) we have
x(t) = t 0 s β+γ α t α β(β + γ) − s α (α + β)(α + β + γ) h(s)s δ−1 ds − t α+β γ t 1 s γ β(α + β) − t γ (β + γ)(α + β + γ) h(s)s δ−1 ds;
clearly x(0) = 0. Taking the α-fractional derivative yields
D α x(t) = 1 β(β + γ) t 0 s β+γ+δ−1 h(s)ds − t β βγ t 1 s γ+δ−1 h(s)ds + t β+γ γ(β + γ) t 1 s δ−1 h(s)ds.
It is easy to see that D α x(0) = 0. Taking the β-fractional derivative of the α-fractional derivative yields
D β D α x(t) = 1 γ t 1 (t γ − s γ ) s δ−1 h(s)ds, so that D β D α x(1) = 0. Next, D γ D β D α x(t) = t 1 s δ−1 h(s)ds,
from which we have D γ D β D α x(1) = 0, and
D δ D γ D β D α x(t) = h(t).
This finishes the proof. Proof. Note that
d dt G(t, s) = t α+β−1 βγ(β+γ) [γs γ + β (s γ − t γ )] : t ≤ s,s β+γ t α−1 β(β+γ)
: s ≤ t. Finally, we will end the present discussion by considering another common set of boundary conditions, namely the so-called Lidstone conditions. Unlike the argument used below, one could also approach this problem as the conjunction of two conjugate problems, whose Green's function is given in (2.4).
D β D α D β D α x(t) = 0 (4.6)
satisfying the Lidstone-type boundary conditions
x(0) = 0 = D β D α x(0), x(1) = 0 = D β D α x(1) (4.7)
is given by
G(t, s) = u(t, s) : t ≤ s, u(s, t) : s ≤ t, (4.8) where u(t, s) = t α αβ(α + β)(2α + β) 2αs α 1 − s β − β (1 − s α ) t α+β + s α+β . (4.9)
Proof. In this proof we construct Green's function from scratch, modifying the classical approach, found for example in [11,Chapter 5]. Let x(t, s) be the Cauchy function associated with (4.3), namely a function satisfying
x(s, s) = D α x(s, s) = D β D α x(s, s) = 0, D γ D β D α x(t, s) = 1.
(Note that we will use γ for now, and take γ = α for symmetry purposes at the conclusion.) Using (1.2) at each step, it is easy to verify that here the Cauchy function is given by
x(t, s) = 1 γ t s τ s (ξ γ − s γ ) ξ β−1 τ α−1 dξdτ,(4.10)
and Green's function takes the form of
G(t, s) = u(t, s) : t ≤ s, u(t, s) + x(t, s) : s ≤ t,
where u(t, s) satisfies (4.3) and the two boundary conditions set at t = 0. Thus u(t, s) = a(s) + b(s)t α + c(s)t α+β + d(s)t α+β+γ ;
the Lidstone boundary conditions force a(s) = c(s) = 0. The two boundary conditions at t = 1 are satisfied by (u + x) for x given in (4.10). In particular, we use
D β D α [u(t, s) + x(t, s)]| t→1 = 0
to solve for d, leading to d(s) = s γ − 1 γ(β + γ)(α + β + γ) , and then u(1, s) + x(1, s) = 0 to solve for b, which yields b(s) = s α+β+γ α(α + β)(α + β + γ) − s β+γ αβ(β + γ)
+ s γ γ 1 β(α + β) − 1 (β + γ)(α + β + γ)
.
Altogether we have u(t, s) = s γ t α s α+β α(α + β)(α + β + γ) + α + 2β + γ β(α + β)(β + γ)(α + β + γ) − s β αβ(β + γ) + (s γ − 1)t α+β+γ γ(β + γ)(α + β + γ) .
To achieve symmetry and to satisfy (4.6), we take γ = α and arrive at (4.9). Then k(0) = α > 0, k(1) = 0, and k ′ (s) = β(α + β)s β−1 (s α − 1) < 0.
Thus, k is strictly decreasing, so that k(s) > 0 for s ∈ [0, 1), forcing u(t, s) ≥ 0 for t, s ∈ [0, 1]. Therefore the result holds.
Corollary 2. 1 (
1Fractional Conjugate and Right-Focal Problems). The corresponding Green's function for the homogeneous problem −D β D α x(t) = 0 satisfying the conjugate boundary conditions x(0) = x(1) = 0 is given by
the previous case, D β D α x(1) = 0 and D γ D β D α x(t) = h(t), proving the claim in this case as well.Theorem 3.2. For G(t, s) given in (3.3), we have that 0 < G(t, s) ≤ G(τ, s) (3.8)for t ∈ (0, 1] and s ∈ (0, 1], provided the condition u(1, τ ) > 0 is met for u in (3.4), that is to say the inequality τ ) = u(t, τ ) + x(t, τ ), ensuring that G(t, s) is a well-defined function; we also see that G(t, s) = 0 if t = 0 or s = 0. From (3.4) specifically, u(0, s) = 0 for all s and
Theorem 3. 3 .
3For all t, s ∈ [0, 1],
Theorem 4. 1 (
1Fractional Cantilever Beam). Let α, β, γ, δ ∈ (0, 1]. The corresponding Green's function for the homogeneous problem
Theorem 4. 2 .
2For all t, s ∈ [0, 1], Green's function given by (4.4) satisfies 0 ≤ G(t, s) ≤ G(1, s).
( 4 . 5 )
45Fix s ∈ [0, 1]. By the first boundary condition G(0, s) = 0, and d dt G(t, s) as given above implies that G(·, s) is monotone increasing on (0, 1]. In particular, G(1, s) ≥ G(t, s) ≥ 0 for all t ∈ [0, 1].
Theorem 4. 3 (
3Fractional Lidstone). Let α, β ∈ (0, 1]. The symmetric Green's function for the homogeneous problem
Theorem 4. 4 .
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| []
|
[
"Dirac eigenvalues and eigenvectors at finite temperature *",
"Dirac eigenvalues and eigenvectors at finite temperature *"
]
| [
"M Göckeler \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"H Hehl \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"P E L Rakow \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"A Schäfer \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"W Söldner \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"T Wettig \nCenter for Theoretical Physics\nRIKEN BNL Research Center\nYale University\n06520-8120New HavenCTUSA\n\nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n"
]
| [
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Center for Theoretical Physics\nRIKEN BNL Research Center\nYale University\n06520-8120New HavenCTUSA",
"Brookhaven National Laboratory\nUpton11973-5000NYUSA"
]
| []
| We investigate the eigenvalues and eigenvectors of the staggered Dirac operator in the vicinity of the chiral phase transition of quenched SU(3) lattice gauge theory. We consider both the global features of the spectrum and the local correlations. In the chirally symmetric phase, the local correlations in the bulk of the spectrum are still described by random matrix theory, and we investigate the dependence of the bulk Thouless energy on the simulation parameters. At and above the critical point, the properties of the low-lying Dirac eigenvalues depend on the Z3-phase of the Polyakov loop. In the real phase, they are no longer described by chiral random matrix theory. We also investigate the localization properties of the Dirac eigenvectors in the different Z3-phases. | 10.1016/s0920-5632(01)00985-9 | [
"https://arxiv.org/pdf/hep-lat/0010049v1.pdf"
]
| 26,620,631 | hep-lat/0010049 | 8eefeb22268f94c6b0286fc895e5b110ae8326c9 |
Dirac eigenvalues and eigenvectors at finite temperature *
26 Oct 2000
M Göckeler
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
H Hehl
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
P E L Rakow
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
A Schäfer
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
W Söldner
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
T Wettig
Center for Theoretical Physics
RIKEN BNL Research Center
Yale University
06520-8120New HavenCTUSA
Brookhaven National Laboratory
Upton11973-5000NYUSA
Dirac eigenvalues and eigenvectors at finite temperature *
26 Oct 20001
We investigate the eigenvalues and eigenvectors of the staggered Dirac operator in the vicinity of the chiral phase transition of quenched SU(3) lattice gauge theory. We consider both the global features of the spectrum and the local correlations. In the chirally symmetric phase, the local correlations in the bulk of the spectrum are still described by random matrix theory, and we investigate the dependence of the bulk Thouless energy on the simulation parameters. At and above the critical point, the properties of the low-lying Dirac eigenvalues depend on the Z3-phase of the Polyakov loop. In the real phase, they are no longer described by chiral random matrix theory. We also investigate the localization properties of the Dirac eigenvectors in the different Z3-phases.
INTRODUCTION
The theoretical understanding of the Dirac spectrum has improved considerably in the past few years. Using a variety of methods such as finite volume partition functions [1], partially quenched chiral perturbation theory [2], and chiral random matrix theory (RMT) [3], it has been shown that in the phase in which chiral symmetry is spontaneously broken, the distribution and the correlations of the low-lying Dirac eigenvalues are described by relatively simple universal functions. This description is valid in a regime in which the zero-momentum modes dominate the effective Lagrangian. The energy scale which limits this regime is known as the Thouless energy. For a review, we refer to Ref. [4].
Recently, two studies [5,6] have appeared in which the Dirac spectrum was investigated in this context for temperatures T close to the critical temperature T c of the chiral phase transition. This is an interesting problem, since the above-mentioned approach only works in the broken phase, and one would like to find out what happens to the universal features as one crosses T c . The present study addresses these and related questions. In addition, we also investigate the properties of the Dirac eigenvectors. * Presented by TW at Lattice 2000, Bangalore, India.
Z 3 ENSEMBLES
We are working in the quenched approximation using the staggered discretization of the Dirac operator. In a study of chiral symmetry restoration in the quenched approximation, an interesting observation was made in Ref. [7]. The gauge action has a Z 3 symmetry (for N c = 3 colors) which is broken in the deconfinement phase. As a result, the phases of the Polyakov loop P cluster around the elements of Z 3 in the complex plane, and one can divide the total ensemble of gauge field configurations into three classes with arg(P ) = 0, ±2π/3. An example is shown in Fig. 1. It was found [7] that the chiral condensate computed from the class of configurations with arg(P ) = 0 vanishes above T c as expected. However, for arg(P ) = ±2π/3 it remains nonzero in a certain range of T above T c . This behavior can be understood qualitatively in NJL-type models [8,9] and in RMT [10]. The point is that the boundary conditions of the Dirac operator are not invariant under Z 3 transformations, and for arg(P ) = ±2π/3, the new boundary conditions lead to a shift of the Dirac eigenvalues to lower values [10]. Using the Banks-Casher relation [11], this implies a nonzero chiral condensate.
In the following analysis, we therefore separate our configurations into two ensembles, those with arg(P ) = 0 (ensemble E1) and those with arg(P ) = ±2π/3 (ensemble E2), respectively. This separation can be done unambiguously. For the purpose of the present analysis, the two classes arg(P ) = ±2π/3 are equivalent and can be combined in E2. In full QCD, the fermion determinant suppresses the E2 configurations. (In Ref. [6], the E2-configurations were Z 3 -rotated before the Dirac operator was diagonalized so that arg(P ) = 0 in their analysis [12].)
Strictly speaking, we should distinguish three critical temperatures, T d for the deconfinement phase transition, and T c1 and T c2 for the chiral phase transitions of E1 and E2, respectively. Here, we assume that T d ≈ T c1 . Since we are mainly interested in the region T ≈ T c1 , we write T c instead of T c1 in the following.
DIRAC SPECTRUM
We have worked on N 3 s ×N t lattices with N t =4 and 6 for which β c (N s → ∞)=5.6925 and 5.8941, respectively [13]. An example for the global spectral density of the Dirac operator near zero for T T c is shown in Fig. 2. For the E1-ensemble, Figure 2. Global spectral density of the Dirac operator at T T c for the two different ensembles.
we find ρ(0) ≈ 0 which, according to the Banks-Casher relation, implies that chiral symmetry is restored. On the other hand, for E2 we observe that ρ(0) = 0 which implies a nonzero chiral condensate, in agreement with [7]. We therefore expect the distribution of the lowlying Dirac eigenvalues to be different in the two ensembles. Here, we concentrate on the smallest positive eigenvalue, λ min . Depending on whether or not ψ ψ = 0, the expectation value λ min scales as follows:
T < T c : λ min ∼ V −1 , T > T c : λ min ∼ V 0 ,(1)T = T c : λ min ∼ V −δ/(δ+1) ,
where δ is one of the universal critical exponents of a second order phase transition. For T < T c , the distribution of λ min is described by the RMT result, [14] are model dependent.
P (λ min ) = (c 2 λ min /2) exp(−(cλ min ) 2 /4) with c = V | ψ ψ |. At T = T c , RMT predictions for P (λ min )
Our results for P (λ min ), along with fits to the RMT prediction in the broken phase, are shown in Fig. 3. As expected, the results for the two ensembles E1 and E2 are very different. The top figure corresponds to T ≈ T c , and the bottom figure to T slightly above T c . In both figures, λ min is much larger in E1 than in E2. This reflects the fact that in the symmetric phase, small eigenvalues are suppressed. Also, P (λ min ) in E1 is clearly not described by the RMT prediction (which is valid for T < T c ). For E2 at T ≈ T c , however, P (λ min ) is still very well described by RMT, consistent with the fact that chiral symmetry is still broken for this ensemble. For T > T c the agreement becomes worse.
Other quantities such as the microscopic density could be analyzed in exactly the same way, and the results and conclusions will be similar.
THOULESS ENERGY FOR T > T c
The Thouless energy [15] is the limiting energy above which the universal description of the Dirac spectrum in terms of RMT is no longer valid. One has to distinguish between the Thouless energy at the hard edge and in the bulk of the spec- trum. The Thouless energy at the hard edge is very well understood, both theoretically [16] and on the lattice [17]. For T > T c the hard edge is no longer described by RMT so that the concept of a Thouless energy no longer exists. However, in the bulk the local spectral correlations are still given by RMT, and it is interesting to study the bulk Thouless energy above T c . A convenient measure of the bulk spectral correlations is the number variance defined by
Σ 2 (L) = n(L) − n(L) 2 ,(2)
where n(L) is the number of levels in an interval of length L after the spectrum has been unfolded. There are several questions related to how the spectrum should be unfolded, see Ref. [18] for a comprehensive discussion. We have used ensemble averaging to construct the average spec- tral density used in the unfolding procedure. The number variance was also computed by ensemble averaging. A typical example for Σ 2 (L), averaged over 600 independent configurations, is shown in Fig. 4, along with the parameter-free prediction of RMT. We observe that for small values of L, the lattice data are nicely described by RMT. We also see that there is a critical scale L c , the Thouless scale, above which nonuniversal behavior sets in. In order to extract this scale from the data, we construct the ratio
ratio(L) = Σ 2 latt (L) − Σ 2 RMT (L) Σ 2 RMT (L)(3)
which should start to deviate strongly from 0 above L c . We find that the numerical value of the Thouless scale depends on where we are in the spectrum, i.e., on the starting point of the interval of length L. This means that spectral averaging must not be used to construct Σ 2 (L) for the purpose of extracting the Thouless energy. In Fig. 5, we show the ratio of Eq. (3) for two different starting points, λ 0 = 0.5 and λ 0 = 0.8, and for various lattice sizes at β = 6.0, which is above β c for both values of N t . Again, we used 600 independent configurations per parameter set. We observe that the Thouless scale seems to be independent of N s , but depends on N t . We are currently investigating the form of this N t dependence.
LOCALIZATION PROPERTIES OF DIRAC EIGENVECTORS
In condensed matter physics, the question of whether or not a disordered mesoscopic sample is a metal or an insulator can be answered by constructing the so-called inverse participation ratio I 2 , which is a measure of how many sites contribute significantly to the wave function. For the case of QCD, we introduce the gauge-invariant definition
I 2 (λ) ≡ V x p λ (x) 2 [ x p λ (x)] 2 ,(4)
where V is the lattice volume and p λ (x) is the gauge-invariant probability density
p λ (x) = Nc α=1 |ψ α λ (x)| 2 .(5)
Here, x denotes a lattice site, α is a color index, and ψ α λ (x) is a component of the eigenvector corresponding to eigenvalue λ. Because of chiral symmetry we have x even p λ (x) = x odd p λ (x). A completely localized state therefore has I 2 = V /2. This case corresponds to uncorrelated eigenvalues and is described by the Poisson ensemble.
(The corresponding mesoscopic sample would be an insulator.) For a completely delocalized state, p λ (x) is the same for all x, and I 2 = 1. In RMT, we find
I 2 = 1 + 1 N c N c V N c V + 2 V →∞ −→ 1 + 1 N c .(6)
This case corresponds to extended wave functions with many significant components and strongly correlated eigenvalues. (The corresponding mesoscopic sample would be a metal.) In QCD, a simple argument (see [4]) shows that if chiral symmetry is broken, the Dirac wave functions must be extended.
Our results for the inverse participation ratio of the low-lying Dirac eigenvectors are shown in Fig. 6. Several features are worth noting: 1. Below T c all eigenvectors are extended, which is consistent with the fact that the eigenvalues are described by RMT both at the hard edge and in the bulk of the spectrum. The eigenvectors corresponding to smaller eigenvalues are slightly more "localized" than those corresponding to larger eigenvalues. The data agree well with the RMT prediction, I 2 = 4/3, for large λ . In contrast to the observation of Ref. [19] for Wilson fermions, we did not find signs of strong localization. 2. The eigenvectors corresponding to eigenvalues in the bulk of the spectrum remain extended for E1 and E2 at all temperatures considered, consistent with the fact that the eigenvalues in the bulk continue to be described by RMT above T c . 3. Most interestingly, the eigenvectors of the E1 ensemble corresponding to the lowest eigenvalues become more localized at and above T c , while in the E2 sector the lowest eigenmodes remain extended.
We are currently studying the topological properties of the low-lying eigenvectors and will present our results in the near future.
T < T c T ≈ T c T > T c Figure 6. Average inverse participation ratio I 2 of the low-lying Dirac eigenvectors vs average eigenvalue in the different Z 3 -phases for temperatures below, at, and above T c .
CONCLUSIONS
The behavior of the Dirac spectrum at and above the critical temperature can help to improve our understanding of the chiral phase transition. In particular, the properties of the Dirac eigenvectors deserve further study. Of course, the quenched approximation and the use of staggered fermions make it difficult to establish contact with continuum QCD. These systematic problems can be circumvented by using dynamical Ginsparg-Wilson fermions which, however, require large computational resources. For the mo-ment, we are restricted to exploratory studies such as the present one.
Figure 1 .
1Scatter plot of the Polyakov loop in the complex plane in the confinement (left) and deconfinement (right) phase.
Figure 3 .
3Distribution of the smallest Dirac eigenvalue (from 2000 configurations) in the two ensembles for T ≈ T c (top) and T T c (bottom).
Figure 4 .
4Number variance of the Dirac eigenvalues in the bulk of the spectrum. λ 0 is the starting point of the interval of length L.
Figure 5 .
5The ratio defined in Eq. (3) for two different values of λ 0 and various lattice sizes.
Acknowledgments. This work was supported by DFG project Scha 458/5-3 and by DOE grants DE-FG02-91ER40608 and DE-AC02-98CH10886.
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| []
|
[
"Status and Perspectives forPANDA at FAIR Elisabetta Prencipe, on behalf of thePANDA Collaboration",
"Status and Perspectives forPANDA at FAIR Elisabetta Prencipe, on behalf of thePANDA Collaboration"
]
| [
"Forschungszentrum Jülich \n52428Strasse, JülichGermany\n",
"Leo Brandt \n52428Strasse, JülichGermany\n"
]
| [
"52428Strasse, JülichGermany",
"52428Strasse, JülichGermany"
]
| [
"Nuclear Physics B Proceedings Supplement"
]
| The Facility for Antiproton and Ion Research (FAIR) is an international accelerator facility which will use antiprotons and ions to perform research in the fields of nuclear, hadron and particle physics, atomic and anti-matter physics, high density plasma physics and applications in condensed matter physics, biology and the bio-medical sciences. It is located at Darmstadt (Germany) and it is under construction. Among all projects in development at FAIR in this moment, this report focuses on thePANDA experiment (antiProton ANnihilation at DArmstadt). Some topics from the Charm and Charmonium physics program of thePANDA experiment will be highlighted, wherePANDA is expected to provide first measurements and original contributions, such as the measurement of the width of very narrow states and the measurements of high spin particles, nowaday undetected. The technique to measure the width of these very narrow states will be presented, and a general overview of the machine is provided. | null | [
"https://arxiv.org/pdf/1410.5201v2.pdf"
]
| 119,198,279 | 1410.5201 | 28b0faf546b6769143d5aa16d3d99da2b74a8729 |
Status and Perspectives forPANDA at FAIR Elisabetta Prencipe, on behalf of thePANDA Collaboration
2014
Forschungszentrum Jülich
52428Strasse, JülichGermany
Leo Brandt
52428Strasse, JülichGermany
Status and Perspectives forPANDA at FAIR Elisabetta Prencipe, on behalf of thePANDA Collaboration
Nuclear Physics B Proceedings Supplement
002014D SCharmSpectroscopyconfinementPANDA
The Facility for Antiproton and Ion Research (FAIR) is an international accelerator facility which will use antiprotons and ions to perform research in the fields of nuclear, hadron and particle physics, atomic and anti-matter physics, high density plasma physics and applications in condensed matter physics, biology and the bio-medical sciences. It is located at Darmstadt (Germany) and it is under construction. Among all projects in development at FAIR in this moment, this report focuses on thePANDA experiment (antiProton ANnihilation at DArmstadt). Some topics from the Charm and Charmonium physics program of thePANDA experiment will be highlighted, wherePANDA is expected to provide first measurements and original contributions, such as the measurement of the width of very narrow states and the measurements of high spin particles, nowaday undetected. The technique to measure the width of these very narrow states will be presented, and a general overview of the machine is provided.
Introduction
The Standard Model of particle physics is well defined and efficient in describing fundamental interactions. However several questions still remain open. For example, the theory describing strong interactions, the Quantum Chromodynamcs (QCD), is still affected by some unsolved fundamental questions, arising in the low energy domain, such as the understanding of confinement and the origin of hadron masses. As a non-Abelian theory, QCD allows the self-interaction of the strong force carriers, e.g. the gluons. In the low enery regime their interactions can only be described exploiting non-perturbative methods. The answer to these questions is a challenge that requires a new generation machine and experiments with higher resolution and better precision, compared to the past. The future experimentPANDA will be located at the HESR at FAIR[1] (High Energy Storage Ring at the Facility for Antiproton and Ion Research), in Germany.
Email address: [email protected] (on behalf of thē PANDA Collaboration) In this report we will put emphasis on the description of thePANDA experiment, in particular on the detector design and the physics program, to motivate the big effort in terms of hardware and software that an international collaboration of 18 countries and more than 500 people are presently going through.
The physics case
The program of thePANDA experiment is wide and ambitious, and covers several areas of interest in nuclear and particle physics [2].
We plan to study with accuracy the mechanism responsible for phenomena like the quark confinement, through the investigation of: • Hadrons in matter.
• Hypernuclei.
PANDA is a fix-target experiment, where a beam of antiprotons will collide against a thick target, e.g. 4·10 15 cm −2 , with a beam life time > 30 minutes. The choice of an antiproton beam is strongly supported from the fact that thepp interactions are gluon rich processes. All quantum numbers will be directly accessible in the annihilation. Several advantages are available in this respect:
• very good mass resolution, which depends basically on the beam resolution and not on that of the detectors. A 100 keV pitch mass scan will be possible inPANDA, which is 20 times better than attained at B factories and more than 2 times better than at the Fermilab experiment E760;
• direct formation of high spin states, forbidden at B factories;
• direct production with very high rate/day.
The detector
The fixed-target experimentPANDA is composed by two main parts: the central and the forward spectrometer, as shown in Fig. 1, inserted in a homogeneous solenoid magnetic field (B = 2T) and a dipole field (B = 2T·m), respectively. Two options are still under investigation for the target: cluster-jet or pellet target.PANDA will span a wide momentum range, from 1.5 up to 15 GeV/c. Focalized through stochastic and electron cooling, the antiproton beam will have excellent momentum resolution. Two operation modes are provided: high resolution mode and high luminosity mode, as reported in Table 1.
PANDA is a 4π-coverage machine. ThePANDA innermost detector is the Micro-Vertex-Detector (MVD), a sophisticated silicon pixel and silicon strip array, which will provide a good vertex reconstruction, essential to reduce the high level of background, and to reconstruct lower momentum particles. A vertex space resolution of 50 µm in x,y, and 100 µm in z is expected. The GEM and the Straw-Tube-Tracker (STT) will allow to track charged particles (∆p T /p T = 1.2% together with the MVD). A Cherenkov detector (DIRC) is planned to discriminate with excellent efficiency K/π. A high performant calorimeter will be equipped with 17200 PbW0 4 crystals, operating at the temperature of -25 0 C; it will be provided of 2 ADPs, so the performances are definitively better than that of the CMS experiment. It will provide an excellent separation between pions and electrons, and excellent photon reconstruction. A forward system is needed to track particles which will be emitted ahead due to the high boost of proton-antiproton in the center of mass. The muon detector, together with a luminosity monitor, are the outmost detectors ofPANDA.
Background and interesting signal events inPANDA will have the same signature; therefore no hardware trigger is able to discriminate a priori background. Online reconstruction can be exploited ("software" trigger): considering that detector acceptance only, the ratio between signal and background (S/B) is expected to be 10 −6 , with 20 MHz average interaction rate. This number makes many searches of thePANDA physics program challenging. Ad hoc techniques to reject the high background level are necessary, depending on the physics channel.
In this report, some simulations performed within the PandaRoot[3, 4] 1 framework will be reported, to show the healthy status of our software project development and to point out which are the potentialities and the original contributions expected fromPANDA in the field of Charm and Charmonium spectroscopy.
Challenges in Charm physics withP AN D A
The study of D mesons is important both for strong and weak interactions. Gluonic excitations and hadrons composed of strange and charm quarks can be abundantly produced inpp interactions and their features will be accessible with unprecedented accuracy, thereby allowing high precision tests of the strong interaction theory in the intermediate energy regime. On the other hand, the search for CP violation in the D sector recently has gained more attention, as a new field of investigation.
Understanding the cs-spectrum (see Fig. 2) is not easy: 11 years after the discovery of the charged state called D * s0 (2317) [5], its mass is known with high precision, but for its width only an upper limit exists. The observation of the D * s0 (2317) represents a break-point, because the existence of combined c,s quark systems
High resolution mode
High luminosity mode cooling e − , 1.5 ≤ p ≤ 8.9 GeV/c stochasting, p ≥ 3.8 GeV/c N. of anti-protons stored 10 10 10 11
Luminosity [cm −2 s −1 ] ≤ 2·10 31 ≤ 2·10 32 ∆p/p 4· 10 −5 2· 10 −4
is theoretically predicted [6,7]; but some experimental observations questioned the potential models, which fairly agree with the observed D meson spectrum, e.g. mesons composed of the light quark u or d and the heavy quark c. Potential models agree also with the observation of several D s states, up to the discovery of the D * s0 (2317). But they cannot explain why the D * s0 (2317) mass was observed more than 100 MeV/c 2 below the predictions. This is an even more complicated issue, as D s mesons are composed by the quarks s and c (see Fig. 2). The same is valid for the observation of the so called D s1 (2460) [9]: its mass was found below the theoretical expectations as well. D * s0 (2317) and D s1 (2460) are both observed below the DK threshold; they are very narrow, and their decays are isospin violating; it is very difficult to predict the cross section ofpp → D +( * ) s D −( * ) s , as we cannot perform perturbative calculations, since they would underestimate the real cross section. The quantum numbers of D * s0 (2317) and D s1 (2460) are, in any case, not fixed yet, although we can exclude J P =0 + for the D s1 (2460) based on experimental observations [10]. In order to answer the important questions related to their interpretation, we need to measure their widths (Γ), because they may allow to discriminate among different theoretical models, which provide and explanation for the D s excited states as pure cs-state (Γ ∼10 keV) [11], or tetraquark (Γ in the range of 10 − 100 keV) [12], or molecular state (Γ ∼130 keV) [13], or chiral partners of the same heavy-light system built with one heavy and one light quarks (c and s, Figure 2: Mass spectrum versus the particle spin J PC for D mesons (top) and D s mesons (down). The horizontal long lines indicate the DK and D * K threshold; the short horizontal lines indicate the predicted states as in Ref. [6,7]. The mass spectra here reported are built using recent experimental results, presented at the conference CHARM2013 and available in Ref. [8]. respectively).
In the past two years the experiment LHCb [8] improved the knowledge of the D s spectrum, and confirmed the previous measurements, with the highest world precision; but it cannot provide the measurement of the width of these very narrow states, so the nature of the D s excited states still remain unclear. With the fine mass scan techniques every 100 keV,PANDA is in a unique position to perform the study of the excitation function of the cross section (see Eq. 1), and discriminate among the theoretical models. The cross section at given energy λ is given by [14]:
σ(λ) = m R Γ · |M 2 | · 1 π λ −∞ λ − x x 2 − 1 dx,(1)σ(0) = m R Γ/2 · |M 2 |(2)
where M is the matrix element, m R is the resonance mass and Γ its width, λ= ( √ s − m R − m Ds )/Γ, √ s = energy in the center of mass for the production e.g. of D * s0 (2317) in the processpp → D − s D * + s0 (2317), that is equal to 4.286 GeV/c 2 at the threshold of this process.
Studies are planned inPANDA for the processpp → D − s D * + s , where D * + s stands for D * s0 (2317), D s1 (2460), and D s1 (2535). There are manifold interests in these decay processes: • study of mixing between D ( * ) s states;
• measure the width of the D * s0 (2317) and D s1 (2460), that will be first observation. The D s1 (2535) is above the DK threshold, and its width is known with large uncertainty [15,16]; therefore it could be repeated inPANDA with higher precision. It would represent an important cross check of our analysis technique. At threshold Eq. 1 reduces to the Eq. 2, so the only observables which we should measure in this case are the mass of the resonant state, and the cross section.
RecentPANDA simulations, shown in Fig. 3, are performed using the MC generator EvtGen [17], within the PandaRoot framework. We use the same model described in Ref. [5], a Dalitz model based on real data, to simulate a realistic case and estimate the run-time needed inPANDA. The high peformance tracking detectors ofPANDA allow excellent track reconstruction and high K/π separation, together with the DIRC; the designed vertex detector allows high background rejection, by setting tight topological selection cuts around the fitted vertex. We reconstruct the D s by using a vertex fit with three charged particles, which are identified as K or π by means of a likelihood PID (Partile Identification) method, that makes use of several variables like energy loss, Cherenkov angle, Zernik momenta. Track finder and fitting procedures in the central spectrometer use the Kalman filter method; wherever the B field is not homogeneous, the Runge Kutta track representation is used. In order to have better mass resolution and higher reconstruction efficiency, the missing mass of the event is exploited: the reconstruction of the D − s is performed, then we extract information on the D * s0 (2317) by evaluating its four-momentum as the difference between the reconstructed D − s four-momentum and the initial state vector. With this technique, we have obtained ∼30% reconstruction efficiency. A full simulation, including electronics and detector material, is being performed. In order to reject naively part of the huge background, originating from very low momentum particles, a preselection cuts on the track momentum p T RACK >100 MeV/c and on the photon momentum p γ >50 MeV/c are applied in our simulations. A challenge of this analysis is the reconstruction of the many low momentum pions. This is the first time a full simulation including the D * s0 (2317) is achieved with PandaRoot: the work in still in progress. Our simulations are based on Geant3 [18]. With the naive selection cuts just described, the ratio S/B improved from 10 −6 to 10 −2 . The optimization of the final selection criteria is presently ongoing. The measurement of the width will be an original and extremely importantPANDA contribution to solve the csspectrum puzzle. In Fig. 4c (right) the plot related to Eq.2 is shown, scanning the D * s0 (2317) + mass in 100 keV steps around its nominal value: the shape of the curve changes, depending on the input width given to the simulation. The minimum momentum needed to produce the D * s0 (2317) in the process mentioned above is p=8.8 MeV/c. In high luminosity mode (see Table 1), with a cross section in the range [1−100] nb, and assuming a luminosity of 8.64 pb/day, with a reconstruction efficiency of 30%, we estimate withPANDA a D * s0 (2317) production rate in the order of (3−300)·10 3 per day, to be scaled by the Branching Fraction (BF) of the reconstructed D s .
Future measurements in the Charmonium sector withP AN D A
In 2003 a new era started not only for the charm, but also for the charmonium sector: the discovery of the X(3872) [20] and its subsequent confirmation by several experiments (at e + e − colliders and pp interaction machines) and in several decay modes. These observations set up the starting point to look after new forms of aggregation of matter: tetraquarks, molecular states, hybrids, etc. [21]. These states were predicted by theory even before the discovery of the X(3872), but in the last decade a plenty of theoretical papers provided new interpretations and new models, that have spurred refreshed interest in spectroscopy. Several new observations and controversial interpretations of the evidence of additional new resonant states suggest that limitations due to statistics and resolution do not allow a unique interpretation, and discriminate among theoretical models. We use to name as generically X, Y, Z these new experimental findings, which do not fit in the spectra forseen by potential models, as in some cases it has been difficult to assign the quantum numbers. Lots of progresses have been made by LHCb; we know that the X(3872) is an isospin violating and very narrow state; its quantum numbers have been found to be 1 ++ [22], and no charged partners are found so far. Clearly the interpretation of the X(3872) as charmonium state is unlikely, and alternative models have to be used to understand its nature. As for the case of the D * s0 (2317), the measurement of the width plays an important role.
PANDA will scan the X(3872) mass in 100 keV steps. In Fig. 4 (left) the simulated X(3872) signal is shown, together with a realistic background; the latter is obtained using DPM [23] MC generator, while the signal events are simulated by EvtGen within the PandaRoot framework: we could reproduce precisely the input values of this simulation in the fit of Fig. 4. A detailed description of this work is provided in [24]. The measurement of the X(3872) width is just an example of the original measurements thatPANDA will be able to perform in the Charmonium sector. One of the challenges ofPANDA, for instance, will be to explore and understand the confinement of quarks. A way to do that is to verify and test the behaviour of the potential in the intermediate and high energy regime with heavy hadrons, e.g. those composed particles whose mass is above the DD threshold, since below it the behaviour of the hadrons is generally well predicted from the potential models. A chance to test the confinement is, for example, to search for a 3 F 4 state [26]. If this state is Figure 4: Full PandaRoot simulations of X(3872) (left), reconstructed in the processpp → X(3872), X(3872) → J/ψπ + π − ; h c (center), reconstructed as recoil of the di-pion mass in the processpp → π + π − h c , h c → D 0D0 * and D 0 → K − π + ; 3 F 4 (right), directly formed in the process pp → ( 3 F 4 ), as radiative J/ψ cascade [25]. found, it could be the proof that the linear term of the potential, in the formula V(r) = − 4 3 α s r + kr, is indeed logarithmic, and not linear as it is believed. In fact, we could consider that the charmonium splitting is roughly the same as for bottomonium states (see Fig. 5). This can be explained assuming that the potential, at high energy, is logarithmic [25]. The search for a high spin particle such as 3 F 4 is also forbidden at B factories (the past BaBar and Belle, or the future Belle II), and it is forbidden at BES III. ButPANDA will have easy access to high spin states in formation, with excellent high production rates. In Fig. 4 preliminary full MC simulations with PandaRoot are performed, to search for two of these states to test the charmonium splitting hypothesis.
A last study illustrated in this report is the simulation of several unpredicted vector resonant states above the DD threshold, with large width and with established quantum numbers (1 −− ), whose nature is still unclear. Fig 6 shows a PandaRoot full simulations for six of them: the VLL PHOTOS [17] model is used. We know that the BF of the J/ψ → e + e − is ∼6%, and that is the narrow charmonium ground state; but the BF of the Yfamily states decaying to e + e − is of order of 10 −6 . This could be explained assuming that the radial quantum number n for those Y-states is larger than 1, fact that supports the idea that they are non-conventional charmonia. However, only few of these vector states seem to decay to e + e − . For example, no measurement exists of BF(Y(4260) → e + e − ). This could be due to the statistics limitation at B factories, or because the vector state Y(4260) is not a charmonium state, as in this case it would decay to e + e − . There are arguments in favor of the Y(4260) as hybrid; but its nature is not fixed, yet. It is worth in any case to look for the decay of all these vector states to e + e − . Simulations inPANDA are very promising in this sense, as the reconstruction efficiency is evaluated to be ∼70% for each state decaying to e + e − , and the background simulations by DPM show that background rejection is easy when the center of mass energy of e + e − is larger that 4 GeV. We expect roughly 16 000 events/day in the high luminosity mode: this is an incredible high rate compared to what was measured in the past experiments. In addition, we could also study interference effects among all these large vector states, which overlap in a narrow energy range: this study was never performed before.
Conclusions
The Standard Model is solid, but it does not give answers to all questions. Several open issues still exist in the Charm and Charmonium physics sector, for example. ThePANDA experiment at FAIR is designed to achieve a mass resolution 20 times better than attained at B factories, essential to perform fine mass scan (every 100 keV).PANDA will have unprecedent PID power, Figure 6: Full simulation of vector states above theDD threshold inPANDA. Background simulation with DPM, with a clear J/ψ and Y(4260) signal (left); reconstruction of known Y-vector states to e + e − with EvtGen, inside the PandaRoot framework, to evaluate the reconstruction efficiency. It is evident that they overlap and inteference effects are expected (right).
indispensable for the measurement of the proton form factors and other measurements. In this report we highlighted the measurement of the width of some Charm and Charmonium states. This task is challenging but the measurement is absolutely needed to clarify and understand the nature of some observed states, such as the D * s0 (2317), D s1 (2460) and X(3872). Understanding the confinement is also a challenge: a new approach could be based on looking for high spin resonant states to test the potential models.PANDA offers a unique opportunity to perform these measurements with high precision. TDRs of several detectors have already been approved, and tests with detector prototypes are ongoing. The official PandaRoot simulation framework is at an advanced stage. Important contributions are expected fromPANDA when it will start to collect data.
•
Hadron spectroscopy: − search for gluonic excitations; − charmonium spectroscopy; − D meson spectroscopy; − baryon spectroscopy; − QCD dynamics. arXiv:1410.5201v2 [hep-ex] 5 Nov 2014 • Nucleon structure: − parton distribution; − time-like form factors of the proton; − transition distribution amplitudes.
Figure 1 :
1General view of thePANDA detector (top). Detailed view of the central spectrometer (down-left) and forward spectrometer (down-right) ofPANDA.
Figure 3 :
3Dalitz plot of m 2 K + K − vs m 2 K + π − on simulatedPANDA events with the MC generator EvtGen: the realistic CLEO model is used for the reconstruction of the D − s → K + K − π − [19] (left). MC simulations of D * s0 (2317) + , reconstructed as missing mass of the event in the process pp → D − s D * s0 (2317) + , D − s → K + K − π − and D * s0 (2317) + → D + s π 0 (center). Excitation function of the cross secton of the same process as in (b): the curve is reconstructed scanning the D * s0 (2317) + every 100 keV, and representing every point of the mass scan in a graph of the cross section as function of the energy difference λ = ( √ s − m Ds(2317) − m Ds )/Γ (right).
Figure 5 :
5Splitting levels of Charmonium (left) and Bottomonium states (right). Similarities are shown in the charm and bottom sector. The dashed lines indicate predicted but undetected resonant states as explained in Ref.[25].
Table 1 :
1Operation modes ofPANDA with HESR.
PandaRoot is the officialPANDA framework in development inside the project FairRoot@GSI.
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| []
|
[
"MATRIX BISPECTRALITY AND NONCOMMUTATIVE ALGEBRAS: BEYOND THE PROLATE SPHEROIDALS",
"MATRIX BISPECTRALITY AND NONCOMMUTATIVE ALGEBRAS: BEYOND THE PROLATE SPHEROIDALS"
]
| [
"F Alberto Grünbaum [email protected] \nDepartment of Mathematics\nUniversity of California\nBerkeleyCA\n",
"Brian D Vasquez \nDepartment of Mathematics\nIMPA\nRio de Janeiro\n",
"Jorge P Zubelli [email protected] \nKhalifa University\n\n"
]
| [
"Department of Mathematics\nUniversity of California\nBerkeleyCA",
"Department of Mathematics\nIMPA\nRio de Janeiro",
"Khalifa University\n"
]
| []
| Keywords bispectral algebras · bispectral triple · presentations of finitely generated algebras · time-and-band limiting · integral and commuting differential operators | null | [
"https://arxiv.org/pdf/2201.08989v3.pdf"
]
| 246,240,738 | 2201.08989 | b4aba60306f334dcabdf1046770ddfd5b57212c7 |
MATRIX BISPECTRALITY AND NONCOMMUTATIVE ALGEBRAS: BEYOND THE PROLATE SPHEROIDALS
1 Feb 2022 February 2, 2022
F Alberto Grünbaum [email protected]
Department of Mathematics
University of California
BerkeleyCA
Brian D Vasquez
Department of Mathematics
IMPA
Rio de Janeiro
Jorge P Zubelli [email protected]
Khalifa University
MATRIX BISPECTRALITY AND NONCOMMUTATIVE ALGEBRAS: BEYOND THE PROLATE SPHEROIDALS
1 Feb 2022 February 2, 2022arXiv:2201.08989v3 [math.FA] DEDICATED TO VAUGHAN JONES, A TRULY INSPIRATIONAL FRIEND, WHO LEFT US SOME GREAT MEMORIES.
Keywords bispectral algebras · bispectral triple · presentations of finitely generated algebras · time-and-band limiting · integral and commuting differential operators
Preamble
The bispectral problem discussed in this paper has its origin in Duistermaat and Grünbaum (1986). It is motivated, as mentioned there, by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admiting a second order differential operator with simple spectrum in its commutator. This property, which gives a good numerical way to compute the eigenfunctions of the integral operator, was put to good use in a series of papers by D. Slepian, H. Landau and H. Pollak at Bells Labs back in the 1960's, see Slepian and Pollak (1961); Landau and Pollak (1962, 1962; Slepian (1964Slepian ( , 1978Slepian ( , 1983Slepian ( , 1976 and is of interest in one other contribution in this issue, see Connes and Consani (2021); Connes and Marcolli (2008); Connes and Moscovici (2021). We are thankful to Luc Haine, of Louvain-la-Neuve, Belgium who alerted one of us (AG) on Nov 2021 about a talk by Alain Connes in the series Mathematicial Picture Language. This talk delivered on Dec 7th. 2021 can be seen in Youtube and covers some of the contents of his work with various collaborators on the zeta function and its relations to the prolate spheroidal functions. For readers interested in this fascinating connection there is no better way of learning about this than watching the lecture. See .
If one tries to extend this property beyond the Fourier case by adding a potential V (x) to the operator − d dx 2 and replacing an expansion in exponentials by an expansion in term of the eigenfunctions ψ(x, z) of − d dx 2 + V (x) it appears plausible that a certain property of these eigenfunctions ψ(x, z) could play a useful role. This property is now known as the bispectral property which will be formulated below. Its solution in the scalar case was the purpose of Duistermaat and Grünbaum (1986).
The version of the bispectral problem that we discuss in this paper is a noncommutative one, obtained by allowing all objects in the original formulation to be matrix valued. The details are given later. Now for a bit of history: one of us (AG) gave a couple of talks, one at Vanderbilt in Nov 2013 and one in IMPA, Rio de Janeiro in March 2014 . The first talk was at the invitation of Vaughan Jones and started by saying that the topic was most likely of no interest to him. At some point in the talk Vaughan said "all of this is about bi-modules and subfactors". The second talk had both Vaughan and another one of us (JZ) in the audience, there was some more mention of these topics and then in Grünbaum (2014), written in April 2014, a reference is made to a future joint paper with Vaughan "On the bimodule structure of the bispectral problem, in preparation". The occasion of this second talk was a visit that Vaughan (and for part of it with Wendy) did to Argentina, Uruguay and Brazil. In Buenos Aires he delivered one lecture, in Rio he delivered a series of talks but in Montevideo besides giving a lecture he was received in a private audience by President Jose Mujica, described as "the world humblest head of state" by Wikipedia. Vaughan whose command of the spanish language was quite high lamented that he had troubles understanding President Mujica.
The examples in this short paper Grünbaum (2014) were discussed with Vaughan who showed some level of interest but we never managed to get him fully on board. Of course we assumed that we always had time to get him onto this project. The short paper Grünbaum (2014) makes three separate conjectures going with three different examples. These conjectures were proved to be correct in the thesis of the third of us (BVC) under the supervision of (JZ).
And now that sadly we have no chance of benefiting from the insights that Vaughan would have brought in, we present the problem and some results to a wider audience in the hope that someone may surmise what it was that Vaughan had in mind. Getting someone involved in this effort would be a nice way of honoring his memory.
Commuting integral and differential operators
The bispectral problem, introduced in the next section is motivated by the following very concrete problem in signal communication: a signal of support in the interval [−T, T ] is transmitted over a channel that has bandwidth [− W, W ] i.e., all frequencies in the signal beyond absolute value of W cannot be sent over. A mathematical formulation is as follows: an arbitrary signal in L 2 (R) is chopped to the interval [−T, T ] and then its Fourier transform is chopped to the interval [−W, W ]. Denoting for simplicity these two chopping operations by T and W we are dealing with the operator E = W F T where F stands for the Fourier transform. The spectral analysis of this operator, i.e., a look at its singular functions and singular values requires the consideration of the operator E * E. It is easy to see that E * E is an integral operator acting in L 2 (−T, T ) whose kernel is given by sin W (t − s) t − s and this bounded operator acts on a function in the space L 2 (−T, T ) by
(Kf )(s) = T −T sin W (t − s) t − s f (t)dt
for f ∈ L 2 (−T, T ) and s ∈ (−T, T ). This K commutes with the operator
(Lf )(x) = − d dx (T 2 − x 2 ) df dx + W 2 x 2 f (x)
defined on C 2 functions.
One can show that this densely defined operator has a unique selfadjoint extension in L 2 (−T, T ) with eigenfunctions and eigenvalues that depend on the parameter W .
Its eigenfunctions are known as the prolate spheroidal wave functions, since this is one of the differential operators resulting in separating varables when solving for the eigenfunctions of the Laplacian on a prolate spheroid.
What other naturally appearing integral operators allow for commuting differential operators? Two other examples are the Bessel and Airy kernels, as in the work of Tracy and Widom, Tracy and Widom (1994a,b) in the context of Random Matrix Theory. For the Bessel case see also Slepian (1964); Grünbaum et al. (1982). There are other examples, but the search is nowhere close to finished. The bispectral property, to be formulated below was put forward in Duistermaat and Grünbaum (1986) as an important ingredient in the search for more examples of this commuting property.
The bispectral problem
The problem was posed and solved in Duistermaat and Grünbaum (1986). It is as follows:
Find all nontrivial instances where a function ψ(x, z) satisfies
L x, d dx ψ(x, z) ≡ − d dx 2 + V (x) ψ(x, z) = zψ(x, z)
as well as
B z, d dz ψ(x, z) ≡ M i=0 b i (z) d dz i ψ(x, z) = Θ(x)ψ(x, z).
All the functions V (x), b i (k), Θ(x) are, in principle, arbitrary except for smoothness assumptions. Notice that here M is arbitrary (finite). The operator L could be of higher order, but in Duistermaat and Grünbaum (1986) attention is restricted to order two.
The complete solution is given as follows: In this case V is a rational solution of the Korteweg-deVries hierarchy of equations. Here A is a first order differential operator.
Theorem If M = 2, then V (x) is (except for translation) either c x 2 or ax, i.e.b) L is obtained from L 0 = − d dx 2 + 1
4x 2 after a finite number of (rational) Darboux transformations.
In all cases we have a solution of the ad-conditions, a complicated system of nonlinear differential equations. These conditions are necesary and sufficient.
Notice that the solutions organize themselves into nice manifolds.
The simplest example of case a) follows from L 0 = − d dx 2 by two Darboux transformations, one gets the operator
L 2 = − d dx 2 + 6(x 4 + 12t 3 x) (x 3 − t 3 ) 2 .
In this case Θ(x) = x 4 − 4t 3 x and the differential operator in the spectral parameter is
B 2 z, d dz = − d dz 2 + 6 z 2 2 + 4it 3 d dz
The potential in the operator
L 2 = − d dx 2 + V (x, t) above satisfies the KdV equation.
It was later observed by Magri and Zubelli, see that in case b) we are dealing with rational solutions of the Virasoro equations (i.e. master symmetries of KdV). The bigger picture became more apparent in the work Zubelli and Valerio Silva (2000) where it is shown that the generic rational potentials that decay at infinity and remain rational by all the flows of the master-symmetry KdV hierarchy are bispectral potentials for the Schrödinger operator.
In case a) the space of common solutions has dimension one and in case b) it has dimension two. One refers to these as the rank one and rank two situations.
Observe that the "trivial cases" when M = 2 are self-dual in the sense that since the eigenfunctions ψ(x, z) are functions either of the product xz or of the sum x + z, one gets B by replacing z for x in L. The bispectral involution introduced in Wilson (1993) shows how this can be adapted in the "higher order cases".
The non-commutative version of the bispectral problem
A first noncommutative (or matrix) version of the bispectral problem was considered in J. Zubelli Ph.D. thesis at Berkeley, see also Zubelli (1990 in the situation where both the physical space and spectral operators act on the same side of the eigenfunction and the eigenvalues are both scalar valued. Later on, several other versions were considered. See Sakhnovich and Zubelli (2001); Kasman (2015); Boyallian and Liberati (2008); Grünbaum and Iliev (2003); Grünbaum (2014); Geiger et al. (2017) and references therein. The noncommutative version of the bispectral problem displayed interesting connections with soliton equations as well. Indeed, in it was shown that a large class of rational solutions to the AKNS hierarchy Ablowitz et al. (1974) led to matrix differential operators that displayed the bispectral property. Among the important equations in Mathematical Physics that are covered by the AKNS hierarchy one finds the modified KdV and the nonlinear Schrödinger equation. The matrix differential operator that appeared in this case was in turn related to Dirac operators. The connection between bispectrality and another important topic in Mathematical Physics, namely Huygens' principle in the strict sense Berest and Veselov (1994) turned out to appear also in the context of Matrix Bispectrality. Indeed, in Chalub and Zubelli (2006Zubelli ( , 2005Zubelli ( , 2004, it was shown that rational solutions to the AKNS hierarchy led to Dirac operators which satisfy Huygens' principle in the stric sense. In other words, the fundamental solutions of the perturbed Dirac equation in a suitably high space-time dimension had its support precisely on the surface of the light cone and not in its interior. Another interesting connection between Matrix bispectrality to soliton equations of Mathematical Physics was explored in Sakhnovich and Zubelli (2001).
In this present paper we take the bi-module structure of the problem into account and let the operators act on different sides as well as allow both eigenvalues to be matrix valued.
We consider triplets (L, ψ, B) satisfying the equations A triplet (L, ψ, B) satisfying (1) is called a bispectral triplet.
Lψ(x, z) = ψ(x, z)F (z) (ψB)(x, z) = θ(x)ψ(x, z) (1) with L = L(x, d dx ), B = B(z, d dz ) linear matrix differential operators, i.e., Lψ = l i=0 a i (x) · d dx i ψ, ψB = m j=0 d dz j ψ · b j (z). The functions a i : U ⊂ C → M N (C), b j : V ⊂ C → M N (C), F : V ⊂ C → M N (C),
The study of the structure of the algebra of possible θ(x) going with a fixed bispectral ψ(x, z) was first raised in Castro and Grünbaum (2006) and analyzed in Tirao (2011); Grünbaum and Tirao (2007). See also Casper (2018) and Zurrián (2017).
We consider now the examples and conjectures given in Grünbaum (2014) as well as their validation and further description given in Vasquez and Zubelli (2021).
For the benefit of the reader we give a few definitions before giving some explicit results in the next section.
Definition 1. Let K be a field, C be a K-algebra, A a subring of C and S ⊂ C. We define
A· < S >= span K n j=1 s j | s 1 , ..., s n ∈ S ∪ A, n ∈ N ,
where the noncommutative product is understood from left to right, i.e., n+1 j=1 s j := ( n j=1 s j )s n+1 , for n = 0, 1, 2, · · · . For completion, 0 j=1 s j := 1.
The set A· < S > is called the subalgebra generated by S over A and we call an element f ∈ A· < S > a noncommutative polynomial with coefficients in A and set of variables S.
Definition 2. Let C be a noncommutative ring and A a subring of C. We say that an element α ∈ C is integral over A if there exists a noncommutative polynomial f with coefficients in A such that f (α) = 0. Furthermore, we say that β ∈ C is integral over α ∈ C if β is integral over A· < α >. Finally, α and β are associated integral if α is integral over β and β is integral over α.
In order to characterize the algebraic structure of bispectrality in the present noncommutative context, we start with the following definitions.
Definition 3. Let K be a field, we denote by K x λ | λ ∈ Λ the free algebra generated by the letters x λ , λ ∈ Λ i.e.,
K x λ | λ ∈ Λ = F ⊂Λ,F finite λ∈F K · x λ .
Definition 4. Let A be a K-algebra. A presentation for an algebra A is a triple (K x λ | λ ∈ Λ , f, I) such that I ⊂ A is an ideal and f : K x λ | λ ∈ Λ /I → A is an isomorphism. Furthermore, we say that A is finitely generated if there exists a presentation with Λ finite and finitely presented if there exists a presentation with Λ finite and the ideal I is generated by finitely many elements.
Three examples
Take for Ψ(x, z) the matrix valued function
Ψ(x, z) = e xz z − 1/x 1/x 2 0 z − 1/x
and consider all instances of matrix-valued polynomials θ(x) and differential operators B (with matrix coefficients b i (z)) such that
ΨB ≡ m i=1 d dz i Ψ b i = θ(x)Ψ(x, z).
In this case one has
LΨ = −z 2 Ψ with L = − d dx 2 + 2 1/x 2 −2/x 3 0 1/x 2 .
In other words for this specific differential operator in the variable x we are asking for all bispectral "partners" of L.
One finds that one such pair (B, θ) is given by
B = d dz 3 − 3 d dz 2 · 1 z + 3 d dz · 1 z 2 + 3 0 1/z 2 0 0 and θ(x) the scalar-valued polynomial θ(x) = x 3 .
The set of all possible θ(x) is given by the following subalgebra A. The complete statement is given by where p ∈ M 2 (C) [x] and all the variables r 11 0 , r 12 0 , r 11 1 , r 12 1 , r 11 2 , r 22 2 , r 11 3 , r 12 3 , r 22 3 ∈ C. Then Γ = A. Moreover, for each θ we have an explicit expression for the operator B.
Furthermore, we have the presentation A = C · α 0 , α 1 | I = 0 with the ideal I given by
I := α 2 0 , α 3 1 + α 0 α 1 α 0 − 3α 1 α 0 α 1 + α 0 α 2 1 + α 2 1 α 0 .
This is an example of an algebra with an integral element over a nilpotent one.
For the next example take for ψ(x, z) the matrix-valued function
ψ(x, z) = d dx − 1/x −1/x 2 1/x 3 0 1/x −1/x 2 0 0 1/x e xz I = e xz z − 1/x 1/x 2 −1/x 3 0 z − 1/x 1/x 2 0 0 z − 1/x .
Here one can see that
Lψ = −z 2 ψ with L = − d dx 2 + 2 1/x 2 −2/x 3 3/x 4 0 1/x 2 −2/x 3 0 0 1/x 2 .
The results in this case about the set of all possible θ(x) are given below.
x 5 + x 6 p(x) , where p ∈ M 3 (C)[x]
and all the variables r 11 0 , r 12 0 , ..., r 33 5 ∈ C are arbitrary. Then, Γ = A and for each θ we have an explicit expression for the operator B.
Furthermore, we have the presentation A = C · α 2 , α 3 | I = 0 with
I = α 3 2 , α 2 3 − α 3 , (α 3 α 2 ) 2 α 3 − 4α 3 α 2 2 α 3 .
This is an example of an algebra with nilpotent and idempotent associated elements.
As the last example we consider a case when both "eigenvalues" F and θ are matrix valued. Let
ψ(x, z) = e xz (x − 2)xz x 3 z 2 −2x 2 z 2 −2x 2 z+3xz+2x−2 xz 1 x xz−2 z x 2 z − 2xz − x + 1 and L = 0 0 0 1 . d dx 2 + 0 1 (x−2)x 2 − 1 x−2 0 . d dx + − 1 x 2 (x−2) 2 x−1 x 3 (x−2) 2 2x−1 x(x−2) 2 − 2x 2 −4x+3 x 2 (x−2) 2 , then Lψ = ψF with F (z) = 0 0 0 z 2 .
It is easy to check that ψB = θψ for
B = d dz 3 . 0 0 1 0 + d dz 2 . 0 0 − 2z+1 z 0 + d dz . 1 0 2(z−1) z 2 1 + −z −1 0 6z −3 z −1 and θ(x) = x 0 x 2 (x − 2) x .
In this case we characterize the algebra A of all polynomial F such that there exist L = L x, d dx with Lψ = ψF as follows Theorem 3. Let Γ be the sub-algebra of M 2 (C)[z] of the form
a 0 b − a b + c c a − b − c −c z + a − b − c c + a − b d e z 2 2 + z 3 p(z),
We close this section by remarking the important role played by Darboux transformations, which goes back to Duistermaat and Grünbaum (1986) and . Indeed, the three examples in this section are instances of rational Darboux transformations from the scalar matrix exponential functions. All such Darboux transformations were shown to be bispectral in Theorems 1.1 and 1.2 of Geiger et al. (2017).
A few extensions of the problems discussed above
Since one of the goals of this paper is to serve as an "invitation" to look at this problem extended to a wider audience, we give a road map with some selected references. Indeed, the bispectral problem has many different incarnations, and in our opinion we are still far from having a unified theory.
In the scalar case here are some early papers that should be mentioned are Kasman and Rothstein (1997); Wilson (1993Wilson ( , 1998; Bakalov et al. (1996Bakalov et al. ( , 1998a; Grünbaum and Yakimov (2002); Geiger et al. (2017).
One natural issue concerns the numerical aspects involving the prolate spheroidal functions. In this case, the reader may want to consult the book Osipov et al. (2013).
Another direction concerns, the purely discrete (actually finite) version of time-band-limiting. In an effort to better understand the commuting property in question, one of us looked at the case when the real line is replaced by the N roots of unity. See Grünbaum (1981) as well as Perlstadt (1984); Perline (1987). Expression (11) in Grünbaum (1981) contains a small typo: the r(2) in the denominator should be replaced by r(1).
Moving on to the discrete-continuous version of the bispectral problem, we have that for the scalar case, involving orthogonal polynomials satisfying differential equations the problem had already been considered by S. Bochner (and even earlier). A very good introduction to this is given in Haine (2001) and its references. See also ; ; Grünbaum et al. (1997); .
For the matrix valued case there are two sources of early examples, one resulting from the theory of matrix valued spherical functions, see Grünbaum et al. ( , 2002aGrünbaum et al. ( ,b, 2003; ; Grünbaum et al. (2005), and another one see Durán and ; Durán and de la Iglesia (2008). See also Grünbaum and Iliev (2003) as well as Boyallian and Liberati (2008); Miranian (2005); Reach (1988).
Solutions of the bispectral problem can be used to obtain integral operators which reflect some ordinary differential operator in the sense of (for instance) Casper et al. (2019). This fact generalizes the commuting property in the scalar case. It would be interesting to see whether this could be extended to the matrix case.
In the (noncommutative) matrix case, by considering operators in the physical and spectral variables acting from opposite directions, we maintain the Ad-conditions that played a substantial role in Duistermaat and Grünbaum (1986).
In this case, this leads to the embedding of the bispectral algebras of eigenvalues into the matrix polynomial algebra M N (C[x]). See Vasquez and Zubelli (2021).
Another natural direction would be to look for a characterization for algebras relating to the spin-Calogero system for matrices of arbitrary size of matrix N . The examples N = 1 and N = 2 were generalized to arbitrary matrix size N and was characterized as a subalgebra of M N (C[x]) using a family of maps {P k } k∈N satisfying some nice properties such as translation and a product rule similar to the Leibniz rule. See Vasquez and Zubelli (2021).
Finally, let us go back to the original problem. We recall that both in the scalar and in the matrix case the motivation behind the bispectral problem was a desire to understand "what is behind" the remarkable commutativity property between the operator of time-and-band limiting in the Fourier, Bessel and Airy cases. It was hard to suspect that this problem would have connections with many of the recent developments in integrable systems. There has been some progress in connecting the bispectral problem with the commutativity property mentioned above, and here againat least in the scalar case-there are some connections with Integrable systems, see for instance Grünbaum (2021); Casper and Yakimov (2020); Casper et al. (2019); Casper et al. (2021a,b). For connections between the bispectral problem and the commuting property see Castro and Grünbaum (2017); Grünbaum et al. ( , 2018.
we have a Bessel or an Airy case. If M > 2, there are two families of solutions a) L is obtained from L 0 = − d dx 2 by a finite number of Darboux transformations (L = AA * →L = A * A).
θ
: U ⊂ C → M N (C) and the nontrivial common eigenfunction ψ : U × V ⊂ C 2 → M N (C) are in principle compatible sized meromorphic matrix valued functions defined in suitable open subsets U, V ⊂ C.
Theorem 1 .
1Let Γ be the sub-algebra of M 2 (C)[x]
Theorem 2 .
2Let Γ the sub-algebra of M 3 (C)[x]
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The Bochner-Krall problem: Some new perspectives. Luc Haine, ISBN 0-7923-7119-4Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute. Tempe, AZ, USA; DordrechtKluwer Academic PublishersLuc Haine. The Bochner-Krall problem: Some new perspectives. In Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute, Tempe, AZ, USA, May 29-June 9, 2000, pages 141-178. Dordrecht: Kluwer Academic Publishers, 2001. ISBN 0-7923-7119-4.
The bispectral problem: An overview. F , Alberto Grünbaum, ISBN 0-7923-7119-4Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute. Tempe, AZ, USA; DordrechtKluwer Academic PublishersF. Alberto Grünbaum. The bispectral problem: An overview. In Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute, Tempe, AZ, USA, May 29-June 9, 2000, pages 129-140. Dordrecht: Kluwer Academic Publishers, 2001. ISBN 0-7923-7119-4.
Associated polynomials, spectral matrices and the bispectral problem. F , Alberto Grünbaum, Luc Haine, 10.4310/MAA.1999.v6.n2.a6Methods Appl. Anal. 62F. Alberto Grünbaum and Luc Haine. Associated polynomials, spectral matrices and the bispectral problem. Methods Appl. Anal., 6(2):209-224, 1999. ISSN 1073-2772. doi:10.4310/MAA.1999.v6.n2.a6.
On the Krall-Hermite and the Krall-Bessel polynomials. F , Alberto Grünbaum, Luc Haine, Emil Horozov, 10.1155/S1073792897000615Int. Math. Res. Not. 19F. Alberto Grünbaum, Luc Haine, and Emil Horozov. On the Krall-Hermite and the Krall-Bessel polynomials. Int. Math. Res. Not., 1997(19):953-966, 1997. ISSN 1073-7928. doi:10.1155/S1073792897000615.
Some functions that generalize the Krall-Laguerre polynomials. F , Alberto Grünbaum, Luc Haine, Emil Horozov, 0377-0427J. Comput. Appl. Math. 1062F. Alberto Grünbaum, Luc Haine, and Emil Horozov. Some functions that generalize the Krall-Laguerre polynomials. J. Comput. Appl. Math., 106(2):271-297, 1999. ISSN 0377-0427.
A matrix-valued solution to Bochner's problem. F A Grünbaum, I Pacharoni, J Tirao, 10.1088/0305-4470/34/48/328J. Phys. A, Math. Gen. 3448F. A. Grünbaum, I. Pacharoni, and J. Tirao. A matrix-valued solution to Bochner's problem. J. Phys. A, Math. Gen., 34(48):10647-10656, 2001. ISSN 0305-4470. doi:10.1088/0305-4470/34/48/328.
Matrix valued spherical functions associated to the complex projective plane. F A Grünbaum, I Pacharoni, J Tirao, 10.1006/jfan.2001.3840J. Funct. Anal. 1882F. A. Grünbaum, I. Pacharoni, and J. Tirao. Matrix valued spherical functions associated to the complex projective plane. J. Funct. Anal., 188(2):350-441, 2002a. ISSN 0022-1236. doi:10.1006/jfan.2001.3840.
Matrix valued spherical functions associated to the three dimensional hyperbolic space. F A Grünbaum, I Pacharoni, J Tirao, 10.1142/S0129167X02001460Int. J. Math. 137F. A. Grünbaum, I. Pacharoni, and J. Tirao. Matrix valued spherical functions associated to the three dimensional hyperbolic space. Int. J. Math., 13(7):727-784, 2002b. ISSN 0129-167X. doi:10.1142/S0129167X02001460.
Matrix valued orthogonal polynomials of the Jacobi type. F A Grünbaum, I Pacharoni, J Tirao, 10.1016/S0019-3577(03)90051-6Indag. Math., New Ser. 143-4F. A. Grünbaum, I. Pacharoni, and J. Tirao. Matrix valued orthogonal polynomials of the Jacobi type. Indag. Math., New Ser., 14(3-4):353-366, 2003. ISSN 0019-3577. doi:10.1016/S0019-3577(03)90051-6.
An invitation to matrix-valued spherical functions: Linearization of products in the case of complex projective space P 2 (C). In Modern signal processing. Alberto Grünbaum, Inés Pacharoni, Juan Tirao, Cambridge University PressCambridgeISBN 0-521-82706-XAlberto Grünbaum, Inés Pacharoni, and Juan Tirao. An invitation to matrix-valued spherical functions: Lineariza- tion of products in the case of complex projective space P 2 (C). In Modern signal processing., pages 147-160. Cambridge: Cambridge University Press, 2004. ISBN 0-521-82706-X.
Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory. F , Alberto Grünbaum, Inés Pacharoni, Juan Alfredo Tirao, http:/www.numdam.org/articles/10.5802/aif.2151/Annales de l'Institut Fourier. 556F. Alberto Grünbaum, Inés Pacharoni, and Juan Alfredo Tirao. Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory. Annales de l'Institut Fourier, 55(6):2051-2068, 2005. doi:10.5802/aif.2151. URL http://www.numdam.org/articles/10.5802/aif.2151/.
Orthogonal matrix polynomials satisfying second-order differential equations. J Antonio, F Durán, Alberto Grünbaum, 10.1155/S1073792804132583Int. Math. Res. Not. 10Antonio J. Durán and F. Alberto Grünbaum. Orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not., 2004(10):461-484, 2004. ISSN 1073-7928. doi:10.1155/S1073792804132583.
Matrix orthogonal polynomials satisfying second-order differential equations: coping without help from group representation theory. J Antonio, F Durán, Alberto Grünbaum, 10.1016/j.jat.2007.02.003J. Approx. Theory. 1481Antonio J. Durán and F. Alberto Grünbaum. Matrix orthogonal polynomials satisfying second-order differential equa- tions: coping without help from group representation theory. J. Approx. Theory, 148(1):35-48, 2007. ISSN 0021- 9045. doi:10.1016/j.jat.2007.02.003.
Second-order differential operators having several families of orthogonal matrix polynomials as eigenfunctions. J Antonio, Manuel D Durán, De La Iglesia, 10.1093/imrn/rnn084Int. Math. Res. Not. 84Antonio J. Durán and Manuel D. de la Iglesia. Second-order differential operators having several families of orthogonal matrix polynomials as eigenfunctions. Int. Math. Res. Not., 2008:24, 2008. ISSN 1073-7928. doi:10.1093/imrn/rnn084. Id/No rnn084.
Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory. L Miranian, 10.1088/0305-4470/38/25/009J. Phys. A, Math. Gen. 3825L. Miranian. Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory. J. Phys. A, Math. Gen., 38(25):5731-5749, 2005. ISSN 0305-4470. doi:10.1088/0305-4470/38/25/009.
Generating difference equations with the Darboux transformation. Michael Reach, 10.1007/BF01218079Communications in Mathematical Physics. 1193Michael Reach. Generating difference equations with the Darboux transformation. Communications in Mathematical Physics, 119(3):385-402, September 1988. doi:10.1007/BF01218079.
Reflective prolate-spheroidal operators and the KP/KdV equations. Riley Casper, F Alberto Grünbaum, Milen Yakimov, Ignacio Zurrián, 10.1073/pnas.1906098116Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA116Riley Casper, F. Alberto Grünbaum, Milen Yakimov, and Ignacio Zurrián. Reflective prolate-spheroidal oper- ators and the KP/KdV equations. Proc. Natl. Acad. Sci. USA, 116(37):18310-18315, 2019. ISSN 0027-8424. doi:10.1073/pnas.1906098116.
Commuting integral and differential operators and the master symmetries of the Korteweg-de Vries equation. Alberto Grünbaum, 10.1088/1361-6420/ac11c6Inverse Probl. 37885010Alberto Grünbaum. Commuting integral and differential operators and the master symmetries of the Korteweg-de Vries equation. Inverse Probl., 37(8):21, 2021. ISSN 0266-5611. doi:10.1088/1361-6420/ac11c6. Id/No 085010.
Integral operators, bispectrality and growth of Fourier algebras. Riley Casper, Milen T Yakimov, 10.1515/crelle-2019-0031J. Reine Angew. Math. 766Riley Casper and Milen T. Yakimov. Integral operators, bispectrality and growth of Fourier algebras. J. Reine Angew. Math., 766:151-194, 2020. ISSN 0075-4102. doi:10.1515/crelle-2019-0031.
Reflective prolate-spheroidal operators and the adelic grassmannian. W Riley Casper, F Alberto Grünbaum, Milen Yakimov, Ignacio Zurrián, Comm. Pure and Applied Mathematics. to appear inW. Riley Casper, F. Alberto Grünbaum, Milen Yakimov, and Ignacio Zurrián. Reflective prolate-spheroidal operators and the adelic grassmannian. to appear in Comm. Pure and Applied Mathematics, 2021a.
Algebras of differential operators for kernels of Airy type. W R Casper, F A Grünbaum, M Yakimov, I Zurrián, in preparationW. R. Casper, F. A. Grünbaum, M. Yakimov, and I. Zurrián. Algebras of differential operators for kernels of Airy type. in preparation, 2021b.
Time-and-band limiting for matrix orthogonal polynomials of Jacobi type. M Castro, F A Grünbaum, 10.1142/S2010326317400019Random Matrices Theory Appl. 641740001M. Castro and F. A. Grünbaum. Time-and-band limiting for matrix orthogonal polynomials of Jacobi type. Random Matrices Theory Appl., 6(4):12, 2017. ISSN 2010-3263. doi:10.1142/S2010326317400019. Id/No 1740001.
Time and band limiting for matrix valued functions: an integral and a commuting differential operator. F A Grünbaum, I Pacharoni, I Zurrián, 10.1088/1361-6420/aa53b8Inverse Probl. 33225005F. A. Grünbaum, I. Pacharoni, and I. Zurrián. Time and band limiting for matrix valued functions: an integral and a commuting differential operator. Inverse Probl., 33(2):14, 2017. ISSN 0266-5611. doi:10.1088/1361-6420/aa53b8. Id/No 025005.
Algebraic Heun operator and band-time limiting. Alberto Grünbaum, Luc Vinet, Alexei Zhedanov, 10.1007/s00220-018-3190-0Commun. Math. Phys. 3643Alberto Grünbaum, Luc Vinet, and Alexei Zhedanov. Algebraic Heun operator and band-time limiting. Commun. Math. Phys., 364(3):1041-1068, 2018. ISSN 0010-3616. doi:10.1007/s00220-018-3190-0.
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| Time-resolved measurements of quantum dynamics are based on the availability of controlled events (e.g. pump and probe pulses) that are shorter in duration than the typical evolution time scale of the dynamical processes to be observed. Here we introduce the concept of noise-enhanced pump-probe spectroscopy, allowing the measurement of dynamics significantly shorter than the average pulse duration by exploiting randomly varying, partially coherent light fields consisting of bunched colored noise. It is shown that statistically fluctuating fields can be superior by more than a factor of 10 to frequency-stabilized fields, with important implications for time-resolved pump-probe experiments at x-ray free-electron lasers (FELs) and, in general, for measurements at the frontiers of temporal resolution (e.g. attosecond spectroscopy). As an example application, the concept is used to explain the recent experimental observation of vibrational wave-packet motion in a deuterium molecular ion on time scales shorter than the average pulse duration. | 10.1103/physrevlett.108.098302 | [
"https://arxiv.org/pdf/1110.5536v2.pdf"
]
| 32,227,971 | 1110.5536 | 51051b9fc603daf91441ba4802b8b4f6166cfe01 |
Noisy pulses enhance temporal resolution in pump-probe spectroscopy
24 Jan 2012
Kristina Meyer
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Christian Ott
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Philipp Raith
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Andreas Kaldun
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Yuhai Jiang
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Arne Senftleben
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Moritz Kurka
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Robert Moshammer
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Joachim Ullrich
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Thomas Pfeifer
Max-Planck-Institut für Kernphysik
Saupfercheckweg 169117HeidelbergGermany
Noisy pulses enhance temporal resolution in pump-probe spectroscopy
24 Jan 2012(Dated: January 25, 2012)1 arXiv:1110.5536v2 [physics.optics]numbers: 8253Hn8253Eb4255Vc0250Ey * Electronic address: tpfeifer@mpi
Time-resolved measurements of quantum dynamics are based on the availability of controlled events (e.g. pump and probe pulses) that are shorter in duration than the typical evolution time scale of the dynamical processes to be observed. Here we introduce the concept of noise-enhanced pump-probe spectroscopy, allowing the measurement of dynamics significantly shorter than the average pulse duration by exploiting randomly varying, partially coherent light fields consisting of bunched colored noise. It is shown that statistically fluctuating fields can be superior by more than a factor of 10 to frequency-stabilized fields, with important implications for time-resolved pump-probe experiments at x-ray free-electron lasers (FELs) and, in general, for measurements at the frontiers of temporal resolution (e.g. attosecond spectroscopy). As an example application, the concept is used to explain the recent experimental observation of vibrational wave-packet motion in a deuterium molecular ion on time scales shorter than the average pulse duration.
Noise, the absence of order and correlation, is ubiquitous in nature. Typically, noise represents a nuisance or even a serious problem in experimental studies of structure or dynamics. With regard to spectroscopy applications, the laser has led to fast-paced progress by exhibiting remarkable coherence properties. Coherence helps to combat noise by inducing structure (fixed phase relations) in time or the spectral domain and thus enables applications such as high-resolution spectroscopy [1], laser cooling [2], and ultrafast probing of quantum dynamics on time scales down to attoseconds [3]. Recent major achievements in several fields based on such high-coherence sources have distracted from the fact that there could be beneficial aspects of temporally noisy light sources.
One example for the benefits of noise is stochastic resonance that was found to be a far-reaching concept in natural systems [4]. It has also been recently shown that noisy light fields can enhance ionization of atoms in moderately strong laser fields [5]. Previous work has studied the influence of noisy pulse shapes on non-resonant autocorrelation measurements [6] and to increase spectral resolution in linear [7] and nonlinear Coherent
Raman scattering experiments [8][9][10], and two-photon absorption with quantum-correlated noise [11]. However, it has not been recognized that noisy pulse shapes can improve the temporal resolution in pump-probe spectroscopy for directly measuring quantum dynamics, such as molecular or electronic wave-packet motion. It is a commonly held belief that in loworder nonlinear processes, the pulse duration limits the temporal resolution for dynamical probing experiments.
Here, we introduce a new concept in time-resolved spectroscopy: using varying but correlated pairs of noisy pulse shapes to enhance temporal resolution in pump-probe experiments far beyond the average-pulse-duration limit. For an example experiment discussed in this letter, the temporal resolution is increased by a factor of 10 from 30 to 3 fs. This finding thus also creates a new paradigm in the current quest for achieving the finest temporal resolution of quantum-dynamical processes: noisy light fields as an accessible alternative if dispersive compression of broadband coherent spectra is not possible. The need to consider and rethink temporal noise has been stimulated by the development of Free-Electron Laser (FEL) Sources as FEL pulses are known to exhibit statistically varying shapes [12,13]. The growing field of FEL science thus provides excellent examples to demonstrate the concept. Here, we chose to describe the conceptual mechanism for the example of a sequential second-order nonlinear process, the probing of induced molecular wave-packet dynamics in a deuterium molecule.
The general physical picture to understand the time-resolved pump-probe experiments with statistically varying fields is also applicable to other dynamical spectroscopy techniques. We also emphasize that the presented mechanism is universal and not even limited to optical molecular spectroscopy. It is a qualitatively new conceptual finding that can be transferred and extended to various applications in physics, technology (communications), chemistry, and the life sciences, and across various time scales and processes, wherever noisy sources of signals are involved.
The experiment discussed here was recently performed at the Free-electron LASer at Hamburg (FLASH) [14]. This XUV-pump-XUV-probe experiment investigated the twophoton double ionization (TPDI) of D 2 using FEL pulses at an energy of 38 eV, with a bandwidth of ∼0.53 eV FWHM, and an average pulse duration of 30 fs. In this study oscillations on a time scale of ∼20 fs were measured, which were initially unexpected given the longer average pulse duration of 30 fs. The variation of the temporal pulse profiles from shot to shot is illustrated in Fig. 1(a).
In the experiment [14], the dynamics of the nuclear wave packet was monitored by measuring the kinetic energy release (KER) of the produced D + ions as a function of a variable time delay τ between the pump and the probe pulses. Both pulses were derived out of the same FEL beam using a two-component split mirror (linearly cut in the center). The XUV-laser electric field experienced by the D 2 molecules in the interaction region is then the sum of the identical electric fields E(t) of pump and time-delayed probe, translating into an intensity I τ (t)
I τ (t) = |E(t) + E(t + τ )| 2 .(1)
In this and any other (electronic or molecular dynamics) sequential TPDI process the molecule can absorb photon 1 and photon 2 at different times t and t . As the absorption of each photon proceeds into a continuum [see Fig. 1
N total (τ ) ∝ ∞ −∞ dt t −∞ dt I τ (t )I τ (t ).(2)
Introducing the time difference t c = t − t of the photon absorption and swapping integration order leads to
N total (τ ) ∝ ∞ 0 dt c ∞ −∞ dt I τ (t − t c )I τ (t ) = ∞ 0 dt c A (2) c (t c , τ ),(3)
where A
c (t c , τ ) = ∞ −∞ dt I τ (t) · I τ (t − t c )(2)
, with t c now interpreted as correlation time, is in the following referred to as the two-dimensional autocorrelation (2dAC) function. It will be shown below that this function is a very general object in the description of sequential second-order pump-probe experiments, irrespective of measuring molecular or electronic dynamics.
In the following, we use a molecular response function M (E KER , t c ) to describe the measured kinetic-energy release (KER) distribution of D + ions as a function of the temporal separation t c between two delta-like intensity pulses. The justification for this response function derives from the following considerations: photon 1 promotes the molecule to the D + 2 1sσ g molecular potential curve by removing one electron, while photon 2 removes the second electron to create the Coulomb-exploding D 2+ 2 state. In the time between the absorption of photon 1 and 2, a molecular wave packet evolves on the 1sσ g state. When the second photon is absorbed, the momentary internuclear-distance distribution of the molecular wave packet will be mapped to KER by Coulomb explosion [15]. The number of measured doubly ionized atoms with a specific kinetic-energy release and for a chosen time delay τ between pump and probe pulse is then given by
N (E KER , τ ) ∝ ∞ 0 dt c M (E KER , t c ) A (2) c (t c , τ ).(4)
While the 2dAC function A
c (t c , τ ) contains the information on the probe light structure (including its noisy properties), the response function M (E KER , t c ) thus contains the physical (quantum-dynamics) information of the system to be studied. For most frequently studied sequential processes (e.g. exciting a state and following its decoherence and decay), it could be calculated for any pump-probe scenario, also for exciting superpositions of electronic states, i.e. time-dependent electron wave packets. There, the response function could also depend on the observed photoelectron energies instead of or in addition to KER.
In the simulation, we used the partial-coherence method (PCM) [13] to generate a set of FEL pulses starting from the average spectrum A(ω) measured in the D 2 experiment [14] and a random spectral phase φ(ω). Looking at N total one can clearly see the signal enhancements for small time delays as seen before in experiments [16]. Fig. 3(d), the dynamics is clearly resolved. Interestingly, although the average pulse duration of the FEL is also 30 fs, the dynamics can be retrieved [see Fig. 3 [14].
The contribution of signal at higher KER is underestimated in our simulation for two reasons:
(1) The cross-section for the ionization step from D + 2 to D 2+ 2 is assumed to be independent of the internuclear distance. (2) The direct (nonresonant) TPDI process is neglected which would lead to a contribution at higher KER.
It should be pointed out that the sum of the 2dACs for 1.12 fs short and 30 fs bandwidth- limited pulses is showing slightly finer "spikes" as the FEL average 2dAC, as can be seen in In conclusion, we presented the concept of noise-enhanced temporal resolution in pumpprobe experiments. We thus also provide the physical mechanism behind the recent surprising observation of sub-pulse-duration dynamics in D 2 measured with statistically varying FEL pulses. We find that dynamical features much shorter than the average pulse duration (on the order of the coherence time of the individual pulses) can be resolved by employing the correlated nature of the noise in the pump and the probe pulses. Importantly, it should be noted that the choice of the molecular response function -also neglecting excitations to all other repulsive potential energy curves that do not result in oscillatory wave-packet dynamics -in this work was only to demonstrate the mechanism and the applicability of the 2dAC functions for well defined and statistical pulse shapes. While a different and better-suited response function can be used for the described example system of D + 2 , the applicability of the presented mechanism is more general: Important consequences arise in the current race for shorter and shorter pulses and better dynamical resolutions in attosecond (and beyond) science. The mechanism could help for the case of extremely broadband highharmonic spectra that are currently generated in experiments [22][23][24][25] and that can likely with other (e.g. spatial superresolution/deconvolution) techniques where the centroid of an intensity distribution can be determined much more accurately (∝ 1/ √ N , N the number of measurements) than its width. In our case, the resolution is defined by the sharp temporal feature in the 2d-autocorrelation functions, which is, however, superimposed on a broader component (resulting in a "background" in the measurement). We show the evolution of this sharp component for averaging over different numbers of pulses in Fig. 1. For single FEL pulses, the 2dAC function varies significantly from pulse to pulse (green curves). A spike is always located at the top of the broad pedestals (t c = τ ), but there are several other spikes whose positions change from pulse to pulse due to the random pulse shapes. If we start averaging over several N FEL pulses, the different peaks are washed out more and more with increasing N except for the narrow spikes on top of the pedestals. Our temporal resolution is given by the locations, intensities, and widths of these spikes, becoming maximal if only the singular spike at t c = ±τ is obtained for averaging over a large-enough number N of pulses.
For the case considered here, N = 100 seems to be sufficient to obtain maximum temporal resolution. This number will vary, however, with the parameters average pulse duration and average spectrum (i.e. coherence time), and needs to be obtained for each given set of parameters, e.g. by performing a convergence analysis as described here.
In order to quantitatively compare our simulation to the experiment the measured ion yield is displayed in Fig. 2 as well as the expected yield derived from our simulation. Thereby, we summed over the KER range of 6 -12 eV (cf. Fig. 3 main paper in Fig. 4(c). For low count numbers, the dynamics in the lower KER range is difficult to identify [ Fig. 3(a)]. The wave-packet-dynamics structures become more and more apparent with increasing number of total counts. For about 4.5·10 6 counts [ Fig. 3(d)], the dynamics can be clearly recovered.
II. GENERAL APPLICABILITY, EXAMPLE FOR ATTOSECOND PULSES
In order to further illustrate that the introduced concept is not only applicable to FEL pump-probe experiments, but more general as we stated in the last paragraph of the paper, we here apply our method to broadband high-harmonic attosecond-pulsed light. As an example, we consider pump-probe experiments using noisy attosecond pulses. Attosecond pump-probe experiments will have a wide-spread applicability throughout the physical and chemical sciences and a proof-of-principle experiment has very recently been published [2].
As attosecond pulses are typically coherent but exhibit a chirp which is not easily compressed by optical elements, it is interesting to study the possibility of using our noise-enhancement method to realize effectively bandwidth-limited resolution also in the attosecond pulse case.
Here, it is important to study the influence of residual coherence existing in the attosecond pulse. Maximum coherence would be obtained if all important laser parameters such as intensity, carrier-envelope phase (CEP) and duration can be kept constant from shot to shot, a case which is difficult to realize in experiments.
For the calculation we use an isolated (gated) attosecond pulse generated by an IR light pulse with a center wavelength of 800 nm, a pulse duration of 5 fs FWHM and a peak electric field of 0.2 a.u. The attosecond pulse exhibits a chirped spectrum with a pulse duration of about 9 a.u. (≈ 220 as) FWHM. The calculation of the two-dimensional autocorrelation (2dAC) function A (2) c (t c , τ ) was performed as described in the paper. Firstly, we add noise to the attosecond pulse original spectral phase, numerically discretized in frequency steps of ∆ω = 0.2 fs −1 . In order to study the influence of the amount of added phase noise, the phase was randomly varied at each frequency with maximum random excursions x between zero and 2π from its original value. Secondly, an average pulse duration of 600 as was assumed by The noisy-pulse scheme with the attainable atomic-unit time resolution would thus be applicable to measurements of electron dynamics based on sequential (i.e. resonant, ionizing) processes, such as creating (multi-)electronic wave-packet motion in (ionic) excited states.
The herein presented calculation shows that the concept of noise-enhanced temporal resolution is not only restricted to experiments with FEL pulses, but also applicable to attosecond pump-probe spectroscopy, paving the way to observing electron dynamics on the few attosecond time scale, especially when high-intensity long-wavelength drivers are able to push high-harmonic cutoffs to several keV and beyond.
As a note towards an experimental realization, the noise could be introduced for instance using non-stationary mirrors with rough surfaces or, possibly more efficiently, employing multilayer-nanofluidic mirrors where different spectral parts of the attosecond pulse are randomly phased or delayed at different penetration depths into such a mirror. It should be noted that the noise-enhanced method is even superior to creating a perfect multilayer mirror for phase compensation (which seems unlikely for future coherent soft-x-ray ranges exceeding several keV), due to the sensitivity of the attosecond pulse chirp on pulse intensity and CEP fluctuations. A noise-enhanced scheme is inherently tolerant to (and even augmented by) such chirps and thus phase variations of the attosecond pulses and robustly provides high
online). The essential elements of the pump-probe light-matter interaction experiment: (a) The temporal pulse shapes of (top to bottom) a 1.12 fs short pulse, a 30 fs pulse and a set of the FEL pulses. (b) A schematic of potential-energy curves of D 2 and its cations, indicating the pathway of the sequential two-photon double-ionization (TPDI) process.
A Gaussian envelope as filter in the time domain accounts for the average experimental FEL pulse duration of 30 fs. For comparison, we consider both a bandwidth-limited Gaussian 30 fs (FWHM) pulse and a very short pulse of 1.12 fs corresponding to the Fourier transform of the average experimental spectrum, i.e. the coherence time of the FEL pulses. Their temporal pulse profiles are shown inFig. 1(a).The 2dAC functions A
cFIG. 2 :
2(t c , τ ) for the three different pulses are displayed inFig. 2(a)-(c) as well as the 2dAC functions integrated over t c which corresponds to the total number of doubly ionized molecules N total [cf. Eq. (3)]. Since typically in the experiment the pump and probe beams do not overlap collinearly, optical-cycle interferences are washed out. We (Color online). Two-dimensional autocorrelation functions (2dAC) A
c
(t c , τ ), representing the intensity autocorrelation of a pair of identical (pump and probe) pulses as a function of their time delay τ : (a) for a statistical average over 2000 30-fs FEL pulses, (b) for 30 fs bandwidth limited (Gaussian) pulses and (c) for 1.12 fs short pulses corresponding to the Fourier transform of the average FEL spectrum assuming a flat spectral phase. Narrow lines for t c = 0 and t c = ±τ are observed in the averaged FEL pulse case resulting from the correlated nature of the noise in pump and probe pulses on a time scale of the coherence time. These narrow temporal features are the origin of the enhanced temporal resolution. The number of doubly ionized molecules N total is also displayed (red curves). A lineout at a time delay of 100 fs is shown in (d) for the FEL (solid blue), the bandwidth limited 30 fs (dotted red) and the 1.12 fs short pulse (dashed black). account for this by convoluting the 2dAC along the τ axis with a Gaussian of width on the order of the optical cycle duration. For the case of the bandwidth-limited 30 fs pulse, three broad lines appear, whereas these lines become very narrow for the 1.12 fs short pulse. For averaged FEL pulses, thin lines surrounded by broader pedestals occur, which are a consequence of the correlated nature of the noise on a time scale of the coherence time.
The 2dAC function for the FEL pulses closely resembles the sum of the 2dACs for the bandwidth limited 30 fs and the short 1.12 fs pulse [Fig. 2(d), see also Fig. 4(a)]. It is important to point out that the time resolution is determined by the time delay τ at which the main and the side peaks become separable in t c , and also by the width of features. As a remark, the displayed 2dAC at a given τ can also be interpreted in the following way: The main peak appearing around t c = 0 fs represents the case that both photons are absorbed in the same pulse whereas for the smaller peaks around t c = τ the photons originate from different pulses. For details on the number of pulses required to obtain a smooth 2dAC function as shown here, please refer to [17]. The model molecular response function M (E KER , t c ) to illustrate the application of the 2dAC concept was chosen in the following way: to describe the evolution of the molecular wave packet after absorption of the first photon, we calculated the time-dependent evolution of the molecular wave packet Ψ(R, t c ) as a function of time t c on the 1sσ g curve with the initial condition of Ψ(R, t c = 0) being equal to the neutral stationary molecular ground-state wave function in D 2 . The time-dependent Schrödinger equation was solved using a split-step operator approach[18][19][20]. The internuclear wave-packet evolution |Ψ(R, t c )| 2 is shown inFig. 3(a), revealing an oscillatory dynamics on a time scale of approximately 20 fs. It is then converted to the molecular response function M (E KER , t c ) by using the Coulomb-explosion mapping KER = 1/R[15].The pump-probe KER spectra N (E KER , τ ) can now be calculated by using Eq. (4). For the case of the bandwidth-limited 30 fs pulse inFig. 3(c), no dynamics can be resolved, as the ≈ 20 fs dynamics is washed out by the long pulse. Employing the very short 1.12 fs pulse,
FIG. 3 :
3(Color online). The internuclear wave-packet evolution |Ψ(R, t c )| 2 is displayed in (a), yielding the molecular response function M (E KER , t c ) by employing the Coulomb-explosion mapping. Using the 2dAC functions A
t c , τ ) shown in Fig. 2, the KER distributions as a function of pump-probe time delay τ (as measurable in the experiment) N (E KER , τ ) can be directly obtained by integration N (E KER , τ ) ∝ +∞ 0 dt c M (E KER , t c )A
t c , τ ). Results for N (E KER , τ ) are shown for an average over 2000 FEL pulses (b), a 30 fs bandwidth-limited pulse (c) and a 1.12 fs pulse (d). While no temporal structure is visible for the 30 fs bandwidth-limited pulse (c), the 30 fs FEL pulses, exhibiting noisy substructure, resolve the dynamics (b).
Fig. 4 (
4a) for the case τ = 100 fs. The obtained temporal resolution of the dynamics will thus always be slightly reduced for the case of statistically fluctuating FEL pulses, compared to the case of extremely short coherence-time duration laser pulses, but still by a factor of 10 better than the 30 fs average pulse duration [seeFig. 4(a)]. It is particularly interesting that the here-discussed phenomenon of noise-enhanced resolution is independent of the average fromFig. 1(d)] for FEL pulses (blue) is compared to the sum of 30 fs bandwidth limited and 1.12 fs short pulses (red). (b): Fourier transform (absolute value) of the averaged FEL KER-vs.-τ trace [Fig. 3(b)]. (c): Pump-probe scans of the FEL pulses after removing the DC component (by Fourier transform): the result approximately recovers the dynamics obtained for the coherence-time limited 1.12 fs pulses [cf.
Fig. 3 (
3d)].pulse duration. Short time-scale spikes will always be present in the 2dACs as long as the coherence time of the source is sufficiently short (the average spectrum is sufficiently broad), just the ratio of spike-to-pedestal area would decrease and thus lower the usable dynamical signal vs. the background from the long pulse.The fast modulations arising by the wave-packet motion can be uncovered by means of a Fourier transform along the time delay τ axis [see Fig. 4(b)], removing the DC contribution and then transforming back into the time-delay domain [see Fig. 4(c)]. The dynamics and time-dependent shape of the molecular wave packet can thus be recovered with very high accuracy, closely resembling the expected signal for the extremely short (coherence-time limited) pulse shown inFig. 3(d). Further detail information on how counting statistics in the experiment influences the resolution of this method can be found in[17].
FIG. 1 :
1not be compressed to their bandwidth-limited few-as duration due to the absence of suitable dispersion-compensating optics. The herein presented results and novel paradigm suggest a new route: if compression is not possible, it is sufficient to statistically vary the spectral phase of the pulses. In [26] we show in detail the example for the attosecond pulse case, including the viability of the method for not fully incoherent spectral phases. This finding also has important consequences for nonlinear spectroscopy in the life sciences, as it enables time-resolved probing through moving quasi-randomly dispersive and scattering soft tissue, and could have other important applications in communications science and biological signal transmission to increase robustness of data transmission in the presence of noise. We gratefully acknowledge funding from the Max-Planck Gesellschaft within the scope of the Max-Planck Research Group (MPRG) program. pulses. Supplemental material to "Noisy pulses enhance temporal resolution in pump-probe spectroscopy" Kristina Meyer, Christian Ott, Philipp Raith, Andreas Kaldun, Yuhai Jiang, Arne Senftleben, Moritz Kurka, Robert Moshammer, Joachim Ullrich, and Thomas Pfeifer * Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany * Electronic address: [email protected] I. COMPARISON TO THE EXPERIMENT AND THE INFLUENCE OF STATIS-TICS In our simulation we averaged over 2000 pulses for the results shown in the main text of our letter. Before quantifying the impact of statistics on the temporal resolution itshould be noted that the noise-enhanced resolution as discussed here should not be confused Lineouts of the 2dAC function A
t c , τ ) at τ = 100 fs for different numbers of averaged pulses, three different averages each: single FEL pulses (green), average over three (blue), ten (red) and 100 pulses (black). For single random pulses several spikes occur at random t c , depending on the individual pulse shape. The larger the number of averaged pulses the more the (2d-)autocorrelation function converges to its final shape: the singular coherence spike on top of the broad pedestals, which results in the increased temporal resolution. Thus, when averaging over only few pulse shapes per time delay in a measurement, the temporal resolution would be less well determined.
FIG. 2 :
2Ion yields depending on the pump-probe time delay for the KER regime 6 -12 eV. The experimental curves (red) taken from[1] and the ion yields obtained from our simulation (black) are displayed. The results agree except for a temporal shift, which often is an unknown parameter in the experiment and needs to be obtained by fitting to a model. The numerical oscillation period of 23 ± 1 fs (obtained by statistics over the peak-to-peak and valley-to-valley delay times) agrees with the experimental period of 22 ± 4 fs.
FIG. 3 :
3(b) of the main text) and determined the period of the oscillation to be 23 ± 1 fs. This is in excellent agreement with the experimentally obtained period of 22 ± 4 fs [1]. Simulation (black curve) and experiment (red curve) agree well except for a shift along the axis of the time delay τ . However, the position of zero time delay in the experiment is often unknown and needs to be matched to the model. In addition, we examined the influence of counting statistics in the experiment, in particular for the reconstruction of the wave-packet motion by Fourier transform as described in the main text. Here, we simulated four different cases of experimental random Poissonian count statistics resulting in a total number of counts of 4.5·10 5 , 9·10 5 , 1.8·10 6 and 4.5·10 6 , and studied their impact on the Fourier-filtered KER-vs.-τ trace (see Fig. 3), shown in the Simulation of Fourier filtered KER-vs.τ traces for different total numbers of counts (Poissonian statistics): (a) 4.5·10 5 , (b) 9·10 5 , (c) 1.8·10 6 and (d) 4.5·10 6 counts.
a
Gaussian envelope as filter in the time domain, in order to model the maximum temporal noise introduced, e.g. by reflecting of rough mirrors for an experimental implementation of the noise source. The lineouts of the 2dAC functions at τ = 60 a.u. for the different values Amplitude [arb. residual spectral coherence and increasing spectral phase noise for the attosecond pump-probe case. Lineouts of the 2dAC function A
t c , τ ) at τ = 600 a.u. for different amounts of spectral phase noise in a range of x added to the original attosecond-pulse spectral phase: x = 0.2π (green), 0.6π (blue), 1.4π (red), and 2π (black). of x are shown inFig. 4. As in the example of FEL pulses, but now starting from a chirped coherent pulse, spikes on top of broader pedestals occur, leading to an enhanced temporal resolution. It can be clearly seen that the result depends on the amount of noise imprinted on the spectral phase. A minimum critical amount of noise is required to achieve enhanced temporal resolution. In our simulation, this occurs near a critical x c = 1.4π. For lower values of x, the spikes at the side peaks at t c = ±60 a.u. disappear. For x = 0.6π, the remaining pedestal does not show a Gaussian, but a rather triangular shape, representing the transition range to the coherent case. For very low x, i.e. 0.2π, the attosecond pulses can be regarded as being nearly fully coherent and the 2dAC lineout is determined by the original attosecond pulse duration (≈ 220 as) and shape. The width of the pedestals for higher x corresponds to the average statistical pulse duration of 600 as.The specific features of the 2dAC function shown above-the very narrow spikes with a FWHM of approximately 1.5 a.u. (≈ 36 as thus corresponding to an effective pulse duration of 25 as)-thus provide substantially (order of magnitude) enhanced temporal resolution over the original 220 as pulse duration. The obtained temporal resolution nearly reaches the bandwidth limit (i.e. assuming flat spectral phase of original attosecond pulse spectrum and no additional temporal filter) of about 0.8 a.u. (≈ 19 as) pulse duration, which is experimentally not feasible due to the lack of compensating optics for dispersion correction.
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| []
|
[
"The B-L/Electroweak Hierarchy in Heterotic String and M-Theory",
"The B-L/Electroweak Hierarchy in Heterotic String and M-Theory"
]
| [
"Michael Ambroso \nDepartment of Physics\nUniversity of Pennsylvania Philadelphia\n19104-6396PA\n",
"Burt A Ovrut \nSchool of Natural Sciences\nThe Institute for Advanced Study\n08540PrincetonNew Jersey\n\nDepartment of Physics\nUniversity of Pennsylvania Philadelphia\n19104-6396PA\n"
]
| [
"Department of Physics\nUniversity of Pennsylvania Philadelphia\n19104-6396PA",
"School of Natural Sciences\nThe Institute for Advanced Study\n08540PrincetonNew Jersey",
"Department of Physics\nUniversity of Pennsylvania Philadelphia\n19104-6396PA"
]
| []
| E 8 ×E 8 heterotic string and M-theory, when compactified on a Calabi-Yau threefold admitting an SU (4) vector bundle with Wilson lines, can give rise to the exact MSSM spectrum with three right-handed neutrino chiral superields, one per family. Rank preserving Wilson lines require that the standard model group be augmented by a gauged U (1) B−L . Since there are no fields in this theory for which 3(B − L) is an even, non-zero integer, the gauged B-L symmetry must be spontaneously broken at a low scale, not too far above the electroweak scale. It is shown that in these heterotic standard models, the B-L symmetry can be broken, with a phenomenologically viable B-L/electroweak hierarchy, by at least one right-handed sneutrino acquiring a vacuum expectation value. This is explicitly demonstrated, in a specific region of parameter space, using a renormalization group analysis and soft supersymmetry breaking operators. The vacuum state is shown to be a stable, local minimum of the potential and the resultant hierarchy is explicitly presented in terms of tanβ. | 10.1088/1126-6708/2009/10/011 | [
"https://arxiv.org/pdf/0904.4509v3.pdf"
]
| 11,729,543 | 0904.4509 | 0122e21d586e5c73a875dcbc740a541a4cb287d1 |
The B-L/Electroweak Hierarchy in Heterotic String and M-Theory
6 Oct 2009
Michael Ambroso
Department of Physics
University of Pennsylvania Philadelphia
19104-6396PA
Burt A Ovrut
School of Natural Sciences
The Institute for Advanced Study
08540PrincetonNew Jersey
Department of Physics
University of Pennsylvania Philadelphia
19104-6396PA
The B-L/Electroweak Hierarchy in Heterotic String and M-Theory
6 Oct 2009
E 8 ×E 8 heterotic string and M-theory, when compactified on a Calabi-Yau threefold admitting an SU (4) vector bundle with Wilson lines, can give rise to the exact MSSM spectrum with three right-handed neutrino chiral superields, one per family. Rank preserving Wilson lines require that the standard model group be augmented by a gauged U (1) B−L . Since there are no fields in this theory for which 3(B − L) is an even, non-zero integer, the gauged B-L symmetry must be spontaneously broken at a low scale, not too far above the electroweak scale. It is shown that in these heterotic standard models, the B-L symmetry can be broken, with a phenomenologically viable B-L/electroweak hierarchy, by at least one right-handed sneutrino acquiring a vacuum expectation value. This is explicitly demonstrated, in a specific region of parameter space, using a renormalization group analysis and soft supersymmetry breaking operators. The vacuum state is shown to be a stable, local minimum of the potential and the resultant hierarchy is explicitly presented in terms of tanβ.
An important goal of heterotic superstrings and M-theory is to show that these higher-dimensional theories can be "compactified" to four-dimensional, phenomenologically realistic particle physics. Specifically, one would like to prove that the minimal N = 1 supersymmetric standard model (MSSM), modified by the addition of three right-handed neutrino chiral supermultiplets, one per family, can arise in this manner. The necessity of having three right-handed neutrino supermultiplets puts strong constraints on heterotic model building. A natural way to achieve this is to compactify on smooth Calabi-Yau threefolds that admit slope-stable, holomorphic vector bundles with structure group SU(4). The non-vanishing connections associated with these bundles then spontaneously break the E 8 group of the heterotic theory down to Spin (10). Each 16 representation of Spin(10) contains a complete family of quarks/leptons plus a right-handed neutrino, exactly as required.
A second requirement is that the four-dimensional theory be symmetric, at least to a low energy scale, under R-parity [1,2] or, equivalently in a supersymmetric theory, matter-parity. This Z 2 symmetry prohibits dangerous baryon and lepton violating processes, such as rapid nucleon decay. The requirement of R-parity, however, also puts additional strong constraints on heterotic model building. While it is difficult in realistic smooth heterotic compactifications to obtain a Z 2 symmetry of the four-dimensional theory, in particular of the soft supersymmetry breaking interactions, it is straightforward to extend the standard model group by a gauged U(1) B−L , which contains matter-parity. Models of this type have been proposed within the context of field theory [3,4,5,6,7] and some string theories, such as heterotic orbifolds [8], in which, in addition to the MSSM matter spectrum, new chiral fields are added for which 3(B − L) is an even, non-zero integer. These new fields can acquire vacuum expectation values (VEVs) which, while spontaneously breaking gauged B-L symmetry at a high scale, preserve the matter-parity subgroup. Unfortunately, this is never possible in realistic smooth compactifications of heterotic theory, since the E 8 decomposition under the vector bundle structure group never has representations satisfying this condition. It follows that, in smooth heterotic compactifications, one is forced to consider the remaining possibility; that is, that U(1) B−L is spontaneously broken by 3(B − L) odd fields at a scale not too far above the electroweak scale. This will play the same role of suppressing baryon and lepton number violating operators.
In fact, a gauged U(1) B−L group arises naturally in the Spin(10) models discussed above. The traditional way to break the rank five Spin(10) to the standard model gauge group is by extending the SU(4) bundle with Abelian Wilson lines. These, however, preserve the rank of the gauge group and, hence, the rank four standard model group must be extended by a product with a rank one group, precisely, it turns out, U(1) B−L . A class of smooth heterotic compactifications of this type were constructed in [9,10]. Specifically, they compactify heterotic theory on elliptically fibered Calabi-Yau threefolds that admit a fixed-point free Z 3 ×Z 3 isometry [11,12]. Slope-stable holomorphic vector bundles with structure group SU(4) were constructed over them which spontaneously break E 8 to Spin (10). The Abelian Z 3 × Z 3 Wilson lines then further break Spin(10) to the low-energy gauge group
G = SU(3) C × SU(2) L × U(1) Y × U(1) B−L .(1)
The manifold and vector bundles being Calabi-Yau and slope-stable, holomorphic respectively, assure that the four-dimensional theory is N = 1 supersymmetric. In addition to the vector superfields corresponding to the gauge group in (1), the low-energy matter spectrum was found, using cohomology techniques introduced in [13,14,15], to be three families of quark and lepton chiral superfields, each family with a right-handed neutrino. They transform under the gauge group in the standard manner as
Q i = (3, 2, 1/3, 1/3), u i = (3, 1, −4/3, −1/3), d i = (3, 1, 2/3, −1/3) L i = (1, 2, −1, −1), ν i = (1, 1, 0, 1), e i = (1, 1, 2, 1)(2)
for the left and right-handed squarks and leptons respectively, where i = 1, 2, 3. The spectrum also contains one pair of Higgs-Higgs conjugate chiral superfields transforming as
H = (1, 2, 1, 0),H = (1, 2, −1, 0).(3)
This is precisely the matter and Higgs spectrum of the MSSM. In addition, the theory contains three Kahler moduli, three complex structure moduli and thirteen vector bundle moduli, all of which are uncharged under the gauge group (1). The supersymmetric potential energy is given by the usual sum over the modulus squared of the F and D-terms. The F -terms are determined from the most general superpotential invariant under the gauge group,
W = µHH + 3 i,j=1 λ u,ij Q i Hu j + λ d,ij Q iH d j + λ ν,ij L i Hν j + λ e,ij L iH e j (4)
Note that the dangerous lepton and baryon number violating interactions
L i L j e k , L i Q j d k , u i d j d k(5)
which generically would lead to rapid nucleon decay, are disallowed by the U(1) B−L gauge symmetry. To simplify the calculations, we will assume a mass-diagonal basis where λ u,ij = λ d,ij = λ ν,ij = λ e,ij = 0 for i = j and denote the diagonal Yukawa couplings by λ ii = λ i , i = 1, 2, 3. A constant, field-independent µ parameter cannot arise in a supersymmetric string vacuum since the Higgs fields are zero modes. However, the HH bilinear can have higher-dimensional couplings to moduli through both holomorphic and non-holomorphic interactions in the superpotential and Kahler potential respectively. When moduli acquire VEVs due to non-perturbative effects, these can induce non-vanishing supersymmetric contributions to µ. A non-zero µ can also be generated by gaugino condensation in the hidden sector. Why this induced µ-term should be small enough to be consistent with electroweak symmetry breaking is a difficult, model dependent problem. In this paper, we will not discuss this "µ-problem", but simply assume that the µ parameter is at, or below, the electroweak scale. In fact, so as to empasize the B-L/electroweak hierarchy and simplify the calculation, we will take µ, while non-zero, to be substantially smaller than the electroweak scale, making its effect sub-dominant. This can be implemented consistently throughout the entire scaling regime. The SU(3) C and SU(2) L D-terms are of the standard form, while
D Y = ξ Y + g Y φ A † (Y/2) AB φ B ,(6)D B−L = ξ B−L + g B−L φ † A (Y B−L ) AB φ B
where the index A runs over all scalar fields φ A . Note that each of these Abelian D-terms potentially has a Fayet-Iliopoulos (FI) additive constant. As with the µ parameter, constant field-independent FI terms cannot occur in string vacua since the low energy fields are zero modes. Field-dependent FI terms can occur in some contexts, see for example [17]. However, since both the hypercharge and B-L gauge symmetries are anomaly free, such field-dependent FI terms are not generated in the supersymmetric effective theory. We include them in (6) since they can, in principle, arise at a lower scale from radiative corrections once supersymmetry is softly broken [18]. Be that as it may, if calculations are done in the D-eliminated formalism, which we use in this paper, these FI parameters can be consistently absorbed into the definition of the soft scalar masses and their beta functions. Hence, we will no longer consider them.
In addition to the supersymmetric potential, the Lagrangian density also contains explicit "soft" supersymmetry violating terms [19]. Those relevant to this paper are V soft = V 2s + V 2f , where V 2s are the scalar quadratic terms
V 2s = 3 i=1 (m 2 Q i |Q i | 2 + m 2 u i |u i | 2 + m 2 d i |d i | 2 + m 2 L i |L i | 2 + m 2 ν i |ν i | 2 +m 2 e i |e i | 2 ) + m 2 H |H| 2 + m 2H |H| 2 − (BHH + hc),(7)
and V 2f contains the gaugino mass terms
V 2f = 1 2 M 3 λ 3 λ 3 + . . . hc.(8)
As above, we have taken the parameters in (7) and (8) to be flavor-diagonal. Cubic scalar interactions as well as the remaining gaugino mass terms can be chosen small enough to be ignored in this calculation, as discussed below. The heterotic compactifications described here satisfy the two criteria discussed above; that is, they give softly broken N = 1 supersymmetric theories with exactly the MSSM matter spectrum with three right-handed neutrinos, and their gauge group extends the standard model group by precisely a factor of U(1) B−L . However, to be realistic, these theories must spontaneously break the U(1) B−L symmetry not too far above the electroweak scale. Clearly, this can only be accomplished if at least one of the right-handed sneutrinos develops a non-vanishing vacuum expectation value. It is straightforward to show using (4), (6) and (7) that, assuming one is free to choose all parameters at the electroweak scale, both U(1) B−L and electroweak symmetry can be broken with a realistic hierarchy between them. Quintessentially, however, one is not free to so choose the parameters. As is well-known, their initial values just below the compactification scale are set by the geometric and bundle moduli expectation values [20,21,22,23,24]. At any lower scale, the parameters are determined by a complicated set of intertwined, nonlinear renormalization group equations (RGEs) [25,26,27,28,29,30,31]. Even if one chooses the initial values arbitrarily, it is unclear that these will allow for an appropriate spontaneous breakdown of both the U(1) B−L and electroweak symmetries. There are many potential problems that can occur. These range, for example, from inducing color or charge breaking expectation values, to not being able to break B-L at all, to breaking B-L but inducing a correlation with electroweak breaking that is unphysical, such as the electroweak scale being much larger than the B-L scale, and so on. All of these scenarios are easily realized. To prove that both U(1) B−L and electroweak symmetries can be broken with an appropriate hierarchy, one must show this explicitly by solving the RGEs for a specific choice of initial parameters. In this paper, we present the results of a quasi-analytic solution of the renormalization group equations valid for a restricted range of parameter space. The detailed calculations will be given elsewhere [32]. It will be shown in [32] that initial parameters can be chosen so that U(1) Y and U(1) B−L kinetic mixing [33,34] is small. Hence, we ignore such mixing in this paper. This solution demonstrates that an appropriate B-L/electroweak hierarchy can indeed be achieved for a range of initial parameters. We have backed up these results with explicit numerical solutions of the RGEs that will be presented elsewhere.
We begin our analysis with the renormalization group solution for the gauge parameters, g a , a = 1, . . . , 4, chosen so as to unify to g(0) ≃ .726 at the scale M u ≃ 3 × 10 16 GeV [35]. This choice of parameters requires the redefinition g Y = 3 5 g 1 , g B−L = 3 4 g 4 . One then finds, at an arbitrary scale t = ln( µ Mu ), that
g a (t) 2 = g(0) 2 1 − g(0) 2 bat 8π 2 , a = 1, . . . , 4 b = ( 33 5 , 1, −3, 12) .(9)
These results will be used throughout the analysis. We note that threshold effects and mass splitting between sleptons/squarks will tend to defocus gauge coupling unification. We will ignore these effects in the present paper. Now consider the gaugino masses or, more specifically, the products g 2 a |M a | 2 , a = 1, . . . , 4 that occur in the beta functions. Denoting the initial values of the gaugino masses by |M a (0)|, one finds
g a (t) 2 |M a (t)| 2 = g(0) 2 |M a (0)| 2 (1 − g(0) 2 bat 8π 2 ) 3 .(10)
Even assuming that the gaugino masses are "unified" at t = 0, making any
ratio ga(0) 2 |Ma(0)| 2 g b (0) 2 |M b (0)| 2 unity,
it is clear that the gluino mass contributions will quickly grow to dominate. For example, at the electroweak scale the ratio of the gluino to the SU(2) L gaugino terms is 25.6. In this paper, so as to simplify the calculation and allow for a quasi-analytic solution, we will not assume unified gaugino masses, instead taking |M 1 (0)| 2 , |M 2 (0)| 2 , |M 4 (0)| 2 ≪ |M 3 (0)| 2 . It follows that in beta functions containing a gluino mass term, the other gaugino terms are sub-dominant everywhere in the scaling regime and can be ignored. Recall that "non-unified" gaugino masses easily occur in string vacua, while unification requires additional "minimal" criteria [27,31]. These are not generically satisfied in our MSSM theory. A similar justification can be made for ignoring soft cubic scalar interactions as sub-dominant.
Next, we make a specific choice for the scalar masses at the unification scale M u . These are taken to be
m H (0) 2 = mH (0) 2 , m Q i (0) 2 = m u j (0) 2 = m d k (0) 2 , m L i (0) 2 = m e j (0) 2 = m ν k (0) 2(11)
for all i, j, k = 1, 2, 3. Note that the sneutrino masses are different than those of the remaining sleptons. This asymmetry is one ingredient in breaking U(1) B−L at an appropriate scale. Other than that, this choice is taken so as to simplify the RGEs as much as possible and to allow a quasi-analytic solution. We point out that soft scalar masses need not be "universal" in string theories, since they are not generically "minimal". We emphasize that a B-L/electroweak hierarchy is possible for a much wider range of initial parameters.
Since the U(1) B−L symmetry should be spontaneously broken by righthanded sneutrinos at energy-momenta larger than the electroweak scale, we begin by restricting the analysis to the slepton sector. This is possible, in part, because initial conditions (11) allow a decoupling of sleptons from the squarks and Higgs fields in the RGEs . These fields will be discussed later. Subject to the initial conditions (11) and associated assumptions, we find that
m L i (t) 2 = m L i (0) 2 + 1 6 (1 − (1 − g(0) 2 b 4 t 8π 2 ) −9/4b 4 )S ′ 1 (0), m e i ,ν i (t) 2 = m e i ,ν i (0) 2 − 1 6 (1 − (1 − g(0) 2 b 4 t 8π 2 ) −9/4b 4 )S ′ 1 (0)(12)
where
S ′ 1 (0) = 3 i=1 (−m L i (0) 2 + m ν i (0) 2 ) = 0 .(13)
Note that in deriving (12), we have assumed |M 1 (0)| 2 , |M 2 (0)| 2 , |M 4 (0)| 2 ≪ S ′ 1 (0). Using (10) and (12), it follows that the hypercharge, SU(2) L and B-L gaugino terms are sub-dominant to g 2 4 S ′ 1 at any scale. Hence, even in the slepton beta functions, which do not have a gluino contribution, the gaugino terms can be ignored..
Given these results, one can now consider U(1) B−L breaking at scales on the order of 10 4 GeV or, equivalently, at t B−L ≃ −28.7 . We begin by discussing the quadratic mass terms near the origin of field space. The relevant part of the scalar potential is V = V 2s + 1 2 D 2 B−L , where V 2s and D B−L are given in (7) and (6) respectively. Recall that the FI term is absorbed into the definition of the soft mass parameters. Expanding this, the slepton quadratic terms at any scale t are
V m 2 sleptons = 3 i=1 (m 2 L i |L i | 2 + m 2 e i |e i | 2 + m 2 ν i |ν i | 2 ) ,(14)
where i = 1, 2, 3 and the slepton masses are given by (12), (13). The first requirement for spontaneous B-L breaking is that at least one of the slepton effective squared masses becomes negative at t B−L . Clearly, this cannot happen for m L i (t B−L ) 2 , which is always positive. However, if the initial squared masses are sufficiently small and S ′ 1 (0) sufficiently large, both m e i (t B−L ) 2 and m ν i (t B−L ) 2 can become negative. Since the e i fields are electrically charged, we do not want them to get a VEV and, hence, we want m e i (t B−L ) 2 to be positive. On the other hand, the ν i fields are neutral in all quantum numbers except B-L. Hence, if they get a nonzero VEV this will spontaneously break B-L at t B−L , but leave the SU(3) C × SU(2) L × U(1) Y gauge symmetry unbroken. This is indeed possible for a wide range of initial parameters. For simplicity, let us choose the initial right-handed slepton masses to satisfy
m ν 1 (0) = m ν 2 (0) = Cm ν (0), m ν 3 (0) = m ν (0), m e 1 (0) = m e 2 (0) = m e 3 (0) = Am ν (0)(15)
which imply, using (11) and (13), that
S ′ 1 (0) = (1 + 2C 2 − 3A 2 )m ν (0) 2 .(16)
Taking, for specificity, A = √ 6 and C ≃ 9.12, then
S ′ 1 (0) = 149 m ν (0) 2(17)
and it follows from (9), (11), (12) and (15) that
m ν 1,2 (t B−L ) 2 ≃ 78.2 m ν (0) 2 , m ν 3 (t B−L ) 2 = −4m ν (0) 2 , m L i (t B−L ) 2 = 11m ν (0) 2 , m e i (t B−L ) 2 = m ν (0) 2(18)
for 1 = 1, 2, 3. We conclude from (18) that, near the origin of field space, there are positive quadratic mass terms in the L i , e i and ν 1,2 field directions for i = 1, 2, 3. However, m ν 3 (t B−L ) 2 is negative, suggesting a non-zero VEV in the ν 3 direction.
To determine this, one must minimize the complete potential V = V 2s + 1 2 D 2 B−L for the slepton fields. Restricted to these scalars, we find that the vacuum specified by
ν 1,2 = 0, ν 3 = 2m ν (0) 3 4 g 4 , L i = e i = 0(19)
with i = 1, 2, 3 is a local minimum of V . The slepton masses at this VEV are
m 2 ν 1,2 ≃ 82.2 m ν (0) 2 , m 2 ν 3 = 8m ν (0) 2 , m 2 L i = 7m ν (0) 2 , m 2 e i = 5m ν (0) 2 .(20)
Vacuum (19) spontaneously breaks the gauged B-L symmetry giving the B-L vector boson a mass,
M A B−L = 2 √ 2m ν (0) ,(21)
while preserving the remaining SU(3) C ×SU(2) L ×U(1) Y gauge group. Note that this result is quite robust, and should be applicable to any theory containing at least two right-handed sneutrinos. We now include the Higgs fields and squarks, and analyze their masses at t B−L around vacuum (19). To the order we are working,
m 2H ≃ m H (0) 2 .(22)
Using the previous assumptions, the hierarchy of Yukawa couplings, choosing
m Q 3 (0) 2 = m H (0) 2 2 ,(23)
and requiring that m 2 H be positive at all relevant scales, we find that
m 2 H ≃ m H (0) 2 e − 3 4π 2 0 t |λu 3 | 2 (1+[ − 2 3π 2 t ′ 0 g 2 3 |M 3 | 2 m 2 H ]) .(24)
Since λ u 3 scales slowly, we take it to be constant with its phenomenological value λ u 3 (0) = 1 .
As discussed previously, a non-vanishing supersymmetric µ parameter can arise from non-perturbative effects in the moduli and hidden sector. To simplify the calculations and focus on the B-L/electroweak hierarchy, we will, henceforth, assume that the µ parameter, while non-zero, is sufficiently smaller than the electroweak scale so that its effects are sub-dominant. Once this is implemented at one scale, it remains true over the entire scaling regime. Then, under the previous assumptions, the quadratic pure Higgs potential arises solely from (7) and is given by V m 2 Higgs = m 2 H |H| 2 + m 2H |H| 2 − B(HH + hc), where m 2 H , m 2H are given in (24), (22) and B satisfies a relatively simple RGE that won't be discussed here. Henceforth, we assume that for t ≪ 0 the coefficient B is such that
4 B m 2H − m 2 H 2 ≪ 1 .(26)
This is easily arranged by adjusting B(0). The Higgs mass matrix can then be diagonalized to V m 2
Higgs = m 2 H ′ |H ′ | 2 + m 2H ′ |H ′ | 2 , where m 2 H ′ ≃ m 2 H − m 2H B m 2H − m 2 H 2 , m 2H ′ ≃ m 2H − m 2 H B m 2H − m 2 H 2(27)
and
H ′ ≃ H − B m 2H − m 2 H H * ,H ′ ≃ B m 2H − m 2 H H * +H .(28)
It follows from (22), (26), and (27) that for any t ≪ 0
m 2H ′ ≃ m 2H = m H (0) 2 > 0 .(29)
Importantly, however, we see from (24), (27) that as t becomes more negative m 2 H can approach, become equal to and finally become smaller than m 2H (B/(m 2H − m 2 H )) 2 . As discussed shortly, our requirement that m 2 H be positive forces m 2 H ′ to vanish below t B−L . We conclude that at the B-L scale and evaluated at vacuum (19), the Higgs masses are
m 2 H ′ > 0, m 2H ′ ≃ m H (0) 2(30)
and, hence, electroweak symmetry is not yet broken. Now include the squarks and analyze their masses at t B−L around vacuum (19). Within the assumptions and approximations discussed earlier, it is straightforward to solve the renormalization group equations for the squarks at arbitrary scale t. The simplest are given by
m 2 Q 1,2 ≃ − 2 3π 2 t 0 g 2 3 |M 3 | 2 + 1 64π 2 t 0 g 2 4 S ′ 1 + m H (0) 2 2 , m 2 u 1,2 , d i ≃ − 2 3π 2 t 0 g 2 3 |M 3 | 2 − 1 64π 2 t 0 g 2 4 S ′ 1 + m H (0) 2 2(31)
where i = 1, 2, 3 and
− 2 3π 2 t 0 g 2 3 |M 3 | 2 = − 8 3b 3 ( 1 (1 − g(0) 2 b 3 t 8π 2 ) 2 − 1)|M 3 (0)| 2 ,(32)− 1 64π 2 t 0 g 2 4 S ′ 1 = − 1 18 ( 1 (1 − g(0) 2 b 4 t 8π 2 ) 9 4b 4 − 1)149 m ν (0) 2 .(33)
Note that both integrals (32) and (33) are positive for t < 0. Somewhat more complicated are
m 2 Q 3 ≃ 1 3 m 2 H − 2 3π 2 t 0 g 2 3 |M 3 | 2 + 1 64π 2 t 0 g 2 4 S ′ 1 + 1 6 m H (0) 2 , m 2 u 3 ≃ 2 3 m 2 H − 2 3π 2 t 0 g 2 3 |M 3 | 2 − 1 64π 2 t 0 g 2 4 S ′ 1 − 1 6 m H (0) 2 . (34)
The masses in (31) and (34) depend, a priori, on three independent initial parameters, M 3 (0), m ν (0) and m H (0). We will relate them as follows. It is clear from (24) that for m 2 H to have the appropriate behavior at the electroweak scale fixes M 3 (0) relative to m H (0). This will be discussed below. Here, we simply use the result that
|M 3 (0)| 2 = .0352 m H (0) 2 .(35)
It is also essential that color and charge be unbroken at the electroweak scale.
If we further require that this be the case for all scales t, then, as will be discussed shortly, one finds
m ν (0) 2 = 0.864 m H (0) 2 .(36)
In both (35) and (36) we present only the leading, (B/(m 2H − m 2 H )) 2 independent results for these quantities.
For these restricted parameters, the m 2 H contributions to (34) are small and can be ignored. Furthermore, at t B−L the Higgs fields have vanishing VEVs. Hence, we can compute the squark masses at (19) using the relevant
terms in V = V 2s + 1 2 D 2 B−L . The squark contribution to the quadratic potential is V m 2 squark = m 2 Q i |Q i | 2 + m 2 u i |u i | 2 + m 2 d i |d i | 2 with m 2 Q i = m 2 Q i + 1 4 g 2 4 | ν 3 | 2 ,(37)m 2 u i ,d i = m 2 u i ,d i − 1 4 g 2 4 | ν 3 | 2 .
Using (31), (34) as well as (9) and (19), we find that at t B−L these squared masses are given by
m 2 Q 1,2 ≃ 0.408 m ν (0) 2 , m 2 Q 3 ≃ 0.0435 m ν (0) 2 , m 2 u 1,2 = m 2 d i ≃ 1.08 m ν (0) 2 , m 2 u 3 ≃ 0.353 m ν (0) 2(38)
for i = 1, 2, 3. Note that they are all positive. It follows from (38) and (30) that (19) is indeed a stable, local minimum with respect to all scalar fields at t B−L . Let us now scale down further to the electroweak scale of order 10 2 GeV or, equivalently, t EW ≃ −33.3. We simplify the notation and implement (26) by taking
T 2 ≡ B m 2H − m 2 H −2 > ∼ 40 ,(39)
Although the mass eigenstate basis H ′ ,H ′ is the most natural for analyzing this vacuum, it is of some interest to express it in terms of the original H andH fields. Using (28), we find
H + = H − = 0(46)
and, to leading order, that
H 0 = 2∆ m H (0) T 3 5 g 2 1 + g 2 2 , H 0 = 1 T H 0 .(47)
Note that the condition H ′0 = 0 in (43) does not imply the vanishing of H 0 . Rather, H 0 is non-zero and related to H 0 through the ratio
H 0 H0 ≡ tanβ = T + O(T −1 ) .(48)
We have indicated the O(T −1 ) contribution to emphasize that although tanβ = T to leading order, this relationship breaks down at higher order in T −1 . We conclude that electroweak symmetry is spontaneously broken at scale t EW by the non-vanishing H ′0 vacuum expectation value in (43). This vacuum has a non-vanishing value of tanβ which, using the assumption for T 2 given in (39), satisfies tanβ > ∼ 6.32 .
As far as the Higgs fields are concerned, the vacuum specified in (43) is a stable local minimum. As a check on our result, choose µ 2 ∼ O(T −4 ) or smaller, that is, non-vanishing but sub-dominant in all equations. Then (29),(39) and (40) satisfy the constraint equations, given, for example, in [27], for the Higgs potential to be bounded below and have a negative squared mass at the origin. Furthermore, to the order in T −1 we are working, (45) and (47) for the Higgs vacuum satisfy the minimization conditions in [27].
To understand the complete stability of this minimum, it is essential to extend this analysis to the entire field space; that is, to include all sleptons and squarks as well as the Higgs fields. The relevant part of the potential energy is the sum of V 2s and the D B−L , D Y and D SU (2) L a contributions. The coefficients in this potential are to be evaluated at t EW . We find a local extremum at
ν 1,2 = 0, ν 3 = (1.05) 2m ν (0) 3 4 g 4 , L i = e i = 0, (50) H ′0 = 2∆ m H (0) tanβ 3 5 g 2 1 + g 2 2 , H ′+ = H ′0 = H ′− = 0
for i = 1, 2, 3. To guarantee that this is a stable local minimum, we must compute all of the scalar squared masses at this VEV. The Higgs masses were given in (44). The slepton masses in (20) and the squark masses in (38) are corrected in two ways, First, they must be scaled down from t B−L to t EW . Secondly, they are altered by the non-zero Higgs VEVs. Finally, in addition to (41) which relates M 3 (0) to m H (0), one must express m ν (0) in terms of m H (0). An overly restrictive but simple way to do this is the following. Demand that, for any choice of tanβ and ∆, all squark and slepton mass squares are non-negative, and, hence, color and electric charge are unbroken, for all values of t. We then find that
m ν (0) 2 = 0.864(1 − 2.25(1 − ∆ 2 ) T 2 )m H (0) 2 .(51)
This justifies (36) where, for simplicity, we dropped the weak T 2 dependence. Using the above approximations and dropping appropriate terms of order T −2 , the slepton and squark mass squares are given by
m 2 ν 1,2 ≃ 82.2 m H (0) 2 , m 2 ν 3 ≃ 8.75 m H (0) 2 ,(52)m 2 N i ≃ m 2 E i ≃ 7.00 m H (0) 2 , m 2 e i ≃ 5.00 m H (0) 2 and m 2 U 3 ≃ m 2 D 3 ≃ 0.132 m H (0) 2 , m 2 U 1,2 ≃ m 2 D 1,2 ≃ 0.465 m H (0) 2 ,(53)m 2 u 1,2 ≃ m 2 d i ≃ 1.04 m H (0) 2 , m 2 u 3 ≃ 0.374 m H (0) 2
for i = 1, 2, 3 respectively. Note that the third family up-squark mass squares receive a positive F-term contribution from their Yukawa interaction in (4). Although this contribution is of order T −2 and, hence, ignored in (53), it can be a sizable correction for smaller values of tanβ. Since all scalar masses in (44), (52) and (53) are positive, we conclude that the vacuum given in (50) is a stable, local minimum of the potential energy. The vacuum state (50) spontaneously breaks both B-L and electroweak symmetry, and exhibits a distinct hierarchy between the two. Using (36), we see that the ratio of the vacuum expectation values is
where the gauge parameters are computed at t EW . Choosing, for specificity, ∆ = 1 √ 2 and evaluating (54) in the region 6.32 ≤ tanβ ≤ 40, we find that
19.9 ≤ ν 3 H ′0 ≤ 126 .(55)
A second measure of the B-L/electroweak hierarchy is given by the ratio of the B-L vector boson mass to the mass of the Z boson. It follows from (21), (36) and (45) that
M A B−L M Z ≃ (1.95) tanβ ∆ .(56)
Again, using ∆ = 1 √ 2 and evaluating this mass ratio in the range 6.32 ≤ tanβ ≤ 40, one finds
17.5 ≤ M A B−L M Z ≤ 110 .(57)
Note that if we take ∆ → 1, the upper bound in our approximation, then
M A B−L M Z
is essentially 2tanβ, whereas if ∆ → 0 this mass ratio becomes arbitrarily large. For typical values of ∆, we conclude that the vacuum (50) exhibits a B-L/electroweak hierarchy of O(10) to O(10 2 ) in a physically interesting range of tanβ.
Finally, let us review the reasons for the existence and magnitude of the B-L/electroweak hierarchy. First, initial conditions (11), (15) give emphasis to the right-handed sneutrinos by not requiring their masses be degenerate with the L i and e i soft masses. This enables the S ′ 1 parameter (16) not only to be non-vanishing but, in addition, to be large enough to dominate all contributions to the RGEs with the exception of the gluino mass terms. This drives m 2 ν 3 negative and initiates B-L breaking at scale m ν . Second, B and M 3 (hence, m H (t EW )) are chosen to satisfy constraints (39) and (40) respectively, with 0 < ∆ 2 < 1. This insures electroweak breaking for positive m 2 H at a scale proportional to ∆m H (0)/T . The large value assumed for T implies that the non-vanishing VEV is largely in the H 0 direction, allowing one to identify T , to leading order, with tan β. Third, equation (51) insures that squark/slepton squared masses are positive at all scales. This guarantees that the electroweak breaking is substantially smaller than the B-L scale, with the B-L/electroweak hierarchy proportional to tan β/∆.
and choose M 3 (0) so thatfor 0 < ∆ 2 < 1. The upper bound on ∆ 2 arises from our requirement that m 2 H be positive for all t ≥ t EW . Using the previous assumptions and a numerical solution of (24), we find that m 2 H satisfies condition (40) if we chooseThis justifies(35)where, for simplicity, we dropped the weak T 2 dependence. It follows from(27),(39)and(40)that at t EWClearly electroweak breaking can only occur for positive ∆ 2 , explaining our lower bound on this parameter. Scaling (42) up to t B−L , we find that the constraint that ∆ 2 be less than unity implies m 2 H ′ > 0, as claimed in(30). To explore the breaking of electroweak symmetry, one must now consider the complete Higgs potential, V = V m 2 Higgs + 1 2 D 2 Y + 1 2 3 a=1 D 2 SU (2) L a , at t = t EW . We express this in terms of the mass diagonal fields H ′ andH ′ defined in(28)where the double bracket indicates the vacuum at t EW . The Higgs masses evaluated at this VEV are found to beThe three non-radial component fields in H ′ are the Goldstone bosons associated with the breakdown of electroweak symmetry. They are eaten by the Higgs mechanism to give mass to the W ± and Z bosons. For example, the Z mass is
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| []
|
[
"Parameter estimation for integer-valued Gibbs distributions",
"Parameter estimation for integer-valued Gibbs distributions"
]
| [
"David G Harris [email protected] \nDepartment of Computer Science\nInstitute of Science and Technology\nUniversity of Maryland\nAustria\n",
"Vladimir Kolmogorov \nDepartment of Computer Science\nInstitute of Science and Technology\nUniversity of Maryland\nAustria\n"
]
| [
"Department of Computer Science\nInstitute of Science and Technology\nUniversity of Maryland\nAustria",
"Department of Computer Science\nInstitute of Science and Technology\nUniversity of Maryland\nAustria"
]
| []
| We consider the family of Gibbs distributions, which are probability distributions over a discrete space Ω given by µ Ω β (x) = e βH(x) Z(β) . Here H : Ω → {0, 1, . . . , n} is a fixed function (called a Hamiltonian), β is the parameter of the distribution, and Z(β) = x∈Ω e βH(x) = n k=0 c k e βk is the normalization factor called the partition function. We study how function Z(·) can be estimated using an oracle that produces samples x ∼ µ β (·) for a value β in a given interval [β min , β max ].Recently, it has been shown how to estimate quantity q = log Z(βmax) Z(βmin) with additive error ε using O( q ε 2 ) samples in expectation. We improve this result toÕ( min{q,n 2 } ε 2 ), matching a lower bound of Kolmogorov (2018) up to logarithmic factors.We also consider the problem of estimating the normalized coefficients c k for indices k ∈ {0, 1, . . . , n} that satisfy max β µ Ω β ({x | H(x) = k}) ≥ µ * , where µ * ∈ (0, 1) is a given parameter. We solve this problem usingÕ( min{q+ √ q µ * ,n 2 + n µ * } ε 2 ) expected samples, and we show that this complexity is optimal up to logarithmic factors. This is improved to roughlyÕ( 1/µ * +min{q+n,n 2 } ε 2 ) for applications in which the coefficients are known to be log-concave (e.g. for connected subgraphs of a given graph). | null | [
"https://arxiv.org/pdf/1904.03139v1.pdf"
]
| 102,351,647 | 1904.03139 | e0f84465d8936f8465042e23835a981569737475 |
Parameter estimation for integer-valued Gibbs distributions
David G Harris [email protected]
Department of Computer Science
Institute of Science and Technology
University of Maryland
Austria
Vladimir Kolmogorov
Department of Computer Science
Institute of Science and Technology
University of Maryland
Austria
Parameter estimation for integer-valued Gibbs distributions
arXiv:1904.03139v1 [math.PR] 5 Apr 2019
We consider the family of Gibbs distributions, which are probability distributions over a discrete space Ω given by µ Ω β (x) = e βH(x) Z(β) . Here H : Ω → {0, 1, . . . , n} is a fixed function (called a Hamiltonian), β is the parameter of the distribution, and Z(β) = x∈Ω e βH(x) = n k=0 c k e βk is the normalization factor called the partition function. We study how function Z(·) can be estimated using an oracle that produces samples x ∼ µ β (·) for a value β in a given interval [β min , β max ].Recently, it has been shown how to estimate quantity q = log Z(βmax) Z(βmin) with additive error ε using O( q ε 2 ) samples in expectation. We improve this result toÕ( min{q,n 2 } ε 2 ), matching a lower bound of Kolmogorov (2018) up to logarithmic factors.We also consider the problem of estimating the normalized coefficients c k for indices k ∈ {0, 1, . . . , n} that satisfy max β µ Ω β ({x | H(x) = k}) ≥ µ * , where µ * ∈ (0, 1) is a given parameter. We solve this problem usingÕ( min{q+ √ q µ * ,n 2 + n µ * } ε 2 ) expected samples, and we show that this complexity is optimal up to logarithmic factors. This is improved to roughlyÕ( 1/µ * +min{q+n,n 2 } ε 2 ) for applications in which the coefficients are known to be log-concave (e.g. for connected subgraphs of a given graph).
Introduction
Given a real-valued function H(·) over some finite set Ω, the Gibbs distribution is defined as the family of distributions {µ Ω β } over Ω parameterized by β, where
µ Ω β (x) = e βH(x) Z(β)
These distributions frequently occur in physics, where the parameter −β corresponds to the inverse temperature, the function H(x) is called the Hamiltonian of the system, and the normalizing constant Z(β) = x∈Ω e βH(x) is called the partition function. They also occur in a number of applications of computer science, particularly sampling and counting algorithms. In this paper we consider a restricted form of the Gibbs distributions, where H(Ω) takes on integer values in the range H def = {0, 1, . . . , n} for some known integer n. If we set c k = |{x ∈ Ω : H(x) = k}|, then the partition function can be written as
Z(β) = k∈H c k e βk(1)
The basic problem we consider is how to estimate various parameters of the Gibbs distribution, given access to an oracle which can return a sample from the distribution for any chosen parameter β ∈ [β min , β max ]. Here the Gibbs distribution may be available as some physical process, in which case the oracle is an experimental run, or it may be available as some computational subroutine. Specifically, we seek to estimate the following parameters:
1. The ratio Q = Z(βmax) Z(β min ) and its logarithm q = log Q.
The coefficients c k (suitably normalized)
For many application to physics and computer science, the parameters q and c k are correlated with underlying problem parameters. For instance, [6] describes how, given some graph G, one can carefully craft a Gibbs distribution for which the parameter q is a pointwise evaluation of the reliability polynomial of G. Similarly, the parameters c k are correlated with counts of certain types of connected subgraphs of G. A number of other problems where the value q is useful are discussed in [13].
One special case of the Gibbs distribution is worth further mention, as it appears in a number of important combinatorial applications: the situation where the coefficients c 0 , . . . , c k are known the be log-concave, that is, they satisfy the bound c 2 k ≥ c k−1 c k+1 for k = 1, . . . , n − 1. We refer to this as the log-concave setting, and a number of results will be specialized for this case. We refer to the situation where coefficients c k are not restricted to be log-concave as the general setting.
Before we state our results, let us state some basic definitions and background assumptions. In this paper the "sample complexity" means the number of calls to the sampling oracle. We always assume for brevity that ε < ε max , n ≥ 2, q ≥ q min for some constant q min > 1, ε max > 0. (We can always increase n by adding dummy coefficients. The algorithms also apply even when q ∈ (0, q min ), but the upper bound on sample complexity will be at most that of the case q = q min ).
For any k ∈ H and β ∈ [β min , β max ], we define
µ β (k) = c k e βk Z(β)(2)
The vector µ β ∈ [0, 1] H is a distribution over H.
Our contribution
We study two computational problems. The first is to estimate the partition function Z(β): Problem P ratio : Given values ε, γ > 0 and interval [β min , β max ], compute valueQ > 0 such that P Q /Q ∈ [e −ε , e ε ] ≥ 1 − γ. Algorithms for P ratio with steadily improving expected sample complexities have been proposed by several authors. For constant γ, these complexities are as follows: O ε (q 2 log 2 n) (Bezáková et al. [3]), O(q log 5 (nq)ε −2 ) (Štefankovič et al. [14]), O(q log n · [log q + log log n + ε −2 ]) (Huber [9]), and O(q log n · ε −2 ) (Kolmogorov [13]). Note that using the standard "median amplification" technique, the success probability can be boosted to 1− γ by repeating the algorithm Θ(log 1 γ ) times. We improve on these results as follows. in the log-concave setting.
Note that [13] also derived a lower bound for a generalization of P ratio . 1 We adapt and strengthen this construction to show a lower bound of Ω min{q,n 2 } log 1 γ ε 2
, even in the log-concave setting. Thus our algorithm in optimal up to logarithmic factors. The second problem we consider is to estimate the coefficients of the Gibbs function. For this we need some preliminary definitions. Given a parameter µ * , let us define 2 For a set K ⊆ H, we say that non-negative vectorĉ ∈ [0, ∞) H is an (ε, K)-estimate of vector c if two conditions hold: (i)ĉ k > 0 for all k ∈ K; (ii) for all pairs k, ℓ ∈ H with c k > 0, c ℓ > 0, we havê c k c ℓ ∈ c k c ℓ [e −ε , e ε ]. We can now state the problem P µ * coef : Problem P µ * coef : Given values ε, γ, µ * > 0 and interval [β min , β max ] compute values {ĉ k } k∈H such that P[ĉ is an (ε, H * )-estimate of c] ≥ 1 − γ.
To our knowledge, problem P µ * coef has not been studied yet in its general form. We prove the following result. ) log n γ ε 2 , (n 2 + 1 µ * ) log n γ ε 2 + n log q, (q log n + n + 1 µ * ) log n γ ε 2 + n log 2 n in the log-concave setting.
We also show a lower bound for this problem of Ω min{q+ √ q µ * ,n 2 + n µ * } log 1 γ ε 2 for the general setting
and Ω
(1/µ * +min{q,n 2 }) log 1 γ ε 2
for the log-concave setting. In the general case, this matches our algorithm up to logarithmic factors in n and q. In the log-concave case, there is an additional additive discrepancy between the upper and lower bounds of orderÕ(n/ǫ 2 ) in the regime when 1/µ * +q = o(n).
As two concrete applications of our sampling algorithm, we obtain faster algorithms to approximate the number of connected subgraphs and number of matchings in a given graph. ). In particular, if G has minimum degree at least |V |/2, then there is an FPRAS for the sequence M i running in timeÕ(|V | 7 /ǫ 2 ).
Notably, Theorem 4 improves by a factor of |V | compared to the FPRAS given in [11]. While other FPRAS algorithms for counting connected subgraphs have been proposed by [7,2], the runtime appears to be very large (and not specifically stated in those works); thus Theorem 3 appears to be the first potentially practical algorithm for this problem.
Overview
The two most important parameters for integer-valued Gibbs distributions are n, the support of the distribution, and q, which measures how "diverse" the distribution is as β varies. There have been a number of previous algorithms whose runtime is primarily determined by q. We will extend and improve these, but we also develop a series of new algorithms whose runtime is primarily determined by n (with only a small logarithmic dependence on q).
We first describe in Section 2 some preliminary material on the behavior of the Gibbs function, as well as some elementary concentration bounds on binomial random variables.
In Section 3, we provide an overview of our algorithms. These use a number of subroutines and data structures, so we just describe these algorithms at a high level. Most of the remainder of the paper is devoted to fleshing out the different parts of these procedures.
The most important data structures used in these algorithms are the schedule and the representative set. We provide an algorithm to find the schedule in Section 4. This algorithm itself has many subroutines, and this is the most technically involved section in the paper. We provide an algorithm to find the representative set in Section 5.
In Section 6, we describe applications to two long-studied estimation problems in graph theory: counting the number of matchings or connected subgraphs of a given graph G. In both of these problems, the relevant Gibbs distribution is log-concave; we state a general result on how to obtain an approximate counting algorithm in such a setting. We use this to transfer known Gibbs sampling algorithms into FPRAS algorithms for the corresponding counting problems.
In Section 7, we show lower bounds on the sample complexity of algorithms which have oracle access to a Gibbs distribution. These are not computational bounds and do not require any complexitytheoretic assumptions. We show that, as a function of n, q, µ * , ε, and γ, the algorithms we develop are nearly optimal, both in the general and log-concave cases.
Computational extensions
For the most part, we focus on the sample complexity, i.e. the number of calls to the Gibbs distribution oracle. There are two mild extensions of this framework worth further discussion. Computational complexity.
The oracle may actually be provided as a randomized sampling algorithm. This is the situation, for example, in our applications to counting connected subgraphs and matchings. In this case we also need to bound our algorithm's computational complexity. In all the algorithms we develop, the time complexity can easily be seen to be a small linear factor times the query complexity. Thus, all our sampling procedures translate directly into efficient sampling algorithms. We will not comment explicitly on time complexity henceforth. Approximate sampling oracles. Many applications have only approximate sampling oraclesμ β , that are close to µ β in terms of the variation distance || · || T V defined via
δ = ||μ β − µ β || T V = max K⊆H |μ β (K) − µ β (K)| = 1 2 k∈H |μ β (k) − µ β (k)|− µ β || T V ≤ O(γ/T )
where T is the sample complexity of the corresponding algorithm. In particular, we have the following result; for completeness, we give a proof in Appendix A.
Theorem 5. Suppose that algorithm A makes in expectation at most T calls to the (exact) sampling oracle and satisfies P[output of A satisfies C] ≥ 1 − γ for some condition C and value γ > 0. Letà be the algorithm obtained from A by replacing calls k ∼ µ β with calls k ∼μ β whereμ β is a distribution
over H satisfying ||μ β − µ β || T V ≤ γ/T . Then P output ofà satisfies C ≥ 1 − 2γ.
For a number of applications, the cost of the approximate sampling oracle is polylogarithmic in the value δ. In such cases, we can use the following crude estimate: Corollary 6. There is an absolute constant C > 0 for which the following holds. Suppose that the sampling oracleμ β has variation distance δ ≤ ( 1 µ * + min{n, q} + 1 ε + log nq γ ) −C with respect to the exact sampling oracle. Then Theorems 1, 2 remain valid for oracleμ β oracle as they do for an exact sampling oracle.
2 Preliminaries
In this section we state several technical results that we will need later. For two real numbers x, y we say that x is an ε-estimate of y if | log x − log y| ≤ ε.
Proposition 7. (a) For each k ∈ H with c k > 0 function log µ β (k) is a concave function of β. (b) For values α, β and k, ℓ ∈ H we have µ α (k)µ β (ℓ) = e (α−β)(k−ℓ) · µ α (ℓ)µ β (k). In particular, if α ≤ β and k ≤ ℓ then µ α (k)µ β (ℓ) ≥ µ α (ℓ)µ β (k). Proof. (a) It is known [15, Proposition 3.1] that log Z(β) is a convex function of β. Therefore, function log µ β (k) = log c k + βk − log Z(β) is concave. (b)
The claim follows immediately from the following equations:
µ α (k) = c k e αk Z(α) µ α (ℓ) = c ℓ e αℓ Z(α) µ β (k) = c k e βk Z(β) µ β (ℓ) = c ℓ e βℓ Z(β)
We can obtain an unbiased estimatorμ β of vector µ β ∈ [0, 1] H for any given value β ∈ [β min , β max ] by taking N ≥ 1 independent samples from µ β (·) and computing the empirical frequencies. We use notationμ β ← Sample(β; N ) for this process. To analyze it, we will use some standard concentration bounds for the binomial distribution which we derive in Appendix B.
Lemma 8. For parameters ε > 0, γ ∈ (0, 1], p • ∈ (0, 1] define the value R(ε, γ, p • ) = 2e ε log(2/γ) (1 − e −ε ) 2 p • = Θ log(1/γ) ε 2 p • Letp ∼ 1 N Binom(N, p) for N ≥ R(ε, γ, p • ).
Then with probability at least 1 − γ we havê
p ∈ [e −ε p, e ε p] if p ≥ e −ε p • [0, p • ) if p < e −ε p •(3)
Observation 9. If (3) holds, and either p ≥ p • orp ≥ p • , thenp is an ε-estimate of p.
Proof. If p ≥ e −ε p • , then this is immediate from (3). Otherwise, if p < e −ε p • , then (3) imply that p < p • , which contradicts the hypothesis.
We use the notationμ β ← Sample(β; ε, γ, p • ) as shorthand forμ β ← Sample(β; R(ε, γ, p • )). Most of our algorithms have the following structure: we executeμ β ← Sample(β; ε, γ, p • ) for various choices of β, ε, γ, p • , and make certain decisions or estimates based on valuesμ β (k). The algorithms succeed as long asμ β (k) does not deviate much from its true value µ β (k), in line with the conditions given above. When we executeμ β ← Sample(β; ε, γ, p • ), we say that this well-estimates some value k if condition (3) holds for p = µ β (k) andp =μ β (k); otherwise it mis-estimates k. This condition depends on the parameters ε, p • and not solely on the values ofμ β (k), µ β (k). Regardless of the true value of µ β (k), Lemma 8 ensures that the any given index k is mis-estimated with probability at most γ.
We will also need the following result, which we show in Appendix C.
Lemma 10. Let a 1 , . . . , a m be a non-negative log-concave sequence satisfying a k ≤ 1 k for each k ∈ [m]. Then a 1 + . . . + a m < e.
Note that without the log-concavity assumption we would have a 1 + . . . + a m ≤ m k=1 1 k ≤ 1+ log m (by a well-known inequality for the harmonic series). Motivated by these facts, we define the following parameter which we use throughout the paper: Γ = 1 + log n in the general setting e in the log-concave setting 5 3 Overview of the algorithms
We begin by describing our algorithm to solve the problem P ratio . In fact, in order to solve P µ * coef later, we will need to solve a more general "batched" version of this problem. Specifically, we are given a set B ⊆ [β min , β max ], and we need to estimate the value Q α := Z(α)/Z(β min ) for all α ∈ B. The problem P ratio is a special case with B = {β max }.
Let us introduce some terminology to explain the key data structure used in the algorithm here.
• A weighted interval is a tuple σ = ([σ − , σ + ], σ weight ) where σ − , σ + ∈ H def = H ∪ {−∞, +∞}, σ − ≤ σ + and σ weight ∈ (0, 1]. We denote Ends(σ) = {σ − , σ + } ∩ H and size(σ) = |[σ − , σ + ] ∩ H|.
• An extended weighted interval is a tuple (β, σ) where β ∈ [β min , β max ] and σ is a weighted interval.
The tuple is called proper if µ β (k) ≥ σ weight for all k ∈ Ends(σ).
• Consider a sequence I = ((β 0 , σ 0 ), . . . , (β t , σ t )) of distinct extended weighted intervals. We will sometimes view it as a set and write (β, σ) ∈ I. We denote Ends(I) = (β,σ)∈I Ends(σ) and
InvWeight(I) = (β,σ)∈I 1 σ weight . • Sequence I of the above form will be called a schedule if it satisfies two conditions:
(i) β min = β 0 < . . . < β t = β max ; (ii) −∞ = σ − 0 < σ + 0 = σ − 1 < σ + 1 = σ − 2 < . . . < σ + t−1 = σ − t < σ + t = +∞. Here t = |I| − 1 ≤ n + 1. We say that I is proper if all tuples (β, σ) ∈ I are proper.
The algorithm for P ratio will be based on the following key result.
Theorem 11. There exists a randomized algorithm that for a given value γ ∈ (0, 1) produces a schedule I such that InvWeight(I) ≤ c(n+1)Γ and P[I is proper ] ≥ 1−γ, where c > 4 is an arbitrary constant. This algorithm has expected sample complexity O(nΓ(log 2 n + log 1 γ ) + n log q).
We prove this later in Section 4, but for now let us describe the main way to use it. By using telescoping products, we obtain the following expression for each i ∈ {0, . . . , t}:
Q β i = Z(β i ) Z(β min ) = C i (I) · µ β 0 (σ + 0 ) · . . . · µ β t−1 (σ + i−1 ) µ β 1 (σ − 1 ) · . . . · µ βt (σ − i ) where C i (I) def = e β 1 σ − 1 · . . . · e β i σ − i e β 0 σ + 0 · . . . · e β i−1 σ + i−1
Now, given set B ⊆ [β min , β max ] and a schedule I, let us consider the following algorithm to "weakly" solve the batched P ratio problem (we will explain more explicitly the guarantees it provides):
Algorithm 1: Weakly solving batched P ratio . Input:
schedule I = ((β 0 , σ 0 ), . . . , (β t , σ t )), parameter ε > 0, set B ⊆ [β min , β max ]. Output: estimatesQ = {Q α } α∈B . 1 foreach i ∈ {0, . . . , t} do 2 setμ β i ← Sample(β i ; max{⌈ N t σ weight i ⌉, R( ε 4 , 1 16 , 1 σ weight i )}) where N = 1 + 16 (1−e −ε/4 ) 2 = Θ( 1 ε 2 ) 3 setQ β i = C i (I) ·μ β 0 (σ + 0 )·...·μ β i−1 (σ + i−1 ) µ β 1 (σ − 1 )·...·μ β i (σ − i ) /* treat division by 0 arbitrarily */ 4 foreach α ∈ B − {β 0 , . . . , β t } do 5 select index i such that α ∈ (β i , β i+1 ) 6 letμ α ← Sample(α; ε 4 , 1 16 , min{σ weight i , σ weight i+1 }) 7 setQ α =μ β i (k) µα(k) e (α−β i )kQ β i where k = σ + i = σ − i+1
Note that . Using standard techniques (e.g. cf. [9]), we can show the following (see Appendix D.1).
6 Theorem 12. If I is proper, then for every α ∈ B, the estimateQ α provided by Algorithm 1 satisfies
P Q α /Q α ∈ [e −ε , e ε ] ≥ 3 4
We may use a technique known as median amplification to increase the success probability to 1 − γ, for any desired parameter γ. To do so, we first apply Theorem 11 with parameter γ/2 to generate a schedule I. We then execute N = Θ(log |B| γ ) independent repetitions of Algorithm 1 with this schedule I, obtaining estimatesQ α . A standard analysis shows that for each α it then holds that
P Q α /Q α ∈ [e −ε , e ε ] > 1 − γ |B| .
We summarize this procedure in the following result.
Corollary 13. There is an algorithm which takes as input a set B ⊆ [β min , β max ], and parameters γ, ε ∈ (0, 1), and returns estimatesQ = {Q α } α∈B such that, with probability at least 1 − γ, we havê
Q α /Q α ∈ [e −ε , e ε ] for all α ∈ B.
The expected sample complexity is O(
(|B|n+n 2 )Γ log |B| γ ε 2 + n log q).
Recall that an alternative algorithm for P ratio with expected sample complexity O q log n log 1 γ ε 2 was proposed in [9,13]. As we discuss in Appendix E, this algorithm can be adapted to batched P ratio with expected sample complexity O
(|B|+q log n) log |B| γ ε 2 .
The parameter q is not known, but we can still combine these algorithms using a technique known as dovetailing. Let us consider running the two algorithm simultaneously in parallel with error parameter γ/2; as soon as either algorithm terminates, we output its answer. This solves the problem with probability at least 1 − γ, for, by the union bound, with probability at least 1 − γ/2 − γ/2, both of the two algorithms will (eventually) return a correct answer. The expected runtime of this procedure is at most twice the expected runtime of either algorithm individually. This gives the following result:
Corollary 14.
There is an algorithm BatchedPratio(B, γ, ε) which takes as input a set B ⊆ [β min , β max ], and parameters γ, ε ∈ (0, 1), and returns estimatesQ = {Q α } α∈B such that, with probability at least 1 − γ, we haveQ α /Q α ∈ [e −ε , e ε ] for all α ∈ B. The expected sample complexity is
O min (|B| + q log n) log |B| γ ε 2 , (|B|n + n 2 )Γ log |B| γ ε 2 + n log q
In particular, with B = {β max }, this gives Theorem 1.
Solving P µ * coef in the general setting
We next turn to proving Theorem 2. Although our definition of an (ε, K)-estimate does not require any condition on individual entries ofĉ k , the algorithms will yield a specific normalization which we refer to as a lower-normalized (ε, K)-estimate of c. Namely, the estimateĉ satisfies that (i)ĉ k > 0 for all k ∈ K; (ii) for all k ∈ H withĉ k > 0 the valueĉ k is an ε/2-estimate of the valuec k = c k Z(β min ) . It is immediate that this is also an (ε, K)-estimate of c.
The key data structure for this problem is a representative set R, which is a subset of the interval [β min , β max ]. It is called proper with respect to parameter ζ ≥ 1 if every k ∈ H satisfies
max α∈R µ α (k) ≥ 1 ζ max β∈[β min ,βmax] µ β (k)
In Section 5, we will discuss the algorithm FindRepresentatives to find a proper representative set. We summarize it as follows:
Theorem 15. The procedure R ← FindRepresentatives(γ) can be implemented to have the following properties:
(a) The expected sample complexity is O(min{ √ q log n, n} log nq γ ). (b) The representative set R is proper for parameter ζ = 256 with probability at least 1 − γ/n. (c) |R| ≤ n + 1 with probability one, and |R| ≤ O( √ q log n) with probability at least 1 − γ/n (and therefore E[|R|] ≤ O(min{ √ q log n, n}) since γ ≤ 1).
Note that procedure FindRepresentatives does not itself depend upon parameter µ * . This leads to a simple algorithm for P µ * coef : Algorithm 2: Solving P µ * coef . Input: parameters γ, µ * , ε > 0
1 call R ← FindRepresentatives(γ/3) 2 callQ ← BatchedPratio(R, γ/3, ε/4) 3 setĉ k ← 0 for all k ∈ H 4 foreach α ∈ R do 5 setμ α ← Sample(α; ε/4, γ 3(n+1) 2 , e −ε/4 µ * /ζ) 6 estimateĉ k ←Q α e −αkμ α (k) for each k ∈ H withμ α (k) ≥ e −ε/4 µ * /ζ andĉ k = 0.
We can calculate the expected sample complexity of Algorithm 2 line by line: With Γ = Θ(log n), we get the expected sample complexity stated in Theorem 2 by considering separately the cases where q log n < n 2 and q log n ≥ n 2 . In Appendix D.2 we show that this algorithm does indeed solve P µ * coef : Theorem 16. P[ĉ is a lower-normalized (ε, H * )-estimate of c] ≥ 1 − γ for the output of Alg. 2.
Line 1: O min{ √ q log n,
Solving P µ *
coef in the log-concave setting
In the log-concave setting, there is an alternative algorithm for P µ * coef which is more efficient than Algorithm 2 in most cases: Algorithm 3: Solving P µ * coef in the log-concave setting. Input: parameters ε, γ, µ * > 0 1 compute schedule I = ((β 0 , σ 0 ), . . . , (β t , σ t )) using the alg. from Theorem 11 with parameter γ . Overall, we get a total sample complexity of O 1/µ * + min{q log n + n, n 2 } ε 2 · log n γ + n log 2 n + n log q
3 2 call BatchedPratio(B, ε/4, γ/3) where B = {β 0 , . . . , β t } 3 update σ weight 0 ← min{σ weight 0 , µ * 1+µ * σ + 0 } and σ weight t ← min{σ weight t , µ * 1+µ * (n−σ − t ) } 4 foreach (β, σ) ∈ I letμ β ← Sample(β; ε/4, γ 3(n+1) 2 , e −ε/4 σ weight ) 5 foreach k ∈ H do 6 pick tuple (β, σ) with k ∈ [σ − , σ + ] 7 setĉ k = Q β e −βk ·μ β (k) ifμ β (k) ≥ e −ε/4 · σ weight
We can separate this into two cases depending on which of q log n or n 2 is larger; in the latter case, the n log q term becomes negligible. This gives the second two terms in the complexity stated in Theorem 2. The first term in the complexity bound is simply copied from the general-case algorithm (which still applies to the log-concave setting).
The correctness of Algorithm 3 is shown in Appendix D. 3. In fact, we show a slightly stronger result:
Theorem 17. P[ĉ is a lower-normalized (ε, H * * )-estimate of c] ≥ 1−γ for the output of Alg. 3, where we define H * * = [min H * , max H * ] ∩ H ⊇ H * .
By dovetailing this algorithm with Algorithm 2, we obtain the second part of Theorem 2.
Computing a schedule: Proof of Theorem 11
We now describe the main algorithm to generate a schedule. Let us fix constants τ ∈ (0, 1 2 ), λ ∈ (0, 1), and denote φ = τ λ 3 /Γ. Thus, φ = Θ( 1 log n ) in the general setting and φ = Θ(1) in the log-concave setting.
The algorithm will maintain a sequence J = ((β 0 , σ 0 ), . . . , (β t , σ t )) of extended weighted intervals satisfying the following invariants:
(I1) β min = β 0 ≤ . . . ≤ β t = β max . (I2) −∞ = σ − 0 ≤ . . . ≤ σ − t ≤ n and 0 ≤ σ + 0 ≤ . . . ≤ σ + t = +∞. (I3) [σ − i−1 , σ + i−1 ] = [σ − i , σ + i ] for i ∈ [t]. Also, if β i−1 = β i then either σ − i−1 = σ − i or σ + i−1 = σ + i . (I4) σ weight ≥ φ size(σ) for each (β, σ) ∈ J .
For a sequence J we define
J ∪ = (β,σ)∈J [σ − , σ + ] ⊆ [−∞, +∞]. A sequence J with J ∪ = [−∞,
+∞] satisfying (I1)-(I4) will be called a pre-schedule. In order to get a proper schedule, our algorithm will also seek to maintain two further invariants. We say that interval (β, σ) is extremal if it satisfies the following conditions:
µ β (k) ≤ 1 λ · size(σ) size(σ) + (σ − − k) · µ β (σ − ) ∀k ∈ {0, . . . , σ − − 1} (4a) µ β (k) ≤ 1 λ · size(σ) size(σ) + (k − σ + ) · µ β (σ + ) ∀k ∈ {σ + +1, . . . , n}(4b)
We say that (β, σ) is left-extremal if it satisfies (4a) and right-extremal if it satisfies (4b). With this notation, we can state the additional invariants (I5), (I6) we hope to maintain.
(I5) Each interval (β, σ) ∈ J is proper. (I6) Each interval (β, σ) ∈ J is extremal.
Note that these conditions (I5), (I6) are defined in terms of the probability function µ, so they cannot be checked directly -only indirectly by sampling. We say that interval (β, σ) is conformant if it obeys all the conditions (I4) -(I6), i.e. it is proper, extremal, and satisfies σ weight ≥ φ size(σ) . We will later show how to convert a pre-schedule satisfying conditions (I1)-(I6) into a proper schedule.
It is convenient to denote L = { 1 2 , 3 2 , . . . , n − 1 2 } and Gaps(J ) = L − J ∪ . Let GapIntervals(J ) = {{k + 1 2 , k + 3 2 , . . . , ℓ − 3 2 , ℓ − 1 2 } ⊆ Gaps(J ) : k, ℓ ∈ J ∪ } be the set of maximal discrete intervals in Gaps(J ). Note that Gaps(J ) = Θ∈GapIntervals(J ) Θ. The following fact can be deduced from (I2): for any θ ∈ Gaps(J ) there exists unique index k ∈ [t] such that σ + i < θ < σ − j for any 0 ≤ i < k ≤ j ≤ t. We will use subroutines with the following specifications (complete details will be provided later):
• FindInterval(β, H − , H + ; i ⋆ , ω ⋆ ): given β ∈ [β min , β max ] and subsets H − , H + ⊆ H with max H − < min H + , it must return weighted interval σ such that σ − ∈ H − , σ + ∈ H + , and σ weight ≥ φ size(σ) .
Ideally, the interval σ should also be proper and extremal. (The parameters (i ⋆ , ω ⋆ ) ∈ (H ∪ {NULL}) × [0, 1] are used in special cases to enforce consistency with the intervals at the margins).
• BinarySearch(β left , β right , θ,γ): given interval [β left , β right ] ⊆ [β min , β max ] and value θ ∈ L, it must return a value β ∈ [β left , β right ]. Ideally, β should satisfy µ β ([0, θ] ∩ H) ≈ 1/2 ≈ µ β ([θ, n] ∩ H). (Here,γ is the requested failure probability.)
The subroutine BinarySearch will also be used later, in our algorithm for FindRepresentatives.
Generating a pre-schedule
We can now formulate the first part of the algorithm which generates a proper pre-schedule.
Algorithm 4: Computing pre-schedule.
1 call σ min ← FindInterval(β min , {−∞}, H; −∞, 1) 2 call σ max ← FindInterval(β max , H, {+∞}; +∞, 1) 3 set J = (σ min , σ max ) 4 while Gaps(J ) = ∅ do 5 pick arbitrary Θ ∈ GapIntervals(J ), let θ ∈ Θ be a median value in Θ 6 let (β left , σ left ), (β right , σ right ) be the unique consecutive pair in J with σ + left < θ < σ − right ; define K − = [σ − left , θ] ∩ H and K + = [θ, σ + right ] ∩ H 7 call β ← BinarySearch(β left , β right , θ, κ 2n ) 8 call σ ← FindInterval(β, {σ − left }, K + ; σ − left , σ weight left ) if β = β left FindInterval(β, K − , {σ + right }; σ + right , σ weight right ) if β = β right FindInterval(β, K − , K + ; NULL, 0) if β left < β < β right 9
insert (β, σ) into J between (β left , σ left ) and (β right , σ right ) 10 return J Using specifications of the subroutines, one can check that the algorithm indeed preserves invariants (I1)-(I4) and produces a pre-schedule upon termination. Furthermore, the loop in lines 4 -9 is executed at most |L| = n times, and thus the algorithm makes at most n + 2 calls to FindInterval and at most n calls to BinarySearch.
To describe the two subroutines, we need introduce a few definitions. We say that the call β ←
BinarySearch(β left , β right , θ,γ) is good if its result satisfies β ∈ Λ τ (β left , β right , θ) where Λ τ (β left , β right , θ) = β ∈ [β left , β right ] : β > β left ⇒ µ β ([0, θ] ∩ H) ≥ τ β < β right ⇒ µ β ([θ, n] ∩ H) ≥ τ The call σ ← FindInterval(β, H − , H + ; i ⋆ , ω ⋆ ) is said to be good if the interval (β, σ) is proper and extremal.
By continuity, it is obvious that Λ τ (β left , β right , θ) = ∅, and so it possible for BinarySearch to have a good output -the challenge is to find one. By contrast, if we execute FindInterval with arbitrary inputs it is very likely that no good output exists. However, the overall structure of Algorithm 4 has been carefully designed so that, as long as invariants (I1)-(I6) have been satisfied so far and calls to BinarySearch have been good, then the output of FindInterval will be good with high probability. More specifically, let us say that a call to FindInterval at line 8 is valid if β ∈ Λ τ (β left , right, θ), interval (β left , σ left ) is conformant, and interval (β right , σ right ) is conformant. We also say that the calls to FindInterval at line 1 and 2 are valid.
Let us fix some constant κ ∈ (0, 1). The following result summarizes FindInterval and BinarySearch. This theorem will be proved later in Sections 4.3 and 4.4.
Theorem 18. (a) The call FindInterval(β, H − , H + ; i ⋆ , ω ⋆ ) can be implemented so that P[the call to FindInterval is good | the call to FindInterval is valid] ≥ 1 − κ 2(n+2) , and its sample complexity is O Γ log n × size([min H − , max H + ]) . (b)
The call BinarySearch(β left , β right , θ,γ) can be implemented so that P[the call is good] ≥ 1 −γ and its expected sample complexity is O(log nq γ ). The next result analyzes Algorithm 4 with the subroutines from Theorem 18.
O(nΓ log 2 n). Let Θ i , θ i , K − i , K + i be the variables at the i th iteration. Define H i = [h − i , h + i ]∩ H where h − i = min K − i and h + i = max K + i .
By Theorem 18(a), the i th iteration of FindInterval has sample complexity O(Γ|H i | log n). We will show next that i |H i | = O(n log n). This will yield the claim about the complexity.
For
k ∈ H define I − (k) = {i : k ∈ H i ∧ θ i < k}, and consider i, j ∈ I − (k) with i < j. It can be seen that θ j ∈ Θ i . (If not then we must have either θ j < h − i or θ j > h + i .
In the former case tuple (β left , σ left ) at the i th iteration would "prevent" going beyond σ + left < θ i < k at the j th iteration, and in the latter case we would have θ j > h + i ≥ k -again a contradiction). Condition θ j ∈ Θ i implies that Θ j ⊆ Θ i . Therefore, |Θ j | ≤ 1 2 |Θ i | (since after the i th iteration integers θ i ± 1 2 are added to J ∪ and θ i is a median of Θ i ). We can now conclude that |I − (k)| ≤ ⌊log 2 |L|⌋ + 1 = O(log n). In a similar way we can prove that |I
+ (k)| = O(log n) where I + (k) = {i : k ∈ H i ∧ θ i > k}. It remains to observe that i |H i | = k∈H |I − (k) ∪ I + (k)|.
Converting the pre-schedule to a schedule
Having formed a pre-schedule, we next need to convert it to a proper schedule. We will use a subroutine FinalizeSchedule(J ,γ) for this. Notably, this procedure FinalizeSchedule also validates its input: it is allowed to output an error code ⊥, and in particular even if J is not proper, it should still be unlikely that FinalizeSchedule returns a non-proper schedule -it should output ⊥ in this case.
Algorithm 5: Computing schedule. Input: parameter γ > 0.
1 while true do 2
call Algorithm 4 with parameter κ = 1/2 to compute pre-schedule J /* transform J to a minimal sequence:
*/ 3 while there exists (β, σ) ∈ J such that (J − (β, σ)) ∪ = [−∞, +∞] do 4 pick any such (β, σ), update J ← J − (β, σ) 5 call I ← FinalizeSchedule(J , γ/4) 6 if I = ⊥ then return I
We will show that FinalizeSchedule(J ,γ) can be implemented to have the following behavior:
Theorem 20. (a) The output I is either a schedule with InvWeight(I) ≤ 2e ν (n + 1)/φ, for an arbitrary constant ν > 0, or is the error code ⊥.
(b) For any input J , with probability at least 1 −γ, the output I is either ⊥ or a proper schedule I. (c) If the input J is proper, then with probability at least 1 −γ, the output I is a proper schedule.
(d) The sample complexity is O(nΓ log ñ γ ). 11
To complete the proof, we prove Theorem 18(a) (i.e. the implementation of FindInterval) in Section 4.3, Theorem 18(b) (i.e. the implementation of BinarySearch) in Section 4.4, and Theorem 20 (analyzing Algorithm 5) in Section 4.5. Assuming these results for the moment, we can combine all our algorithmic results to show Theorem 11.
Proof of Theorem 11. Proposition 19 and Theorem 20 show that each iteration of Algorithm 5 terminates with probability at least (1 − κ)(1 − γ/4) ≥ Ω(1). Therefore, the expected number of runs is O(1). Each call to FinalizeSchedule has sample complexity O(nΓ log n γ ). Each iteration of Algorithm 4 has sample complexity O(n log q + nΓ log n). Thus, the overall expected sample complexity of Algorithm 5 is O(nΓ log n γ + n log q + nΓ log 2 n). By Theorem 20(a), we have InvWeight(I) ≤ 2e ν (n + 1)/φ = 2Γ(n + 1) × e ν τ λ 3 . The term e ν τ λ 3 gets arbitrarily close to 2 for constants ν, λ, τ sufficiently close to 0, 1, 1 2 respectively. Finally, let us show that the output I of Algorithm 5 is a proper schedule with probability at least 1 − γ. LetÎ denote the value obtained at line 5 of any given iteration. Since the iterations of Algorithm 5 are independent, the distribution of I is the same as the distribution ofÎ, conditioned onÎ = ⊥. Thus P[ I is an improper schedule] = P Î is an improper schedule |Î = ⊥ .
By Theorem 20(b), the probability thatÎ is an improper schedule is at most γ/4, even conditional on any fixed value for the pre-schedule J . By Proposition 19(b), in any given iteration the pre-schedule J is proper with probability at least κ = 1/2; in such case, by Theorem 20(b), we haveÎ = ⊥ with probability at least 1 − γ ≥ 1/2. Overall, we have P Î = ⊥ ≥ 1/4. We therefore have
P Î is an improper schedule |Î = ⊥ ≤ P Î is an improper schedule P Î = ⊥ ≤ γ/4 1/4 = γ 4.3 Proof of Theorem 18(a): Procedure FindInterval(β, H − , H + ; i ⋆ , ω ⋆ )
To describe the algorithm, let us define h − = min H − ∈ H∪{−∞}, a − = max H − +1, a + = min H + −1, and h + = max H + ∈ H ∪ {+∞}. We also denote S = |[h − , h + ] ∩ H| and set γ = κ 2(n+2) .
Algorithm 6: FindInterval(β, H − , H + ; i ⋆ , ω ⋆ ). 1 letμ β ← Sample(β; 1 2 log 1 λ , γ S , φ S ) 2 for each i ∈ H − ∪ H + set α(i) = λ 3/2 ·μ β (i) if i ∈ H − {h − , h + , i ⋆ } λ 1/2 ·μ β (i) if i ∈ {h − , h + } − {i ⋆ } ω ⋆ if i = i ⋆ 3 set k ← arg max i∈H − (a − − i)α(i) and ℓ ← arg max i∈H + (i − a + )α(i) 4 set σ = ([k, ℓ], min{α(k), α(ℓ)}) 5 if size(σ) · σ weight ≥ φ then return σ 6 else return ([h − , h + ], 1) /*Whenever i ⋆ = NULL we have either H − = {i ⋆ } or H + = {i ⋆ }.
These cases are handled very differently, as there there is no "free choice" for the left margin k = σ − or right-margin ℓ = σ + respectively. Desired properties of σ (namely, extremality and properness) then follow from the corresponding properties of σ left or σ right .
If H − , H + = {i ⋆ } then we rely on sampling outputs to choose k or ℓ. We give a slight bias to the endpoints h − or h + ; this helps preserve the slack factor 1 λ in the definition of extremality (4a),(4b) (Without such bias this factor would become worse and worse as the algorithm progresses, and we would need many more samples to make sure that it does not grow too much). Next, we proceed with the formal algorithm analysis. The sample complexity is O(SΓ log n) (bearing in mind that λ is a constant and γ ≥ 1 poly(n) ). Because of the check at line 6, this always returns an interval satisfying size(σ)σ weight ≥ φ. The non-trivial thing to check is that if the call it valid, it returns an extremal proper interval with probability at least 1 − γ.
For the remainder of this section, let us therefore suppose that the call is valid. So either we are executing FindInterval at line 1 or 2 in Algorithm 4, or β ∈ Λ τ (β left , β right , θ), and intervals (β left , σ left ) and (β right , σ right ) are both conformant. When FindInterval is executed at line 1 or 2, we say that the execution of FindInterval is degenerate. The behavior of FindInterval in the degenerate case is quite similar to the non-degenerate case, but simpler.
Let us first state a useful formula.
Lemma 21. There holds
µ β (i) ≤ 1 λ · j − h − j − i · µ β (h − ) ∀i ∈ {0, . . . , h − − 1}, ∀j ≥ a − (5a) µ β (i) ≤ 1 λ · h + − j i − j · µ β (h + ) ∀i ∈ {h + + 1, . . . , n}, ∀j ≤ a + (5b)
Proof. We only show (5a); the proof of (5b) is analogous. We assume that h − ≥ 1 and the call is non-degenerate, otherwise the claim is vacuous.
Now consider i ∈ {0, . . . , h − − 1} and j ≥ a − . Since (β left , σ left ) is left-extremal and h − = σ − left , we have µ βleft (i) ≤ 1 λ · size(σ left ) size(σ left ) + (h − − i) · µ βleft (h − ) (6) Since i < h − and β ≥ β left , Proposition 7(a) gives µ βleft (i)µ β (h − ) ≥ µ βleft (h − )µ β (i). Combined with Eq. (6), this yields µ β (i) ≤ 1 λ · size(σ left ) size(σ left ) + (h − − i) · µ β (h − ) It remains to notice that size(σ left ) ≤ j − h − (since j ≥ a − ≥ σ + left + 1) and therefore size(σ left ) size(σ left ) + (h − − i) ≤ j − h − (j − h − ) + (h − − i) = j − h − j − i
We need another existential result on some values of µ β . Note that this is the only place in the analysis that we need to distinguish between the general setting (where φ = Θ(1/ log n)) and the log-concave setting (where φ = Θ(1)).
Lemma 22. In either the general or log-concave settings, the following holds:
(a) If H − = {i ⋆ } then there exists k ∈ H − with (a − − k) · µ β (k) ≥ τ λ/Γ = φ/λ 2 . (b) If H + = {i ⋆ } then there exists ℓ ∈ H + with (ℓ − a + ) · µ β (ℓ) ≥ τ λ/Γ = φ/λ 2
Proof. The two claims are completely analogous, so we only prove (a). Denote A = {0, . . . , a − −1} ⊆ H (recall that a − − 1 ∈ H). We make the following claim:
µ β (A) ≤ Γδ if a − ≤ n 2Γδ if a − = n + 1 where δ = max k∈A (a − − k) · µ β (k)(7)Indeed, denote b i = µ β (a − −i) δ for i ∈ [a − ], then δ = max k∈A (a − − k) · (δb a − −k ) implying that b i ≤ 1 i for all i ∈ [a − ]. Also, we have µ β (A) δ = a − i=1 b i . Now consider two possible cases. 13
• Log-concave setting (with Γ = e). If coefficients c k are log-concave then so is the sequence
b 1 , . . . , b a − (since µ β (k) ∝ c k e βk ). Lemma 10 then gives a − i=1 b i ≤ e = Γ.
• General setting (with Γ = 1 + log n). We have a − i=1 b i ≤ 1 + log a − by the well-known inequality for the harmonic series. It remains to observe the following:
(i) if a − ≤ n then 1 + log a − ≤ Γ; (ii) if a − = n + 1 then a − i=1 b i ≤ 1 + log(n + 1) ≤ 2(1 + log n) = 2Γ for any n ≥ 1. From now on we assume that (a) is false, i.e. (a − − k) · µ β (k) < τ λ Γ for all k ∈ H − . If the execution is degenerate, then a − = n + 1 and H − = A = H, implying that δ < τ λ Γ . From (7) we obtain µ β (H) < 2Γ · τ λ Γ = 2τ λ < 2 · 1 2 · 1,
which is a contradiction since µ β (H) = 1. Now suppose that the execution is non-degenerate. We claim that the following holds:
µ β (k) < τ Γ · 1 a − − k for all k ∈ A(8)
Indeed, we already know that a stronger inequality
µ β (k) < τ λ Γ · 1 a − −k holds for all k ∈ H − . In particular, we have µ β (h − ) < τ λ Γ · 1 a − −h − . It remains to prove the claim for k ∈ {0, . . . , h − − 1}. Eq. (5a) with (i, j) = (k, a − ) gives µ β (k) ≤ 1 λ · a − − h − a − − k µ β (h − )
Using our bound on µ β (h − ), we now get the desired claim:
µ β (k) < 1 λ · a − − h − a − − k × τ λ Γ · 1 a − − h − = τ Γ · 1 a − − k Eq. (8) implies that δ < τ Γ .
We have a − = θ + 1 2 ≤ n, so from (7) we get µ β (A) < Γ · τ Γ = τ . On the other hand, condition H − = {i ⋆ } means that β > β left . We assumed that β ∈ Λ τ (β left , β right , θ), and therefore µ β ([0, θ] ∩ H) ≥ τ . This is a contradiction, since [0, θ] ∩ H = A.
We are now ready to show that FindInterval is good with probability at least 1 − γ. We have already assumed that the call is valid; let us also suppose that line 1 well-estimates every k ∈ [h − , h + ]∩H. By construction, this holds with probability at least 1 − γ. We will show that under this condition, the output is extremal and proper. We denote p • = φ/S to be the parameter used at line 1. Throughout, we let k, ℓ be the parameters selected at line 3.
Proposition 23. (a) If H − = {i ⋆ }, then (a − − k) · α(k) ≥ φ and µ β (k) ≥ √ λ ·μ β (k). (b) If H + = {i ⋆ }, then (ℓ − a + ) · α(ℓ) ≥ φ and µ β (ℓ) ≥ √ λ ·μ β (ℓ).
Proof. We only prove (a); the case (b) is completely analogous.
By Lemma 22, there exists k ′ ∈ H − with (a − − k ′ )µ β (k ′ ) ≥ φ/λ 2 . Note that µ β (k ′ ) ≥ φ λ 2 (a − −k ′ ) ≥ φ λ 2 S > p • ; since line 1 well-estimates k ′ , this implies thatμ β (k ′ ) ≥ √ λ · µ β (k ′ ) ≥ φ λ 3/2 (a − −k ′ ) . Therefore α(k ′ ) ≥ φ a − −k ′ . Since k is chosen as the argmax, this means that (a − − k)α(k) ≥ (a − − k ′ )α(k ′ ) ≥ φ. This further implies thatμ β (k) ≥ α(k) √ λ ≥ φ √ λ·(a − −k) ≥ p • . Since k is well-estimated, this implies that µ β (k) ≥ √ λμ β (k).
Proposition 24. FindInterval does not reach line 6; i.e. the interval σ determined at line 4 has size(σ) · σ weight ≥ φ.
Proof. We need to show that size(σ)α(k) ≥ φ and size(σ)α(ℓ) ≥ φ, since σ weight = min{α(k), α(ℓ)}. Let us prove the first inequality; the second one is completely analogous. Two cases are possible:
• H − = {i ⋆ }.
We have size(σ) = ℓ − k + 1 ≥ a − − k and thus size(σ)α(k) ≥ (a − − k)α(k). By Proposition 23, this is at least φ as desired.
• H − = {i ⋆ }, implying that k = i ⋆ , α(k) = ω ⋆ and size(σ) ≥ size(σ left ). If FindInterval is called from line 1 of Algorithm 4 then ω ⋆ = 1 and size(σ)α(k) ≥ 1 · 1 ≥ φ, as desired. If FindInterval is called from line 8 then ω ⋆ = σ weight left and so size(σ)α(k) ≥ size(σ left )·σ weight left , which is at least φ since (β left , σ left ) is conformant.
We now know that FindInterval returns interval σ = ([k, ℓ], min{α(k), α(ℓ)}).
Proposition 25. Interval σ is proper.
Proof. Let us define µ β (−∞) = µ β (+∞) = +∞. We need to show that min{µ β (k), µ β (ℓ)} ≥ min{α(k), α(ℓ)}. We will show that µ β (k) ≥ α(k); the case for ℓ is completely analogous. We can assume that k = −∞, otherwise the claim is trivial.
First suppose that H − = {i ⋆ }, and so the call to FindInterval is non-degenerate. Then
k = i ⋆ = σ − left and α(k) = ω ⋆ = σ weight left and β = β left . Since interval (β left , σ left ) is proper, we have µ β (k) = µ β (σ − left ) ≥ σ weight left = α(k). Next suppose that H − = {i ⋆ }.
Then by Proposition 23, we have µ β (k) ≥ √ λ ·μ β (k). By definition, this is always at least α(k).
Proposition 26. Interval σ is extremal.
Proof. We only verify that the interval is left-extremal; the proof of right-extremality is completely analogous. We can assume that k ≥ 1, otherwise there is nothing to show. Let us denote ℓ + = max([σ − , σ + ] ∩ H) + 1 ≥ a − , so that size(σ) = ℓ + − k. We thus need to prove that
µ β (i) ≤ 1 λ · ℓ + − k ℓ + − i · µ β (k) ∀i ∈ {0, . . . , k − 1}(9)
Two cases are possible. Case 1: k = h − . Then Eq. (5a) with j = ℓ + gives Eq. (9).
Case 2: k > h − . We must have H − = {i ⋆ } since k, h − ∈ H − . For i ∈ {h − , . . . , k} define ρ i = λ 1/2 i = h − λ 3/2 i > h − so that α(i) =μ β (i)ρ i . Since k was chosen as the argmax, we have (a − − i)α(i) ≤ (a − − k)α(k), i.e. µ β (i) ≤ (a − − k)α(k) ρ i (a − − i)(10)
We can show that the RHS here is at least p • . For, by Proposition 23, we have (
a − − k)α(k) ≥ φ and so (a − −k)α(k) ρ i (a − −i) ≥ φ λ 1/2 ρ i S ≥ φ λS > p • .
Since line 1 well-estimates i, this in turn implies that
µ β (i) ≤ (a − − k)α(k) ρ i λ 1/2 (a − − i)
Proposition 23 shows thatμ β (k) ≤ µ β (k)/ √ λ. Since k = h − , we have α(k) = λ 3/2μ β (k). We also have ℓ + ≥ a − . Combining all these bounds, we have shown that
µ β (i) ≤ (ℓ + − k)λ 1/2 µ β (k) ρ i (ℓ + − i)(11)
For i ∈ {h − + 1, . . . , k − 1}, we have ρ i = λ 3/2 , and so Eq. (11) shows that µ β (i) ≤ (ℓ + −k)µ β (k) λ(ℓ + −i) , which establishes Eq. (9). For i = h − , we have ρ i = λ 1/2 and so Eq. (11) shows
µ β (h − ) ≤ (ℓ + − k)µ β (k) ℓ + − h −(12)
which again establishes Eq. (9). In this section we will use the following notation:
Now consider i ∈ {0, . . . , h − − 1}. Eq. (5a) with j = ℓ + gives µ β (i) ≤ 1 λ · ℓ + − h − ℓ + − i · µ β (h − )H − = [0, θ] ∩ H H + = [θ, n] ∩ H Λ τ = Λ τ (β left , β right , θ) Z(β) = k∈H c k e βk µ β (k) = c k e βk Z(β) p(β) = µ β (H + ) Z − (β) = k∈H − c k e βk = (1 − p(β))Z(β) Z + (β) = k∈H + c k e βk = p(β)Z(β)
It will be easy to verify that BinarySearch succeeds with probability one if µ β (H − ) = 0 or µ β (H + ) = 0. Hence we assume for the remainder of this section that p(β) ∈ (0, 1) for all values β ∈ R. Before we begin our algorithm analysis, we record a few elementary properties about these parameters.
Lemma 27. p(β) is a strictly increasing function of β.
Proof. For any β ∈ R and δ > 0 we have Z − (β + δ) < Z − (β) · e δθ and Z + (β + δ) > Z + (β) · e δθ , and thus Z − (β+δ)
Z + (β+δ) < Z − (β) Z + (β) . Therefore, 1 p(β) − 1 = Z − (β) Z + (β)
is a strictly decreasing function of β, and accordingly p(β) is a strictly increasing function of β.
Since p(β) is an increasing function, it has an inverse p −1 . We use this to define parameter β crit :
β crit = β left if p(β left ) > 1/2 β right if p(β right ) < 1/2 p −1 (1/2) if p(β left ) ≤ 1/2 ≤ p(β right )
Proposition 28. There holds β right − β crit ≤ q + 1.
Proof. Let β 1 = β right − q − 1 and p 1 = p(β 1 ). If β 1 ≤ β left , then β right − β left ≤ q + 1 and we are done. Otherwise, we can write
Z(β right ) ≥ Z + (β right ) ≥ Z + (β 1 ) · e βright−β 1 = p 1 Z(β 1 ) · e βright−β 1
where the second inequality holds since min H + ≥ 1. Now since β 1 ≥ β left ≥ β min , there holds
q ≥ log Z(β right ) Z(β 1 ) ≥ β right − β 1 + log p 1 = q + 1 + log p 1
This implies that log p 1 ≤ −1, which in turn implies that
p 1 ≤ 1/2. So β 1 ≤ β crit .
The starting point for our algorithm is a sampling procedure of Karp & Kleinberg [12] for noisy binary search. We summarize their algorithm as follows:
Theorem 29 ( [12]). Suppose we have oracle access to draws from Bernoulli random variables X 1 , . . . , X N , wherein each X i has mean x i , and we know 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x N ≤ 1 but the values x 1 , . . . , x N are unknown. Let us also write x 0 = 0, x N +1 = 1.
Then there is a sampling procedure which takes as input two parameters α, ∆ ∈ (0, 1), and uses O( log N ∆ 2 ) oracle queries to the variables X i in expectation. With probability at least 3/4, it returns an
index v ∈ {0, . . . , N } such that [x v , x v+1 ] ∩ [α − ∆, α + ∆] = ∅.
Theorem 29 is used to search a discrete set. By quantization, we can adapt it to weakly solve BinarySearch; we will afterward describe the limitations of this preliminary algorithm and how to get the full result.
Theorem 30. Let τ ′ ∈ (0, 1/2) be an arbitrary constant. There is a sampling procedure with the following properties:
(i) It takes as input an interval [β ′ left , β ′ right ] ⊆ [β left , β right ] and returns a valueβ ∈ [β ′ left , β ′ right ]. (ii) If β ′ left ≤ β crit ≤ β ′ right , then with probability at least 3/4 the outputβ satisfiesβ ∈ Λ τ ′ . (iii) The expected sample complexity is O(log(n(1 + β ′ right − β ′ left )).
Proof. Let us define parameters
δ = 2 n log (1 − τ ′ ) · (1 − 2τ ′ ) τ ′ · (3 − 2τ ′ ) > 0 N = β ′ right − β ′ left δ + 1 = O(n(β ′ right − β ′ left ) + 1) Let us define values u 1 , . . . , u N by u i = β ′ left + i−1 N −1 (β ′ right − β ′ left )
. Note that we simulate access to a Bernoulli variable X i with rate x i = p(u i ) by drawing k ∼ µ u i and checking if k < θ.
Our algorithm is to apply Theorem 29 with respect to the variables X 1 , . . . , X N and with parameters α = 1/2, ∆ = 1/2−τ ′ 2 ; let v ∈ {0, . . . , N } denote the resulting return value. If 1 ≤ v ≤ N − 1, then we outputβ = uv+u v+1 2 . If v = 0, then we outputβ = β ′ left . If v = N , then we outputβ = β ′ right . This has expected sample complexity O( log N ∆ 2 ) = O(log(n(1 + β ′ right − β ′ left )) (bearing in mind that ∆ is constant). This shows property (iii).
To show property (ii), suppose that v satisfies [x v , x v+1 ] ∩ [1/2 − ∆, 1/2 + ∆] = ∅, which occurs with probability at least 3/4; we will show that thenβ ∈ Λ τ ′ as desired. There are a number of cases.
• Suppose that 1 ≤ v ≤ N − 1. Then we need to show that τ ′ ≤ p(β) ≤ 1 − τ ′ . We will show only the inequality p(β) ≥ τ ′ ; the complementary inequality is completely analogous.
Choose arbitrary x ∈ [x v , x v+1 ] such that x ≥ 1/2 − ∆ (this exists because of our hypothesis that the algorithm of Theorem 29 returned a good answer). We write
u = p −1 (x) ∈ [u v , u v+1 ]. If u ≤β, then p(β) ≥ p(u) ≥ 1/2 − ∆ ≥ τ ′ .
Otherwise, suppose that u >β. Since max H − ≤ θ − 1 2 , min H + ≥ θ + 1 2 and max H + ≤ n, we can then write
p(β) 1 − p(β) = Z + (β) Z − (β) ≥ Z + (u)e −n(β−u) Z − (u) = e −n(β−u) p(u) 1 − p(u) ≥ e −n(β−u) (1/2 − ∆) 1/2 + ∆ We know that u v+1 − u v = 1 N −1 (β ′ right − β ′ left )
≤ δ, and since u ≥β = (u v + u v+1 )/2, this implies thatβ ≥ u − δ/2. So we have shown that
p(β) 1 − p(β) ≥ e −nδ/2 (1/2 − ∆) 1/2 + ∆ = τ ′ 1 − τ ′
This in turn implies that p(β) ≥ τ ′ as desired.
• Suppose that v = 0 and p(β left ) ≤ 1/2. Again, we need to show that τ ′ ≤ p(β) ≤ 1 − τ ′ . Sincê β = β ′ left ≤ β crit , we know that p(β) ≤ 1/2 ≤ 1 − τ ′ . So we only need to show the lower bound p(β) ≥ τ ′ .
As in the first case, let x ∈ [x 0 , x 1 ] be such that
x ≥ 1/2 − ∆. Write u = p −1 (x) ≤ u 1 . Since x 0 = 0 and x 1 = p(β ′ left ), we know that u ≤ β ′ left , so that p(β ′ left ) ≥ p(u) ≥ 1/2 − ∆ ≥ τ ′ . • Suppose that v = 0 and p(β left ) > 1/2. In this case, since β left ≤ β ′ left ≤ β crit , we know that β ′ left = β left .
The algorithm returns valueβ = β ′ left = β left and soβ ∈ Λ τ ′ . • Suppose that v = N . This is completely analogous to the cases where v = 0.
Theorem 30 does not directly solve BinarySearch on its own, for two reasons. First, the success probability is only a constant 3/4, not the desired value 1 − γ. Second, the runtime depends on the size β ′ right − β ′ left , which may be unbounded. We use an exponential back-off strategy to address both of these issues simultaneously.
Let us choose arbitrary constant τ ′ ∈ (τ, 1/2). We formulate the final algorithm:
Procedure BinarySearch(β left , β right , θ, γ).
1 set i 0 = ⌈log 2 log 2 n γ ⌉ 2 for i = i 0 , i 0 + 1, i 0 + 2, . . . , do 3 set β ′ i = max{β left , β right − 2 2 i } 4 let β be the output of the alg. of Theorem 30 with β ′ left = β ′ i , β ′ right = β right 5 setμ β ← Sample(β; 1 2 log τ ′ τ , γ/2 i−i 0 +2 , τ ) 6 if β = β left ∨μ β (H − ) ≥ √ τ τ ′ ) ∧ β = β right ∨μ β (H + ) ≥ √ τ τ ′ then return β
Note that the loop in line 2 runs indefinitely, starting at index value i = i 0 .
Proposition 31. The expected sample complexity of BinarySearch is O(log nq γ ).
Proof. We claim that the expected sample complexity of iteration i (if it is reached) is O(2 i ). Indeed, the complexities at lines 4 and 5 are respectively O(log(n(β right − β ′ i )) + 1) ≤ O(log(n2 2 i )) = O(2 i + log n) and O(log(2 i−i 0 +2 /γ)) ≤ O(i + log 1 γ ), which together give O(2 i + log n γ ). By observing that 2 i ≥ 2 i• ≥ log 2 n γ we get the desired claim. Let s be the minimal integer such that β right − 2 2 s ≤ β crit . By Proposition 28, we have s ≤ log 2 log 2 (q + 1). Let t = max{i 0 , s}.
We first consider the sample complexity due to iterations i = i 0 , i 0 + 1, . . . , t. In each such iteration i, the expected sample complexity is O(2 i ). Summing over i = i 0 , . . . , t, the overall expected sample complexity is O(2 t ).
We next claim that in each iteration i > t, there is a probability of at least 9/16 that the algorithm BinarySearch terminates. Indeed, since i ≥ s, we have β ′ i ≤ β crit , and thus by Theorem 30 there is a probability of at least 3/4 that the resulting value β is in Λ τ ′ . In such a case, if line 5 wellestimates the sets H + and H − , then the algorithm will return value β and terminate. This occurs with probability at least 1 − 2 · γ/2 1+2 ≥ 3/4. Overall, the probability of termination at this iteration is at 3/4 × 3/4 = 9/16. This in turn implies that the probability that BinarySearch reaches iteration i = t + 1 + j is at most (7/16) j . If it does reach this iteration, the expected sample complexity is O(2 i ) = O(2 t+j ). Thus, the overall expected sample complexity due to iteration i = t + 1 + j is O((7/16) j 2 t+j ).
So the expected sample complexity due to iterations i > t is at
most ∞ j=0 O((7/8) j 2 t ) = O(2 t ). The total expected sample complexity of the algorithm is O(2 t ) = O(max{2 s , 2 i 0 }) = O(log nq γ ). 18
Proposition 31 implies, in particular, that BinarySearch terminates with probability 1.
Proposition 32. If BinarySearch returns value β then P[β ∈ Λ τ ] ≥ 1 − γ.
Proof. By construction, line 5 at iteration i well-estimates H − and H + with probability at least 1 − γ/2 i−i 0 +1 . Thus, sets H − and H + are well-estimated at all iterations with probability at least 1 − i≥i 0 γ/2 i−i 0 +1 = 1 − γ. It remains to observe that if such event happens and BinarySearch returns value β then β ∈ Λ τ .
Proof of Theorem 20: Procedure FinalizeSchedule(J , γ)
In this section we assume that J is a minimal pre-schedule of the form J = ((β 0 , σ 0 ), . . . , (β t , σ t )), which satisfies conditions (I1) -(I4). Such a minimal pre-schedule J has the following properties:
• σ − i ∈ [σ − i−1 , σ + i−1 ] and σ + i−1 ∈ [σ − i , σ + i ] for each i ∈ {1, . . . , t}. • σ + i−1 < σ − i+1 for each i ∈ {1, .
. . , t − 1} (otherwise (β i , σ i ) could have been removed from J ). Thus, J is similar to a schedule, except that the intervals may cross each other. Since J is minimal, each k ∈ H is covered in at most two intervals. So t i=0 size(σ i ) ≤ 2(n + 1). By (I4), we have 1 σ weight ≤ size(σ) φ for each interval in J , and thus
InvWeight(J ) ≤ i size(σ) φ ≤ 2(n + 1) φ
The next algorithm shows how to uncross a minimal proper pre-schedule J to obtain a proper schedule I with InvWeight(I) ≤ e ν InvWeight(J ), for an arbitrary constant ν > 0.
Algorithm 7: FinalizeSchedule(J , γ) for minimal pre-schedule J = ((β 0 , σ 0 ), . . . , (β t , σ t )). We need to argue that the output is good with high probability. Let us suppose that each iteration
1 foreach i ∈ {0, . . . , t} letμ β i ← Sample(β i ; ν/2, γ 4(t+1) , e −ν/2 σ weight i ) 2 foreach i ∈ {0, . . . , t − 1} do 3 if ∃ k ∈ {σ + i , σ − i+1 } s.t.μ β i (k) ≥ e −ν/2 σ weight i andμ β i+1 (k) ≥ e −ν/2 σ weight i+1i of line 1 well-estimates σ + i , σ − i , σ − i+1 , σ + i+1 .
Since we use error parameter γ 4(t+1) , this has probability at least 1 − γ. We show that, under this condition, we output either a proper schedule or ⊥; furthermore, if J is proper, then we output a proper schedule I.
We first claim that the schedule I at line 6 is proper. We need to show that the updated values σ i satisfy µ β i (σ − i ) ≥ e −ν σ weight for i ≥ 1 and µ β i (σ + i ) ≥ e −ν σ weight for i ≤ t − 1. These are exactly analogous, so we just show the former one. To see this, note that in iteration i − 1, we update
σ − i = k where k satisfiesμ β i (k) ≥ e −ν/2 σ weight i
. Since line 1 well-estimates k, this implies that µ β i (k) ≥ e −ν σ weight i as required.
Next, we argue that if J is proper then we do not output ⊥. Suppose we do so at iteration i, and
let k = σ − i+1 ≤ σ + i = ℓ. Since J is proper, we have µ β i (ℓ) ≥ σ weight i and µ β i+1 (k) ≥ σ weight i+1
. Since line 1 well-estimates k, ℓ this implies thatμ β i (ℓ) ≥ e −ν/2 σ weight i andμ β i+1 (k) ≥ e −ν/2 σ weight i+1 . Since we reached line 5, this implies that we must have hadμ β i+1 (ℓ) < e −ν/2 σ weight i+1
andμ β i (k) < e −ν/2 σ weight i . Since k, ℓ are well-estimated, this implies that µ β i (k) < σ weight i and µ β i+1 (ℓ) < σ weight i+1 . Therefore, we have µ β i (k)µ β i+1 (ℓ) < σ weight i σ weight i+1 ≤ µ β i (ℓ)µ β i+1 (k),1 initialize i ← n and R ← ∅ 2 while true do 3 set α ← BinarySearch(β min , β max , i + 1 2 , γ 4(n+1) 2 ) 4 setμ α ← Sample(α; log 2, γ 2(n+1) 3 , 1 256 ) 5 set j to be the maximum value such thatμ α ([0, j − 1]) ≤ 1 128 and j ≤ i 6 insert α into R 7 update i ← j − 1 8 if j = −1 or α = β min then return R.
In analyzing this, we let α t , i t , j t denote the values of those variables at iteration t. We let T be the stopping time, i.e. the final iteration count t. We write R for the final set returned by this procedure, and we also set r = min{ √ q log n, n} throughout. Finally, we write Z t = Z(α t ) for t = 1, . . . , T . Note that |R| = T . Also, we have i t+1 < j t ≤ i t , and thus T ≤ n + 1. We now turn to proving Theorem 15(b) and (c). To do so, we will assume that each execution of BinarySearch is good and each iteration of line 5 well-estimates every interval {0, . . . , k}. By the specification of these subroutines, this has probability at least 1 − γ n . We then argue that the resulting value R satisfies the required conditions, namely that R is proper with respect to constant ζ = 256 and that |R| ≤ T ≤ O( √ q log n). For the remainder of this section, we assume without further comment that these conditions all hold.
Proposition 33. For any k ∈ H there exists α ∈ R such that
µ α (k) ≥ max β∈[β min ,βmax] µ β (k)/256
In particular, R is proper with respect to constant parameter ζ = 256.
Proof. Let β = arg max β∈[β min ,βmax] µ β (k). Due to the termination condition of FindRepresentatives, there are two possibilities: either there is some iteration t with j t ≤ k ≤ i t , or k < j T and α T = β min . Let us first consider the former case, and write α = α t , i = i t , j = j t . First, suppose that β > α. In this case, α = β max , and so since the call to BinarySearch at line 3 is good, we have µ α ([i + 1, n]) ≥ τ = 1/4. Since k ≤ i, we have:
µ β (k) ≤ c k e βk n ℓ=i+1 c ℓ e βℓ ≤ c k e αk n ℓ=i+1 c ℓ e αℓ = µ α (k) µ α ([i + 1, n]) ≤ 4µ α (k)
If β = α then the condition obviously holds. Otherwise, suppose that β < α. In this case, α = β min , and so since the call to BinarySearch at line 3 is good, we have µ β ([0, i]) ≥ τ = 1/4. By the definition of j, it holds that either j = i orμ α ([0, j]) > 1/128. In either case, since line 4 well-estimates interval [0, j], it must be that µ α ([0, j]) ≥ 1/256. Since k ≥ j, we have:
µ β (k) ≤ c k e βk j ℓ=0 c ℓ e βℓ ≤ c k e αk j ℓ=0 c ℓ e αℓ = µ α (k) µ α ([0, j]) ≤ 256µ α (k)
Finally, suppose that k < j T and α T = β min < β. Let us set i = i T . Since α T = β max and the call to BinarySearch at line 3 is good, it holds that µ α T ([i + 1, n]) ≥ τ = 1/4. We then compute:
µ β (k) ≤ c k e βk n ℓ=i+1 c ℓ e βℓ ≤ c k e α T k n ℓ=i+1 c ℓ e α T ℓ = µ α T (k) µ α T ([i + 1, n]) ≤ 4µ α T (k)
This proves Theorem 15(b). We will next bound T .
Proposition 34. For t = 1, . . . , T − 1 we have α t > α t+1
Proof. First, note that α t = β min , as otherwise the algorithm would have terminated at iteration t.
If α t+1 = β min , the claim therefore follows immediately. Otherwise, suppose that α t+1 < β min . Since BinarySearch is good it then holds that µ α t+1 ([0, i t+1 ]) ≥ 1/4. On the other hand, since line 4 wellestimate every interval, we must have
µ αt ([0, j t − 1]) ≤ 1/64. Since i t+1 = j t − 1, this implies that µ αt ([0, i t+1 ]) ≤ 1/64. In particular, µ αt ([0, i t+1 ]) ≤ µ α t+1 ([0, i t+1 ])
. By Lemma 27, this implies that α t > α t+1 strictly.
Proposition 35. For t = 1, . . . , T − 2 we have the bounds:
Z t Z t+1 ≥ 8 exp( j t+1 2(i t − j t+1 ) )
Proof. Because the algorithm terminates whenever α t = β min , we know that α t > β min for t < T . Since BinarySearch is good it holds that µ αt ([0, i t ]) ≥ 1/4. Also, since line 4 well-estimates every interval, we have µ αt [0, j t − 1] ≤ 1/64. Thus, if we define the interval V t = [j t , i t ], we have
µ αt (V t ) ≥ 1/4 − 1/64 ≥ 1/8 and µ αt (V t+1 ) ≤ µ αt ([0, i t+1 ]) ≤ µ αt ([0, j t − 1]) ≤ 1/64.
We can estimate:
Z t Z t+1 = µ α t+1 (V t+1 ) µ αt (V t+1 ) × k∈V t+1 c k e αtk k∈V t+1 c k e α t+1 k ≥ 1/8 1/64 × k∈V t+1 c k e αtk k∈V t+1 c k e α t+1 k ≥ 8e (αt−α t+1 )j t+1(13)
the last inequality here comes from the fact that j t+1 is the least element of V t+1 and that α t > α t+1 . Alternatively, we can estimate: (14) where again the last inequality comes from the fact that i t is the largest element of V t and that α t > α t+1 .
Z t Z t+1 = µ α t+1 (V t ) µ αt (V t ) × k∈Vt c k e αtk k∈Vt c k e α t+1 k ≤ 1 1/4 × k∈Vt c k e αtk k∈Vt c k e α t+1 k ≤ 4e (αt−α t+1 )it
Putting these two inequalities together, we conclude that 8e (αt−α t+1 )j t+1 ≤ 4e (αt−α t+1 )it , which implies that (α t − α t+1 )(i t − j t+1 ) ≥ log 2 ≥ 1/2. Substituting this bound into Eq. (13) we get for t = 1, . . . , T − 2:
Z t Z t+1 ≥ 8e j t+1 2(i t −j t+1 )
We can use this to estimate q in terms of T .
Proposition 36. We have T ≤ O(q) and T ≤ O( √ q log n).
Proof. Since β max ≥ α 1 > α 2 > · · · > α T ≥ β min , we can compute:
q = log Z(β max ) Z(β min ) ≥ T −2 t=1 log Z(α t ) Z(α t+1 ) ≥ (T − 2) log 8 + 1 2 T −2 t=1 j t+1 i t − j t+1
This immediately shows that T ≤ O(q). If q ≤ log n, this in turn shows that T ≤ O( √ q log n) and we are done. So, let us suppose that q > log n. For notational convenience, let us suppose that T is even (the case that T is odd is nearly the same), and define L = {2, 4, 6, . . . , T − 2}. We can lower-bound the sum here by
T −2 t=1 j t+1 i t − j t+1 ≥ T −2 t=2 j t+1 j t−1 − j t+1 ≥ ℓ∈L j ℓ+1 j ℓ−1 − j ℓ+1
For ℓ ∈ L, let us set z ℓ = log( j ℓ−1 j ℓ+1 ). We note that j 1 ≤ n and j T −1 > i T ≥ 0, so we can compute:
n ≥ j 1 j T −1 = ℓ∈L j ℓ−1 j ℓ+1 = exp( ℓ∈L z ℓ )
and so ℓ∈L z ℓ ≤ log n. We can also compute:
ℓ∈L j ℓ+1 j ℓ−1 − j ℓ+1 = ℓ∈L 1 e z ℓ − 1
The function f (z) = 1 e z −1 is decreasing concave-up, and so by Jensen's inequality we have:
ℓ∈L 1 e z ℓ − 1 = ℓ∈L f (z ℓ ) ≥ |L| × f ℓ∈L z ℓ |L| ≥ |L| × f log n |L| = (T − 2)/2 e log n (u−2)/2 − 1
Now recall that we have assumed that q > log n. If T ≤ 6 + 2 log n, then this shows that T ≤ O( √ q log n) and we are done. Otherwise, for T ≥ 6 + 2 log n, we have e log n (T −2)/2 − 1 ≤ e log n (T −2)/2 , and therefore
ℓ∈L j ℓ+1 j ℓ−1 − j ℓ+1 ≥ (T − 2)/2 × (T − 2)/2 e log n ≥ Ω(T 2 / log n)
which further implies that q ≥ Ω(T 2 / log n), i.e. that T ≤ O( √ q log n) as desired.
Applications
There is a pervasive close connection between sampling and counting algorithms. Consider the following scenario: we have a collection of objects of various sizes, and we would like to estimate the number C i of objects of size i. If we can sample from the Gibbs distribution on these objects (weighted by their size), then our algorithm allows us to convert this sampling procedure into a counting procedure.
In an number of combinatorial applications, we further know that the counts C i are log-concave; for example, the matchings in a graph [8], or the number of independent sets in a matroid [1]. One main motivation for our focus on the case of log-concave coefficients is indeed to handle these combinatorial situations. In the context of log-concave coefficients, there are natural choices for certain parameters for our algorithm which lead to particularly clean bounds:
Theorem 37. Suppose coefficients {c k } k∈H are known to be log-concave. If we select appropriate values β min ≤ log c 0 c 1 , µ * = 1 n+1 and β max ≥ log c n−1 cn , then when we execute Theorem 2, we get the following results where F := max{β max , log c 1 c 0 , 1}: (a) We obtain a (ε, H)-estimate of c with probability at least 1 − γ. Proof. Let us define b i = c i−1 /c i for i = 1, . . . , n. Since c i is log-concave, the sequence b 1 , . . . , b n is non-decreasing. Let us first show the following fact: for each i, k ∈ H, we have the bound
c i e i log b i ≥ c k e k log b i(15)
To show this for k > i, we use the fact the sequence b j is non-decreasing to compute:
c i e i log b i c k e k log b i = e (i−k) log b i k−1 j=i c j c j+1 = exp( k−1 j=i log b j+1 − log b i ) ≥ 1 A similar calculation applies for k < i. Since µ β (k) ∝ c k e αk , Eq. (15) shows that µ log b i (i) ≥ 1 n+1 . Also, since sequence b ℓ is non-decreasing, we have log b i ∈ [log b 0 , log b n ] ⊆ [β min , β max ] for i ≥ 1.
By similar reasoning, we have µ log b 0 (0) ≥ 1 n+1 . Therefore, with µ * = 1 n+1 , we have H * = H, and so we have shown (a).
We next turn to part (b). To begin, we can lower-bound Z(β min ) as
Z(β min ) = c i e β min i ≥ c 0 e β min ×0 = c 0
To upper-bound Z(β max ), we observe that for every k ≤ n, we have c n e nβmax c k e kβmax = c n e nbn c k e kbn × e (βmax−bn)(n−k)
By Eq. (15), we have cne nbn c k e kbn ≥ 1 and by hypothesis we have β max ≥ b n . Therefore, c n e nβmax ≥ c k e kβmax for every k ≤ n, and so we bound Z(β max ) as Z(β max ) = i c i e βmaxk ≤ nc n e βmaxn ≤ e F n (n + 1)c n Thus we estimate Q as:
Q = Z(β max ) Z(β min ) ≤ e F n × (n + 1)c n b n n c 0 = e F n × (n + 1) c n−1 c n n−1 c n−1 c 0
The sequence b ℓ is non-decreasing, so c n−1 /c 0 telescopes as:
c n−1 c 0 = n−1 i=1 c i c i−1 = n−1 i=1 (1/b i ) ≤ n−1 i=1 (1/b 1 ) = c 1 c 0 n
so Q ≤ e F n × (n + 1) c n−1 cn n−1 c 1 c 0 n and q ≤ nF + log(n + 1) + (n − 1) log c n−1 cn + n log c 1 c 0 ≤ nF . With this value of q and µ * , we get the stated sample complexity.
Counting connected subgraphs
Consider a connected graph G = (V, E). For each i = |V | − 1, . . . , |E| let N i denote the number of connected subgraphs of G with i edges; these quantities are essentially the coefficients of the reliability polynomial for G.
In [7], Guo & Jerrum described an algorithm to sample a connected subgraph
G ′ = (V, E ′ ) with probability proportional to f ∈E ′ (1 − p(f )) f ∈E−E ′ p(f ), for some weighting function p : E → [0, 1].
This has a probabilistic interpretation, wherein each edge f fails independently with probability p(f ), and we wish to condition on the resulting subgraph remaining connected. Here E − E ′ is the set of failed edges. If we set p(f ) = 1 1+e β for all edges f , then the resulting distribution on connected subgraphs is a Gibbs distribution, with rate β and with coefficient sequence given by c i = N |E|−i .
The runtime of this algorithm was subsequently improved by [6], which we rephrase in our notation as follows:
Theorem 38 ([6], Corollary 10). There is an algorithm to sample from the Gibbs distribution with coefficients c i = N |E|−i for any value of β > 0; the expected runtime is O(|E| + |E||V |e β ).
The sequence N i here counts the number of independent sets in the co-graphic matroid. By the result of [1], this implies that sequence N i (and hence the coefficient sequence c i ) is log-concave.
Proof of Theorem 3. Observe that N |E| = 1, and so if we can estimate the coefficients c i , then this immediately allows us to estimate N i as well. The number of coefficients in the Gibbs distribution is n = |E| − |V | + 1.
In order to apply Theorem 37, we need to bound the quantities c n−1 /c n and c 1 /c 0 . These are both at most |E|, since to enumerate a connected graph with |V | edges we may select a spanning tree and any other edge in the graph, and to enumerate a graph with |E| − 1 edges we simply select an edge of G to delete. Therefore, we set β max = log |E|. This gives F = log |E|, and by Theorem 38 the cost of each call to the sampling oracle is O(|E| 2 |V |).
So Theorem 37 shows that we need to take O(n log |E| log 2 n log n γ /ε 2 ) samples. It is traditional in analyzing FPRAS to take γ = O(1), and since n = |E| we overall O(|E| log 3 |E|/ε 2 ) samples.
The work [7] sketches an FPRAS for this problem as well; the precise complexity is unspecified and appears to be much larger than Theorem 3. We also note that Anari et al. [2] provide a general FPRAS for counting the number of independent sets in arbitrary matroids, which would include the number of connected subgraphs. This uses a very different sampling method, which is not based on the Gibbs distribution. They also do not provide concrete complexity estimates for their algorithm.
Counting matchings
Consider a graph G = (V, E) with |V | = 2v nodes which has a perfect matching. For i = 0, . . . , n = v, let M i denote the number of i-edge matchings. Since G has a perfect matching these are all non-zero. As originally shown in [8], the sequence M i is log-concave.
In [10], Jerrum & Sinclair described an algorithm based on Markov Chain Monte Carlo to approximate the Gibbs distribution on matchings. They improved their analysis further in [11]. To rephrase their result in our terminology:
Theorem 39 ( [11]). There is an algorithm to approximately sample from the Gibbs distribution with coefficients c i = M i for any value of β > 0; the expected runtime isÕ(|E||V | 2 (1 + e β ) log 1 δ ) to get within a total variation distance of δ.
There remains one complication to applying Theorem 37: for general graphs, the ratio between the number of perfect and near-perfect matchings, i.e. the ratio M v−1 /M v , could be exponential in n. This would cause the parameter F to be too large in applying Theorem 37. This is the reason for our assumption of a known bound on the ratio M v−1 /M v . With this stipulation, we prove Theorem 4:
Proof of Theorem 4. Let us first show the general result in terms of parameter f . Observe that M 0 = 1, and so if we can estimate the coefficients c i , then this immediately allows us to estimate M i as well. The number of coefficients in the Gibbs distribution is given by n = |V |/2 = v.
We first determine the sample complexity needed when applying Theorem 37. To do so, we need to bound the quantities c n−1 /c n and c 1 /c 0 . The first is bounded by f by assumption, and the latter is clearly bounded by |E|. Therefore, we set β max = log f , and F ≤ max{log |E|, log f }. So Theorem 37 shows that we need O(nF log n log n γ /ε 2 ) samples. By Corollary 6, we must take δ = poly(1/n, 1/f, ε, γ) to ensure that the sampling oracle is sufficiently close to the Gibbs distribution. It is traditional in analyzing FPRAS algorithms to take γ = O(1). With these choices, Theorem 39 requires O(|E||V | 2 f polylog(|V |, f, 1/ε)) time per sample. Overall, our FPRAS has runtime ofÕ(|E||V | 3 f /ε 2 ).
To show the second result, [10] showed that if the graph has minimum degree at least |V |/2, then
M v > 0 and M v−1 Mv ≤ f = O(|V | 2 ). Furthermore, it is clear that |E| = Θ(|V | 2 ).
Lower bound on sample complexity
In [13], Kolmogorov showed lower bounds on the sample complexity of a generalization of P ratio . In this section, we adapt this construction to show lower bounds on the problems P ratio and P µ * coef for a wider variety of parameters. Note that our lower bound for P µ * coef applies to obtaining any (ε, H * )-estimate of c, not just a lower-normalized estimate as is provided by our algorithm.
The main strategy of [13] is based on an "indistinguishability" lemma. Here, a target distribution c (0) (a coefficient vector) is surrounded by an envelope of alternate probability distributions c (1) , . . . , c (d) , which all use the same ground set H = {0, . . . , n} and the same values β min , β max . The lemma establishes a lower bound on the samples needed to distinguish between Gibbs distributions with these different coefficients.
Let us define µ β (k | c (r) ) to be the Gibbs distributions with parameter β under the coefficient vectors c (r) . We also define q (r) to be the corresponding value of q for distribution c (r) . For some parameter µ * (which will common to all distributions c (0) , . . . , c (d) ), we likewise define H * (r) to the set H * with respect to distribution c (r) .
For any k ∈ H, let us define
U β (k) = d r=1 µ β (k | c (0) ) µ β (k | c (r) ) = d r=1 c (0) k Z(β | c (r) ) c (r) k Z(β | c (0) )
and let us define the key parameter
∆ = max β∈[β min ,βmax] k∈H log U β (k)
Lemma 40 ( [13]). Let A be a randomized algorithm which generates a set of queries β 1 , . . . , β T ∈ [β min , β max ] and receives values K 1 , . . . , K T , wherein each K i is drawn from distribution µ β i . At some point the procedure stops and either outputs either TRUE or FALSE. The queries β i may be adaptive and may be randomized, and the stopping time T may also be randomized. Suppose that, with probability at least 1 − γ algorithm A outputs TRUE on input c (0) , whereas with probability at least 1 − γ it outputs FALSE on inputs c (1) , . . . , c (d) , for some parameter γ < 1/4.
Then the expected sample complexity of A on instance c (0) is Ω( d log(1/γ) ∆ ).
This lemma implies lower bounds on the sampling problems P ratio and P µ * coef :
Corollary 41. (a) Suppose that |q (0) − q (r) | > 2ε for all r = 1, . . . , d. Then any algorithm to solve P ratio must have expected sampling complexity Ω( d log(1/γ) ∆ ) on problem instance c (0) .
(b) Fix some parameter µ * . Suppose that for each r = 1, . . . , d there exists parameters i, j ∈ H * (0) such that | log(c
(0) i /c (0) j ) − log(c (r) i /c (r) j )| > 2ε.
Then any algorithm to solve P µ * coef must have expected sampling complexity Ω( d log(1/γ) ∆ ) on problem instance c (0) . Note that i, j may depend on the value r.
Proof. (a) Whenever P ratio succeeds on problem instance c (0) , the estimateq is within ±ε of q (0) . Whenever P ratio succeeds on problem instance c (r) , the estimateq is within ±ε of q (r) , and consequently it is not within ±ε of q (0) . Thus, solving P ratio allows us to distinguish c (0) from c (1) , . . . , c (d) .
(b) Let us run P µ * coef , obtaining estimateĉ. If there exists any pair i, j ∈ H * (0) such that either
c i = 0,ĉ j = 0, or | log(ĉ i /ĉ j ) − log(c (0) i /c (0)
j )| > ε then we output FALSE; otherwise we output TRUE. When run on problem instance c (0) , it holds with probability at least 1 − γ that the vectorĉ is an (ε, H * (0) ) estimate of c. In this case, by definition, this procedure will output TRUE.
When run on problem instance c (r) , again with probability at least 1−γ the vectorĉ is an (ε, H * (r) ) estimate of c. In this case, let i, j be the pair guaranteed by the hypothesis. By definition, in this case we eitherĉ i = 0,ĉ j = 0, or the valueĉ i /ĉ j is an ε-estimate of the true value c
(r) i /c (r)
j . In all three of these cases, the procedure will output false.
Thus, solving P µ * coef allows us to distinguish c (0) from c (1) , . . . , c (d) . By constructing appropriate problem instances and applying Corollary 41, we will show the following lower bounds on the sampling problems:
Theorem 42. Let n ≥ 2, ε < ε max , γ < γ max , q ≥ q min , µ * ≤ µ * ,max , where µ * ,max , ε max , γ max , q min are some universal constants.
(a) Any algorithm to solve P ratio on log-concave problem instances with these parameters must have expected sample complexity
Ω( min{q, n 2 } log 1 γ ε 2 )
(b) Any algorithm to solve P µ * coef on log-concave problem instances with these parameters must have expected sample complexity
Ω( ( 1 µ * + min{q, n 2 }) log 1 γ ε 2 )
(c) Any problem to solve P µ * coef on general problem instances with these parameters must have expected sample complexity
Ω( min{q + √ q µ * , n 2 + n µ * } log 1 γ ε 2 )
7.1 Bounds for P µ * coef in terms of µ * in the log-concave setting
The construction here is very simple: we set β min = 0, and n = 1. We have three choices for the coefficients, namely c
(0) 0 = 2µ * , c(1)0 = 2µ * e −3ε , c(2)0 = 2µ * e 3ε .
In all three cases, we set c
(i) 1 = 1.
We can also add dummy extra coefficients c i = 0 for i = 2, . . . , n. Note that c (0) has log-concave coefficients.
Since Z(β max ) is a continuous function of β max with Z(+∞) = +∞, we can ensure that this problem instance has the desired value of q by setting β max sufficiently large.
This allows us to show one of the lower bounds of Theorem 42:
Proposition 43. Under the conditions of Theorem 42, any procedure to solve P µ * coef for log-concave problem instances must have expected sample complexity Ω( log(1/γ) µ * ε 2 ) Proof. We will show this using Corollary 41 with parameters i = 0, j = 1. It is clear that
| log(c (0) i /c (0) j ) − log(c (r) i /c (r)
j )| > 2ε, and that 0, 1 ∈ H * (0) with respect to parameter µ * . We need to compute the parameter ∆. We begin by computing Z(β | c (r) ) as:
Z(β | c (0) ) = 2µ * + e β , Z(β | c (1) ) = 2µ * e −3ε + e β , Z(β | c (2) ) = 2µ * e 3ε + e β
and thus, after some algebraic simplifications, we get
U β (0) = U β (1) = (2µ * e −3ε + e β )(2µ * e 3ε + e β ) (2µ * + e β ) 2
Simple calculus shows that this is a decreasing function of β for β ≥ 0. So its maximum value in the interval [β min , β max ] occurs at β = 0 and ∆ = log U 0 (k) = log (2µ * e −3ε + 1)(2µ * e 3ε + 1)
(2µ * + 1) 2 = log 1 + 2µ * (e 3ε + e −3ε − 2) (1 + 2µ * ) 2 ≤ 2µ * (e 3ε + e −3ε − 2) ≤ O(µ * ε 2 )
By Corollary 41, solving P µ * coef on problem instance c (0) requires expected sample complexity of
Ω( log(1/γ) ∆ ) = Ω( log(1/γ) µ * ε 2 ).
7.2 Bounds for P µ * coef in terms of µ * in the general setting
In this construction, let us set a parameter t (which we will determine later). We set c We select β min = 0; the parameter β max will be specified later.
Proposition 44. For ν ≤ O(1), the problem instances c (0) , . . . , c (d) have ∆ ≤ O(µ * ν 2 ).
Proof. Given value β ∈ [β min , β max ], let us compute U β (k) as:
U β (k) = d r=1 c (0) k Z(β | c (r) ) c (r) k Z(β | c (0) ) = d r=1 Z(β | c (r) ) Z(β | c (0) ) = t−1 i=0 (e ν − 1)2 −i−i 2 × 8µ * e (2i+1)β + Z(β | c (0) ) (e −ν − 1)2 −i−i 2 × 8µ * e (2i+1)β + Z(β | c (0) ) Z(β | c (0) ) 2 = t−1 i=0 1 + (e ν − 1)2 −i−i 2 × 8µ * e (2i+1)β Z(β | c (0) ) 1 + (e −ν − 1)2 −i−i 2 × 8µ * e (2i+1)β Z(β | c (0) ) ≤ exp (e ν + e −ν − 2) × 8µ * Z(β | c (0) ) t−1 i=0 2 −i−i 2 e (2i+1)β Let us define S i = 2 −i−i 2 e (2i+1)β and S = t−1 i=0 S i . We claim that S Z(β | c (0) ) ≤ O(1) (16) Note that Z(β | c (0) ) ≥ t i=0 c 2i e (2i)β ≥ 1 2 t−1 i=0 Z i , where we define Z i = 2 −i 2 e (2i)β + 2 −(i+1) 2 e (S i Z i = 2 −i−i 2 e (2i+1)β 2 −i 2 e (2i)β + 2 −(i+1) 2 e (2i+1)β = 2 −i e β 1 + 2 −2i−1 e 2β = x 1 + x 2 /2 ≤ 1/ √ 2 where x = 2 −i e β 27
Thus, we have shown Eq. (16), and as ν ≤ O(1) this shows that
log U β (k) ≤ ((e ν + e −ν − 2) × 8µ * ) ≤ O(µ * ν 2 )
Proposition 45. Given some parameter ν ≤ ν max , where ν max is a sufficiently small constant, it is possible to select the parameter t ≥ Ω(min{n, √ q}) so that the problem instance c (0) has the required values of q and n and so that {0, 1, 3, 5, . . . , 2t − 1} ⊆ H * (0) .
Proof. We will set β max ≥ t log 2, for some parameter t to be chosen. By taking t ≤ n/2, we ensure that the coefficients are in the range {0, . . . , n}. We need to select β max , t to ensure that problem instance has q = q • , where q • > q min is some given target value.
When β max = t log 2, the problem instance c (0) has
Q = Z(β max ) Z(β min ) = t i=0 2 −i 2 e 2iβmax + t−1 i=0 2 −i−i 2 e (2i+1)βmax × 8µ * 1
Simple calculus shows that these summands are increasing at a super-constant rate, and thus the sums can be bounded by their value at maximum index,
Q ≤ O(2 −t 2 e 2βmaxt + 2 −t 2 +t e (2t−1)βmax × 8µ * ) ≤ O(2 t 2 + 2 t 2 × µ * × (2/e) t ) ≤ O(2 t 2 )
So q ≤ t 2 log 2 + O(1). This implies that, by selecting t ≤ a √ q • for some sufficiently small constant a, we can ensure that q ≤ q • for β max = t log 2. By continuity, this in turn implies that we get q = q • for some choice β max ≥ t log 2. Suppose now we have fixed such t and β max . Let us show that a given coefficient 2k + 1 is in H * (0) . To witness this, take β = k log 2 ∈ [0, β max ]. For this, we have:
Z(β | c (0) ) = t i=0 2 −i 2 e 2iβ + t−1 i=0 2 −i−i 2 e (2i+1)β × 8µ * = t i=0 2 2ik−i 2 + 8µ * t−1 i=0 2 −i−i 2 +(2i+1)k
It is easy to see that in the first sum, the summands of the first sum decay at rate at least 1/2 away from the peak value i = k, while the in the second sum the summands decay at rate least 1/4 from their peak values at i = k, k − 1. So Z(β | c (0) ) ≤ 3 × 2 k 2 + 8µ * × 8 3 2 k 2 , which is smaller than 2 k 2 +2 for µ * sufficiently small. So we get
µ β (2k + 1 | c (0) ) = c (0) k+1 e (2k+1)β Z(β | c (0) ) ≥ 2 −k−k 2 e (2k+1)β × 8µ * 2 k 2 +2 ≥ µ *
A similar analysis with β = 0 shows that 0 ∈ H * (0) as well.
Proposition 46. Under the conditions of Theorem 42, any procedure to solve P µ * coef for general problem instances must have expected sample complexity Ω(
log(1/γ) min{n, √ q} µ * ε 2 )
Proof. Construct the problem instance with t = Ω(min{ √ q, n}) which has the desired parameters n, q and where we set ν = 3ε, for ε ≤ ε max sufficiently small. Consider some r ∈ {1, . . . , d}. For this instance, we have | log(c 7.3 Bounds for P ratio and P µ * coef in terms of n, q in the log-concave setting
For this case, we adapt a construction of [13], with some slightly modified parameters and definitions. This construction will be based on Lemma 40 with d = 2. To simplify the notation, we write c, c − , c + instead of c (0) , c (1) , c (2) . The vectors c − , c + will be derived from c by setting
c − k = c k e −kν , c + k = c k e kν
for some parameter ν > 0. We define the values c 0 , . . . , c n to be the coefficients of the polynomial g(x) = n−1 k=0 (e k + x); equivalently, we have Z(β | c) = n−1 k=0 (e k + e β ) for all values β. Since this polynomial g(x) is real-rooted, the coefficients c 0 , . . . , c n are log-concave [4].
There is another way to interpret the coefficients c i which is useful for us. Consider independent random variables X 0 , . . . , X n−1 , wherein X i is Bernoulli-p i for p i = e β e i +e β . Then µ β is the probability distribution on random variable X = X 0 + · · · + X n−1 . In particular, coefficient c k is a scaled version of the probability µ 0 (k), which in turn is the probability that X = k at β = 0.
We will fix β min = 0. By a simple continuity argument, it is possible to select value β max ≥ 0 to ensure that the problem instance c has any desired value of q > 0. Let us fix such β max . Because it recurs frequently in our calculations, we define ψ = min{n, β max }. Also, we define
z(β) = log Z(β | c) = n−1 k=0 log(e k + e β ).
We recall a result of [13] calculating various parameters of the problem instances c, c − , c + .
Lemma 47 ( [13]). Suppose that ν ≤ ν max for some constant ν max . Define the parameters κ, ρ by
κ = sup β∈R z ′′ (β), ρ = |z ′ (β max ) − z ′ (β min )|
Then the problem instances c − , c + , c have their corresponding values q − , q + bounded by
|q ± − q| ∈ [ρν − κν 2 , ρν + κν 2 ]
Furthermore, the triple of problem instances c, c − , c + has ∆ ≤ O(κν 2 ).
We next estimate some parameters of these problem instances.
Proposition 48. For q min sufficiently large, we have q ≤ O(β max ψ) and ψ ≥ Ω(min{ √ q, n}).
Proof. We compute q as:
q = z(β max ) − z(0) = n−1 k=0
log(e k + e βmax ) − log( e k + e βmax e k + 1 )
For k ≤ β max , note that e k +e βmax
e k +1
is a decreasing function of k, hence it can be upper bounded by its limit as k → −∞, namely e βmax . The total number of such summands is at most min{n, β max } = ψ, hence all such terms contribute a total of O(β max ψ).
For k ≥ β max , we upper bound the term by e k +e βmax e k = 1 + e βmax−k . The total contribution of these terms is then at most k≥βmax log(1 + e βmax−k ) ≤ k≥βmax e βmax−k ≤ O(1).
In total, we have shown that q ≤ O(β max ψ + 1). For q min sufficiently large, this implies that q ≤ O(β max ψ). For the bound on ψ, observe that if β max ≥ n, then obviously ψ = n ≥ Ω(min{ √ q, n}).
Otherwise, we have shown that q ≤ O(β 2 max ) = O(ψ 2 ).
Proposition 49. For q min sufficiently large, we have
z ′ (0) = Θ(1), z ′ (β max ) = Θ(ψ), ρ = Θ(ψ), κ ≤ 4
Proof. Differentiating the function z, we have z ′ (β) = n k=0 e β e k +e β . So z ′ (0) = n−1 k=0 1 e k +1 , which is easily seen to be constant.
Likewise, we have z ′ (β max ) = k e βmax e k +e βmax . For 1 ≤ k ≤ β max , simple calculations show that the summand is Θ(1). The total number of such terms is at min{n, β max }, up to some additive constant of value at most 1. Hence the number of such terms is Θ(ψ), and the total contribution of all terms with k ≤ β max is Θ(ψ). The contribution of all terms with k > β max is given by k>βmax e βmax e k +e βmax − 1 e k +1 ≤ k>βmax e βmax−k ≤ O(1), which for q min sufficiently large is at most 1/2 the value of the terms with k ≤ β max .
Finally By Proposition 48, we have β max ≥ Ω(q/ψ) ≥ Ω(q min ). So for q min sufficiently large, the bounds on z ′ (β max ) and z ′ (0) establish that ρ = Θ(ψ).
We can now prove Theorem 42 part (a) and (b).
Proposition 50. Under the conditions of Theorem 42, any algorithm to solve P ratio on log-concave problem instances with given values n, q must have expected sample complexity Ω( min{q,n 2 } log(1/γ)
ε 2 ).
Proof. Let us set ν = 3ε/ρ. Then by Lemma 47, the values q, q − , q + are separated by at least ρν − κν 2 = 3ε − 3κε 2 /ρ 2 . By selecting q min sufficiently large, we can ensure that β max ≥ C for any constant C. Since ρ = Θ(ψ) ≥ Ω(1) and κ ≤ 4, this in turn ensures that the term 3κε 2 /ρ 2 is below ε as long as ε max is sufficiently small. Overall, the separation between q, q − , q + is at least 2ε.
By Lemma 47, these problem instances have ∆ = O(κν 2 ) = O(κε 2 /ρ 2 ). By Propositions 49 and 48, this is O(ε 2 / min{q, n 2 }). Therefore, by Corollary 41, the expected sample complexity of P ratio on c is Ω( min{q,n 2 } log(1/γ)
ε 2 ).
Proposition 51. Under the conditions of Theorem 42, any algorithm to solve P µ * coef on log-concave problem instances with given parameters n, q must have expected sample complexity Ω( min{q,n 2 } log 1 γ ε 2 ).
Proof. Let us first show that there is some value L = Θ(ψ) such that µ βmax (L) ≥ Ω(1). To show this, we first compute the mean of the random variable X:
E[X] = n−1 k=0 E[X k ] = n−1 k=0 e βmax e βmax + e k
It is straightforward to check that for k ≤ β max , the summand is of order Θ(1) while for k > β max the summands converge exponentially to zero, and hence E[X] = Θ(ψ).
We likewise compute the variance of X. Since variables X 0 , . . . , X k are independent, we have
Var[X] = n−1 k=0 Var[X k ] = n−1 k=0
e βmax e k e βmax + e k Calculations similar to Proposition 49 show that this is O(1). By Chebyshev's inequality, there is constant probability that X is within O(1) of its mean value, and so there is some value L = Θ(ψ) such that P[X = L] ≥ Ω(1). Equivalently, this value has µ βmax (L) ≥ Ω(1).
Similarly, we calculate µ 0 (0) = P[X 0 = · · · = X n−1 = 0] = n−1 k=0 e k e k +1 . Routine calculations show that this is Ω(1).
C Proof of Lemma 10
Let X ⊆ R m be the set of vectors (a 1 , . . . , a m ) satisfying preconditions of the lemma. Since X is compact, we may pick vector (a 1 , . . . , a m ) ∈ X maximizing a 1 + . . . + a m . Clearly, the vector is strictly positive, and a k = 1 k for at least one k ∈ [m] (otherwise we would have (λa 1 , . . . , λa m ) ∈ X for some λ > 1). Furthermore, we can assume that k ≥ 2 (if k = 1 is the only index k with a k = 1 k then we would have (a 1 , λa 2 , . . . , λ m−1 a m ) ∈ X for some λ > 1). Denote x i = log a i ≤ log 1 i , then x 1 , . . . , x m is a discrete concave sequence. Let us define c = log k−1 k , d = log k k+1 and
y i = x k + c(i − k) if i < k x k + d(i − k) if i ≥ k ∀i ≥ 1
Since y k = x k = log 1 k , we get y k−1 = log 1 k−1 ≥ x k−1 and y k+1 = log 1 k+1 ≥ x k+1 (assuming that k + 1 ≤ m). Note that y i is linear on i ∈ {1, . . . , k} and on i ∈ {k, . . . , m}, so from concavity we obtain that x i ≤ y i for i = 1, . . . , m. We can thus write
− e d = 1 k k−1 k 1−k − 1 1 − k−1 k + 1 1 − k k+1 = k k − 1 k−1 + 1 k
The claim will now follow from the result below.
Proposition 52. Let f (x) = ( x x−1 ) x−1 + 1
x . Then f (x) < e for all x ≥ 2. Proof. It is clear that lim x→∞ f (x) = e, so it suffices to show that f (x) is non-decreasing for x ≥ 2.
The derivative of f (x) is given by
x x−1 (x−1) x−1 x 2 log x x−1 − x − 1 x 2 so we need to show that x x−1 (x − 1) x−1 x 2 log x x − 1 − x ≥ 1 for x ≥ 2.
The quantity x x−1 (x−1) x−1 can be easily seen to be an increasing function in x, and since x ≥ 2 we have x x−1 (x−1) x−1 ≥ 2. Therefore, in order to show that f ′ (x) ≥ 0, it suffices to show that x 2 log x x−1 − x ≥ 1/2 for x ≥ 2. This in turn follows from the elementary bound log x x−1 ≥ 1/x. Throughout this section we assume that schedule I provided to Algorithm 1 is proper. Let us define the following notation: if X is a random variable then S[X] def = E[X 2 ] (E[X]) 2 = Var(X) (E[X]) 2 + 1 (the relative variance of X plus 1). The following fact is known (and easy to derive).
Lemma 53 ([5, page 136]). For P = i P i where the P i are independent,
E[P ] = i E[P i ], S[P ] = i S[P i ]
Proposition 54. The following holds for each index i ∈ {0, . . . , t}: (a)μ β i (k) is an ε/4-estimate of µ β i (k) with a probability of at least 1/16 for each k ∈ Ends(σ i ); (b)Q β i is an ε/2-estimate of Q β i with a probability of at least 1/8.
Proof. Part (a) follows directly from Lemma 8 and properness of I, so we focus on part (b). Denote λ = γ·ε 2 2 whereε = 1 − e −ε/4 and γ = 1 8 , and consider (β, σ) ∈ I. The number of samples from each µ β satisfies N β ≥ ⌈ N t σ weight ⌉ = ⌈ (1+λ)t λσ weight ⌉. For each k ∈ Ends(σ) we haveμ β (k) = 1 N β N β j=1 X (j) , where X (j) ∈ {0, 1} are iid Bernoulli random variables with mean µ β (k) ≥ σ weight . Thus, Proof of Theorem 12. If α ∈ Ends(I), we have already shown this in Proposition 54. Otherwise, consider some α ∈ B − Ends(I), where α ∈ (β i , β i+1 ). Let k = σ + i = σ − i+1 . By Lemma 8, Proposition 54 and the union bound, the following events hold with probability at least 1 − 1/16 − 1/8 − 1/16 = 3/4: (1)μ β i (k) is an ε/4-estimate of µ β i (k); (2)Q β i is an ε/2-estimate of Q β i ; (3) line 6 well-estimates k. We show next that these events and properness of I imply that Q α is an ε-estimate of Q α .
E[μ β (k)] = µ β (k) S[μ β (k)] = 1 + 1 − µ β (k) N β · µ β (k) ≤ 1 + 1 − 0 (1+λ)t λσ weight · σ weight = 1 + λ (1 + λ)t (17) Denote W = i−1 j=0 µ β j (σ + j ),Ŵ = i−1 j=0μ β j (σ + j ), V = i j=1 µ β j (σ − j ),V =
Observe that µ β i (k) ≥ ω and µ β i+1 (k) ≥ ω, where ω = min{σ weight i , σ weight i+1 }. By the log-concavity of function µ β as a function of β (Proposition 7(a)), this implies that µ α (k) ≥ ω as well. Sinceμ α well-estimates k, this implies thatμ α (k) is an ε/4-estimate of µ α (k).
Since Q α = µ β i (k) µα (k)e (α−β i )k Q β i , these facts together imply thatQ α is an ε-estimate of Q α .
D.2 Proof of Theorem 16 (correctness of Algorithm 2)
Let us suppose that R is proper, the call at line 2 estimates every Q α correctly for α ∈ R, and line 4 well-estimates every k ∈ H. Since |R| ≤ n + 1 with probability one, this overall has probability at least 1 − γ. We then show that the resulting valuesĉ k are lower-normalized (ε, H * )-estimates of c. First, supposeĉ k is set to a non-zero value at line 5. Soμ α (k) ≥ e −ε/4 µ * /ζ, and soμ α (k) is an ε/4-estimate of µ α (k). Also,Q α is an ε/4-estimate of Q α . Sincec k = Q α e −αk µ α (k), this implies that c k is an ε/2-estimate ofc k .
Next, consider k ∈ H * . Since R is proper, there is α ∈ R with µ α (k) ≥ µ * /ζ. This value of α haŝ µ α (k) ≥ e −ε/4 µ * /ζ, and so line 5 setsĉ k to a non-zero value (if it was not already set in an earlier iteration).
D.3 Proof of Theorem 17 (correctness of Algorithm 3)
We assume here that we are in the log-concave setting. We will need the following result.
Lemma 55. Let I = ((β 0 , σ 0 ), . . . , (β t , σ t )) be a proper schedule satisfying σ weight 0 ≤ µ * 1+µ * σ + 0 and σ weight t ≤ µ * 1+µ * (n−σ − • σ − , σ + ∈ H. Then µ β (k) ≥ min{µ β (σ − ), µ β (σ + )} ≥ σ weight where the first inequality follows from log-concavity of the coefficients and the second inequality holds since I is proper.
• (β, σ) = (β 0 , σ 0 ). Suppose the claim is false: µ β (k) < σ weight . Denote H − = [0, k − 1] ∩ H and H + = [k, n]∩H. We have µ β (k) < σ weight ≤ µ β (σ + ), so by log-concavity µ β (ℓ) ≤ µ β (k) < σ weight for all ℓ ∈ H − . Therefore, µ β (H − ) < |H − | · σ weight = k · σ weight ≤ σ + · σ weight .
Since k ∈ H * * , there exists k * ≤ k such that k * ∈ H * . So let β * be chosen such that µ β * (k * ) ≥ µ * . As observed in the previous paragraph, we have µ β (k * ) ≤ µ β (k) < σ weight . Also note that β * ≥ β = β min . By Proposition 7, for each ℓ ∈ H + with c ℓ > 0 we have
µ β (ℓ) µ β * (ℓ) ≤ µ β (k * )
µ β * (k * ) < σ weight µ * , and therefore µ β (H + ) ≤ σ weight µ * ·µ β * (H + ) ≤ σ weight µ * . We can now obtain a contradiction as follows: 1 = µ β (H − ) + µ β (H + ) < σ + · σ weight + σ weight µ * ≤ (σ + + 1 µ * ) · µ * 1+µ * σ + = 1.
• (β, σ) = (β t , σ t ). This case is completely analogous to the previous one.
Proof of Theorem 17. By construction, the schedule I in line 1 is proper with probability at least 1−γ/3. By the specification of BatchedPratio the valueQ β i is an ε/4-estimate of Q β i with probability at least 1 − γ/3. With probability at least 1 − γ/3, every iteration of line 4 well-estimates every value ℓ ∈ H. Let us assume that all these events occur, which has overall probability at least 1 − γ, and then show that the resulting valuesĉ satisfy the required property. Consider k ∈ H * * and corresponding tuple (β, σ) chosen at line 6 with k ∈ [σ − , σ + ]. From Lemma 55 we get µ β (k) ≥ σ weight . Sinceμ β well-estimates k with respect to parameters ε/4, σ weight , we get thatμ β (k) ≥ e −ε/4 µ β (k) ≥ e −ε/4 σ weight , and soĉ k > 0.
Ifĉ k > 0 for index k, thenμ β (k) ≥ e −ε/4 σ weight for some tuple (β, σ), soμ β (k) is an ε/4-estimate of µ β (k). SinceQ β is an ε/4-estimate of Q β , this implies thatĉ k is an ε/2-estimate ofc k . E Alternate algorithms for P ratio As mentioned in the introduction, an alternative algorithm for problem P ratio with expected sample complexity O q log n log 1 γ ε 2 was proposed in [9,13]. We begin by reviewing this algorithm and showing how to extend it to solve the batched P ratio problem.
Let us define a cooling schedule to be a sequence α = (α 0 , α 1 , . . . , α ℓ ) with β min = α 0 < . . . < α ℓ = β max . We define the length of α as |α| = ℓ + 1. We also define
κ(α) = ℓ i=1 (z(α i−1 ) − 2z( α i−1 +α i 2 ) + z(α i ))
where z(β) = log Z(β). Note that z(·) is an increasing convex function, and therefore κ(α) ≥ 0. The algorithms in [9,13] first compute a cooling schedule α with small values of |α| and κ(α). The following result of [13] summarizes this:
Theorem 56 ([13, Theorem 8]). Fix a constant γ 1 ∈ (0, 1). There is a randomized algorithm that for given values κ • > 0 and λ ∈ (0, 1) produces a cooling schedule α such that P[κ(α) ≤ κ • ] ≥ 1 − γ 1 and E[|α|] = O(mq + 1) where m = 1+log n λ κ•+log(1−λ) . Its expected sample complexity is O(mq + 1).
Theorem 1 .
1There is an algorithm to solve P ratio with expected sample complexity log q where Γ = Θ(log n) in the general setting and Γ = Θ(1)
H
* = {k ∈ H | max β∈[β min ,βmax] µ β (k) ≥ µ * } 1 The work [13] considered the case when H = {0} ∪ [1, n] instead of H = {0, . . . , n}, and proved lower and upper bounds on the expected sample complexity of respectively Ω
Theorem 2 .
2There exists an algorithm to solve P µ * coef with expected sample complexity
Theorem 3 .
3Let G = (V, E) be a connected graph and for i = |V | − 1, . . . , |E| let N i denote the number of connected subgraphs of G with i edges. There is an FPRAS for the sequence N i running in time O( |E| 3 |V | log 3 |E| ε 2).
Theorem 4 .
4Let G = (V, E) be a graph with |V | = 2v nodes and for i = 0, . . . , v let M i denote the number of matchings of G with i edges. Suppose M v > 0 and M v−1 /M v ≤ f for some known parameter f . There is an FPRAS for the sequence M i running in timeÕ( |E||V | 3 f ε 2
≤
InvWeight(I) for all i, so Algorithm 1 has sample complexity O (t+|B|)
ε 2 ·
2InvWeight(I) = O (|B|n+n 2 )Γ ε 2
i = 1, . . . , N . For each α ∈ B, our final estimateQ α is the median of the valuesQ
2 .
2compute the complexity of Algorithm 3. Lines 1 and 2 have expected sample complexity of respectively O(n(log 2 n + log q + log 1 γ )) and O min{n+q log n,n 2 } log n γ ε The update in line 3 increases InvWeight(I) by at most 2 µ * + 2n, therefore line 4 has sample complexity O (n + 1 µ * ) ·
Proposition 19 .
19(a) The pre-schedule J produced by Alg. 4 is proper with probability at least 1 − κ. (b) The expected sample complexity of Alg. 4 is O(n log q + nΓ log 2 n). Proof. (a) If all calls to BinarySearch and FindInterval are good, then the resulting pre-schedule J maintains properties (I5) and (I6). In particular, due to property (I5), it is proper. Since BinarySearch or FindInterval fail with probability at most κ 2n and κ 2(n+2) respectively, a simple union bound shows that properties (I5) and (I6) are maintained with probability at least 1 − κ. (b) By Theorem 18(b), and bearing mind that κ = O(1), the subroutines BinarySearch have expected sample complexity O(n log(nq)). Let us show that subroutines FindInterval have sample complexity
Combined with Eq. (12), this immediately establishes Eq. (9).4.4 Proof of Theorem 18(b): Procedure BinarySearch(β left , β right , θ, γ)
such k, update J by setting σ + i = σ − i+1 = k 5 else output ⊥ 6 return the schedule I = {(β 0 , ((σ − 0 , σ + 0 ), e −ν σ weight 0 )), . . . , (β t , ((σ − t , σ + t ), e −ν σ weight t ))}Let us show that this algorithm satisfies Theorem 20. The sample complexity is O(InvWeight(J ) log n γ ) = O(nΓ log n γ ), and the algorithm returns either ⊥ or a schedule I with InvWeight(I) ≤ e ν InvWeight(J ). (The updates in line 4 ensure that the right-margin of interval i is aligned with the left-margin of interval i + 1, and thus I is indeed a schedule.)
Let us first show how parts (b) and (c) of Theorem 15 imply part (a). The expected sample complexity of each iteration t of this algorithm is O(log nq γ ). Note that this expectation holds even conditioned on the entire past history, i.e., on all the randomness at iterations 1, . . . , t − 1. Because of this, the overall expected sample complexity is E[T ] × O(log nq γ ). The bound on E[|R|] implies that E[T ] = O(r), which yields the claimed sample complexity.
2 −i−i 2 × 8µ * for i = 0, . . . , t − 1.The remaining coefficients c set to zero. We will define d = 2t related problem instances; for each index i = 0, . . . , t − 1, we construct a problem instance where we set c ν , and all other coefficients agree with c (0) ; we also create a problem instance where we set c −ν , and all other coefficients agree with c (0) .
2i+1)β . Thus in order to show Eq. (16), it suffices to show that S i ≤ O(Z i ) for all i = 0, . . . , t − 1. For this, we compute:
j
)| = ν > 2ε where i = 0, j = 2r + 1. Furthermore, i, j ∈ H * (0) .Applying Corollary 41, we see that P µ * coef has expected sample complexity of Ω( d log 1 γ ∆ ). Here, we have ∆ = O(µ * ν 2 ) = O(µ * ε 2 ). Also, we have d = 2t = Ω(min{n, √ q}).
=
e x k +c(1−k) 1 − e c(k−1) 1 − e c + e x k 1 1 − e d = e x k e c(1−k)
(σ − j ). Here, Q β i = C i (I)· W V andŴ ,V are random variables satisfyingQ β i = C i (I) ·Ŵ V . From Lemma 53 and Eq. (17) we getE[Ŵ ] = W S[Ŵ ] 's inequality, P(|Ŵ /W − 1| ≥ε) ≤ (S[Ŵ ] − 1)/ε 2 < γ 2 . Similarly, P(|V /V − 1| ≥ε) < γ 2 .By the union bound, there is a probability of at least 1 − γ that |Ŵ /W − 1| ≤ε and |V /V − 1| ≤ε. This in turn implies thatŴ /W ∈ [e −ε/4 , e ε/4 ] andV /V ∈ [e −ε/4 , e ε/4 ] and thusQ β i Q β i ∈ [e −ε/2 , e ε/2 ].
Something went wrong. In practice, we would restart Alg. 4 from scratch; to simplify the presentation, we instead return an incorrect result consistent with property (I4) */
We now describe the algorithm FindRepresentatives. This algorithm uses the subroutine BinarySearch; for it, we fix the constant value τ = 1/4 throughout.contradicting Proposition 7.
19
5 Proof of Theorem 15: Procedure FindRepresentatives
Algorithm 8: FindRepresentatives(γ)
, we calculate κ. Differentiating twice, we have z ′′ (β) = n−1 k=0 e k e β (e k +e β ) 2 . Summing over k ≤ β contributes at most k≤β e k e β e 2β ≤ ⌊β⌋ k=−∞ e k−β ≤ e e−1 . Likewise, summing over k ≥ β contributes at most k≥β e k e β e 2k ≤ ∞ k=⌈β⌉ e β−k ≤ e e−1
t ) . If (β, σ) ∈ I and k ∈ [σ − , σ + ] ∩ H * * then µ β (k) ≥ σ weight . Proof. Three cases are possible.
AcknowledgementsThe author David Harris thanks Heng Guo for helpful explanations of the algorithms for sampling connected subgraphs and matchings.TheNow let us set ν = 3ε/L to construct the problem instances c + , c − . We will now apply Corollary 41; for either of the problem instances c − , c + , let us set i = 0, j = L. We have shown that i, j ∈ H * with respect to problem instance c, for some sufficiently small constant µ * .Next, we observe that | log(c i /c j ) − log(c + i /c + j )| = | log(c i /c j ) − log(c − i /c − j )| = Lν = 3ε. Therefore, the hypotheses of Corollary 41 are satisfied. So the expected sample complexity of P µ * coef is Ω( log(1/γ) ∆ ). By Lemma 47, we have here ∆ = O(κν); by Proposition 49 and with our definition of ν, this is O(ε 2 /L 2 ). Since we have shown that L = Θ(ψ), we therefore have ∆ ≤ O(ε 2 /ψ 2 ). By Proposition 48 in turn, this shows that ∆ ≤ O(ε 2 / min{q, n 2 }). This completes the proof.A Proof of Theorem 5 (correctness with approximate oracles)One can construct a coupling between µ β andμ β such that samples k ∼ µ β and k ∼μ β are identical with probability at least 1 − ||μ β − µ β || T V ≥ 1 − γ T . Assume that the k th call to µ β in A is coupled with the k th call toμβ inà when β =β. We say that the k th call is good if the produced samples are identical.We can now write P output ofà satisfies C ≥ P[output of A satisfies C ∧ all calls are good] ≥ 1−2γ where the last inequality is by the union bound (recall that P[output of A satisfies C] ≥ 1 − γ).B Proof of Lemma 8 (properties of the binomial distribution)We first consider the case where p ≥ e −ε p • . For this, we use two variants of the Chernoff bound for binomials:Setting x = (e ε − 1)p and x = (1 − e −ε ) respectively, these give us the boundsThese terms are both below γ/2 as long as N ≥ 2e ε log(2/γ) (1−e −ε ) 2 p• . The union bound now gives the claim. Next, consider the case where p < e −ε p • . For fixed values p • and N ≥ 2e ε log(2/γ) (1−e −ε ) 2 p• , let us define the function f :. We need to show that f (p) ≤ γ for p < e −ε p • . Clearly, f (z) is a non-decreasing function of z. Also f (e −ε p • ) ≤ γ/2, as shown previously. The claim follows.Algorithm 9: Paired product estimator. Input: cooling schedule α = (α 0 , . . . , α ℓ ), integer r.Theorem 57 ([9,13]). Let r > 0 be an integer, γ 2 > 0 be a real number, and let α be a cooling schedule with κ(α) ≤ log(1 + 1 2 γ 2 r(1 − e −ε/2 ) 2 ). Then the output of Alg. 9 satisfies P Q /Q ∈ [e −ε , e ε ] ≥ 1 − γ 2 .These two results give the following algorithm to estimate Q: fix positive constants κ • , λ, γ 1 , γ 2 with (1 − γ 1 )(1 − γ 2 ) =3 4, run the algorithm from Theorem 56 and then Algorithm 9 with the obtained schedule α and r = 2(e κ• −1)3 4, and the expected sample complexity is O( q log n ε 2 ). The success probability can be boosted to 1 − γ by repeating the algorithm Θ(log 1 γ ) times and taking the median of estimates. This gives the following result:Theorem 58 ([13]). There is an algorithm for P ratio with expected sample complexity O q log n log 1 γ ε 2.Used directly, this estimates Z(β) for a single value β = β max . With small modifications, it can be used in a batch mode. We use the following observation.Lemma 59. Consider cooling schedule α = (α 0 , . . . , α ℓ ) and value β ∈ (β min , β max ), and let L(β) denote the unique index i with β ∈ (α L(β) , α L(β)+1 ] Then κ(α 0 , . . . , α L(β) , β) ≤ κ(α).Proof. Denote f (x) = z(α L(β) ) − 2z≥ 0, i.e. function f (x) is also non-decreasing on [α L(β) , α L(β)+1 ]. In particular, f (β) ≤ f (α L(β)+1 ). We can now write κ(α 0 , . . . , α L(β) , β) ≤ κ(α 0 , . . . , α L(β) , α L(β)+1 ) ≤ κ(α).This motivates Algorithm 10 below. As in the previous section, we fix positive constants κ • , λ, γ 1 , γ 2 with (1 − γ 1 )(1 − γ 2 ) = 3 4 . Algorithm 10: Estimating Q β . Input: subset B ⊆ [β min , β max ], parameter ε > 0.1 call the algorithm of Theorem 56 to get cooling schedule α = (α 0 , . . . , α ℓ )For each fixed β ∈ B, Algorithm 10 can be viewed as a special case of Algorithm 9 with the cooling schedule (α 0 , . . . , α L(β) , β), and thus shares its guarantees. This implies the following result:Proposition 60. Algorithm 10 has expected sample complexity O( |B|+q log n ε 2 ), and its outputs satisfy3 4for each β ∈ B.35By using a simple median-amplification technique, we can immediately get the following:Theorem 61. There is an algorithm which takes as input a set B ⊆ [β min , β max ], and returns estimateŝ Q β for β ∈ B such that, with probability at least γ, we haveQ β /Q β ∈ [e −ε , e ε ] for all β ∈ B. The expected sample complexity is O().
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Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning. M J Wainwright, M I Jordan, 1M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1(1-2):1-305, December 2008.
| []
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[
"Shoulder Surfing attack in graphical password authentication",
"Shoulder Surfing attack in graphical password authentication"
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| [
"Arash Habibi Lashkari ",
"Samaneh Farmand ",
"Dr Rosli Saleh ",
"\nComputer Science and Data Communication (MCS)\nComputer Science and Information Technology (IT), University Malaya (UM) Kuala Lumpur\nUniversity Malaya (UM) Kuala Lumpur\nMalaysia, Malaysia\n",
"\nComputer Science and Data Communication (MCS), University of Malaya (UM)\nKuala LumpurMalaysia\n",
"\nComputer Science and Data Communication (MCS), University of Malaya (UM)\nKuala LumpurMalaysia\n"
]
| [
"Computer Science and Data Communication (MCS)\nComputer Science and Information Technology (IT), University Malaya (UM) Kuala Lumpur\nUniversity Malaya (UM) Kuala Lumpur\nMalaysia, Malaysia",
"Computer Science and Data Communication (MCS), University of Malaya (UM)\nKuala LumpurMalaysia",
"Computer Science and Data Communication (MCS), University of Malaya (UM)\nKuala LumpurMalaysia"
]
| [
"IJCSIS) International Journal of Computer Science and Information Security"
]
| Information and computer security is supported largely by passwords which are the principle part of the authentication process. The most common computer authentication method is to use alphanumerical username and password which has significant drawbacks. To overcome the vulnerabilities of traditional methods, visual or graphical password schemes have been developed as possible alternative solutions to text-based scheme. A potential drawback of graphical password schemes is that they are more vulnerable to shoulder surfing than conventional alphanumeric text passwords. When users input their passwords in a public place, they may be at risk of attackers stealing their password. An attacker can capture a password by direct observation or by recording the individual's authentication session. This is referred to as shouldersurfing and is a known risk, of special concern when authenticating in public places. In this paper we will present a survey on graphical password schemes from 2005 till 2009 which are proposed to be resistant against shoulder surfing attacks. | null | [
"https://arxiv.org/pdf/0912.0951v1.pdf"
]
| 1,050,958 | 0912.0951 | a7e302754e8e66cf15bb84a1e75322ab6e16c5de |
Shoulder Surfing attack in graphical password authentication
2009
Arash Habibi Lashkari
Samaneh Farmand
Dr Rosli Saleh
Computer Science and Data Communication (MCS)
Computer Science and Information Technology (IT), University Malaya (UM) Kuala Lumpur
University Malaya (UM) Kuala Lumpur
Malaysia, Malaysia
Computer Science and Data Communication (MCS), University of Malaya (UM)
Kuala LumpurMalaysia
Computer Science and Data Communication (MCS), University of Malaya (UM)
Kuala LumpurMalaysia
Shoulder Surfing attack in graphical password authentication
IJCSIS) International Journal of Computer Science and Information Security
622009Graphical PasswordShoulder SurfingAuthentication SchemePasswordsGraphical AuthenticationPassword Attacks
Information and computer security is supported largely by passwords which are the principle part of the authentication process. The most common computer authentication method is to use alphanumerical username and password which has significant drawbacks. To overcome the vulnerabilities of traditional methods, visual or graphical password schemes have been developed as possible alternative solutions to text-based scheme. A potential drawback of graphical password schemes is that they are more vulnerable to shoulder surfing than conventional alphanumeric text passwords. When users input their passwords in a public place, they may be at risk of attackers stealing their password. An attacker can capture a password by direct observation or by recording the individual's authentication session. This is referred to as shouldersurfing and is a known risk, of special concern when authenticating in public places. In this paper we will present a survey on graphical password schemes from 2005 till 2009 which are proposed to be resistant against shoulder surfing attacks.
I. INTRODUCTION
Current authentication systems suffer from many weaknesses. The vulnerabilities of the textual password have been well known. Users tend to pick short passwords or passwords that are easy to remember, which makes the passwords unprotected for attackers to break. Furthermore, textual password is vulnerable to guessing, dictionary attack, key-loggers, and social engineering, shouldersurfing, hidden-camera and spyware attacks. To conquer the limitations of text-based password, techniques such as two-factor authentication and graphical password have been put in use. Other than that, applications and input devices such as mouse, stylus and touch-screen that permit make the appearance of the graphical user authentication techniques possible. However, they are mostly vulnerable to shoulder-surfing as well. Passwords possess many useful properties as well as widespread legacy deployment; consequently we can expect their use for the foreseeable future. Unfortunately, today's standard methods for password input are subject to a variety of attacks based on observation, from casual eavesdropping (shoulder surfing), to more exotic methods. Shoulder-surfing attack occurs when using direct observation techniques, such as looking over someone's shoulder, to get passwords, PINs and other sensitive personal information. As well as when a user enters information using a keyboard, mouse, touch screen or any traditional input device, a malicious observer may be able to acquire the user's password credentials. This is a problem that has been difficult to overcome II. SHOULDER SURFING In this part, we explain sixteen articles from shoulder surfing section of graphical password by focusing on problems, solutions, findings and their future work. Problem 1: The most common computer authentication method is to use alphanumerical usernames and passwords. This method has been shown to have significant drawbacks. For example, users tend to pick passwords that can be easily guessed. On the other hand, if a password is hard to guess, then it is often hard to remember [1]. Methodology used: To address this problem, some researchers have developed authentication methods that use pictures as passwords. The past decade has seen a growing interest in using graphical passwords as an alternative to the traditional text-based passwords. In this paper, they conducted a comprehensive survey of the existing graphical password techniques till 2005. They classified these techniques into two categories: recognition-based and recall based approaches. They discussed the strengths and limitations of each method and pointed out the future research directions in this area. They also tried to answer two important questions: "Are graphical passwords as secure as text-based passwords?"; "What are the major design and implementation issues for graphical passwords?" This survey will be useful for information security researchers and practitioners who are interested in finding an alternative to text-based authentication methods [1].
Findings/Outcome: A comparison of current graphical password techniques was presented. Although the main argument for graphical passwords is that people are better at memorizing graphical passwords than text-based passwords, the existing user studies are very limited and there is not yet convincing evidence to support this argument. Their preliminary analysis suggests that it is more difficult to break graphical passwords using the traditional attack methods such as brute force search, dictionary attack or spyware. However, since there is not yet wide deployment of graphical password systems, the vulnerabilities of graphical passwords are still not fully understood. Overall, the current graphical password techniques are still immature. Much more research and user studies are needed for graphical password techniques to achieve higher levels of maturity and usefulness [1].
Problem 2:
To overcome the shoulder-surfing attack issue without adding any extra complexity into the authentication procedure [2].
Methodology used:
In line with the recent call for technology on Image Based Authentication (IBA) in JPEG committee, they presented a novel graphical password design in this paper. It rests on the human cognitive ability of association-based memorization to make the authentication more user-friendly, comparing with traditional textual password. Based on the principle of zero-knowledge proof protocol, they further improved their primary scheme to overcome the shoulder-surfing attack issue without adding any extra complexity into the authentication procedure. System performance analysis and comparisons were presented to support their proposals [2].
Problem 3:
The advantages of pass-thought over many of the existing authentication technologies include changeability, shoulder surfing resistance, and protection against theft and user non-compliance. Disadvantages of pass-thought authentication include the requirement for a new hardware component (including electrodes) to record the user's brain signals, and the associated performance. For this reason, a pass-thought system may not be accepted for widespread use, but perhaps for high-value or high-importance applications or environments (e.g. within banks and governments) [3].
Methodology used: Recent advances in Brain-Computer Interface (BCI) technology indicate that there is potential for a new type of human-computer interaction: a user transmitting thoughts directly to a computer. BCI technology to date has been focused on interpreting brain signals for communication and control for the disabled. The BCI requirements of a pass-thought system are entirely different: they require no interpretation of the brain signals, but the use of as much signal information as possible [3].
The presented novel idea for user authentication called pass-thoughts, whereby a user authenticates to a device by "transmitting" a thought. This transmission would occur through a Brain Computer Interface (BCI), tailored specifically for this purpose. The goal of a passthought system would be to extract as much entropy as possible from a user's brain signals upon "transmitting" a thought which has the opposite goal from the filtering and many-to-one signal translation that must occur for interpretation of brain signals. Provided that these brain signals can be recorded and processed in an accurate and repeatable way, a pass-thought system might provide a quasi two-factor, changeable, authentication method resistant to shoulder-surfing. The potential size of the space of a pass-thought system would seem to be unbounded in theory, although in practice it will be finite due to system constraints. In this paper, they discussed the motivation and potential of pass-thought authentication, the status quo of BCI technology, and outline the design of what they believed to be a currently feasible passthought system. They also briefly mention the need for general exploration and open debate regarding ethical considerations for such technologies [3].
Findings/Outcome: There are many unknowns to resolve before pass-thoughts might become the method they envisioned. It is a hope that this idea for a pass-thought system will inspire research into the area of signal processing and translation algorithms that retain as much repeatable information as possible. If the recording and processing of brain signals can be accurate and repeatable, pass-thoughts might become a viable and useful new form of authentication [3].
Problem 4:
An attacker can capture a password by direct observation or by recording the individual's authentication session while inserting passwords in public. This is referred to as shoulder-surfing [4].
Methodology used: Until recently, the only defence against shoulder-surfing has been vigilance on the part of the user. This paper reports on the design and evaluation of a game-like graphical method of authentication that extends the challenge response paradigm to resist shoulder-surfing. In doing so it aims to motivate the user with a fun, game-like visual environment designed to develop positive user affect and counterbalance the drawback of the longer time to input the password [4].
The Convex Hull Click (CHC) Scheme is an effort to develop security innovations with people in mind. As such, it is an example of "usable security," an approach to design of security systems that is gaining increasing attention. This scheme allows a user to prove knowledge of the graphical password safely in an insecure location because users never have to click directly on their password images [4].
Findings/Outcome: Usability testing of the CHC scheme showed that novice users were able to enter their graphical password accurately and to remember it over time. However, the protection against shoulder-surfing comes at the price of longer time to carry out the authentication. The user study and interviews supported the overall concept but identified areas of improvement needed to enhance usability and reduce risks [4].
Contextual changes have to do with how the user thinks about the system. Most of the novice users felt the time was acceptable, although it was objectively long compared to a traditional alphanumeric password. Factors that potentially increase its acceptability to users are multiple: high security which warrants taking more time to login, use of CHC in contexts that do not entail logging in at frequent intervals, ease of remembering the passicons and inputting the password accurately, and importantly the "fun factor" of a game-like environment [4].
Future Work: Future work should target increasing the speed of input of the password. There is no single solution to this problem. Instead, several incremental changes, human, technical, visual, and contextual, will improve the system. Humans can speed up with practice, the system can be tweaked to improve efficiency, and the icons can be improved [4].
Further directions for CHC are to improve the current icons, create additional icon sets, make the security settings more fully realistic, and then test it in a longitudinal study of everyday use. This longitudinal study could be carried out in a research or teaching lab where users log in to computers daily. Investigating the entropy issue of pass-icons and study in more depth the motivational aspects of the game-like approach were also planned [4].
Problem 5: Previous research has found graphical passwords to be more memorable than non-dictionary or "strong" alphanumeric passwords. Participants in a prior study expressed concerns that this increase in memorability could also lead to an increased susceptibility of graphical passwords to shoulder-surfing. The seminal question still remains: Can we have both usable and secure authentication systems? In particular, are graphical passwords the leading candidates to address this long-standing challenge, or do the very characteristics that make graphical passwords more memorable and usable lead to increased security vulnerabilities like shoulder-surfing? [5] Methodology used: This appears to be yet another example of the classic trade-off between usability and security for authentication systems. This paper explores whether graphical passwords' increased memorability necessarily leads to risks of shoulder-surfing. To date, there are no studies examining the vulnerability of graphical versus alphanumeric passwords to shouldersurfing [5].
This paper examined the real and perceived vulnerability to shoulder-surfing of two configurations of a graphical password, Passfaces, compared to nondictionary and dictionary passwords. A laboratory experiment with 20 participants asked them to try to shoulder surf the two configurations of Passfaces (mouse versus keyboard data entry) and strong and weak passwords. Data gathered included the vulnerability of the four authentication system configurations to shouldersurfing and study participants' perceptions concerning the same vulnerability. An analysis of these data compared the relative vulnerability of each of the four configurations to shoulder-surfing and also compared study participants' real and perceived success in shouldersurfing each of the configurations. Further analysis examined the relationship between study participants' real and perceived success in shoulder-surfing and determined whether there were significant differences in the vulnerability of the four authentication configurations to shoulder-surfing [5].
Findings/Outcome: Findings indicate that configuring data entry for Passfaces through a keyboard is the most effective deterrent to shoulder-surfing in a laboratory setting and the participants' perceptions were consistent with that result. While study participants believed that Passfaces with mouse data entry would be most vulnerable to shoulder-surfing attacks, the empirical results found that strong passwords were actually more vulnerable [5].
Despite the common belief that non-dictionary passwords are the most secure type of password-based authentication; the results demonstrate that it is in fact the most vulnerable configuration to shoulder-surfing. This result is unexpected, but possibly explainable. A major finding from the study is that secure and usable authentication might be possible when considering shoulder-surfing risks, but that configuration for data entry (i.e., mouse versus numeric keypad) is an important consideration for graphical passwords like Passfaces. Finally, these findings call into question the notion that non-dictionary passwords are universally "better" than dictionary passwords. The risk mitigation from password choice clearly depends on the nature of the attack [5].
Future Work: The non-dictionary passwords, being highly vulnerable to shoulder-surfing attacks is a finding that calls for further investigation. Future studies may investigate shoulder-surfing methods used by real hackers (for example multiple cameras or other equipment) as well as investigation of circumstances for most popular shoulder-surfing environments (work, public access points, etc.) Moreover, further studies may focus on typing speed and possible training effects from long-term use of passwords (both dictionary and non-dictionary) to better establish the impact of long-term use of passwords on their shoulder-surfing vulnerability [5].
Problem 6:
A potential drawback of graphical password schemes is that they are more vulnerable to shoulder surfing than conventional alphanumeric text passwords [6].
Methodology used: They presented a variation of the Draw-a-Secret scheme originally proposed by Jermyn et al. that is more resistant to shoulder surfing through the use of a qualitative mapping between user strokes and the password, and the use of dynamic grids to both obfuscate attributes of the user secret and encourage them to use different surface realizations of the secret. The use of qualitative spatial relations relaxes the tight constraints on the reconstruction of a secret; allowing a range of deviations from the original. They described QDAS (Qualitative Draw-A-Secret), an initial implementation of this graphical password scheme, and the results of an empirical study in which they examined the memorability of secrets, and their susceptibility to shoulder-surfing attacks, for both Draw-A-Secret and QDAS [6].
Findings/Outcome: In their preliminary empirical study QDAS proved to be more resistant to shoulder surfing than its DAS counterpart [6].
Future Work: In future they planned to further analyze QDAS by running more studies, and in particular they hope to accurately simulate the context of shouldersurfing scenario to improve the ecological validity of their findings [6].
Problem 7:
Shoulder-surfing is a problem that has been difficult to overcome [7].
Methodology used: An EyePassword, a system that mitigates the issues of shoulder surfing via a novel approach to user input was presented which is an alternative approach to password entry, based on gaze that deters or prevents a wide range of these attacks. They demonstrated through user studies that their approach requires marginal additional entry time and has accuracy similar to traditional keyboard input, while providing an experience preferred by a majority of users. With EyePassword, a user enters sensitive input (password, PIN, etc.) by selecting from an on-screen keyboard using only the orientation of their pupils (i.e. the position of their gaze on screen), making eavesdropping by a malicious observer largely impractical. They presented a number of design choices and discussed their effect on usability and security. They conducted user studies to evaluate the speed, accuracy and user acceptance of their approach [7].
Findings/Outcome: Results demonstrated that gazebased password entry requires marginal additional time over using a keyboard, error rates are similar to those of using a keyboard and subjects preferred the gaze-based password entry approach over traditional methods [7].
A password can be strengthening by extracting a few additional entropy bits from the gaze path that the user follows while entering the password. Supposedly, the user will follow a similar path, with similar dwell times, every time. A different user, however, may use completely different dwell times. As a result, stealing the user's password is insufficient for logging in and the attacker must also mimic the user's gaze path. A similar technique was previously used successfully to enhance the entropy of passwords entered on a keyboard. While their results showed that the trigger-based mechanism had considerably higher error rates due to eye-hand coordination, it is conceivable that this can be accounted for algorithmically by examining the historical gaze pattern and correlating it with trigger presses [7].
Problem 8:
To gain access to computer systems, users are required to be authenticated. This is usually accomplished by having the user enter an alphanumeric username and password. Users are usually required to remember multiple passwords for different systems and this poses such problems as usability, memorability and security. Passwords are usually difficult to remember and users have developed their own methods some of which are not secure of selecting passwords which are easy to remember. The main weakness of graphical password systems is shoulder surfing [8].
Methodology used: In this research a secure and usable password system which addresses the memorability problem was developed. ToonPasswords is an alternative to traditional text passwords. It draws on the best usability features of existing systems, but provides enhanced security. It reduces the memory load on students by giving them familiar cartoon characters which are demonstrate and are easier to recall than a typical secure text password. Unlike some systems these images are system generated. This avoids users selecting images which might be familiar to an attacker who knows the user personally. They increased the number of images on a screen thus making the probability of a lucky guess as low as 1/64,000. They locked the user out after ten attempts to thwart the most determined and patient of attackers. Giving the user up to ten chances should alleviate frustration when an incorrect password is guessed since the user has more chances. With ToonPasswords the problem of shoulder surfing was overcame by randomizing the location of images at each login [8].
Findings/Outcome: This system was shown to be secure based on the probability of guessing a password and on the likelihood of an observer "shoulder surfing" the password and on the difficulty of launching a brute force attack against a graphical image system. Their work demonstrated that security and usability can be achieved simultaneously. It lays the foundation for developing a class of similar password systems, differing only in the degree of security required. Their password system with its low memory requirements can be used in a wide array of applications [8].
Future Work: For future work the proposed password system will be implemented and tested for security and usability with real users. Eventually the size of the grid will be increased and more screens will be added to offer more security. ToonPasswords will be compared with text passwords and eventually they want to implement the system on mobile devices [8].
Problem 9:
Textual password is vulnerable to shouldersurfing, hidden-camera and spyware attacks. Graphical password schemes have been proposed as a possible alternative to text-based scheme. However, they are mostly vulnerable to shoulder-surfing as well [9].
Methodology used: In this paper, they proposed a Scalable Shoulder-Surfing Resistant Textual-Graphical Password Authentication Scheme (S3PAS). This model seamlessly integrates both graphical and textual password schemes and provides nearly perfect resistant to shouldersurfing, hidden-camera and spyware attacks. It can replace or coexist with conventional textual password systems without changing existing user password profiles. Moreover, it is immune to brute-force attacks through dynamic and volatile session passwords. S3PAS can accommodate various lengths of textual passwords, which requires zero-efforts for users to migrate their existing passwords to S3PAS. Further enhancements of S3PAS scheme are proposed and briefly discussed. Theoretical analysis of the security level using S3PAS is also investigated [9].
Findings/Outcome: However, there are still some minor drawbacks in this system similar to other graphical password schemes. The major issues in S3PAS schemes include slightly more complicated and longer login processes. They planned to design a simplified version of S3PAS with a little lower security level to ease its adoption [9]. Problem 10: Previous efforts involving picture-based passwords have not focused on maintaining a measurably high level of entropy. Since password systems usually allow user selection of passwords, their true entropy remains unknown [10].
Methodology used:
A protocol for ignoring duplicate inputs was presented. A shoulder-surfing resistant input method was also evaluated, with six out of 15 users performing an insecure behaviour. A 23-participant study was performed in which picture and character-based passwords of equal strength were randomly assigned. Memorability was tested with up to one week between sessions [10].
In this case, the picture-password group performed better than the character group after one week: 100% recall and 67% respectively, with all 15 picture-group participants correctly selecting only their password items within two tries. This appeared to be a confirmation of the picture superiority effect, and may also be attributable to the "multiple encodings" of each password item (each item was represented by a picture as well as a keyboard key and location in the home grid). However, when ordered passwords with a full 50 bits of entropy were considered, performance for both picture and character passwords was quite poor: 67% recall and 50% respectively. Serial order information either does not benefit from the multiple encodings of picture-password items or passwords at this entropy level are too difficult to remember. A couple observations about user behaviour were also made. Most importantly, the fact that users repeat incorrect inputs is likely based on the fact that users do not receive adequate feedback when entering a password (they cannot see the actual text submitted to the system). Since this is unavoidable for security reasons, the duplicate inputs should be discarded by the authentication server and not counted against the user. This does not compromise the security of the system, since attackers have nothing to gain from duplicating inputs. User behaviour during the SSR task was unexpected. The purpose of the task had been explained immediately before it was performed, yet six out of 15 participants revealed their password through an insecure behaviour. This highlights the importance of usability testing in security applications [10].
Findings/Outcome: The study found that both character and picture passwords of very high entropy were easily forgotten. However, when password inputs were analyzed to determine the source of input errors, serial ordering was found to be the main cause of failure. This supports a hypothesis stating that picture-password systems which do not require ordered input may produce memorable, high-entropy passwords. Input analysis produced another interesting result, that incorrect inputs by users are often duplicated. This reduces the number of distinct guesses users can make when authentication systems lock out users after a number of failed logins. Across all conditions, picture passwords were more memorable than character passwords, though the difference was not significant due to the small sample size of the study. It was marginally significant when input data was analyzed to determine how well participants would have performed at an unordered input task [10].
Future Work: Picture passwords are a relatively new area of study, so the possibilities for future work are extensive. Based on the results presented here, the most promising future work is in the area of unordered, randomly-assigned passwords. Research into insecure user behaviours and training methods is also extremely important [10].
Problem 11:
One common practice in relation to alphanumeric passwords is to write them down or share them with a trusted friend or colleague. Graphical password schemes often claim the advantage that they are significantly more secure with respect to both verbal disclosure and writing down. In this paper they investigated the reality of this claim in relation to the Passfaces graphical password scheme [11].
Methodology used: By collecting a corpus of naturalistic descriptions of a set of 45 faces, they explored participants' ability to associate descriptions with faces across three conditions in which the decoy faces were selected: (1) at random; (2) on the basis of their visual similarity to the target face; and (3) on the basis of the similarity of the verbal descriptions of the decoy faces to the target face [11].
They conducted an informal pilot study using their implementation of the passpoints system. 5 users ability to select click points in response to verbal descriptions were investigated. The experiment involved the listener sitting in front of a computer screen with the passpoints software loaded. A male experiment moderator stood behind the participant and described the location of each click point in turn. No gesticulation was allowed and descriptions were not permitted to include reference to the current position of the mouse pointer, only points on the image. The listener was allowed to request more information or clarification to which the describer could respond. The results of this pilot study showed 4 out of 5 participants were able to correctly interpret the descriptions into the correct sequence of click points. Participants were found to perform significantly worse when presented with visual and verbally grouped decoys, suggesting that Passfaces can be further secured for description. Subtle differences in both the nature of male and female descriptions, and male and female performance were also observed [11].
Findings/Outcome: This study has in part demonstrated the degree to which Passfaces can be verbally described, but also how through judicious choice of decoys we can reduce the vulnerability of Passfaces to description. Moreover, this empirical study has highlighted the reality that contrary to common wisdom users can share Passfaces graphical passwords. They also anticipated that the vulnerability of graphical password schemes to description could have impact on issues of both memorability and shoulder surfing. If decoy faces in Passfaces employed a grouping policy that means that they were in some way similar to the target face, it is likely to impact on the ability of a user to select the target face. Likewise, in a shoulder-surfing scenario, attackers have a limited amount of time to form a quick memory association with components of an authentication secret [11].
Problem 12:
User authentication is one of the important topics in information security. Traditional strong password schemes could provide with certain degree of security; however, the fact that strong passwords being difficult to memorize often leads their owners to write them down on papers or even save them in a computer file. As a result, security becomes greatly compromised [12].
Methodology used: On the other hand, knowing that human beings are predominant visual creatures, many researchers have investigated or developed graphical password schemes recently. In this paper, they proposed a graphical password scheme for user authentication using images with random tracks of geometric shapes. This method is not only more secure than most of the existing graphical password schemes, it also solves problems like requiring a large image database, uneasy to repeat mouse clicking at the same position, as well as images being too simple to cause collisions on points selected for different users [12].
Findings/Outcome: They justified that graphical password random geometric graphical password (RGGPW) indeed is robust against common security attacks like brute-force search, spyware, shoulder surfing, social engineering, and forgery. They also showed the images to demonstrate user friendliness in both recognition and selection of pass-objects from the given images. In addition, RGGPW is storage-efficient as all images are created when needed [12].
Future Work: Working on how to create images with more complex tracks and easier recognizable objects and implementing a website to test the acceptance of this technique is in their future work [12].
Problem 13: Alphanumeric passwords are widely used in computer and network authentication to protect users' privacy. However, it is well known that long, text-based passwords are hard for people to remember, while shorter ones are susceptible to attack [13].
Methodology used: Graphical password is a promising solution to this problem. Draw-A-Secret (DAS) is a typical implementation based on the user drawing on a grid canvas. Currently, too many constraints result in reduction in user experience and prevent its popularity. A novel graphical password strategy Yet Another Graphical Password (YAGP) inspired by DAS is proposed in this paper. In a 48×64 grid, the secret drawings can be described in detail. The users can concentrate on the drawing to improve user experience because exact positions are not required in YAGP. Meanwhile, the algorithm proposed in YAGP is trend-sensitive which actually reflects drawing trends. The proposal has the advantages of free drawing positions, strong shoulder surfing resistance and large password space. Experiments illustrated the effectiveness of YAGP. Furthermore, user personalities have a great influence on the drawings and therefore make it harder for others to imitate. Additionally, users can draw the secrets small enough to resist shoulder surfing [13].
Findings/Outcome: Some preliminary experiments are carried out. The results showed that YAGP achieves an encouraging performance in usability and security and possesses a high resistance to shoulder surfing [13].
The main drawback of YAGP is that it's hard to redraw the password precisely. The legal user cannot always be assured to login successfully because the gaps between user drawings are uncertain while the similarity threshold value is fixed [13].
Future Work: Future research will concentrate on improving YAGP as well as developing a comparison algorithm of higher efficiency in distinguishing the legal user from attackers [13].
Problem 14:
Threats such as key-loggers, weak password, and shoulder surfing, forcing the user to memorize different passwords or carrying around different tokens, "familiarization" or a lengthy "password setup" process are today's drawbacks of authentication systems [14].
Methodology used: In this paper, they proposed a new authentication scheme based on graphical password and multifactor authentication. To that end, they employed the user's personal handheld device as the password decoder and the second factor of authentication. In these methods, a service provider challenges the user with an image password. To determine the appropriate click points and their order, the user needs some hint information transmitted only to her handheld device. They showed that this method can overcome threats such as keyloggers, weak password, and shoulder surfing [14].
Their approach can be effectively and securely used as user-friendly authentication mechanism for public and un-trusted terminals. The proposed solution is unique in many ways: 1. It is the first graphical password solution that employs two-factor authentication. 2. They never assume the handheld device is trusted. 3. This solution resists screen recording attacks. 4. This method doesn't need a "familiarization" or a lengthy "password setup" process. 5. Lost or stolen handheld doesn't expose a security risk [14].
Findings/Outcome: With the increasing popularity of handheld devices such as cell phones, this approach can be leveraged by many organizations without forcing the user to memorize different passwords or carrying around different tokens. This system can be applied to more than just authentication mechanisms: their system is applicable anywhere that there is a need to enter sensitive or private data. For instance, Social Security Number can be entered via this system without leaking or revealing any directly usable information to the terminal or even the handheld device [14].
Problem 15: Two Factor authentication mechanisms are considered to be secure for authenticating a user in Internet based environment. As the number of services provided online is day by day increasing, users intending to use various online services are also increasing. With each service requiring the user to register separately, the overhead of remembering many ID/password pairs has lead to the problem of memorability. To address this, researchers have proposed mechanisms for multi-server environment where in the user needs to register with a single registration centre using one ID/password pair and thereby access all the services registered through that server. But, as these mechanisms employ textual passwords, they suffer from many inherent drawbacks [15].
Methodology used: In this paper they proposed a two factor password authenticated key agreement mechanism using graphical password where in the user needs to recognise his secret image presented to him as challenge. The scheme allows the user to choose images from a given set of image categories. The scheme is based on simple collision resistant hash functions. The protocol was designed such that it does not maintain verification table at the server for authentication and to resist replay attack it does not employ the time concurrency mechanism which has various weaknesses. Instead, it uses random nonce. In addition, the protocol provides secure low computation mutual authentication and session key agreement [15].
Findings\Outcome: It is secure against several ID thefts, Insider attack, Replay attack, Shoulder surfing attack, Reconnaissance attack, Server spoofing attack and guessing attack. The proposed scheme will work efficiently on wired network scenario. Their system is secure, efficient and user friendly authentication scheme which has several unique features [15].
Future Work: They intend to explore further, the same concept and protocol for the wireless domain where there are bandwidth constraints [15].
Problem 16:
To overcome the vulnerabilities of traditional methods, visual or graphical password schemes have been developed. Because simply adopting graphical password authentication also has some drawbacks, some hybrid schemes based on graphic and text were developed [16].
Methodology Used: In this paper, a stroke-based textual password authentication scheme was proposed. It uses shapes of strokes on the grid as the origin passwords and allows users to login with text passwords via traditional input devices. The method provides strong resistant to hidden-camera and shoulder-surfing. Moreover, the scheme has flexible enhancements to secure the authentication process. The analysis of the security of this approach was also discussed [16].
Findings/Outcome:
The scheme has salient features as a secure system for authentication immune to shouldersurfing, hidden camera and brute force attacks. It also has variants to strengthen to security level through changing the login interface of the system [16]. However, the system still has some drawbacks. Firstly, this method is relativity unfamiliar to the general people so that the users may adopt the simple and weak strokes as their passwords. Secondly, the process of creating original password is more vulnerable than the login step. Thirdly, the login process is longer than other graphical schemes [16].
Future Work: Designing more advanced authentication system to improve this method [16].
III. COMPARISON TABLE
In our study, we found twelve methodologies which overcome shoulder surfing attack in graphical password. We tried to highlight them in a table and demonstrated drawbacks and future works of each scheme briefly. The below Table indicates the result of our study on sixteen articles on shoulder surfing. N/A signal processing and translation algorithms, extract as much entropy as possible from a user's brain signals [3] not accurate and repeatable recording and processing of brain signals, requirement for a new hardware component and the associated performance game-like graphical method of authentication that extends the challenge response paradigm [4] longer time to carry out the authentication qualitative mapping between user strokes and the password, use of dynamic grids to both obfuscate attributes of the user secret, use different surface realizations of the secret [6] N/A user enters sensitive input by selecting from an on-screen keyboard using only the orientation of their pupils [7] similar error rates to those of using a keyboard and needs marginal additional time over using it randomizing the location of images(familiar system generated cartoon characters)at each login [8] increasing the size of the grid, adding more screens to offer more security integrates both graphical and textual password schemes without changing existing user password profiles [9] Slightly more complicated and longer login processes.
using images with random tracks of geometric shapes for authentication [12] creating images with more complex tracks and easier recognizable objects, implementing a website to test acceptance of this technique inspired by DAS, algorithm reflects drawing trends, free drawing positions, users can draw the secrets small enough, users personalities have a great influence on the drawings [13] hard to redraw the password precisely, the gaps between user drawings are uncertain while the similarity threshold value is fixed using handheld device as a password decoder and sending hint information to determine appropriate click points and their order, second factor of authentication [14] N/A images are given by the system, simple collision resistant hash functions, no verification table at the server for authentication [15] N/A using shapes of strokes on the grid as the origin passwords, changing the login interface of the system [16] method relativity unfamiliar to the general people, longer login process and more secure than registration phase
IV. CONCLUSION
In this paper we studied on more than 30 graphical password designs then selected 16 algorithms resistant to shoulder surfing. We emphasized the problem, methodology and future works of them and brought out a summery table of our work at the end.
To have a good system high security and good usability are both needed and cannot be separated. Shoulder surfing attack is under security provision. There are few proposed methods to shoulder surfing problem but they still need to be improved.
In our next paper in graphical password algorithms, we will propose an enhancement on one of graphical password algorithm in recognition-based category to solve limitations of graphical password scheme.
We hope this study be useful for those who have new ideas on secure and useable graphical authentication system.
TABLE 1 :
1COMPARISON TABLE ON SHOULDER SURFING ATTACK METHODOLOGIESProblem
Methodology
Drawbacks/Future
Work
Shoulder-Surfing Attack
association-based
memorization, zero-
knowledge proof
protocol [2]
http://sites.google.com/site/ijcsis/ ISSN 1947-5500
http://sites.google.com/site/ijcsis/ ISSN 1947-5500
http://sites.google.com/site/ijcsis/ ISSN 1947-5500
http://sites.google.com/site/ijcsis/ ISSN 1947-5500
ACKNOWLEDGMENTWe would like to express our appreciation to our parents, all the teachers and lecturers who help us to understand the importance of knowledge and show us the best way to gain it.
Graphical passwords: a survey. Xiaoyuan Suo, Ying Zhu, G Scott, Owen, 21st Annual Computer Security Applications ConferenceXiaoyuan Suo, Ying Zhu G. Scott. Owen, 2005, 'Graphical passwords: a survey', 21st Annual Computer Security Applications Conference.
An Association-Based Graphical Password Design Resistant to Shoulder-Surfing Attack. Zhi Li, Qibin Sun, Yong Lian, D D Giusto, IEEE International Conference on Multimedia and Expo (ICME). Zhi Li, Qibin Sun, Yong Lian, and D. D. Giusto, 2005, 'An Association-Based Graphical Password Design Resistant to Shoulder- Surfing Attack', IEEE International Conference on Multimedia and Expo (ICME).
Passthoughts: authenticating with our minds. Julie Thrope, P C Van Oorschot, Proceedings of the 2005 workshop on New security paradigms. the 2005 workshop on New security paradigmsACMJulie Thrope, P. C. van Oorschot, Anil Somayaji, 2005, 'Pass- thoughts: authenticating with our minds', Proceedings of the 2005 workshop on New security paradigms, ACM.
Design and Evaluation of a Shoulder-Surfing Resistant Graphical Password Scheme. Susan Wiedenbeck, Jim Waters, Leonardo Sobrado, Jean-Camille Birget, Proceedings of Advanced Visual Interfaces (AVI2006). Advanced Visual Interfaces (AVI2006)Susan Wiedenbeck, Jim Waters, Leonardo Sobrado, Jean-Camille Birget, 2006, 'Design and Evaluation of a Shoulder-Surfing Resistant Graphical Password Scheme',Proceedings of Advanced Visual Interfaces (AVI2006).
A comparison of perceived and real shoulder-surfing risks between alphanumeric and graphical passwords. Tari Furkan, A Ozok, Stephen H Holden, Proceedings of the second symposium on Usable privacy and security. the second symposium on Usable privacy and securityACMFurkan, Tari, A. Ant Ozok, Stephen H. Holden, 2006, 'A comparison of perceived and real shoulder-surfing risks between alphanumeric and graphical passwords', Proceedings of the second symposium on Usable privacy and security, ACM.
Graphical passwords & qualitative spatial relations. Di Lin, Paul Dunphy, Patrick Olivier, Jeff Yan, Proceedings of the 3rd symposium on Usable privacy and security. the 3rd symposium on Usable privacy and securityACMDi Lin, Paul Dunphy, Patrick Olivier, Jeff Yan, 2007, 'Graphical passwords & qualitative spatial relations',Proceedings of the 3rd symposium on Usable privacy and security, ACM.
Reducing shoulder-surfing by using gaze-based password entry. Manu Kumar, Tal Garfinkel, Dan Boneh, Terry Winograd, Proceedings of the 3rd symposium on Usable privacy and security. the 3rd symposium on Usable privacy and securityACMManu Kumar, Tal Garfinkel, Dan Boneh, Terry Winograd, 2007, 'Reducing shoulder-surfing by using gaze-based password entry',Proceedings of the 3rd symposium on Usable privacy and security, ACM.
Increasing security and usability of computer systems with graphical passwords. Hinds Cheryl, Chinedu Ekwueme, Proceedings of the 45th annual southeast regional conference. the 45th annual southeast regional conferenceACMCheryl, Hinds and Chinedu Ekwueme, 2007, 'Increasing security and usability of computer systems with graphical passwords',Proceedings of the 45th annual southeast regional conference, ACM.
S3PAS: A Scalable Shoulder-Surfing Resistant Textual-Graphical Password Authentication Scheme. Huanyu Zhao, Xiaolin Li, 21st International Conference on Advanced Information Networking and Applications Workshops. AINAWHuanyu Zhao and Xiaolin Li, 2007, 'S3PAS: A Scalable Shoulder- Surfing Resistant Textual-Graphical Password Authentication Scheme', 21st International Conference on Advanced Information Networking and Applications Workshops (AINAW).
Order and entropy in picture passwords. Saranga Komanduri, Dugald R Hutchings, Proceedings of graphics interface. graphics interfaceCanadian Information Processing SocietySaranga Komanduri and Dugald R. Hutchings, 2008, 'Order and entropy in picture passwords', Proceedings of graphics interface, Canadian Information Processing Society.
Securing passfaces for description. Paul Dunphy, James Nicholson, Patrick Oliver, Proceedings of the 4th symposium on Usable privacy and security. the 4th symposium on Usable privacy and securityACMPaul Dunphy, James Nicholson, Patrick Oliver,2008, 'Securing passfaces for description', Proceedings of the 4th symposium on Usable privacy and security, ACM.
Graphical Passwords Using Images with Random Tracks of Geometric Shapes. Phen-Lan Lin, Li-Tung Weng, Po-Whei Huang, Congress on Image and Signal Processing. CISPPhen-Lan Lin, Li-Tung Weng, Po-Whei Huang, 2008, 'Graphical Passwords Using Images with Random Tracks of Geometric Shapes', Congress on Image and Signal Processing (CISP).
YAGP: Yet Another Graphical Password Strategy. Haichang Gao, Xuewu Guo, Xiaoping Chen, Liming Wang, Xiyang Liu, Annual Computer Security Applications Conference (ACSAC). Haichang Gao, Xuewu Guo, Xiaoping Chen, Liming Wang, and Xiyang Liu, 2008, 'YAGP: Yet Another Graphical Password Strategy', Annual Computer Security Applications Conference (ACSAC).
Universal Multi-Factor Authentication Using Graphical Passwords. Alireza Pirayesh Sabzevar, Angelos Stavrou, IEEE International Conference on Signal Image Technology and Internet Based Systems (SITIS). Alireza Pirayesh Sabzevar, Angelos Stavrou, 2008, 'Universal Multi-Factor Authentication Using Graphical Passwords', IEEE International Conference on Signal Image Technology and Internet Based Systems (SITIS).
A user friendly password authenticated key agreement for multi server environment. P Mohammed Misbahuddin, A Premchand, Govardhan, Proceedings of the International Conference on Advances in Computing, Communication and Control. the International Conference on Advances in Computing, Communication and ControlACMMohammed Misbahuddin, P. Premchand, A.Govardhan, 2009, 'A user friendly password authenticated key agreement for multi server environment', Proceedings of the International Conference on Advances in Computing, Communication and Control, ACM.
A Stroke-Based Textual Password Authentication Scheme. Ziran Zheng, Xiyu Liu, Lizi Yin, Zhaocheng Liu, First International Workshop on Education Technology and Computer Science (ETCSZiran Zheng, Xiyu Liu, Lizi Yin, Zhaocheng Liu, 2009, 'A Stroke- Based Textual Password Authentication Scheme', First International Workshop on Education Technology and Computer Science (ETCS).
| []
|
[
"Multipolar gravitational waveforms and ringdowns generated during the plunge from the innermost stable circular orbit into a Schwarzschild black hole",
"Multipolar gravitational waveforms and ringdowns generated during the plunge from the innermost stable circular orbit into a Schwarzschild black hole"
]
| [
"Antoine Folacci \nEquipe Physique Théorique\nFaculté des Sciences\nUMR 6134 du CNRS et de l'\nSPE\nUniversité de Corse\nUniversité de Corse\nBP 52F-20250CorteFrance\n",
"Mohamed Ould \nEquipe Physique Théorique\nFaculté des Sciences\nUMR 6134 du CNRS et de l'\nSPE\nUniversité de Corse\nUniversité de Corse\nBP 52F-20250CorteFrance\n",
"El Hadj \nEquipe Physique Théorique\nFaculté des Sciences\nUMR 6134 du CNRS et de l'\nSPE\nUniversité de Corse\nUniversité de Corse\nBP 52F-20250CorteFrance\n"
]
| [
"Equipe Physique Théorique\nFaculté des Sciences\nUMR 6134 du CNRS et de l'\nSPE\nUniversité de Corse\nUniversité de Corse\nBP 52F-20250CorteFrance",
"Equipe Physique Théorique\nFaculté des Sciences\nUMR 6134 du CNRS et de l'\nSPE\nUniversité de Corse\nUniversité de Corse\nBP 52F-20250CorteFrance",
"Equipe Physique Théorique\nFaculté des Sciences\nUMR 6134 du CNRS et de l'\nSPE\nUniversité de Corse\nUniversité de Corse\nBP 52F-20250CorteFrance"
]
| []
| We study the gravitational radiation emitted by a massive point particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole. We consider both evenand odd-parity perturbations and describe them using the two gauge-invariant master functions of Cunningham, Price, and Moncrief. We obtain, for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle, the regularized multipolar waveforms, i.e., the waveforms constructed by summing over of a large number of modes, and their unregularized counterparts constructed from the quasinormal-mode spectrum. They are in excellent agreement and our results permit us to especially emphasize the impact on the distortion of the waveforms of (i) the harmonics beyond the dominant ( = 2, m = ±2) modes and (ii) the direction of observation, and therefore the necessity to take them into account in the analysis of the last phase of binary black hole coalescence.CONTENTS | 10.1103/physrevd.98.084008 | [
"https://arxiv.org/pdf/1806.01577v2.pdf"
]
| 67,797,324 | 1806.01577 | 24db7584994a8fc5a3527375829e9ad10bfe3a5c |
Multipolar gravitational waveforms and ringdowns generated during the plunge from the innermost stable circular orbit into a Schwarzschild black hole
Antoine Folacci
Equipe Physique Théorique
Faculté des Sciences
UMR 6134 du CNRS et de l'
SPE
Université de Corse
Université de Corse
BP 52F-20250CorteFrance
Mohamed Ould
Equipe Physique Théorique
Faculté des Sciences
UMR 6134 du CNRS et de l'
SPE
Université de Corse
Université de Corse
BP 52F-20250CorteFrance
El Hadj
Equipe Physique Théorique
Faculté des Sciences
UMR 6134 du CNRS et de l'
SPE
Université de Corse
Université de Corse
BP 52F-20250CorteFrance
Multipolar gravitational waveforms and ringdowns generated during the plunge from the innermost stable circular orbit into a Schwarzschild black hole
(Dated: June 6, 2018)
We study the gravitational radiation emitted by a massive point particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole. We consider both evenand odd-parity perturbations and describe them using the two gauge-invariant master functions of Cunningham, Price, and Moncrief. We obtain, for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle, the regularized multipolar waveforms, i.e., the waveforms constructed by summing over of a large number of modes, and their unregularized counterparts constructed from the quasinormal-mode spectrum. They are in excellent agreement and our results permit us to especially emphasize the impact on the distortion of the waveforms of (i) the harmonics beyond the dominant ( = 2, m = ±2) modes and (ii) the direction of observation, and therefore the necessity to take them into account in the analysis of the last phase of binary black hole coalescence.CONTENTS
I. Introduction 1
II. Gravitational waves generated by the plunging massive particle 2 A. The Schwarzschild BH and the plunging massive particle 2 B. Gravitational perturbations induced by the plunging particle 3 C. Construction of the partial amplitudes ψ . Regularization of the partial waveform amplitudes (even and odd parity) 16
I. INTRODUCTION
In this article, we shall obtain and analyze in terms of quasinormal modes (QNMs) the multipolar gravitational waveforms generated by a massive "point particle" plunging from slightly below the innermost stable circular orbit (ISCO) into a Schwarzschild black hole (BH). Here, it is important to note that, by multipolar waveforms, we intend waveforms constructed by superposition of a large number of modes. We shall assume an extreme mass ratio for the physical system considered, i.e., that the BH is much heavier than the particle, such a hypothesis permitting us to describe the emitted radiation in the framework of BH perturbations [1][2][3][4][5][6][7]. In the context of gravitational wave physics and with the first direct gravitational-detection of a binary black hole coalescence by LIGO [8], the problem we study is of fundamental importance and there exists a large literature concerning it more or less directly (see, e.g., Refs. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). Indeed, the "plunge regime" from the ISCO is the last phase of the evolution of a stellar mass object orbiting near a supermassive BH or it can be also used to describe the latetime evolution of a binary BH and, moreover, the waveform generated during this regime encodes the final BH fingerprint. It should be recalled that, in this context, a multipolar description of the gravitational signal will be of fundamental interest with the enhancement of the sensitivity of laser-interferometric gravitational wave detectors (see, e.g., Refs. [28][29][30] and references therein). In our work, by taking into account a large number of higher harmonics, we shall show that the waveform is strongly distorted and that the distortion highly depends on the direction of observation. It should be noted that our work extends the study of Hadar and Kol [19] (see also the analysis of Hadar, Kol, Berti, and Cardoso in Ref. [20]) but we will not limit ourselves solely to the quasinormal response observed in the orbital plane of the plunging particle. It also extends our recent work concerning the electromagnetic radiation emitted by a charged particle plunging from the ISCO into a Schwarzschild black hole [31].
Our paper is organized as follows. In Sec. II, after a brief overview of gravitational perturbation theory in the Schwarzschild spacetime, we establish theoretically the expression of the waveforms emitted by a massive point particle plunging from the ISCO into the BH. More precisely, we consider both the even-and odd-parity gravitational perturbations for arbitrary ( , m) modes and describe them using the two gauge-invariant master functions of Cunningham, Price, and Moncrief [3][4][5][6][7]. We then solve, in the frequency domain and by using standard Green's function techniques, the Regge-Wheeler equation [1] governing the odd-perturbations as well as the Zerilli-Moncrief equation [2,3] governing the evenperturbations. This is achieved after having constructed the sources for these two equations from the closed-form expression of the plunge trajectory. In Sec. III, we extract from the results of Sec. II the QNM counterpart of the waveforms corresponding to the gravitational ringing (or ringdown) of the BH. We gather all our numerical results and their analysis in Sec. IV where we display the regularized multipolar waveforms emitted [i.e., the waveforms constructed by summing over of a large number of ( , m) modes] and compare them with their unregularized counterparts constructed solely from the QNM spectrum. Both are obtained for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle. It should be noted that, in the late phase of the signals, they are in excellent agreement. Moreover, our results especially emphasize the impact on the distortion of the waveforms of the harmonics beyond the dominant ( = 2, m = ±2) modes and of the direction of observation. In the Conclusion, we briefly summarize the main results obtained in this article and, in an Appendix, we carefully examine the regularization of the partial amplitudes from both the theoretical and numerical point of view. Indeed, the exact waveforms theoretically obtained in Sec. II are integrals over the radial Schwarzschild coordinate which are strongly divergent near the ISCO. For even as well as for odd perturbations, they can be numerically regularized by using the Levin's algorithm [32] but only after having reduced the degree of divergence of these integrals by a succession of integrations by parts, i.e., by extending the method we developed in our work concerning the charged particle plunging from the ISCO into a Schwarzschild black hole where we encountered a similar problem [31] Throughout this article, we adopt units such that G = c = 1 and we use the geometrical conventions of Ref. [33].
II. GRAVITATIONAL WAVES GENERATED BY THE PLUNGING MASSIVE PARTICLE
In this section, we shall obtain theoretically the expression of the even-and odd-parity waveforms emitted by a massive point particle plunging from slightly below the ISCO into the BH. This will be achieved by working in the frequency domain and using the standard Green's function techniques. Moreover, we shall fix the notations and conventions used throughout the whole article.
A. The Schwarzschild BH and the plunging massive particle
We recall that the exterior of the Schwarzschild BH of mass M is defined by the metric
ds 2 = −f (r)dt 2 + f (r) −1 dr 2 + r 2 dσ 2 2 (1)
where f (r) = (1 − 2M/r) and dσ 2 2 = dθ 2 + sin 2 θdϕ 2 denotes the metric on the unit 2-sphere S 2 and with the Schwarzschild coordinates (t, r, θ, ϕ) which satisfy t ∈] − ∞, +∞[, r ∈]2M, +∞[, θ ∈ [0, π] and ϕ ∈ [0, 2π]. In the following, we shall also use the so-called tortoise coordinate r * ∈] − ∞, +∞[ defined in terms of the radial Schwarzschild coordinate r by dr/dr * = f (r) and given by r * (r) = r + 2M ln[r/(2M ) − 1]. We recall that the function r * = r * (r) provides a bijection from ]2M, +∞[ to ] − ∞, +∞[. We denote by t p (τ ), r p (τ ), θ p (τ ) and ϕ p (τ ) the coordinates of the timelike geodesic γ followed by the plunging particle (here τ is the proper time of the particle) and by m 0 its mass. Without loss of generality, we can consider that its trajectory lies in the BH equatorial plane, i.e., we assume that θ p (τ ) = π/2. The geodesic equations defining γ are given by [34] f (r p )
dt p dτ = E,(2a)r 2 p dϕ p dτ = L (2b)
and
dr p dτ 2 + L 2 r 2 p f (r p ) − 2M r p = E 2 − 1.(2c)
Here E and L are, respectively, the energy and angular momentum per unit mass of the particle which are two conserved quantities given on the ISCO, i.e., at r = r ISCO with
r ISCO = 6M,(3)
by
E = 2 √ 2 3 and L = 2 √ 3M.(4)
By substituting (4) into the geodesic equations (2a)-(2c), we obtain after integration
t p (r) 2M = 2 √ 2 (r − 24M ) 2M (6M/r − 1) 1/2 − 22 √ 2 tan −1 (6M/r − 1) 1/2 +2 tanh −1 1 √ 2 (6M/r − 1) 1/2 + t 0 2M(5)
and
ϕ p (r) = − 2 √ 3 (6M/r − 1) 1/2 + ϕ 0(6)
where t 0 and ϕ 0 are two arbitrary integration constants. From (6), we can write the spatial trajectory of the plunging particle in the form
r p (ϕ) = 6M [1 + 12/(ϕ − ϕ 0 ) 2 ] .(7)
We have displayed this trajectory in Fig. 1. . Here, we assume that the particle starts at r = rISCO(1 − ) with = 10 −3 and we take ϕ0 = 0. The red dashed line at r = 6M and the red dot-dashed line at r = 2M represent the ISCO and the horizon, respectively, while the black dashed line corresponds to the photon sphere at r = 3M .
B. Gravitational perturbations induced by the plunging particle
Gravitational waves emitted from the Schwarzschild BH excited by the plunging particle can be characterized by the field h µν which satisfies the wave equation
h µν − h ρ µ;νρ − h ρ ν;µρ + h ;µν +g µν (h ρσ ;ρσ − h) = −16πT µν(8)
where T µν , which is the stress-energy tensor associated with the massive particle, is given by
T µν (x) = m 0 γ dτ dx µ p (τ ) dτ dx ν p (τ ) dτ δ 4 (x − x p (τ )) −g(x) (9a) = m 0 dx µ p dτ (r) dx ν p dτ (r) dr p dτ (r) −1 × δ[t − t p (r)]δ[θ − π/2]δ[ϕ − ϕ p (r)] r 2 sin θ . (9b)
In this last equation, t p (r) and ϕ p (r) are respectively given by (5) and (6). The resolution of the problem defined by (8) and (9b) and, more generally, the topic of gravitational perturbations of BHs, have been the subject of lots of works since the pioneering articles by Regge and Wheeler [1] and Zerilli [2]. So, because gravitational perturbations of the Schwarzschild BH are very well described in the article by Martel and Poisson [6] as well as in the review by Nagar and Rezzolla [7], we just briefly recall some of the results we need for our particular work. The gravitational signal emitted can be described in terms of the two gauge-invariant master functions of Cunningham, Price, and Moncrief [3][4][5] denoted by ψ
V (e) (r) = f (r) × Λ 2 (Λ + 2)r 3 + 6Λ 2 M r 2 + 36ΛM 2 r + 72M 3 (Λr + 6M ) 2 r 3(11)
and we have for the Regge-Wheeler potential
V (o) (r) = f (r) Λ + 2 r 2 − 6M r 3 .(12)
In Eqs. (11) and (12), we have introduced
Λ = ( − 1)( + 2) = ( + 1) − 2.(13)
We note that only the functions ψ We recall that the functions S m (t, r) is given by Eq. (24) of Ref. [7]). After having checked these two results, we have used them to construct the sources corresponding to the stress-energy tensor (9b). By using the orthonormalization properties of the (scalar, vector and tensor) spherical harmonics [6,7], we have obtained
S (e) m (t, r) = 8πm 0 [Y m (π/2, 0)] * √ 2π(Λ + 2)(Λr + 6M ) f (r) 3Λ − 2 − 64M Λr+6M + 72M 2 (Λ+2−m 2 ) r 2 + 216M 3 (Λ+2−2m 2 ) Λr 3 (6M/r − 1) 3/2 − 8 (6M/r − 1) 5/2 − im 4 √ 3M r 1 + 8 (6M/r − 1) 3 δ [t − t p (r)] − 12 √ 2 r 2 + 12M 2 r (6M/r − 1) 3 δ [t − t p (r)] exp[−imϕ p (r)](14)
and
S (o) m (t, r) = 16πm 0 [X m ϕ (π/2, 0)] * √ 2πΛ(Λ + 2) f (r) − 8 √ 6M r 2 (6M/r − 1) 3/2 + 36 √ 6M 2 r 3 (6M/r − 1) 5/2 +im 72 √ 2M 2 r 3 (6M/r − 1) 3 δ [t − t p (r)] + 18 √ 3M r 2 + 12M 2 r 3 (6M/r − 1) 3 δ [t − t p (r)] exp[−imϕ p (r)].(15)
In the last equation, we have introduced the vector spherical harmonic
X m ϕ = − sin θ ∂ ∂θ Y m .(16)
Furthermore, we note that the coefficients Y m (π/2, 0) and X m ϕ (π/2, 0) appearing respectively in Eqs. (14) and (15) are given by
Y m (π/2, 0) = 2 m √ π 2 + 1 4π ( − m)! ( + m)! × Γ[ /2 + m/2 + 1/2] Γ[ /2 − m/2 + 1] cos [( + m)π/2] .(17)
and
X m ϕ (π/2, 0) = 2 m+1 √ π 2 + 1 4π ( − m)! ( + m)! × Γ[ /2 + m/2 + 1] Γ[ /2 − m/2 + 1/2] sin [( + m)π/2] . (18)
Here, it is important to remark that Y m (π/2, 0) and hence the source (14) vanish for + m odd while X m ϕ (π/2, 0) and hence the source (15) vanish for + m even.
It is also important to recall that the partial amplitudes ψ (e/o) m (t, r) of Cunningham, Price and Moncrief permit us to obtain the gravitational wave amplitude observed at spatial infinity (i.e., for r → +∞). In the transverse traceless gauge [33], the two circularly polarized components (h + , h × ) of the emitted gravitational wave are given by [7] h + = h (e)
+ + h (o) + and h × = h (o) × + h (o) × (19a) with h (e) + = 1 r +∞ =2 + m=− ψ (e) m 2 ∂ 2 ∂θ 2 + ( + 1) Y m , (20a) h (e) × = 1 r +∞ =2 + m=− ψ (e) m 2 sin θ ∂ 2 ∂θ∂ϕ − cos θ sin θ ∂ ∂ϕ Y m , (20b) h (o) + = 1 r +∞ =2 + m=− ψ (o) m 2 sin θ ∂ 2 ∂θ∂ϕ − cos θ sin θ ∂ ∂ϕ Y m , (20c) h (o) × = 1 r +∞ =2 + m=− ψ (o) m − 2 ∂ 2 ∂θ 2 + ( + 1) Y m .(20d)
It should be noted that, due to Eqs. (17) and (18) [see also the remark following these equations], we have to only consider the couples ( , m) with + m even in the superpositions (20a) and (20b) and the couples ( , m) with + m odd in the superpositions (20c) and (20d).
C. Construction of the partial amplitudes ψ
(e/o) m (t, r)
In order to solve the Zerilli-Moncrief and Regge-Wheeler equations (10), we shall work in the frequency domain by writing
ψ (e/o) m (t, r) = 1 √ 2π +∞ −∞ dω ψ (e/o) ω m (r)e −iωt(21)
and
S (e/o) m (t, r) = 1 √ 2π +∞ −∞ dω S (e/o) ω m (r)e −iωt .(22)
Then, these two wave equations reduce to
d 2 dr 2 * + ω 2 − V (r) ψ (e/o) ω m (r) = S (e/o) ω m (r)(23)
where the new source terms, which are obtained from (14) and (15), are given by
S (e) ω m (r) = 8πm 0 [Y m (π/2, 0)] * √ 2π(Λ + 2)(Λr + 6M ) f (r) iω 12 √ 2 r 2 + 12M 2 r (6M/r − 1) 3 − im 4 √ 3M r 1 + 8 (6M/r − 1) 3 − 8 (6M/r − 1) 5/2 + 3Λ − 2 − 64M Λr+6M + 72M 2 (Λ+2−m 2 ) r 2 + 216M 3 (Λ+2−2m 2 ) Λr 3 (6M/r − 1) 3/2 exp[i(ωt p (r) − mϕ p (r))](24)
and
S (o) ω m (r) = 16πm 0 [X m ϕ (π/2, 0)] * √ 2πΛ(Λ + 2) f (r) −iω 18 √ 3M r 2 + 12M 2 r 3 (6M/r − 1) 3 + im 72 √ 2M 2 r 3 (6M/r − 1) 3 + 36 √ 6M 2 r 3 (6M/r − 1) 5/2 − 8 √ 6M r 2 (6M/r − 1) 3/2 exp[i(ωt p (r) − mϕ p (r))].(25)
We have checked that our results (24) and (25) are in agreement with the corresponding results obtained by Hadar and Kol in Ref. [19]. It should be however noted that we do not use the same conventions for the definition of the sources, for the Fourier transform as well as the same normalization for the partial amplitudes ψ (e/o) m (t, r). Furthermore, it seems to us that our expression for the even-parity source is much simpler than theirs. It is also important to note that, due to the relation
Y −m = (−1) m [Y m ] * ,(26a)
we have
X −m ϕ = (−1) m [X m ϕ ] * ,(26b)
and therefore, as a direct consequence of (26a) and (26b), we can easily observe that
S (e/o) ω −m = (−1) m [S (e/o) −ω m ] * .(27)
The Zerilli-Moncrief and Regge-Wheeler equations (23) can be solved by using the machinery of Green's functions (see Ref. [35] for generalities on this topic and, e.g., Ref. [36] for its use in the context of BH physics). We consider the Green's functions G (e/o) ω (r * , r * ) defined by
d 2 dr 2 * + ω 2 − V (r) G (e/o) ω (r * , r * ) = −δ(r * − r * ) (28)
which can be written as
G (e/o) ω (r * , r * ) = − 1 W (e/o) (ω) × φ in (e/o) ω (r * ) φ up (e/o) ω (r * ), r * < r * , φ up (e/o) ω (r * ) φ in (e/o) ω (r * ), r * > r * .(29)
Here W
d 2 dr 2 * + ω 2 − V (e/o) (r) φ (e/o) ω = 0.(30)
The functions φ in (e/o) ω are defined by their purely ingoing behavior at the event horizon r = 2M (i.e., for
r * → −∞) φ in (e/o) ω (r) ∼ r * →−∞ e −iωr * (31a)
while, at spatial infinity r → +∞ (i.e., for r * → +∞), they have an asymptotic behavior of the form
φ in (e/o) ω (r) ∼ r * →+∞ A (−,e/o) (ω)e −iωr * + A (+,e/o) (ω)e +iωr * .
(31b) Similarly, the functions φ up (e/o) ω are defined by their purely outgoing behavior at spatial infinity
φ up (e/o) (r) ∼ r * →+∞ e +iωr *(32a)
and, at the horizon, they have an asymptotic behavior of the form
φ up (e/o) ω (r) ∼ r * →−∞ B (−,e/o) (ω)e −iωr * + B (+,e/o) (ω)e +iωr * .
(32b) In the previous expressions, the coefficients A (−,e/o) (ω),
A (+,e/o) (ω), B (−,e/o) (ω) and B (+,e/o) (ω) are complex
amplitudes. By evaluating the Wronskians W (e/o) (ω) at r * → −∞ and r * → +∞, we obtain
W (e/o) (ω) = 2iωA (−,e/o) (ω) = 2iωB (+,e/o) (ω). (33)
Here, it is worth noting some important proper-ties of the coefficients A (±,e/o) (ω) and of the functions φ in (e/o) ω (r) that we will use extensively later. They are a direct consequence of Eqs. (30) and (31) and they are valid whether ω is real or complex. We have
φ in (e/o) −ω (r) = [φ in (e/o) ω (r)] * (34a)
and
A (±,e/o) (−ω) = [A (±,e/o) (ω)] * .(34b)
It is important to also recall that the solutions of the homogeneous Zerilli-Moncrief and Regge-Wheeler equations (30) are related by the Chandrasekhar-Detweiler transformation [34,37] [
Λ(Λ + 2) − i(12M ω)] φ (e) ω = Λ(Λ + 2) + 72M 2 r(Λr + 6M ) f (r) + 12M f (r) d dr φ (o) ω(35)
and, as a consequence, the coefficients A (±,e/o) (ω) satisfy the relations
A (−,e) (ω) = A (−,o) (ω)(36a)
and
A (+,e) (ω) = Λ(Λ + 2) + i(12M ω) Λ(Λ + 2) − i(12M ω) A (+,o) (ω).(36b)
Using the Green's functions (29) Here it is important to note that these partial waveforms satisfy
ψ (e/o) −m = (−1) m [ψ (e/o) m ] * .(40)
This is a direct consequence of the definition (21) and of the relation
ψ (e/o) ω −m = (−1) m [ψ (e/o) −ω m ] *(41)
which is easily obtained from (38) and (27)
III. QUASINORMAL RINGINGS DUE TO THE PLUNGING MASSIVE PARTICLE
In this section, we shall construct the quasinormal ringings associated with the gravitational wave amplitudes (20). Of course, they can be obtained by summing over the ringings associated with all the partial amplitudes ψ (e/o) m (t, r). In order to extract from these partial amplitudes the corresponding quasinormal ringings ψ QNM (e/o) mn (t, r), the contour of integration over ω in Eq. (39) may be deformed (see, e.g., Ref. [38]). This deformation permits us to capture the zeros of the Wronskians (33) lying in the lower part of the complex ω plane and which are the complex frequencies ω n of the ( , n) QNMs. We note that, for a given , n = 1 corresponds to the fundamental QNM (i.e., the least damped one) while n = 2, 3, . . . to the overtones. We also recall that the spectrum of the quasinormal frequencies is symmetric with respect to the imaginary axis, i.e., that if ω n Table I.
In the previous expression, C (1) Thanks to Chandrasekhar and Detweiler, we know that the zeros of the Wronskians (33), i.e., the quasinormal frequencies ω n (and −ω * n ), do not depend on the parity sector. This is a consequence of the relation (36a).
(2) The excitation factors (45) depend on the parity sector because their expressions involve the coefficients A (+,e/o) (ω) which are parity dependent [see Eq. (36b)]. It is interesting to recall that this was not the case for the problem of the electromagnetic field generated by charged particle plunging into the Schwarzschild BH [31]. By combining (36) with the definition (45), we obtain
B (e) n = Λ(Λ + 2) + i(12M ω n ) Λ(Λ + 2) − i(12M ω n ) B (o) n .(46)
(3) The excitation coefficients (44a) and (44b) depend on the parity sector because they are constructed from the excitation factors as well as from wave equations with sources which are parity dependent [see Eq. (23)].
(4) In our problem, the spherical symmetry of the Schwarzschild BH is broken due to the asymmetric plunging trajectory. It is this dissymmetry which, in connection with the presence of the azimuthal number m, forbids us to gather the two terms in Eq. (43). (t, r) do not provide physically relevant results at "early times" due to their exponentially divergent behavior as t decreases. It is necessary to determine, from physical considerations (see below), the time beyond which these quasinormal waveforms can be used, i.e., the starting time t start of the BH ringing.
IV. MULTIPOLAR WAVEFORMS AND QUASINORMAL RINGDOWNS
A. Numerical methods
In order to construct the gravitational wave amplitudes (20), it is first necessary to obtain numerically the partial amplitudes ψ (e/o) m (t, r) given by (39). For that purpose, using Mathematica [41]:
(1) We have determined the functions φ in (e/o) ω as well as the coefficients A (−,e/o) (ω). This has been achieved by integrating numerically the homogeneous Zerilli-Moncrief and Regge-Wheeler equations (30) with the Runge-Kutta method. We have initialized the process with Taylor series expansions converging near the horizon and we have compared the solutions to asymptotic expansions with ingoing and outgoing behavior at spatial infinity that we have decoded by Padé summation. Our numerical calculations have been performed independently for the two parities and we have checked their robustness and internal consistency by using the relations (35) and (36).
(2) We have regularized the partial amplitudes ψ (e/o) ω m (r) given by (38), i.e., the Fourier transform of the partial amplitudes ψ (e/o) m (t, r). Indeed, these amplitudes as integrals over the radial Schwarzschild coordinate are strongly divergent near the ISCO. This is due to the behavior of the sources (24) and (25) in the limit r → 6M . The regularization process is described in the Appendix. It consists in replacing the partial amplitudes (38) by their counterparts (A.11) and to evaluate the result by using Levin's algorithm [32]. (20). We have constructed the even components from the ( , m) modes with = 2, . . . , 10 and m = ± which constitute the main contributions. Similarly, we have constructed the odd components from the ( , m) modes with = 2, . . . , 10 and m = ±( − 1). In fact, it is not necessary to take higher values for because, in general, they do not really modify the numerical sums (20) (see also the discussion in Sec. IV B).
In order to construct the quasinormal ringings associated with the gravitational wave amplitudes (20), it is necessary to obtain numerically the partial amplitudes ψ QNM (e/o) m (t, r) given by (42) and, as a consequence, we need the quasinormal frequencies ω n , the excitation factors B mn . The quasinormal frequencies ω n can be determined by using the method developed by Leaver [42]. We have implemented numerically this method by using the Hill determinant approach of Majumdar and Panchapakesan [43]. The excitation coefficients C of the gravitational wave observed at infinity in the direction (θ = π/3, ϕ = 0) (for even components) and (θ = π/6, ϕ = 0) (for odd components). We emphasize the impact of the harmonics beyond the dominant ( = 2, m = ±2) modes on the multipolar waveforms and their ringdowns (zoom in on the waveforms).
ing behavior at spatial infinity. Our numerical results are in agreement with the theoretical relations (36) and (46). The evaluation of the integrals in Eqs. (44a) and (44b) is rather elementary because we do not have to regularize them. It should be noted that, for a given , it is possible to consider only the fundamental QNM (n = 1) which is the least damped one. Moreover, we need only the excitation coefficients C Tables I and II, we provide the various ingredients permitting us to construct the quasinormal ringings associated with the gravitational wave amplitudes (20). It should be finally recalled that it is necessary to select a starting time t start for the ringings. By taking t start = t p (3M ), i.e., the moment the particle crosses the photon sphere, we have obtained physically relevant results.
B. Results and comments
In Figs. 2-8, we have considered the components h (e/o) +/× of the gravitational waves observed at infinity. The multipolar waveforms have been obtained by assuming that the particle starts at r = r ISCO (1 − ) with = 10 −4 and, furthermore, in Eqs. (5) and (6), we have taken ϕ 0 = 0 and chosen t 0 /(2M ) in order to shift the interesting part of the signal in the window t/(2M ) ∈ [0, 245]. Without loss of generality, we have constructed only the signals for directions above the orbital plane of the plunging particle. Indeed, we could obtain those observed below that plane by using the symmetry properties of the vector spherical harmonics in the antipodal transformation on the unit 2-sphere S 2 . Moreover, we have assumed that the observer lies in the plane ϕ = 0. In fact, for any other value of ϕ, the behavior of the signals is very similar. The results corresponding to arbitrary values of θ and ϕ are available to the interested reader upon request.
In Figs. 2 and 3, we have focused our attention on the construction of the multipolar waveforms by summing the expressions (20) over the harmonics beyond the dominant ( = 2, m = ±2) modes. Of course, the necessity to take higher harmonics into account clearly appears but we can also note that the sums truncated at = 5 already provide strong results.
The distortion of the multipolar waveforms and of the associated quasinormal ringdowns clearly appears in Figs. 4-7. It can be observed in the adiabatic phase corresponding to the quasicircular motion of the particle near the ISCO (see Fig. 1) as well as in the ringdown phase. It is due to the "large" number of ( , m) modes taken into account in the sums (20) and is strongly dependent on the direction of the observer.
The multipolar waveforms and the associated quasinormal ringdowns are in excellent agreement as can be seen in Figs. 4-7 or, more clearly, in Fig. 8 where we work with semi-log graphs. Here, it is important to recall (see Sec. IV A) that it has been necessary to regularize the former while the latter are unregularized.
V. CONCLUSION
In this article, we have described the gravitational radiation emitted by a massive "point particle" plunging + observed at infinity for various directions above the orbital plane of the plunging particle. We consider ϕ = 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θ varies between −π/2 and +π/2. We note that, for θ = 0, only the ( = 2, m = ±2) modes contribute to the signal. × observed at infinity for various directions above the orbital plane of the plunging particle. We consider ϕ = 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θ varies between −π/2 and +π/2. We note that h + observed at infinity for various directions above the orbital plane of the plunging particle. We consider ϕ = 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θ varies between −π/2 and +π/2. We note that, for θ = 0, only the ( = 3, m = ±2) modes contribute to the signal. × observed at infinity for various directions above the orbital plane of the plunging particle. We consider ϕ = 0 and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when θ varies between −π/2 and +π/2. We note that h from slightly below the ISCO into a Schwarzschild BH. In order to do this, we have constructed the associated multipolar waveforms and analyzed their late-stage ringdown phase in terms of QNMs. We have noted the excellent agreement between the "exact" waveforms we had to carefully regularize and the corresponding quasinormal waveforms which have not required a similar treatment.
Our results have been obtained for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle. They have permitted us to emphasize more particularly the impact on the distortion of the waveforms of (i) the higher harmonics beyond the dominant ( = 2, m = ±2) modes and (ii) the direction of observation and, as a consequence, the necessity to take them into account in the analysis of the last phase of binary black hole coalescence.
Here, we fall on the result previously obtained in the context of the regularization of the partial amplitude ψ (e) ω m describing the electromagnetic radiation generated by a charged particle plunging into the Schwarzschild BH [see Eqs. (A.6)-(A.8) in Ref. [31]]. That is a direct consequence of the redefinition (A.5)-(A.6) of the functions φ in (e/o) ω
FIG. 1 .
1The plunge trajectory obtained from Eq.(7)
m
(t, r) [here, and in the following, the symbols (e) and (o) are respectively associated with even (polar) and odd (axial) objects according they are of even or odd parity in the antipodal transformation on the unit 2-sphere S 2 ] which satisfy respectively the Zerilli-Moncrief and ReggeZerilli-Moncrief potential is given by
. . . and m = − , − + 1, . . . , + are physically relevant for our study.
m
(t, r) and S
m
(t, r) are source terms which depend on the components, in the basis of tensor spherical harmonics, of the stress-tensor inducing the perturbations of the Schwarzschild spacetime. Their expressions can be found in the review by Nagar and Rezzolla: S
.
These functions are linearly independent solutions of the homogeneous Zerilli-Moncrief and Regge-Wheeler equations
, we can show that the solutions of the Moncrief-Zerilli and Regge-Wheeler equations with source (23) is a consequence of Eqs. (29), (32a) and (33). We can now obtain the solutions of the Zerilli-Moncrief and Regge-Wheeler equations (10) by inserting (38) into (21) and we have, in the time domain, for the ( , m) waveforms ψ (e/o) m (t, r) = 1 √ 2π +∞ −∞ dω e −iω[t−r * (r)] 2iωA (−,e/o) (ω) × 6M 2M dr f (r ) φ in (e/o) ω (r ) S (e/o) ω m (r ). (39)
by noting that the solutions φ in (e/o) ω of the problem (30)-(31) and the associated coefficients A (−,e/o) (ω) satisfy the relations (34). The relations (40) and (26a) permit us to check that the gravitational wave amplitudes (20) are purely real.
10 −5 − 1.1413 × 10 −5 i 1.2331 × 10 −7 + 1.6133 × 10 −8 i 1.3027 × 10 −5 − 3.7333 × 10 −6 i 1.7606 × 10 −8 − 9.7647 × 10 −7 i(3, 1) −3.4129 × 10 −6 + 3.7527 × 10 −7 i −8.9300 × 10 −10 − 1.6665 × 10 −9 i 1.5710 × 10 −6 + 2.3923 × 10 −7 i −1.1817 × 10 −8 + 4.4988 × 10 −9 i (4, 1) 3.4146 × 10 −7 − 8.3671 × 10 −7 i 5.0989 × 10 −11 + 2.4768 × 10 −11 i −2.3418 × 10 −7 + 3.0764 × 10 −7 i 2.4210 × 10 −10 − 2.4821 × 10 −10 i (5, 1) −2.5796 × 10 −9 + 3.2195 × 10 −7 i −1.4574 × 10 −12 + 1.4770 × 10 −12 i 3.3738 × 10 −8 − 1.2541 × 10 −7 i 8.0274 × 10 − 12 + 2.2079 × 10 −11 i (6, 1) −2.1803 × 10 −9 − 1.3600 × 10 −7 i −1.1924 × 10 −13 + 5.2103 × 10 −14 i −1.1875 × 10 −8 + 5.0871 × 10 −8 i −3.3844 × 10 −12 + 5.2475 × 10 −12 i(7, 1) −1.0692 × 10 −8 + 6.3204 × 10 −8 i 4.0607 × 10 −14 − 6.8884 × 10 −14 i 9.2800 × 10 −9 − 2.1679 × 10 −8 i −3.0697 × 10 −12 + 1.3254 × 10 −12 i (8, 1) 1.5158 × 10 −8 − 2.8867 × 10 −8 i −2.9165 × 10 −14 + 3.4046 × 10 −14 i −7.5475 × 10 −9 + 8.7255 × 10 −9 i −2.0486 × 10 −12 − 3.6520 × 10 −13 i (9, 1) −1.3623 × 10 −8 + 1.1053 × 10 −8 i 2.0391 × 10 −14 − 2.3094 × 10 −14 i 5.3721 × 10 −9 − 2.6496 × 10 −9 i −1.0116 × 10 −12 − 9.2597 × 10 −13 i (10, 1) 9.6352 × 10 −9 − 2.0768 × 10 −9 i −1.4482 × 10 −14 + 1.2662 × 10 −14 i −3.2557 × 10 −9 − 1.9158 × 10 −11 i −2.6317 × 10 −13 − 9.0578 × 10 −13 iis a quasinormal frequency lying in the fourth quadrant, −ω * n is the symmetric quasinormal frequency lying in the third one. We then easily obtain e +iω * n [t−r * (r)] .
even and odd excitation factors associated with the ( , n) QNM of complex frequency ω n . The first term in the r.h.s. of Eq. (43) is the contribution of the quasinormal frequency ω n lying in the fourth quadrant of the ω plane while the second one is the contribution of −ω * n , i.e., its symmetric with respect to the imaginary axis. In front of the bracket in the r.h.s. of Eq. (44b), the coefficients B (e/o) n * are nothing else than the even and odd excitation factors associated with the ( , n) QNM of complex frequency −ω * n . They are obtained from (45) by using the properties (34b). A few remarks are in order:
( 5 )
5It is however important to note that the excitation coefficients (44) , r) are then purely real as a consequence of the relations (48) and (26a).(6) The ringing amplitudes ψ QNM (e/o) mn (t, r) and ψ QNM (e/o) m
be "easily" calculated. Indeed, we first obtain the excitation factors B(e/o) n , as well as the functions φ in (e/o) ω n (r) and the coefficients A (+,e/o) (ω n ) by integrating numerically the homogeneous Zerilli-Moncrief and Regge-Wheeler equations (30)(for ω = ω n ) with the Runge-Kutta method and then by comparing the solutions to asymptotic expansions with ingoing and outgo-
Fig. 2 .
2Semi-log graphs highlighting the impact on the ringdowns of the harmonics beyond the dominant ( = 2, m = ±2) modes.and m = ± and the excitation coefficients C with = 2, . . . , 10 and m = ±( − 1). In
, φ = 0 FIG. 4 .
04Multipolar gravitational waveforms h (e)
, φ = 0 FIG. 5 .
05Multipolar gravitational waveforms h (e)
×
vanishes for θ = ±π/2 and that, for θ = 0, only the ( = 2, m = ±2) modes contribute to the signal.
, φ = 0 FIG. 6 .
06Multipolar gravitational waveforms h (o)
FIG. 7 .
7Multipolar gravitational waveforms h (o)
×
vanishes for θ = ±π/2 and that, for θ = 0, only the ( = 3, m = ±2) modes contribute to the signal.
FIG. 8 .
8Semi-log graphs of some multipolar waveforms showing the dominance of the quasinormal ringing at intermediate times and the agreement of the regularized waveforms with the unregularized quasinormal responses.
TABLE I :
IThe first quasinormal frequencies ω n and the associated excitation factors B(e/o)
n
.
( , n)
2M ω n
B
(e)
n
B
(o)
n
(2, 1)
0.747343 − 0.177925i
0.120928 + 0.070666i
0.126902 + 0.0203151i
(3, 1)
1.198890 − 0.185406i
−0.088969 − 0.061177i
−0.093890 − 0.049193i
(4, 1)
1.618360 − 0.188328i
0.062125 + 0.069099i
0.065348 + 0.065239i
(5, 1)
2.024590 − 0.189741i
−0.036403 − 0.074807i
−0.038446 − 0.073524i
(6, 1)
2.424020 − 0.190532i
0.011847 + 0.075196i
0.013129 + 0.074877i
(7, 1)
2.819470 − 0.191019i
0.010478 − 0.069864i
0.009689 − 0.069924i
(8, 1)
3.212390 − 0.191341i
−0.029331 + 0.059378i
−0.028863 + 0.059573i
(9, 1)
3.603590 − 0.191565i
0.043628 − 0.044836i
0.043370 − 0.045060i
(10, 1)
3.993576 − 0.191728i
−0.052616 + 0.027669i
−0.052494 + 0.027875i
TABLE II :
IIThe excitation coefficients C
ACKNOWLEDGMENTSWe gratefully acknowledge Thibault Damour for drawing, some years ago, our attention to the plunge regime in gravitational wave physics. We wish also to thank Yves Decanini and Julien Queva for various discussions and Romain Franceschini for providing us with powerful computing resources.Appendix: Regularization of the partial waveform amplitudes (even and odd parity)In this appendix, we shall explain how to regularize the partial amplitudes ψ (e/o) ω m . Indeed, the exact waveforms(38)as integrals over the radial Schwarzschild coordinate are strongly divergent near the ISCO. This is due to the behavior of the sources(24)and(25)in the limit r → 6M . It should be noted that we have encountered a similar problem in our study of the electromagnetic radiation generated by a charged particle plunging into the Schwarzschild BH[31]. We recall that we have addressed this problem by combining a theoretical and a numerical approach: for even electromagnetic perturbations, we have reduced the degree of divergence of the integrals involved by successive integrations by parts and then we have"numerically regularized" them by using Levin's algorithm. Here, we can proceed identically and the reader will be assumed to have "in hand" a copy of Ref.[31]where the electromagnetic case is treated in great details in the Appendix. Moreover, we shall use a trick that will allow us to quickly provide the expected results from those obtained in Ref.[31].The trick we use is based on the fact, that the expressions(38), which are constructed from the source terms(24)and(25), can be written in the formandas well asWe now remark that the amplitudes A (e) (r) and A (o) (r) can be split into a divergent and a regular part in the formand that the divergent part, which is obtained by the Taylor expansion of A (e/o) (r) at r = 6M , is independent of the parity and given byWe note that the terms A (e/o) reg in the r.h.s. of (A.11) are obtained from (A.7)-(A.10). Moreover, the function Θ reg (r) is constructed from the phase (A.3). This is explained in Appendix of Ref.[31]. We just recall thatIt is finally important to point out that the result (A.11) has to be "numerically regularized". Indeed [see also the discussion in Sec. (A.2) of Ref.[31]], the integrands in the r.h.s. of (A.11) belong to a particular family of rapidly oscillatory functions whose amplitudes diverge as 1/(6M − r) 3/2 in the limit r → 6M and whose the phase Φ(r) behaves as 1/(6M − r) 1/2 in the same limit. As a consequence, it is possible to neutralize the divergences remaining in the amplitudes from the oscillations induced by the phase term. This has been achieved by using Levin's algorithm[32]which is implemented in Mathematica[41]. It is this last step which permits us to obtain, in Sec. IV, stable numerical results for the partial amplitudes ψ
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| []
|
[
"Theory of Interacting Bloch Electrons in a Magnetic Field",
"Theory of Interacting Bloch Electrons in a Magnetic Field"
]
| [
"Takafumi Kita ",
"Masao Arai \nNational Institute for Materials Science\nNamiki 1-1305-0044TsukubaIbaraki\n",
"\nDivision of Physics\nHokkaido University\n060-0810Sapporo\n"
]
| [
"National Institute for Materials Science\nNamiki 1-1305-0044TsukubaIbaraki",
"Division of Physics\nHokkaido University\n060-0810Sapporo"
]
| []
| We study interacting electrons in a periodic potential and a uniform magnetic field B taking the spin-orbit interaction into account. We first establish a perturbation expansion for those electrons with respect to the Bloch states in zero field. It is shown that the expansion can be performed with the zero-field Feynman diagrams of satisfying the momentum and energy conservation laws. We thereby clarify the structures of the self-energy and the thermodynamic potential in a finite magnetic field. We also provide a prescription of calculating the electronic structure in a finite magnetic field within the density functional theory starting from the zerofield energy-band structure. On the basis of these formulations, we derive explicit expressions for the magnetic susceptibility of B → 0 at various approximation levels on the interaction, particularly within the density functional theory, which include the result of Roth [J. Phys. Chem. Solids 23 (1962) 433] as the non-interacting limit. We finally study the de Haas-van Alphen oscillation in metals to show that quasiparticles at the Fermi level with the many-body effective mass are directly relevant to the phenomenon. The present argument may be more transparent than that by Luttinger [Phys. Rev. 121 (1961) 1251] of using the gauge invariance and has an advantage that the change of the band structure with the field may be incorporated. | 10.1143/jpsj.74.2813 | [
"https://arxiv.org/pdf/cond-mat/0510381v1.pdf"
]
| 119,425,495 | cond-mat/0510381 | 4d395ab75fa71b14099ded5f01a9aa350ec85c69 |
Theory of Interacting Bloch Electrons in a Magnetic Field
14 Oct 2005
Takafumi Kita
Masao Arai
National Institute for Materials Science
Namiki 1-1305-0044TsukubaIbaraki
Division of Physics
Hokkaido University
060-0810Sapporo
Theory of Interacting Bloch Electrons in a Magnetic Field
14 Oct 2005(Received November 19, 2018)Typeset with jpsj2.cls <ver.1.2> Full PaperBloch electronsmagnetic fieldcorrelation effectsmagnetic susceptibilityde Haas-van Alphen effectdensity functional theory
We study interacting electrons in a periodic potential and a uniform magnetic field B taking the spin-orbit interaction into account. We first establish a perturbation expansion for those electrons with respect to the Bloch states in zero field. It is shown that the expansion can be performed with the zero-field Feynman diagrams of satisfying the momentum and energy conservation laws. We thereby clarify the structures of the self-energy and the thermodynamic potential in a finite magnetic field. We also provide a prescription of calculating the electronic structure in a finite magnetic field within the density functional theory starting from the zerofield energy-band structure. On the basis of these formulations, we derive explicit expressions for the magnetic susceptibility of B → 0 at various approximation levels on the interaction, particularly within the density functional theory, which include the result of Roth [J. Phys. Chem. Solids 23 (1962) 433] as the non-interacting limit. We finally study the de Haas-van Alphen oscillation in metals to show that quasiparticles at the Fermi level with the many-body effective mass are directly relevant to the phenomenon. The present argument may be more transparent than that by Luttinger [Phys. Rev. 121 (1961) 1251] of using the gauge invariance and has an advantage that the change of the band structure with the field may be incorporated.
Introduction
Extensive theoretical studies have been carried out on Bloch electrons in a magnetic field B which display many exciting phenomena. Among them are investigations on fundamental quantities and phenomena such as the effective Hamiltonian, 1-9 the static magnetic susceptibility, 1, 4, 5, 10-35 and the de Haas-van Alphen (dHvA) oscillation. 15,[36][37][38][39][40][41] Since the relevant energy scale is µ B B 1meV with µ B the Bohr magneton, it seems natural to start from the energy-band structure in zero field and try to include the field effects perturbatively. However, a uniform magnetic field in quantum mechanics necessarily accompanies a vector potential with a non-periodic linear spatial dependence. Hence the procedure is not at all an easy task to perform as it apparently looks, particularly when the electron-electron interaction is taken into account. For example, we still do not have a satisfactory calculation of the total magnetic susceptibility of metals, even for Li and Na.
The purposes of the present paper are threefold. We first establish a definite theoretical prescription to calculate properties of interacting Bloch electrons in a uniform magnetic field on the basis of the energy-band structure in zero field. This includes an extension of the density functional theory [42][43][44][45][46][47][48][49][50][51][52][53][54] to a finite magnetic field. We then derive explicit expressions of the total magnetic susceptibility χ(B → 0) at various approximation levels on the electron-electron interaction. Finally, the formulation is used to study many-body effects on the dHvA oscillation in metals.
Let us briefly summarize the relevant works on the effective Hamiltonian, the density functional theory, the magnetic susceptibility and the dHvA oscillation together with the extensions considered here.
We now have a well-established effective Hamiltonian in a uniform magnetic field at the one-particle level; [1][2][3][4][5]8 see also Ref. 9 on mathematical aspects. Let t (0) (R−R ′ ) denote the transfer integral in zero field between two unit cells specified by R and R ′ ; it completely determines the energy-band structure. 55 Then the transfer integral in a finite field is obtained by the replacement:
t (0) (R−R ′ ) −→ e iI RR ′ t(R−R ′ , B) ,(1a)
where I RR ′ is the Peierls phase 1 and t(R−R ′ , B → 0) → t (0) (R−R ′ ). The relation was first obtained by Peierls 1 for the tight-binding model without the B dependence in t. The extension beyond the tight-binding model is due to Luttinger. 2 The B dependence in t was taken into account by Kohn, 3 Blount, 4 and Roth. 5 As shown by Luttinger, 2 eq. (1a) provides a microscopic justification for the procedure: E (0) (k) → E −i∇− e c A with E (0) (k) an energy eigenvalue in zero field, k a wave vector in the first Brillouin zone, and A the vector potential, which was a key assumption in the Onsager-Lifshitz-Kosevich theory for the dHvA oscillation. 36,37 Here, we also consider interaction effects on the basis of the treatments by Luttinger 2 and Roth. 5 It is thereby shown that the self-energy also experiences the change:
Σ (0) (ε n , R−R ′ ) −→ e iI RR ′ Σ(ε n , R−R ′ , B) ,(1b)
with ε n the Matsubara frequency. We will provide a definite prescription of how to calculate Σ(ε n , R−R ′ , B) for an arbitrary finite field B. Thus, our approach is more powerful than those by the expansion in A 19, 23 that is effective only in the zero-field limit. With an application to superconductors in mind, the whole formulation will be carried out without assuming a specific gauge for the vector potential in such a way that an extension to a non-uniform magnetic field may be performed easily. The density functional theory (DFT) is regarded now as one of the most efficient and reliable methods for the quantitative understanding of solids. 52,54 Its extension to a finite magnetic field has been performed by Vignale and Rasolt 51 using the current density j(r) as the relevant extra variable, and also by Grayce and Harris 53 choosing B instead of j(r). In these formulations, however, one has to calculate the electronic structure in B from the beginning by treating the field and the periodic potential on an equal footing. Thus, practical calculations for solids have never been carried out. Here, we will provide a prescription of obtaining the electronic structure in B within DFT based on the known zero-field energy-band structure. This two-step procedure may be regarded as one of the main advantages of the present formulation over the previous ones. 51,53 We choose the average flux density B ≡ ∇ × A as the relevant variable. The total moment may be obtained by M = −(∂Ω/∂B), and the external field is found by H = B −4πM /V with V the volume.
The static magnetic susceptibility χ(B → 0) has been a matter of extensive theoretical investigations, 1,4,5, and we now have several apparently different but essentially equivalent expressions of χ for non-interacting Bloch electrons. 5,13,14,20 Among them, Roth 5 gave a complete expression including the spin-orbit interaction with a clear derivation process. We closely follow her procedure to extend the consideration into interacting Bloch electrons. It should be noted that χ of interacting Bloch electrons were already calculated for the orbital part by by Fukuyama,19 Phillipas and McClure, 21 and Fukuyama and McClure, 23 and for both the orbital and spin parts by Buot 28 and Misra et al. 31 However, most of them provided only approximate treatments of χ: either the vertex corrections were neglected 19,23,31 or the frequency dependence of the self-energy were not considered. 21,31 Also, the treatment by Buot 28 fails to incorporate vertex corrections explicitly in the general expression. We here present a complete framework to calculate χ at various approximation levels on the interaction. Particularly, we derive an explicit expression of χ within DFT. Calculations of the susceptibility by DFT have been performed only for the spin part, 24-26, 29, 34, 35 for the orbital part neglecting vertex corrections, 27,30,32 and for the both parts but neglecting vertex corrections. 33 Thus, there is no available expression within DFT for the total susceptibility with vertex corrections. The formula obtained here, which incorporates the effects of the spin-orbit interaction, core polarizations, interband transitions, and vertex corrections, is expected to enhance our understanding on the total magnetic susceptibility of solids. It will be shown that the vertex corrections of the spin part in our formula naturally include the Stoner enhancement factor. 10,56 The dHvA oscillation in metals provides unique information on the Fermi-surface structures and interaction effects. The classic theory at the one-particle level is due to Lifshitz and Kosevich, 37 who applied Onsager's semiclassical quantization scheme 36 to non-interacting band electrons. The interaction effects were considered by Luttinger 38 and Bychkov and Gor'kov. 39 However, Bychkov and Gor'kov considered only an isotropic Fermi liquid. Also, the work by Luttinger 38 is based on a gauge-invariance argument E (0) (k) + Σ (0) (ε n , k) → E −i∇− e c A +Σ ε n , −i∇− e c A without clarifying the structure of the self-energy explicitly. Hence it is not clear from Luttinger's argument whether it is the momentum p or the crystal momentum k in the first Brillouin zone that is really relevant. The replacement procedure for the self-energy needs to be established microscopically in the same sense as eq. (1a) had to. His argument also has an ambiguity as to the "non-oscillatory part" of the self-energy, which later caused an interpretation 40, 41 that the self-energy does not participate in making up the quantized energy levels, i.e., one only has to replace the one-particle part as
E (0) (k) → E −i∇− e c A .
To remove the confusion and also to analyze experiments unambiguously, it will be well worth placing the theory on a firm ground. On the basis of the formulation to derive eq. (1b), we will present a hopefully clearer argument for the many-body effects on the dHvA oscillation. This argument also has an advantage that the change of the energy-band structure with the field can be taken into account.
This paper is organized as follows. Section 2 provides an alternative derivation of the effective one-particle Hamiltonian with the spin-orbit interaction. We combine the advantages in the treatments of Luttinger 2 and Roth 5 to formulate the problem so that extensions (i) to interacting systems and (ii) to a non-uniform magnetic field may be performed easily. Section 3 takes the electron-electron interaction into account. We establish a perturbation expansion for the thermodynamic potential and the self-energy in terms of the energy-band structure in zero field. It is shown that only the zero-field Feynman diagrams are necessary. We also construct a DFT in a finite magnetic field. Section 4 derives expressions of the magnetic susceptibility at various approximation levels, including that of DFT. Section 5 studies manybody effects on the dHvA oscillation in metals. Section 6 summarizes the paper. We put k B = 1 throughout.
Effective Hamiltonian
Hamiltonian and basis functions
We consider Bloch electrons in a uniform magnetic field described by a Hamiltonian with the spin-orbit interaction and Pauli paramagnetism:
H =P 2 2m + V(r) − g e 2m 2 c 2 (ŝ × ∇V) ·P − g e 2mcŝ · B − µ .
(2) Here m and e(< 0) are the electron mass and charge, respectively,P is defined bŷ
P ≡p − e c A(r) ,(3)
withp the momentum operator and A(r) the vector potential, V(r) denotes the periodic lattice potential, g is the electron g factor,ŝ is the dimensionless spin operator, B ≡ ∇ × A(r) is the magnetic field, and µ is the chemical potential.
The eigenfunctions of eq. (2) at A = 0 are the Bloch spinors:
ψ bk (r) = e ik·r u bk (r) ,(4)
where k is a wave vector in the first Brillouin zone and b denotes a set of quantum numbers for the band and spin. They are orthonormal as ψ bk | ψ b ′ k ′ = δ bb ′ δ kk ′ and form a complete set. For our consideration, however, the wave functions (4) may not necessarily be the eigenfunctions of eq. (2) at A = 0. Thus, we will proceed by assuming only the completeness and orthonormality of eq. (4). Note that, relaxing the conditions in this way, ψ bk (r) may be chosen analytic in k throughout the Brillouin zone, as shown by des Cloizeaux 57,58 and Nenciu. 9,59 Those basis functions, which are analytic in k but not necessarily the eigenfunctions ofĤ, were named quasi-Bloch functions by des Cloizeaux. 57, 58 A set of alternative basis functions was introduced by Wannier, 60 which are more suitable for the present purpose. They are defined as a Fourier transform of ψ bk (r) by
w bR (r) ≡ 1 √ N c k e −ik·R ψ bk (r) ,(5)
where R specifies a unit cell and N c denotes the number of unit cells in the system. It hence follows that the Wannier functions are also complete bR | w bR w bR | = 1 and orthonormal as
w bR | w b ′ R ′ = δ bb ′ δ RR ′ with w bR (r) = w b0 (r−R).
To describe Bloch electrons in a finite magnetic field, we adopt the basis functions introduced by Luttinger. 2 They differ from eq. (5) by only a phase factor as
w ′ bR (r) = e iIrR w bR (r) ,(6a)
where I rR is defined by
I rR ≡ e c r R A(r ′ ) · dr ′ ,(6b)
with dr ′ taken along the straight line path from R to r. We assume that the functions { w ′ bR } form a complete set, although they are not orthonormal in a finite magnetic field. This latter inconvenience can be removed by considering the linear combination:
ϕ bR (r) = b ′ R ′ S b ′ R ′ ,bR w ′ b ′ R ′ (r) ,(7)
and orthonormalize them as ϕ bR | ϕ b ′ R ′ = δ bb ′ δ RR ′ ; this will be performed shortly below. We now transform eq. (2) into a matrix form by using eq. (7). Let us introduce the matrices:
S = (S bR,b ′ R ′ ) ,(8a)O = ( w ′ bR | w ′ b ′ R ′ ) ,(8b)H ′ = ( w ′ bR |Ĥ| w ′ b ′ R ′ ) .(8c)
Note that O is positive-definite Hermitian and reduces to a unit matrix as B → 0. Now, the orthonormality
ϕ bR | ϕ b ′ R ′ = δ bb ′ δ RR ′ reads S † O S = 1 ,(9)
where 1 is the unit matrix: 1 = (δ bb ′ δ RR ′ ). Equation (9) is solved easily by choosing S as Hermitian S † = S as
S = O −1/2 .(10)
We next express the eigenstate of eq. (2) as a linear combination of eq. (7) as ψ(r) = bR c b (R) ϕ bR (r). Then the eigenvalue problem of eq. (2) is transformed into
b ′ R ′ H bR,b ′ R ′ c b ′ (R ′ ) = E c b (R) ,(11)
with
H ≡ S H ′ S .(12)
A couple of comments are in order on eqs. (7) and (10) in connection with the treatment by Roth. 5 First, our basis functions (7) with the phase integral (6b) have an advantage over Roth's basis functions that an extension to a non-uniform magnetic field can be performed easily. Whereas Roth had to assume that the field is uniform from the beginning, there is no limitation at this stage on the form of A(r) in eq. (6b). Second, eq. (10) is different from Roth's choice. Indeed, Roth expressed S as a sum of Hermitian and anti-Hermitian matrices and fixed the anti-Hermitian part by imposing that H be semidiagonal up to the order of B 2 . However, eq. (10) has the advantage that the formulation becomes more transparent. It helps to avoid unnecessary complications at the one-particle level and make the extension to interacting systems easier. Once the orthonormality is endowed as eq. (10), the semi-diagonalization may be performed by a similarity transformation. It should be noted, however, that the physical results are irrelevant of whether one carries out such a similarity transformation at an intermediate step. Indeed, our procedure without further similarity transformations leads exactly the same expression for the non-interacting magnetic susceptibility as that of Roth.
Overlap integral
We first concentrate on the overlap w ′ bR | w ′ b ′ R ′ , where the phase factor in eq. (6) yields a line integral from R ′ to R via r. We transform it as follows:
I Rr + I rR ′ = I RR ′ + e c C A(r ′ ) · dr ′ = I RR ′ + e c (r R ×r R ′ ) · 1 0 du 1−u 0 dv B = I RR ′ + e 2 c B · (r R ×r R ′ ) .(13)
Here C denotes closed path R ′ → r → R → R ′ along the triangle, r R is defined by
r R ≡ r − R ,(14)
and the second line follows via Stokes' theorem by writing the vector area element as dS = (R ′ −R)du×(r−R)dv. The transformation (13) was used previously in deriving a transport equation for superconductors with Hall terms. 61 Although we have assumed a uniform magnetic field here, an extension to a non-uniform field can be performed easily as eqs. (31) and (33)
h αβ = ǫ αβγ e 2 c B γ ,σ αβ = ǫ αβγ g 2 2mŝ γ ,(15)
where ǫ αβγ (α, β, γ = x, y, z) is the third-rank completely antisymmetric tensor, and summations over the repeated index γ are implied. The quantity h αβ is essentially inverse of the magnetic length squared. Also useful is the following identity for an arbitrary function f (r R , r R ′ ):
f (r R , r R ′ ) w † bR (r) w b ′ R ′ (r) = 1 N c kk ′ f (r R , r R ′ ) u † bk (r) u b ′ k ′ (r) e −ik·rR+ik ′ ·r R ′ = 1 N c kk ′ u † bk (r) u b ′ k ′ (r)f (i∇ k , −i∇ k ′ ) e −ik·rR+ik ′ ·r R ′ = 1 N c kk ′ f (−i∇ k , i∇ k ′ ) u † bk (r) u b ′ k ′ (r) ×e −ik·rR+ik ′ ·r R ′ ,(16)
where we substituted eqs. (4) and (5) and performed partial integrations over k and k ′ . Using eqs. (13)- (16) and the periodicity u bk (r + R) = u bk (r), we obtain an expression for eq. (8b) as
O bR,b ′ R ′ = e iI RR ′ exp(ih αβ r α R r β R ′ ) w † bR (r) w b ′ R ′ (r) dr = e iI RR ′ N c k e ik·(R−R ′ ) O bb ′ (k, B) ,(17)
where O bb ′ (k, B) is given by
O bb ′ (k, B) ≡ e ih αβ ∇ α k ∇ β k ′ u † bk (r) u b ′ k ′ (r) k ′ =k
dr .
(18a) The expression (17) has a general form for any matrix element between eq. (6), i.e., the Peierls phase factor 1 e iI RR ′ times a summation over k for the product of e ik·(R−R ′ ) and some function of k.
It is useful for practical purposes to expand eq. (18a) in terms of B as
O bb ′ (k, B) = δ bb ′ + ∞ j=1 O (j) bb ′ (k, B) ,(18b)
where O
(j)
bb ′ is the quantity of the order B j . The terms for j = 1, 2 are given explicitly by
O (1) bb ′ = ih αβ ∇ α k u bk |∇ β k u b ′ k = ih αβ b ′′ x α bb ′′ (k)x β b ′′ b ′ (k) ,(19a)O (2) bb ′ = − 1 2 h αβ h α ′ β ′ ∇ α ′ k ∇ α k u † bk |∇ β ′ k ∇ β k u b ′ k , (19b) with x α bb ′ (k) ≡ i u bk |∇ α k u b ′ k = −i ∇ α k u bk | u b ′ k = R e ik·R w b0 | r α R | w b ′ R .(20)
The second expressions of eqs. (19a) and (20) have been derived by noting u bk (r + R) = u bk (r), reducing the integral into a unit cell, and using the completeness and orthonormality of { u bk } over the unit cell. 5 The last expression of eq. (20) is given entirely with respect to the Wannier functions of eq. (5). Hence it may be suitable for the evaluation of the matrix element x α bb ′ (k) between the core states. Equations (18a) and (19) are exactly those obtained by Roth. 5 A couple of comments are in order before closing the section. First, the expansion of eq. (18) is convergent if we choose ψ bk (r) as the quasi-Bloch functions of des Cloizeaux 9, 57-59 which are analytic in k. This property can be satisfied for a simple band by the eigenfunctions of eq. (2), but not for a complex band with band crossings. This statement holds for every expansion we shall encounter below. Second, matrix elements like eqs. (19) and (20) cannot be determined uniquely due to the gauge degree of freedom:
u bk (r) → e iφ bk u bk (r) .
However, obsevable quantities such as the magnetic susceptibility do not depend on the choice of the phase φ bk . See also the discussions of Roth 5 on this point, and those by Resta 62 for the electronic polarization.
Expression of S
We next derive an expression of S in powers of B starting from eq. (9) and S † = S. Equation (17) suggests that we may expand the matrix element as
S bR,b ′ R ′ = e iI RR ′ N c k e ik·(R−R ′ ) S bb ′ (k, B) ,(21)
with S † (k, B) = S(k, B). Let us substitute eqs. (17) and (21) into eq. (9). We then encounter a summation over R ′′ with both the Peierls phase factor e iI RR ′′ +iI R ′′ R ′ and the Bloch phase factor e ik·(R−R ′′ )+ik ′ ·(R ′′ −R ′ ) . This summation can also be transformed with the procedures of eqs. (13) and (16). For example, we obtain
b ′′ R ′′ S bR,b ′′ R ′′ O b ′′ R ′′ ,b ′ R ′ = e iI RR ′ N c b ′′ k e ik·(R−R ′ ) S bb ′′ (k)⊗O b ′′ b ′ (k) ,(22)
where the operator ⊗ is defined by
S(k)⊗O(k) ≡ e ih αβ ∇ α k ′ ∇ β k S(k ′ ) O(k) k ′ =k .(23)
Equation (23) is exactly the multiplication theorem of Roth. 5 It is worth summarizing the properties of the operator ⊗. First, we realize from eq. (22) that it is associative as
A(k)⊗[B(k)⊗C(k)] = [A(k)⊗B(k)]⊗C(k) . (24a) Second, it satisfies k A(k)⊗B(k) = k A(k) B(k) ,(24b)
as can be shown with partial integrations over k and the antisymmetry h αβ = −h βα . By using eqs. (17), (21) and (22), eq. (9) with S † = S is transformed into
S(k, B) ⊗ O(k, B) ⊗ S(k, B) = 1 , (25a) with 1 ≡ (δ bb ′ ). Let us expand S in powers of B as S(k, B) = 1 + ∞ j=1 S (j) (k, B) . (25b)
We then find from eqs. (18b), (25a), and (25b) that S (j) for j = 1, 2 are given by
S (1) = − 1 2 O (1) ,(26a)S (2) = − 1 2 O (2) + 3 8 O (1) O (1) . (26b)
Matrix element of Hamiltonian
We next consider H ′ bR,b ′ R ′ ≡ w ′ bR |Ĥ| w ′ b ′ R ′ withĤ and w ′
bR given by eqs. (2) and (6), respectively. The following identity is useful for this purpose: 2
∇I rR = e c A(r) + e c r R × 1 0 B u du = e c A(r) − e 2 c B×r R .(27)
It is worth noting that this identity can also be extended easily to a non-uniform magnetic field. 2 Using eqs. (13), (16) and (27) as well as u bk (r +R) = u bk (r), the above matrix element is transformed into
H ′ bR,b ′ R ′ = e iI RR ′ N c k e ik·(R−R ′ ) H ′ bb ′ (k, B) ,(28)
where
H ′ bb ′ is defined by H ′ bb ′ (k, B) ≡ e ih αβ ∇ α k ′ ∇ β k u † bk ′ (r) Ĥ (0) k − h α ′ β ′ (i∇ α ′ kπ β ′ k +σ α ′ β ′ ) − 2 2m h α ′ β ′ h α ′ γ ′ ∇ β ′ k ∇ γ ′ k u b ′ k (r) k ′ =k dr ,(29a)withĤ (0) k ≡Ĥ(p →p + k, A = 0),π k ≡ ∂Ĥ (0) k /∂k = m p+ k−g e
2mc 2ŝ ×∇V(r) , andσ αβ given in eq. (15). We now expand eq. (29a) formally in terms of B as
H ′ bb ′ (k, B) = H (0) bb ′ (k) + ∞ j=1 H ′(j) bb ′ (k, B) ,(29b)
with
H (0) bb ′ (k) ≡ u bk |Ĥ (0) k | u b ′ k .(30a)
To write down H
′(j)
bb ′ (k, B) in eq. (29b) explicitly, we introduce the following matrices:
v bb ′ (k) ≡ 1 ∂H (0) bb ′ (k) ∂k , (30b) π bb ′ (k) ≡ u bk m p + k − g e 2mc 2ŝ ×∇V(r) u b ′ k , (30c) σ αβ bb ′ (k) ≡ u bk |σ αβ | u b ′ k . (30d) Note ṽ α = π α + i(x α H (0) − H (0) x α ), as can be shown fromπ k = ∂Ĥ (0) k /∂k. Next, we expressĤ (0) k | u b ′ k and ih α ′ β ′ ∇ α ′ kπ β ′ k | u b ′ k in eq. (29a) aŝ H (0) k | u b ′ k = b ′′ | u b ′′ k H (0) b ′′ b ′ (k) , ih α ′ β ′ ∇ α ′ kπ β ′ k | u b ′ k = h α ′ β ′ b ′′ | u b ′′ k π β ′ (k) x α ′ (k) b ′′ b ′ .
We then expand eq. (29a) in powers of B and perform straightforward calculations order by order to obtain H ′(j) in eq. (29b). The first-order term can be written down easily. Averaging the resulting expression with its Hermitian conjugate, we obtain
H ′(1) = {O (1) , H (0) } − h αβ {x α , ṽ β +π β }+σ αβ , (31a)
with O (1) given by eq. (19a) and {A, B} ≡ 1 2 (A B+B A). To obtain H ′(2) compactly, on the other hand, we carry out partial integrations using the antisymmetry h αβ = −h βα . For example, one of the relevant terms are transformed as
− 1 2 h αβ h α ′ β ′ ∇ α ′ k ∇ α k u bk |∇ β ′ k ∇ β kĤ (0) k u b ′ k = b ′′ O (2) bb ′′ H (0) b ′′ b ′ − ∇ β k ∇ α ′ k ∇ α k u bk | u b ′′ k ṽ β ′ b ′′ b ′ + 1 2 ∇ α ′ k ∇ α k u bk | u b ′′ k ∇ β ′ k ∇ β k H (0) b ′′ b ′ (k) h αβ h α ′ β ′ , with ∇ α ′ k ∇ α k u bk | u b ′ k = i∇ α ′ k x α bb ′ − b ′′ x α bb ′′ x α ′ b ′′ b ′ = i∇ α k x α ′ bb ′ − b ′′ x α ′ bb ′′ x α b ′′ b ′ , and h αβ ∇ β k [(∇ α k x α ′ bb ′′ ) ṽ β ′ b ′′ b ′ ] = h αβ (∇ α k x α ′ bb ′′ )∇ β k ∇ β ′ k H (0) b ′′ b ′ .
Similar calculations are performed for the other terms to make H ′(2) compact and symmetric. We finally perform the averaging with the Hermitian conjugate. We thereby obtain
H ′(2) = {O (2) , H (0) } + h αβ h α ′ β ′ ∇ β k {x α , x α ′ }, ṽ β ′ + 1 2 (x α π β ′ x α ′ +x α ′ π β ′ x α ) + {x α , σ α ′ β ′ } + 1 2 {x α , x α ′ }, 2 m δ ββ ′ 1 − ∇ β k ∇ β ′ k H (0) + i 4 ∇ β k ∇ β ′ k [ x α ′ , ṽ α ] ,(31b)
Effective Hamiltonian
We are ready to derive an effective one-particle Hamiltonian in a finite magnetic field. Let us substitute eqs. (21) and (28) into eq. (12). Using eq. (22) twice, we obtain
H bR,b ′ R ′ = e iI RR ′ N c k e ik·(R−R ′ ) H bb ′ (k, B) ,(32)
with
H(k, B) = S(k, B)⊗H ′ (k, B)⊗S(k, B) . (33a)
Let us expand eq. (33a) in powers of B as
H(k, B) = H (0) (k) + ∞ j=1 H (j) (k, B) ,(33b)
with H (0) (k) given by eq. (30a). The expressions of H (j) can be obtained order by order by substituting eqs. (25b), (29b) and (33b) into eq. (33a) and using eqs. (26) and (31). The first-order term is obtained from
H (1) = H ′(1) + 2{S (1) , H (0) } as H (1) (k) = −h αβ {x α , ṽ β +π β }+σ αβ , (34a) with {A, B} ≡ 1 2 (A B + B A). The second-order contri- bution is calculated from H (2) = H ′(2) + 2{S (2) , H (0) } + 2{S (1) , H ′(1) } + S (1) H (0) S (1) − ih αβ [∇ β k S (1) , ∇ α k H (0) ] as H (2) (k) = h αβ h α ′ β ′ ∇ β k {x α , x α ′ }, ṽ β ′ + 1 2 (x α π β ′ x α ′ +x α ′ π β ′ x α ) + {x α , σ α ′ β ′ } + 1 2 {x α , x α ′ }, 2 m δ ββ ′ 1 −∇ β k ∇ β ′ k H (0) + i 4 ∇ β k ∇ β ′ k [x α ′ , ṽ α ] − O (1) , H (1) + 1 8 O (1) , O (1) , H (0) + i 2 h αβ ∇ β k O (1) , ṽ α , (34b) with {A, B} ≡ 1 2 (A B +B A) and [ A, B ] ≡ A B−B A.
Schrödinger equation and Green's function
We finally write down an effective Schrödinger equation and the corresponding Green's function for noninteracting Bloch electrons in a magnetic field. To this end, it is useful to introduce the transfer integral in terms of eq. (33) as
t(R, B) ≡ 1 N c k e ik·R H(k, B) .(35)
Substituting eq. (32) and using eq. (35), the Schrödinger equation (11) is transformed into
b ′ R ′ e iI RR ′ t bb ′ (R − R ′ , B) c b ′ (R ′ ) = E c b (R) .(36)
Thus, the Hamiltonian is given as a product of the transfer integral and the Peierls phase factor. Note that the transfer integral also depends on the magnetic field here. We can provide an alternative expression to eq. (36). Let us change the summation in the above equation
from R ′ to R ′′ ≡ R − R ′ , express c b ′ (R − R ′′ ) = e −R ′′ ·∇R c b ′ (R),
and use the following identity proved in Appendix II of Ref. 2:
exp(iI R,R−R ′′ ) exp −R ′′ ·∇ R = exp −R ′′ ·∂ R , with ∂ R defined by ∂ R ≡ ∂ ∂R − i e c A(R) .(37)
Equation (36)
is thereby transformed into b ′ H bb ′ (−i∂ R , B) c b ′ (R) = E c b (R) ,(38)
where
H bb ′ (−i∂ R , B) is an operator symmetrized with respect to ∂ R as H(−i∂ R , B) ≡ R ′′ t(R ′′ , B) e iR ′′ · (−i∂R) .(39)
Equation (38)
b ′′ [iε n δ b ′ b ′′ −H b ′ b ′′ (−i∂ R ′ , B)] g b ′′ R ′ ,bR (ε n ) = δ b ′ b δ R ′ R , (40) where g b ′′ R ′ ,bR (ε n ) is the non-interacting Matsubara
Green's function with ε n ≡ (2n + 1)πT (n = 0, ±1, · · · ). Suggested by eq. (36), we write the Green's function as
g b ′ R ′ ,bR (ε n ) = e iI R ′ Rg b ′ R ′ ,bR (ε n ) ,(41a)
and substitute it into eq. (40). Using eq. (27), we then find an equation forg which is also given as eq. (40) with a replacement:
∂ R ′ −→∂ R ′ ≡ ∂ ∂R ′ − i e 2 c B×(R ′ −R) .
It hence follows thatg b ′ R ′ ,bR (ε n ) depends only on R ′ −R and can be expanded as
g b ′ R ′ ,bR (ε n ) = 1 N c k e ik·(R ′ −R) g b ′ b (ε n , k) . (41b) Let us substitute eq. (41b) into the equation forg, ex- press (R ′ −R)e ik·(R ′ −R) = −i∇ k e ik·(R ′ −R)
, and perform partial integrations over k. We thereby obtain an equation for g(ε n , k) as
k ′ δ kk ′ [iε n 1 − H(κ, B)] g(ε n , k ′ ) = 1 ,(42)
where the operator κ is defined by
κ ≡ k − i e 2 c B ×∇ k ′ .(43)
Note that eq. (42) may be written alternatively by using the operator of eq. (23) as [iε n 1 − H(k, B)] ⊗ g(ε n , k) = 1. This latter result may have been obtained directly by writing Dyson's equation (40) in terms of the Hamiltonian in eq. (36), substituting eqs. (35) and (41) into it, and using eq. (22).
Interaction Effects
We now include the two-body interaction:
U(r−r ′ ) = 1 V q U q e iq·(r−r ′ )(44)
into our consideration, where V is the volume of the system.
The total HamiltonianĤ tot is given in second quantization by using the basis function (7) and the notation ν ≡ bR aŝ
H tot = νν ′ H νν ′ c † ν c ν ′ + 1 2 ν1ν2ν ′ 1 ν ′ 2 U ν1ν2;ν ′ 1 ν ′ 2 c † ν1 c † ν2 c ν ′ 2 c ν ′ 1 .
(45) Here c ν is the fermion operator and H νν ′ is given by eq.
(32). The quantity U ν1ν2;ν ′ 1 ν ′ 2 is defined by U ν1ν2;ν ′ 1 ν ′ 2 ≡ ν3ν4ν ′ 3 ν ′ 4 S ν1ν3 S ν2ν4 U ′ ν3ν4;ν ′ 3 ν ′ 4 S ν ′ 3 ν ′ 1 S ν ′ 4 ν ′ 2 ,(46)where S νν ′ is given by eq. (21) and U ′ ν3ν4;ν ′ 3 ν ′ 4 denotes U ′ ν3ν4;ν ′ 3 ν ′ 4 ≡ dr dr ′ w ′ † ν3 (r) w ′ ν ′ 3 (r)U(r−r ′ ) w ′ † ν4 (r ′ ) w ′ ν ′ 4 (r ′ ) , (47) with w ′ ν (r) defined by eq. (6).
Bare vertex
As shown in Appendix A, eq. (46) can be written alternatively with respect to the Bloch states {ψ bk (r)} in zero field as
U ν1ν2;ν ′ 1 ν ′ 2 = e iI R 1 R ′ 1 +iI R 2 R ′ 2 N 2 c kk ′ e ik·(R1−R ′ 1 )+ik ′ ·(R2−R ′ 2 ) × 1 V q e iq·(R1−R2) U q Λ b1b ′ 1 (k, q, B) ×Λ b2b ′ 2 (k ′ , −q, B) .(48)
Here I RR ′ and U q are given in eqs. (6b) and (44), respectively. The quantity Λ(k, q, B) may be called a vertex function which is defined in terms of the operator ⊗ in eq. (23), S in eq. (25) and Λ ′ of eq. (A·3) by
Λ(k, q, B) ≡ S(k+q, B)⊗Λ ′ (k, q, B)⊗S(k, B) , (49a)
with k+q the wave vector in the first Brillouin zone corresponding to k+q in the extended zone scheme. Thus, k in Λ(k, q, B) belongs to the incoming electron, q is an additional wave vector from the interaction, and k+q specifies the outgoing electron. In zero field without the Peierls phase factors, the interaction (48) in k space may be expressed diagrammatically as Fig. 1. We shall see below that this diagram in zero field suffices for the diagrammatic calculations of the properties in a finite magnetic field. It is also possible to expand eq. (49a) in powers of B as (48) in the absence of the Peierls phase factors. The vertex Λ(k, q) is given by eq. (49), and k+q denotes the wave vector in the first Brillouin zone corresponding to k + q in the extended zone scheme.
Λ(k, q, B) = Λ (0) (k, q) + ∞ j=1 Λ (j) (k, q, B) . (49b) k k k+q k-q q Λ(k,q) Λ(k,-q) ' ' ' Fig. 1. A diagrammatic representation of the interaction
The expressions of Λ (j) may be obtained with the same procedure as that of deriving eq. (34). The quantity Λ (0) is given by (26) and (A·4a) as
Λ (0) bb ′ (k, q) = u bk+q |e iK k+q ·r | u b ′ k , (50a) with K k+q ≡ k + q − k+q a reciprocal lattice vec- tor. The first-order term is obtained from Λ (1) (k, q) = Λ ′(1) (k, q) + S (1) (k+q)Λ (0) (k, q) + Λ (0) (k, q)S (1) (k) to- gether with eqs.Λ (1) (k, q) = − 1 2 h αβ x α (k+q) ∇ β k Λ (0) (k, q) − 1 2 h αβ ∇ β k Λ (0) (k, q) x α (k) .(50b)
Note that the argument of x α is k (k+q) when it appears to the right (left) of Λ (0) (k, q); keeping this rule in mind and dropping the argument of x α , eq. (50b) may be written exactly as the ṽ β contribution of eq. (34a). This rule also applies to higher-order terms. The second-order term is calculated similarly. We obtain an expression analogous to the ṽ contributions of eq. (34b) as
Λ (2) (k, q) = 1 2 h αβ h α ′ β ′ ∇ β k x α (k+q), x α ′ (k+q) ∇ β ′ k Λ (0) (k, q) + ∇ β ′ k Λ (0) (k, q) x α (k), x α ′ (k) − 1 4 h αβ h α ′ β ′ x α (k+q), x α ′ (k+q) ∇ β k ∇ β ′ k Λ (0) (k, q) + ∇ β k ∇ β ′ k Λ (0) (k, q) x α (k), x α ′ (k) + i 4 h αβ h α ′ β ′ ∇ β k ∇ β ′ k x α ′ (k+q)∇ α k Λ (0) (k, q) −∇ α k Λ (0) (k, q) x α ′ (k) − 1 2 O (1) (k+q)Λ (1) (k, q) + Λ (1) (k, q)O (1) (k) + 1 8 O (1) (k+q)O (1) (k+q)Λ (0) (k, q) J. Phys. Soc. Jpn. Full Paper Takafumi Kita and Masao Arai −2O (1) (k+q)Λ (0) (k, q)O (1) (k) +Λ (0) (k, q)O (1) (k)O (1) (k) + i 2 h αβ ∇ β k O (1) (k+q)∇ α k Λ (0) (k, q) − ∇ α k Λ (0) (k, q) O (1) (k) . (50c)
Perturbation expansion
With these preliminaries, we proceed to the calculation of the thermodynamic potential Ω = −T ln Tr e −Ĥtot/T , the self-energy Σ νν ′ and Green's function G ν ′ ν in the framework of the conserving approximation. 64,65 Let us define the Matsubara Green's function by
G ν ′ ν (ε n ) = − 1/T 0 T τ c ν ′ (τ ) c † ν e iεnτ dτ ,(51)
which as U → 0 reduces to g ν ′ ν (ε n ) of eq. (40). As shown by Luttinger and Ward, 63 Ω can be written as a functional of G as
Ω = −T n Tr ln H + Σ − iε n 1 + Σ G e iεn0+ + Φ[G] .(52)
Here 0 + is an infinitesimal positive constant and Φ denotes contributions of all the skeleton diagrams in the bare perturbation expansion for Ω with G used as the propagator. The self-energy is obtained from Φ by
Σ νν ′ (ε n ) = 1 T δΦ δG ν ′ ν (ε n ) .(53a)
With this relation, Ω is stationary with respect to a variation in G satisfying Dyson's equation:
[iε n 1 − H − Σ(ε n )] G(ε n ) = 1 .(53b)
The conserving approximation denotes approximating Φ by some selected diagrams and determining G and Σ selfconsistently by eq. (53). 64,65 Its advantages may be summarized as follows: (i) both the equilibrium and dynamical properties can be described within the same approximation with various conservation laws automatically satisfied; (ii) the Laudau Fermi-liquid corrections, 66, 67 or vertex corrections in a different terminology, are automatically included. For example, the lowest-order conserving approximation is nothing but the Hartree-Fock approximation where Φ HF and Σ HF νν ′ are given by
Φ HF = T 2 2 n1,n2 ν1ν2ν ′ 1 ν ′ 2 U ν1ν2;ν ′ 1 ν ′ 2 [G ν ′ 1 ν1 (ε n1 )G ν ′ 2 ν2 (ε n2 ) −G ν ′ 1 ν2 (ε n1 )G ν ′ 2 ν1 (ε n2 )]e iεn 1 0++iεn 2 0+ ,(54a)Σ HF νν ′ = T n1 ν1ν ′ 1 (U νν1;ν ′ ν ′ 1 − U νν1;ν ′ 1 ν ′ )G ν ′ 1 ν1 (ε n1 )e iεn 1 0+ ,(54b)
respectively.
It follows from eqs. (32) and (48) that Σ νν ′ (ε n ) and G ν ′ ν (ε n ) can be expanded as
Σ bR,b ′ R ′ (ε n ) = e iI RR ′ N c k e ik·(R−R ′ ) Σ bb ′ (ε n , k) , (55a) k k ' k k+q K q (a) k+q k+q k+q+q ' ' ' k q q (b)G b ′ R ′ ,bR (ε n ) = e iI R ′ R N c k e ik·(R ′ −R) G b ′ b (ε n , k) . (55b)
This is proved by induction as follows: First, the noninteracting Green's function g ν ′ ν (ε n ) is already given in the form of eq. (55b) as eq. (41). We next assume the expression (55b) and substitute it into eq. (53a) to calculate Σ νν ′ (ε n ) order by order by using eq. (22). We then find that the self-energy can also be written as eq. (55a), i.e., the same expression as the non-interacting Hamiltonian (32). It hence follows that G ν ′ ν (ε n ) may be written as eq. (55b). This completes the proof.
To see how to calculate Σ(ε n , k) practically, we present expressions of Σ HF (k) obtained from eq. (54b), see also Fig. 2(a), and Σ 2e (ε n , k) which corresponds to the diagram of Fig. 2(b):
Σ HF (k) = T V n K U K Λ(k, K) k ′ TrΛ(k ′ , −K)G(ε n , k ′ ) − q U q Λ(k+q, −q)⊗G(ε n , k+q)⊗Λ(k, q) e iεn0+ ,(56a)Σ 2e (ε n , k) = T 2 V 2 n1n2 qq ′ U q U q ′ Λ(k+q ′ , −q ′ ) ⊗ G(ε n+n2−n1 , k+q ′ ) ⊗ Λ(k+q+q ′ , −q) ⊗ G(ε n2 , k+q+q ′ ) ⊗ Λ(k+q, q ′ ) ⊗ G(ε n1 , k+q) ⊗ Λ(k, q) ,(56b)
where ⊗ operates on the k ′ or k dependences of the adjacent functions. Note that removing the operator ⊗ in the above expressions yields the self-energies at B = 0. We hence have a simple rule to calculate the k-space self-energy for B > 0 using only the zero-field Feynman diagrams as follows: (i) Label each of the connected electron lines with the same wave vector, like k or k ′ in the above examples, and insert the operator ⊗ of eq. (23) between every adjacent Λ and G. (ii) We can remove any single operator ⊗ within every closed electron loop, as in the first term of eq. (56a), because of the absence of the Peierls phase factor in the final calculation of using eq. (22); this may also be proved directly in k space by using eq. (24b). The rules (i) and (ii) apply also to the calculations of Ω and G(ε n , k). We now provide k-space expressions of eqs. (52) and (53). Noting the rule (ii) above, we realize that Φ in eq. (52) is given alternatively as a functional of G(ε n , k) as
Φ = T n ∞ ℓ=1 1 2ℓ k Tr Σ ℓ (ε n , k, B) G(ε n , k, B)e iεn0+ ,(57)
with Σ ℓ the contribution of the ℓth-order skeleton diagrams to Σ. We hence have
Σ bb ′ (ε n , k, B, {G}) = 1 T δΦ δG b ′ b (ε n , k, B)
.
+ k Σ(ε n , k, B) G(ε n , k, B) e iεn0+ + Φ[G] ,(59)
with Tr k denoting the trace in k space. It satisfies
δΩ δG b ′ b (ε n , k, B) = 0 ,(60)
as shown by using eq. (58). Finally, eqs. (57) and (58a) may be expanded with respect to the explicit B dependences originating from the vertex Λ and the operator ⊗ as
Φ(B, {G}) = Φ (0) ({G}) + ∞ j=1 Φ (j) (B, {G}) ,(61)Σ(ε n , k, B, {G}) = Σ (0) (ε n , k, {G}) + ∞ j=1 Σ (j) (ε n , k, B, {G}) ,(62)
where Φ (j) and Σ (j) are quantities of the order B j . Note that Φ (0) and Σ (0) are different from those at B = 0 due to the implicit B dependences through G. It follows from eq. (57) that Φ (j) is connected with Σ (j) as
Φ (j) = T nℓ 1 2ℓ k TrΣ ℓ(j) (ε n , k, B)G(ε n , k, B) e iεn0+ .(63)
Hartree-Fock approximation
We here concentrate on the Hartree-Fock approximation. Since there is no ε n dependence in the self-energy Σ HF (k), eqs. (57)-(60) can be simplified further by performing every summation over n with the Fermi distribution function f (ε) ≡ 1/(e ε/T + 1). 63 The potentials Ω and Φ are thereby transformed into the functionals of the occupation number defined by
n(k, B) ≡ T n G(ε n , k, B) e iεn0+ .(64)with Φ HF = 1 2V kk ′ K U K Tr[ n(k)Λ(k, K)] Tr[ n(k ′ )Λ(k ′ , −K)] − kq U q TrΛ(k+q, −q) ⊗ n(k+q) ⊗ Λ(k, q) ⊗ n(k) .(65)
The self-energy Σ HF (k) of eq. (56a) is obtained from Φ HF by
Σ HF bb ′ (k) = δΦ HF δn b ′ b (k) .(67a)
It hence follows that Ω HF is stationary with respect to a variation in n(k) satisfying n(k) = 1 e [H(k)+Σ HF (k)]/T + 1 .
Equations (67a) and (67b) constitute a closed set of selfconsistent equations. Finally, Φ HF and Σ HF can be expanded with respect to the explicit B dependences as eqs. (61) and (62), respectively. The expressions of Φ HF(j) and Σ HF(j) are given in Appendix B to clarify some of their fundamental properties.
Density functional theory
Extensive theoretical studies have been carried out for more than three decades to describe atoms and solids quantitatively based on the DFT. 52,54 It is thereby established now as one of the most efficient and reliable methods for the quantitative understanding of solids. Hence it is well worth applying the present method to the density functional theory.
We consider the cases where the exchange-correlation energy E xc is given as a functional of the spin density:
n ± (r) = T n Tr 1 2 (1 ± τ 3 ) G(r, r, ε n ) e iεn0+ ,(68)
where 1, τ 3 , and G are 2 × 2 matrices corresponding to the spin degrees of freedom with τ 3 denoting the third Pauli matrix. Let us expand G(r, r, ε n ) in eq. (68) with respect to the basis functions of eq. (7) and transform the resulting expression with the procedures of eqs. (13), (16) and (22). We thereby obtain where n(k) is given by eq. (64) and ρ ± (k, r) is defined by
n ± (r) = 1 N c k Tr n(k) ρ ± (k, r) ,(69)ρ ± (k, r) ≡ S(k) ⊗ ρ ′ ± (k, r) ⊗ S(k) ,(70)
with
ρ ′ ± bb ′ (k, r) ≡ e ih αβ ∇ α k ∇ β k ′ u † bk (r) N c 2 (1 ± τ 3 ) u b ′ k ′ (r) k ′ =k .(71)
The factor N c is introduced in eq. (71) to make ρ ′ ± bb ′ (k, r) finite in the thermodynamic limit. Note that ρ ± (k, r) in eq. (69) is just a matrix element with no information on the occupation number. It hence follows that the DFT as a functional of n ± (r) can be written alternatively as a functional of n(k).
The thermodynamic potential of the DFT is given in a form similar to eq.
The quantity Φ DFT is defined by
Φ DFT = Φ H [n(r)] + E xc [n σ (r)] ,(73)
where Φ H [n(r)] and E xc [n σ (r)] are the classic Coulomb energy and the exchange-correlation energy, respectively, with n(r) ≡ n + (r)+n − (r) and σ = ±. The former is given explicitly by
Φ H [n(r)] = e 2 2 n(r)n(r ′ ) |r−r ′ | drdr ′ ,(74)
which is equivalent to the Hartree term in eq. (66) with U K = 4πe 2 /K 2 ; the K = 0 term should be removed due to the charge neutrality of the system. The self-energy Σ DFT (k) is obtained from Φ DFT by
Σ DFT bb ′ (k) = δΦ DFT δn b ′ b (k) .(75a)
The condition δΩ DFT /δn b ′ b (k) = 0 for the thermodynamic equilibrium then yields
n(k) = 1 e [H(k)+Σ DFT (k)]/T + 1 ,(75b)
which is the Kohn-Sham equation 44 in disguise. Equations (75a) and (75b) should be solved self-consistently for a given E xc . The equivalence between eqs. (59) and (72) for the exact functional E xc can be established by an argument of using the coupling-constant integral. Suppose we change the interaction of eq. (44) as U → λU and express the corresponding thermodynamic potential as Ω λ . The two expressions (59) and (72) satisfy the same differential equation 52, 63 ∂Ω λ /∂λ = Ĥ λ int /λ and the same initial condition Ω λ=0 = Ω 0 , where Ĥ λ int is the thermodynamic average of the interaction in eq. (45) and Ω 0 denotes the non-interacting thermodynamic potential. It hence follows that eqs. (59) and (72) are equivalent.
It is also possible to expand Φ DFT and Σ DFT with respect to the explicit B dependences as eqs. (61) and (62), respectively. The expressions of Φ DFT(j) and Σ DFT(j) are given in Appendix C.
Susceptibility
We here study the susceptibility χ α ′ α of B → 0 based on eq. (59). We first establish a general procedure to calculate χ α ′ α for a given functional Φ. We then specialize to the Hartree-Fock approximation and the densityfunctional theory to derive explicit expressions of χ α ′ α within those approximations. Unlike the treatments by Buot 28 and Misra et al. 31 where they jumped directly to the expression χ α ′ α ≡ −∂ 2 Ω/∂B α ′ ∂B α , our consideration will proceed in two stages by first calculating the magnetization M α = −∂Ω/∂B α and then performing another differentiation with respect to B α ′ . This approach has an advantage that vertex corrections can be incorporated explicitly in the formula.
Exact expression
To derive a formally exact expression of χ α ′ α , let us start with classifying various B dependences of the thermodynamic potential Ω in Eq. (59) into three groups. The first category is that through κ in eq. (43). The second category is the explicit B dependences in H(k, B) and Σ(ε n , k, B, {G}). Those in Σ originate from the B dependence in the vertex Λ of eq. (49) and the operator ⊗ of eq. (23); see eq. (56), for example. The third category is the implicit B dependence through G, which is responsible for the Fermi-liquid corrections. 66,67 When calculating M ≡ −∂Ω/∂B, however, we need not consider this third category because of eq. (60).
The dependence through κ needs a special treatment due to the property (24b). As shown in Appendix D, it does not mix with the other categories for B → 0, yielding the Landau-Peierls diamagnetic susceptibility 1, 68 as
χ LP α ′ α = ǫ α ′ β ′ γ ′ ǫ αβγ 6 e 2 c 2 T n e iεn0+ k Tr G 2 (ε n , k) ×[∇ β ′ k ∇ β k G −1 (ε n , k)][∇ γ ′ k ∇ γ k G −1 (ε n , k)] . (76)
This expression agrees exactly with eq. (4.15) of Buot. 28 We next consider the second category. Differentiating eq. (59) with respect to the explicit B dependences in H(k, B), Σ(ε n , k, B, {G}), and Φ(B, {G}), we obtain the relevant magnetization as
M ′ α = −T n e iεn0+ k Tr G(ε n , k, B) ∂H(k, B) ∂B α − ∂Φ(B, {G}) ∂B α .(77)
We further differentiate eq. (77) with respect to B α ′ and put B = 0, where we also need to consider the implicit B dependence through G. Using eq. (58a), we obtain
χ ′ α ′ α = −T n e iεn0+ k Tr G ∂ 2 H (2) ∂B α ′ ∂B α − ∂Φ (2) ∂B α ′ ∂B α −T n e iεn0+ k Tr ∂H (1) ∂B α + ∂Σ (1) ∂B α dG dB α ′ ,(78)
where H (j) , Φ (j) , and Σ (j) (j = 1, 2) are quantities of the order B j with respect to the explicit B dependences as given by eqs. (34), (61), and (62), respectively. To find an expression for dG/dB α ′ in eq. (78), we differentiate G −1 G = 1 with respect to B α ′ . We thereby obtain a leading-order equation for dG/dB α ′ given in terms of
dG −1 /dB α ′ as dG dB α ′ = G ∂H (1) ∂B α ′ + ∂Σ (1) ∂B α ′ + dΣ (0) dB α ′ G ,(79)
with G = G(B = 0) on the right-hand side. The last term in the bracket of eq. (79) originates from the implicit B dependence though G, giving rise to the Fermiliquid (or vertex) corrections. 66,67 To find an equation for dΣ (0) /dB α ′ , we introduce a vertex function as
Γ (0) b1b ′ 1 ,b2b ′ 2 (1, 2) ≡ N c T 2 δ 2 Φ (0) δG b ′ 1 b1 (1)δG b ′ 2 b2 (2) ,(80)
with 1 ≡ (ε n1 , k 1 ). Noting eq. (58a), we then obtain an expression for dΣ (0) /dB α ′ in terms of dG/dB α ′ as
dΣ (0) b1b ′ 1 (1) dB α ′ = T N c b2b ′ 2 n2k2 Γ (0) b1b ′ 1 ,b2b ′ 2 (1, 2) dG b ′ 2 b2 (2) dB α ′ .
(81) Equations (79) and (81) form a closed set of equations to determine dG/dB α ′ . Introducing the matrix I by
I b ′ 1 b11,b ′ 2 b22 ≡ T N c b3b ′ 3 G b ′ 1 b3 (1)G b ′ 3 b1 (1)Γ (0) b3b ′ 3 ;b2b ′ 2 (1, 2) ,(82)
the coupled equations are solved formally as
dG b ′ 1 b1 (1) dB α ′ = b2b ′ 2 n2k2 (1 − I) −1 b ′ 1 b11,b ′ 2 b22 × G(2) ∂H (1) (k 2 ) ∂B α ′ + ∂Σ (1) (2) ∂B α ′ G(2) b ′ 2 b2 ,(83)with 1 ≡ (δ b ′ 1 b ′ 2 δ b1b2 δ k 1 k 2 δ n1n2χ α ′ α = χ LP α ′ α + χ ′ α ′ α ,(84)
with χ LP α ′ α given by eq. (76). The factor (1 − I) −1 in eq. (83) originates from vertex corrections for both the orbital and spin parts, which have been derived naturally in our treatment. It includes the Stoner enhancement factor as the intra-band contribution of the spin part.
Hartree-Fock approximation
We now specialize to the Hartree-Fock approximation. Due to the absence of ε n dependence in the self-energy Σ HF (k), every summation over n can be performed 63 by using the Fermi distribution function:
f (ε) ≡ 1 e ε/T + 1 .(85)
Thus, it is possible to simplify the expression of χ α ′ α further.
Let us adopt the representation of diagonalizing the Hartree-Fock energy in zero field as
H bb ′ (k, B = 0) + Σ HF bb ′ (k, B = 0) = δ bb ′ ξ bk .(86)
Then the Landau-Peierls susceptibility of eq. (76) is transformed into
χ LP−HF α ′ α = 1 6 e 2c 2 bk ∂f (ξ bk ) ∂ξ bk ǫ α ′ β ′ γ ′ ǫ αβγ m * β ′ β (bk) m * γ ′ γ (bk) , (87) with m * β ′ β (bk) ≡ 2 /(∇ β ′ k ∇ β k ξ bk )
. Thus, only the states near the Fermi energy are relevant to the Landau-Peierls susceptibility with the effective mass in place of the bare electron mass. However, this simple result no longer holds in those approximations where the self-energy has ε n dependence. Equation (87) may be regarded as an extension of the Philippas-McClure result for the electron gas 21 to Bloch electrons.
Next, eq. (78) can be written with respect to n(k) of eq. (64) as
χ ′HF α ′ α = − k Tr n(k) ∂ 2 H (2) (k) ∂B α ′ ∂B α − ∂Φ HF(2) ∂B α ′ ∂B α − k Tr ∂H (1) (k) ∂B α + ∂Σ HF(1) (k) ∂B α dn(k) dB α ′ .(88)
The quantity dn/dB α ′ is obtained from eq. (79) by adopting the representation (86) and carrying out the summation over n. We thereby arrive at the expression:
dn b ′ b (k) dB α ′ = f (ξ b ′ k )−f (ξ bk ) ξ b ′ k −ξ bk ∂H (1) b ′ b (k) ∂B α ′ + ∂Σ HF(1) b ′ b (k) ∂B α ′ + dΣ HF(0) b ′ b (k) dB α ′ .(89)
To find an expression of dΣ
HF(0) b ′ b
(k)/dB α ′ , we introduce a vertex function in terms of eq. (66) as
Γ HF(0) b1b ′ 1 ,b2b ′ 2 (k 1 , k 2 ) ≡ N c δ 2 Φ HF(0) δn b ′ 1 b1 (k 1 )δn b ′ 2 b2 (k 2 ) = N c V K U K Λ (0) b1b ′ 1 (k 1 , K)Λ (0) b2b ′ 2 (k 2 , −K) − U k 2 −k1+K ×Λ (0) b1b ′ 2 (k 2 , k 1 −k 2 −K)Λ (0) b2b ′ 1 (k 1 , k 2 −k 1 +K) . (90)
Noting eq. (67a), we then obtain
dΣ HF(0) b ′ b (k) dB α ′ = 1 N c b1b ′ 1 k1 Γ HF(0) b ′ b,b1b ′ 1 (k, k 1 ) dn b ′ 1 b1 (k 1 ) dB α ′ .
(91) Equations (89) and (91) form a closed set of equations to determine dn b ′ b (k)/dB α ′ . Defining the matrix:
I HF b ′ 1 b1k1,b ′ 2 b2k2 ≡ 1 N c f (ξ b ′ 1 k1 )−f (ξ b1k1 ) ξ b ′ 1 k1 −ξ b1k1 Γ HF(0) b ′ 1 b1,b2b ′ 2 (k 1 , k 2 ) ,(92)
the coupled equations are solved formally as
dn b ′ b (k) dB α ′ = b ′ 1 b1k1 (1 − I HF ) −1 b ′ bk,b ′ 1 b1k1 × f (ξ b ′ 1 k1 )−f (ξ b1k1 ) ξ b ′ 1 k1 −ξ b1k1 ∂H (1) b ′ 1 b1 (k 1 ) ∂B α ′ + ∂Σ HF(1) b ′ 1 b1 (k 1 ) ∂B α ′ , (93) with 1 ≡ (δ b ′ b ′ 1 δ bb1 δ kk1 )
. Equations (88) and (93) enables us to calculate χ ′HF α ′ α . The total susceptibility is then obtained by
χ HF α ′ α = χ LP−HF α ′ α + χ ′HF α ′ α ,(94)
with χ LP−HF α ′ α given by eq. (87).
Density functional theory
From a practical viewpoint, it is perhaps most useful at present to derive an expression of the magnetic susceptibility within the density functional theory. We hence consider it in most detail. Since the self-energy does not depend on the Matsubara frequency, however, we can exactly follow the procedure of the previous Hartree-Fock treatment.
Let us adopt the representation of diagonalizing the Kohn-Sham single-particle energy in zero field as
H (0) bb ′ (k) + Σ H(0) bb ′ (k) + Σ xc(0) bb ′ (k) = δ bb ′ ξ bk ,(95)
where
H (0) bb ′ , Σ H(0)
bb ′ , and Σ xc(0) bb ′ are given by eqs. (30a), (B·2), and (C·9), respectively, with n bb ′ (k) = δ bb ′ f (ξ bk ). The vertex function corresponding to eq. (90) is defined in terms of eq. (73) by
Γ DFT(0) b1b ′ 1 ,b2b ′ 2 (k 1 , k 2 ) ≡ N c δ 2 Φ DFT(0) δn b ′ 1 b1 (k 1 )δn b ′ 2 b2 (k 2 ) = N c V K =0 Λ (0) b1b ′ 1 (k 1 , K) 4πe 2 K 2 Λ (0) b2b ′ 2 (k 2 , −K) + 1 N c σ,σ ′ =± ρ σ(0) b1b ′ 1 (k 1 , r)F (2) σσ ′ (r)ρ σ ′ (0) b2b ′ 2 (k 2 , r) dr ,(96)
with ρ
σ(0) bb ′ and F(2)
σσ ′ given by eqs. (C·3a) and (C·7b), respectively. We also introduce a matrix:
I DFT b ′ 1 b1k1,b ′ 2 b2k2 ≡ 1 N c f (ξ b ′ 1 k1 )−f (ξ b1k1 ) ξ b ′ 1 k1 −ξ b1k1 Γ DFT(0) b ′ 1 b1,b2b ′ 2 (k 1 , k 2 ) ,(97)
with f the Fermi distribution function of eq. (85). Table I. Quantities necessary to calculate the magnetic susceptibility χ DFT α ′ α of the density functional theory given by eq. (103). Those with underlines are matrices with respect to the band-spin index b.
u bk (r) w bR (r) ξ bk v α (k) m * ββ ′ (bk) ǫ αβγ (4) (5) (95) (106a) (106b)(15)x α (k) π α (k) σ α (k) O (α) H DFT(α) Γ DFT(0) (20) (30c) (30d) (98) (99) (96) I DFT Λ (0) ρ ±(0)Λ (α) Kñ (α) σ (r) F (2) σσ ′ (r) (97) (50a) (C·3a) (101a) (101b) (C·7b)
It is also useful to define a couple of quantities in terms of eqs. (19a), (34a), (B·5), and (C·11) as
O (α) (k) ≡ ∂O (1) (k) ∂B α = iǫ αβγ e 2 c x β (k)x γ (k) ,(98)H DFT(α) (k) ≡ ∂ ∂B α H (1) (k)+Σ H(1) (k)+Σ xc(1) (k) = −ǫ αβγ e 2 c {x β , v γ + π γ } + σ βγ + N c V K =0 Λ (0) (k, K) 4πe 2 K 2Λ (α) −K + 1 N c σ,σ ′ =± ρ σ(0) (k, r) F (2) σσ ′ (r)ñ (α) σ ′ (r) dr . (99)
Here v α is the renormalized velocity:
v α bb ′ ≡ δ bb ′ 1 ∂ξ bk ∂k α ,(100)
and Λ (0) (k, K), ρ ±(0) (k, r), and F
σσ ′ (r) are given by eqs. (50a), (C·3a) and (C·7b), respectively. The quan-titiesΛ
(α) K andñ (α) ± (r) are defined bỹ Λ (α) K ≡ 1 N c bk f (ξ bk ) Λ ′(α) bb (k, K) −{O (α) (k), Λ (0) (k, K)} bb , (101a) n (α) ± (r) ≡ 1 N c bk f (ξ bk ) ρ ′±(α) bb (k, r) −{O (α) (k), ρ ±(0) (k, r)} bb ,(101b)
with
Λ ′(α) (k, K) ≡ iǫ αβγ e 2 c x β (k)Λ (0) (k, K)x γ (k) , (102a) ρ ′±(α) (k, r) ≡ iǫ αβγ e 2 c x β (k)ρ ±(0) (k, r)x γ (k) . (102b)
Note Λ ′(α) = ∂Λ ′(1) /∂B α and ρ ′±(α) = ∂ρ ′ ±(1) /∂B α with Λ ′(1) and ρ ′ ±(1) given by eqs. (A·4a) and (C·4a), respectively. With these preliminaries, the susceptibility χ DFT α ′ α can be calculated easily by eq. (94) with the replacement of the superscript HF→DFT. It may be written as a sum of three contributions as
χ DFT α ′ α = χ LP−DFT α ′ α + χ PvV−DFT α ′ α + χ (2)−DFT α ′ α .(103)
The first term denotes the Landau-Peierls diamagnetism corresponding to eq. (87), given explicitly by
χ LP−DFT α ′ α = 1 6 e 2c 2 bk ∂f (ξ bk ) ∂ξ bk ǫ α ′ β ′ γ ′ ǫ αβγ m * β ′ β (bk) m * γ ′ γ (bk) , (104a) with m * β ′ β (bk) ≡ 2 /(∇ β ′ k ∇ β k ξ bk ).
The second term in eq. (103) comes from the last term in eq. (88), i.e., the second-order perturbation with respect to H (1) +Σ DFT (1) . Its intra-and inter-band contributions yield the Pauli and van Vleck paramagnetism, respectively. It is calculated as
χ PvV−DFT α ′ α = − b1b ′ 1 k1 b2b ′ 2 k2 H DFT(α ′ ) b1b ′ 1 (k 1 ) ×[(1 − I DFT ) −1 ] b ′ 1 b1k1,b ′ 2 b2k2 f (ξ b ′ 2 k2 )−f (ξ b2k2 ) ξ b ′ 2 k2 −ξ b2k2 ×H DFT(α) b ′ 2 b2 (k 2 ) ,(104b)
where I DFT and H DFT(α) are given by eqs. (97) and (99), respectively, and [f (
ξ b ′ 2 k2 )−f (ξ b2k2 )]/(ξ b ′ 2 k2 −ξ b2k2 ) should be replaced by ∂f (ξ b2k2 )/∂ξ b2k2 for ξ b ′ 2 k2 = ξ b2k2
. Thus, the factor (1 − I DFT ) −1 from vertex corrections is explicitly included in our formula.
The third term in eq. (103) is due to the first two terms in eq. (88). It is calculated from eq. (34b), (B·6), and (C·10b) as
χ (2)−DFT α ′ α = ǫ α ′ β ′ γ ′ ǫ αβγ e 2 2c 2 bk ∂f (ξ bk ) ∂ξ bk {x β ′ , x β } bb v γ ′ b v γ b + 1 4 (x β π γ ′ x β ′ +x β ′ π γ ′ x β ) bb v γ b + 1 2 {x β , σ β ′ γ ′ } bb v γ b + 1 4 (x β ′ π γ x β +x β π γ x β ′ ) bb v γ ′ b + 1 2 {x β ′ , σ βγ } bb v γ ′ b − 1 2m f (ξ bk ){x β ′ , x β } bb δ γ ′ γ − m m * γ ′ γ (bk) + bk f (ξ bk ) {O (α) (k), H (α ′ ) (k)} bb +{O (α ′ ) (k), H (α) (k)} bb + 1 4 bb ′ k f (ξ bk )(ξ b ′ k −ξ bk )(O (α ′ ) bb ′ O (α) b ′ b + O (α) bb ′ O (α ′ ) b ′ b ) − N 2 c V K =0Λ (α ′ ) K 4πe 2 K 2Λ (α) −K −2 σ,σ ′ =± ñ (α) σ (r)F (2) σσ ′ (r)ñ (α ′ ) σ ′ (r) dr .(104c)
Equation (103) with eq. (104) is one of the main results of the paper. It extends the result of Roth 5 to take both the Coulomb and exchange-correlation effects into account with appropriate vertex corrections. Indeed, one can check that χ DFT reduces in the non-interacting limit to the expression obtained by Roth. 69 The formula enables us to perform non-empirical calculations of the magnetic susceptibility in solids based on the densityfunctional theory. Table I summarizes the quantities necessary for the calculation of χ DFT α ′ α . For practical purposes, it may be worth transforming eq. (100) and m * β ′ β (bk) ≡ 2 /(∇ β ′ k ∇ β k ξ bk ) in eq. (104a) into expressions without differentiations with respect to k. To this end, let us define a transfer integral in terms of eq. (95) ast
bR ≡ 1 N c k e ik·R ξ bk .(105)
Then v α b (k) can be calculated by either of the two expressions:
v α b (k) = 1 ∂ξ bk ∂k α = −i R e −ik·Rt bR R α . (106a) Also 1/m * β ′ β (bk) is obtained by 1 m * β ′ β (bk) = 1 2 ∂ 2 ξ bk ∂k β ∂k β ′ = − 1 2 R e −ik·Rt bR R β R β ′ .
(106b) Together with eq. (20), one may perform a calculation of χ DFT α ′ α without recourse to numerical differentiations in k space.
Three comments are in order before closing the section. First, our susceptibility is defined in terms of the mean flux density B by M α ′ = χ α ′ α B α , which is different from the conventional definition M α ′ = χ H α ′ α H α of using the external field H. However, χ H α ′ α is found easily from the thermodynamic relation
H = B−4πM/V . Indeed, χ H α ′ α
in the crystallographic coordinates is given by
χ H α ′ α = δ α ′ α χ αα 1 − 4πχ αα /V .(107)
However, the difference between χ αα and χ H αα is negligible for most of the non-magnetic materials. It is also worth noting that choosing B as an independent variable is more favorable when extending the theory to superconductors. 70,71 Secondly, an expression of the spin susceptibility within the density functional theory was already derived by Vosko and Perdew 24 incorporating vertex corrections, which has been a basis of the number of theoretical works on the spin susceptibility of metals. 25,26,29 Compared with it, however, the present formula has a couple of advantages that (i) the orbital contribution can be calculated on an equal footing with the spin part and (ii) vertex corrections are incorporated explicitly in the formula without any further approximations. Third, any calculations of the orbital susceptibility based on a model Hamiltonian, such as the Hubbard model, completely fail to incorporate the orbital contributions to χ PvV α ′ α and χ (2) α ′ α . This is because the key quantity x α (k), which originates from the change of the basis functions by the magnetic field, is necessarily set equal to zero in those calculations. Thus, we need to include the field effect on the basis functions for any practical calculations of the orbital magnetism.
De Haas-Van Alphen Oscillation
We finally study many-body effects on the dHvA oscillation in metals, limiting our consideration to clean systems without impurity scatterings. A definite advantage here over the previous studies [38][39][40][41] is that the structure of the thermodynamic potential is known explicitly as eq. (59).
Following the original work by Luttinger and Ward, 63 let us regard eq. (59) as a functional of Σ(ε n , k) instead of G(ε n , k). It is also stationary with respect to a variation in Σ satisfying Dyson's equation (58b). We then write Σ as a sum of the monotonic and oscillatory parts as
Σ = Σ 0 + Σ osc .(108)
As shown by Luttinger within the Hartree-Fock approximation 38 and discussed more generally by Bychkov and Go'kov, 39 the oscillatory part Σ osc for the spherical Fermi surface is smaller than the monotonic part Σ 0 by the order ( ω c /µ) 3/2 (ω c : the cyclotron frequency). We assume that the statement holds up to the infinite order, as expected for three-dimentional Fermi surfaces. Let us expand Ω(Σ) at Σ = Σ 0 as
Ω(Σ) = Ω(Σ 0 ) + 1 2 b1b ′ 1 b2b ′ 2 k1k2 δ 2 Ω δΣ b1b ′ 1 (k 1 )δΣ b2b ′ 2 (k 2 ) 0 ×Σ osc b1b ′ 1 (k 1 )Σ osc b2b ′ 2 (k 2 ) + · · · ,(109)
where the term linear in Σ osc vanishes due to Dyson's equation (58b) for Σ = Σ 0 and G = G 0 . The second term on the right-hand side of eq. (109) is of the order ( ω c /µ) 3 . On the other hand, the first term also has an oscillatory contribution originating from the operator κ in the logarithm of eq. (59); it is of the order ( ω c /µ) 5/2 relative to the nomotonic part of Ω, as seen from the result of non-interacting systems. 37 It hence follows that we may neglect the second contribution in eq. (109). The term relevant to the dHvA oscillation in Ω(Σ 0 ) is given by
Ω osc = −T n Tr Tr k ln H(κ, B)+Σ 0 (ε n , κ, B)−iε n 1 ×e iεn0+ = −Tr Tr k ∞ −∞ f (ε) ln H(κ, B)+Σ 0A (ε, κ, B)−ε1 − ln H(κ, B)+Σ 0A * (ε, κ, B)−ε1 dε 2πi ,(110)
where Σ 0A (ε, κ, B) ≡ Σ 0 (ε n → −iε − 0 + , κ, B), and we have used eq. (85) to transform the summation over n into an integration on the real energy axis. Since the oscillation is due to the states near the Fermi level, we only need to consider the region ε ∼ 0 in the integral where ImΣ 0A (ε) is infinitesimal positive definite and may be approximated by 0 + 1. The double trace in eq. (110) is thereby simplified into
Ω osc = − b Tr k ∞ −∞ f (ε) ln h b (ε, κ, B)−µ−ε + i0 + − ln h b (ε, κ, B)−µ−ε − i0 + dε 2πi ,(111)
where h b (ε, k, B) denotes a characteristic value of the Hermitian matrix H(k, B)+ReΣ 0A (ε, k, B)+µ1. Next, we diagonalize h b (ε, κ, B) by solving the eigenvalue problem:
h b (ε, κ, B)ϕ λ (ε, k, B) = g λ (ε, B)ϕ λ (ε, k, B) , (112a)
where ϕ λ is an eigenfunction and g λ is its eigenvalue. The subscript λ is composed of the band-spin index b, the Landau level N together with an additional quantum number distinguishing its degeneracy, and the wave vector k z parallel to the magnetic field B z. We then define the energy E λ = E λ (B) as the solution of the equation:
E λ = g λ (E λ −µ, B) .
(112b)
Note that the quantity 1− ∂g λ ∂ε ε=0
for the state E λ = µ is inverse of the discontinuity in the single-particle occupation at the Fermi level so that it is positive. 72 It hence follows that ε − g λ (ε) + µ is a monotonically increasing function for ε ∼ 0 and E λ ∼ µ. Equation (111) is thereby transformed into
Ω osc = − λ ∞ −∞ f (ε) θ(ε + µ − g λ (ε)) dε , = −T λ ln 1 + e −(E λ −µ)/T ,(113)
where θ is the step function. Equation (113) includes the main oscillatory part of the thermodynamic potential.
Since E λ ∼ µ for the oscillatory part of eq. (113), E λ may be calculated accurately by a two-step semiclassical quantization scheme as follows. We first determine the energy E bk = E bk (B) by solving
E bk = h b (E bk −µ, k, B) .(114)
We then adopt the Onsager-Lifshitz-Kosevich procedure: 36,37 S
(E λ ) = 2π(N +γ) |e|B c ,(115)
with N an integer, γ a constant of the order 1, and S(E λ ) denoting the area in k ′ space perpendicular to B specified by E bk ′ ≤ E λ and k ′ z = k z . Let us substitute those quantized energy levels into eq. (113) and follow the procedure of Lifshitz and Kosevich. 37 We thereby obtain a theory of the dHvA oscillation where the Fermi surface by E bk (B) replaces the non-interacting Fermi surface of Lifshitz and Kosevich. 37 There may be an alternative way to calculate E λ semiclassically. We first determine the eigenvalue g λ (ε, B) in eq. (112a) as a function of ε by
S(ε, g λ ) = 2π(N +γ) |e|B c ,(116)
where S(ε, g λ ) denotes the area in k ′ space perpendicular to B z specified by h b (ε, k ′ , B) ≤ g λ (ε, B) and k ′ z = k z . We then obtain the quasiparticle energy E λ by solving eq. (112b). Since the characteristic energy of κ is ω c , however, the Fermi-surface structures determined by this latter procedure will not differ substantially from those of the first one. It hence follows that we may adopt the first procedure which is certainly more convenient.
Three comments are in order before closing the section. First, the operator κ appears naturally in the argument of Σ 0 as well as in H of eq. (110). This fact shows unambiguously the necessity of considering the self-energy to make up the quantized energy levels. Equation (110) thus removes the confusion on the many-body effects of the dHvA oscillation mentioned in Introduction; it supports Luttinger's original argument, refuting the quantization procedure without the self-energy. 40,41 Second, the present theory can also treat changes of the energy band structure with B by incorporating the explicit B dependences in H and Σ. Indeed, H(k, B) can be calculated by eq. (34), and Σ(ε n , k, B) may be obtained by the procedure of §3.2 with dropping the oscillatory contribution of G(ε n , k, B). This effect was neglected by Luttinger 38 but can have a substantial importance, particularly when approaching the magnetic breakdown. 73 It should be noted that this magnetic breakdown is beyond the description of the semiclassical quantization procedure and we have to solve eq. (112) exactly by taking the relevant multiple bands into account. Third, we have used in eq. (111) a characteristic value h b (ε, k, B) and replaced k by the operator κ. This procedure is well defined for a simple band where h b (ε, k, B) is analytic in k. 9 For a complex band with band crossings, however, h b (ε, k, B) may not be analytic in k so that the use of h b (ε, κ, B) may fail to describe some important effects such as band mixings due to κ.
In this situation, one may be required to use a representation where H(k, B) + ReΣ 0A (ε, k, B) + µ1 is analytic in k and directly solve the eigenvalue problem for H(κ, B)+ReΣ 0A (ε, κ, B)+µ1.
Summary
We have constructed a many-body perturbation theory for Bloch electrons in a magnetic field on the basis of the energy band structure in zero field. We have thereby clarified the structures of the thermodynamic potential and the self-energy in a finite magnetic field, and provided a microscopic foundation for the replacement procedure on the self-energy: Σ (0) (ε n , k) → Σ ε n , −i∇− e c A . This perturbation theory is then applied to obtain explicit expressions of the magnetic susceptibility χ at various approximation levels on the interaction. The result for the density functional theory is given by eq. (103) together with eq. (104) and Table I. It incorporates vertex corrections as well as interband transitions and core polarizations. The expression enables us non-empirical calculations of χ. Thus, it will be useful to improve our quantitative understanding of the magnetic susceptibility in solids. Finally, we have presented a many-body theory on the dHvA oscillation in metals to show unambiguously that the Fermi surface structure with interaction effects in zero field are indeed relevant to the phenomenon.
The present formulation may be extended easily to a non-uniform magnetic field. Hence an application to superconductors will be fairly straightforward. It is also desired to make up a many-body theory on the transport phenomena of Bloch electrons in a magnetic field.
× u b ′ k ′ (r) dr .
NoteΛ ′ bb ′ (k, k ′ , q) ∝ δ −k+k ′ +q,K with K a reciprocal lattice vector. It hence follows that k inΛ ′ bb ′ (k, k ′ , q) can be written alternatively as k = k ′ +q with k ′ +q denoting the wave vector in the first Brillouin zone corresponding to k ′ +q in the extended zone scheme. Using this notation and noting e i(k ′ +q)·R = e i(k ′ +q)·R , eq. (A·1) is simplified into
U ′ ν3ν4;ν ′ 3 ν ′ 4 = e iI R 3 R ′ 3 +iI R 4 R ′ 4 N 2 c V kk ′ e ik·(R3−R ′ 3 )+ik ′ ·(R4−R ′ 4 ) × q e iq·(R3−R4) U q Λ ′ b3b ′ 3 (k, q, B) ×Λ ′ b4b ′ 4 (k ′ , −q, B) ,(A·2)
where Λ ′ bb ′ (k, q, B) is defined by Λ ′ bb ′ (k, q, B)
≡ e iK k+q ·r e ih αβ ∇ α
k ′ ∇ β k u † bk ′ (r) u b ′ k (r) k ′ =k+q dr = e ih αβ ∇ α k ′ ∇ β k u † bk ′ (r) e iK k+q ·r u b ′ k (r) k ′ =k+q dr ,(A·3a)
with K k+q ≡ k+q−k+q. The third line follows from the fact that the discrete reciprocal vector K k+q is independent of the infinitesimal changes in k or k+q. The wave vector k in Λ ′ (k, q, B) belongs to the incoming electron, q is an additional wave vector from the interaction, and k+q specifies the outgoing electron. Equation (A·3a) can be expanded in powers of B as Λ ′ bb ′ (k, q, B) = Λ
= k Tr n(k) h αβ h α ′ β ′ ∇ β k {x α , x α ′ }, ∇ β ′ k Σ H(0) − 1 2 {x α , x α ′ }, ∇ β k ∇ β ′ k Σ H(0) + i 4 ∇ β k ∇ β ′ k x α ′ , ∇ α k Σ H(0) +h αβ O (1) , x α , ∇ β k Σ H(0) + 1 8 O (1) , O (1) , Σ H(0) + i 2 h αβ ∇ β k O (1) , ∇ α k Σ H(0) + 1 2V
Kkk ′ U K Tr n(k)Λ (1) (k, K)Tr n(k ′ )Λ (1) (k ′ , −K) .
(B·6)
The terms with Σ H(0) (k) in the above expression have the effect of renormalizing the non-interacting energy H (0) and the velocity ṽ. Indeed, we can find the correspondents to them in eq. (34b). On the other hand, the last term cannot be expressed as the renormalization effect. The Fock part may be calculated similarly.
Appendix C: Φ DFT(j) and Σ DFT (1) We here derive expressions of Φ DFT(j) for j = 1, 2 and Σ DFT (1) . Among the two contributions in eq. (73), the Hartree part has already been treated in Appendix B. The quantities Φ H(1) and Φ H(2) are given by eqs. (B·1) and (B·6), respectively, and Σ H(1) is obtained as eq. (B·5) with U K = 4πe 2 /K 2 (K = 0).
As for the exchange-correlation part, we here consider the cases where E xc is given by 50 E xc [n σ (r), ∇n σ (r), △n σ (r)] .
(C·1a)
Let us express n σ (r) as eq. (69) and regard E xc as a functional of n(k) instead of n σ (r). We then expand E xc with respect to the explicit B dependences through ρ ± (k, r, B) as
E xc = E (0) xc + ∞ j=1 E (j) xc .(C·1b)
To obtain the expressions of E (j) xc , we expand ρ ± (k, r, B) of eq. (70) in powers of B as ρ ± (k, r, B) = ρ ±(0) (k, r) + The expansion coefficients are found easily as ρ ±(0) bb ′ (k, r) ≡ u † bk (r)
N c 2 (1 ± τ 3 ) u b ′ k (r) ,(C·3a)ρ ±(1) = ρ ′ ±(1) + 2 S (1) , ρ ±(0) ,(C·3b)
ρ ±(2) = ρ ′ ±(2) + 2 S (2) , ρ ±(0) + 2 S (1) , ρ ′ ±(1)
+ S (1) ρ ′ ±(0) S (1) − ih αβ ∇ β k S (1) , ∇ α k ρ ±(0) ,(C·3c)
with ρ ′ ±(j) obtained from eq. (71) as
ρ ′ ±(1) bb ′ (k, r, B) ≡ ih αβ [∇ α k u † bk (r)] N c 2 (1 ± τ 3 )∇ β k u b ′ k (r) , (C·4a) ρ ′ ±(2) bb ′ (k, r, B) ≡ − 1 2 h αβ h α ′ β ′ [∇ α k ∇ α ′ k u † bk (r)] N c 2 (1 ± τ 3 ) ×∇ β k ∇ β ′ k u b ′ k (r) .(C·4b)
It follows from eq. (C·1) with eqs. (69) and (C·2) that E
F (1) σ (r) ≡ 4 ν=0 (−1) pν D ν δE xc δ[D ν n σ (r)] ,(C·6)F (2) σσ ′ (r) ≡ νν ′ (−1) pν +p ν ′ D ν D ν ′ δ 2 E xc δ[D ν n σ (r)]δ[D ν ′ n σ ′ (r)] ,(C·7a)
with (D ν , p ν ) = (1, 1), (∇ x , −1), (∇ y , −1), (∇ z , −1), and (△, 1) for ν = 0, · · · , 4, respectively.
Noting F (1) σ (r + R) = F (1) σ (r), we can further simplify the first term in the square bracket of eq. (C·5) by using the completeness of { u b ′′ k (r)} over the unit cell. To be more specific, we expand σ F (1) σ (r) Nc 2 (1 +στ 3 ) u b ′ k (r) in σ F (1) σ (r)ρ σ(j) (k, r, B) as
σ F (1) σ (r) N c 2 (1+στ 3 ) u b ′ k (r) = b ′′ u b ′′ k (r)Σ xc(0) b ′′ b ′ (k) , (C·8) where Σ xc(0) bb ′ (k) ≡ δE (0) xc δn b ′ b (k) = 1 N c σ F (1) σ (r)ρ σ(0)
bb ′ (k, r) dr (C·9) denotes the exchange-correlation self-energy in zero field. With the same procedure as that of deriving eqs. (31a) and (34a), eq. (C·5) for j = 1 is then transformed into E (1) xc = −h αβ k Tr n(k) x α (k), ∇ β k Σ xc(0) (k) .
(C·10a) Equations (B·1) and (C·10a) have the effect of turningṽ in eq. (34a) into the renormalized velocity v.
The second-order term E
xc can be transformed similarly with the procedure of deriving eqs. (31b) and (34b).
− k (∇ β k ∇ β ′ k E ℓ )G 2 ℓ .
We thereby obtain a simple expression for eq. (D·3) as
S = ℓ ln E ℓ + 1 12 h αβ h α ′ β ′ (∇ α k ∇ α ′ k E ℓ )(∇ β k ∇ β ′ k E ℓ ) G 2 ℓ .
(D·9) The case ε n < 0 can be treated similarly. Indeed, we only need to change the range of integration in eq. (D·3) into (−∞, 0), replace 1 − by 1 + , and remove the minus signs in front of the integrals. We thereby arrive at the same expression as eq. (D·9). Differentiating the second contribution of eq. (D·9) twice with respect to B and going back to the original k representation, we arrive at eq. (76).
with
{A, B} ≡ 1 2 (A B + B A) and [ A, B ] ≡ A B − B A. Equation (31b) is in agreement with eq. (57) of Roth. 5
Fig. 2 .
2Some typical diagrams for the self-energy: (a) the Hartree and Fock terms; (b) the second-order exchange contribution.
Dyson's equation of eq. (53b) is transformed into [iε n 1−H(k, B)−Σ(ε n , k, B)]⊗G(ε n , k, B) = 1, as shown by using eqs. (32), (55), and (22). It may be written alternatively as k ′ δ kk ′ [iε n 1−H(κ, B)−Σ(ε n , κ, B)] G(ε n , k ′ , B) = 1 , (58b) where κ is defined by eq. (43). Hence the functional (52) is given in k space by Ω = −T n Tr Tr k ln{[ H(κ, B)+Σ(ε n , κ, B)−iε n 1 ]}
Ω
HF = −T Tr Tr k ln 1 + e −[H(κ)+Σ HF (κ)]/T − k Tr n(k) Σ HF (k) + Φ HF ,
(65) of the Hartree-Fock theory as Ω = −T Tr Tr k ln 1 + e −[H(κ)+Σ DFT (κ)]/T − k Tr n(k) Σ DFT (k) + Φ DFT .
( 0 )
0bb ′ (k, q) + ∞ j=1 Λ ′(j) bb ′ (k, q, B) ,
∞ j=1 ρ
j=1±(j) (k, r, B) . (C·2)
Tr n(k) ρ σ(j) (k, r, B) ,
extends the well-known rule E(k) → E(−i∂ R ) in a finite magnetic field 2, 36 to include the change of the energy-band structures in B.Next, Dyson's equation corresponding to eq. (38) is given by
). Substituting eq. (83) into eq. (78), one can check that the symmetry χ ′ α ′ α = χ ′αα ′
is satisfied as required.
Equation (78) with eqs. (80), (82), and (83) enables
us to calculate the relevant susceptibility χ ′
α ′ α once the
functional Φ is given explicitly. The total susceptibility
is then obtained by
AcknowledgementsThis work is supported by the 21st century COE program "Topological Science and Technology," Hokkaido University.Appendix A: Derivation of eq. (48)We here transform eq. (46) into eq.(48). To start with, let us consider eq. (47) and rewrite w ′ † ν3 (r) w ′ ν ′ 3 (r) in the integral with the procedure of eqs.(13)and(16)as(r ′ ) may be expressed similarly. Equation(47)is thereby transformed intoeqs. (31a) and (31b). To start with, let us rewriteor equivalently,where use has been made of the completeness 1 =over the unit cell. We then substitute the above identities into eq. (A·3a), expand it in powers of B to obtain Λ (j) (k, q), and average the resulting two different expressions. We thereby obtain the first-order term asNote that the arguments of O (1) and x α are k (k+q) when they appear to the right (left) of Λ (0) (k, q). Keeping this rule in mind and dropping the arguments of O (1) and x α , we can write eq. (A·4a) exactly as the O(1)and ṽ β contributions of eq. (31a). This rule also applies to higher-order terms. The second-order term is transformed in the same way into Λ ′(2) (k, q)Let us substitute eqs.(21)and (A·2) into eq. (46) and carry out the procedure of eq. (22) repeatedly. We thereby obtain eq. (48).Appendix B: Φ HF(j) and Σ HF(1)In this Appendix we derive explicit expressions for Φ HF(j) (j = 1, 2) and Σ HF(1). First of all, the Hartree contribution to Φ HF(1) is calculated from eq. (66) aswhere we have used eq. (50b) to obtain the second expression with Σ H(0) (k) given by(B·2) On the other hand, the Fock contribution has extra terms from the operator ⊗ asThe last expression has been obtained through a calculation of using the symmetry h αβ = −h βα and partial integrations with The corresponding self-energy is obtained easily by Σ HF(1) (k) = δΦ HF(1) /δn(k). The Hartree part is calculated from eq. (B·1) asThus, the Hartree self-energy already has a term which can not be expressed in terms of ∇ γ k Σ H(0) (k). The Fock self-energy may be obtained similarly from eq. (B·3). It also has extra terms besides the one with ∇ γ k Σ F(0) (k). The second-order functional Φ HF(2) may be calculated similarly. The Hartree part is obtained from eqs.(50)We thereby obtain an expression of E(2)xc aswhich is analogous with eq. (B·6). The terms with Σ xc(0) (k) in the above expression have the effect of renormalizing the non-interacting energy H (0) and velocity ṽ. Indeed, we can find the correspondents in eq. (34b). On the other hand, the last term cannot be expressed as the renormalization effect. The first-order self-energy is obtained from ΣThus, Σ xc(1) (k) also has a term which can not be expressed in terms of ∇ γ k Σ xc(0) (k). Appendix D: Derivation of eq. (76)We here derive eq. (76) valid for B → 0 by expanding the logarithmic term in eq. (59) with respect to the B dependence in κ up to the second order. Our method is a slight modification of the one developed by Sondheimer and Wilson 74 and refined byRoth. 5It enables us to treat the correlation effects exactly, as seen below.Let us define the matrix H(ε n , k) byIt is connected with the Green's function as H = −G −1 and appears in the logarithm of eq. (59) as H(ε n , κ). Let us denote the eigenvalues of H(ε n , k) and H(ε n , κ) as E ℓ and E λ , respectively. Specifically, ℓ is given by ℓ = bk, and λ is composed of the band-spin index b, the Landau level N accompanied by an additional quantum number distinguishing its degenerate states, and the wave vector k z perpendicular to the magnetic field. Both E ℓ ≡ E ℓ (ε n ) and E λ ≡ E λ (ε n ) are complex in general and can be written asfor example, where E ′ λ and E ′′ λ are some real numbers. We will proceed by assuming that E ′′ λ has the same sign as ε n , i.e., E ′′ λ (ε n ) = |E ′′ λ (ε n )|sgn(ε n ), which is expected from the analytic properties of Green's function.Let us concentrate on the case ε n > 0 where E ′′ λ > 0. Then the trace of the logarithmic term in eq. (59) is transformed asThe final expression enables us to expand e −iH(κ)t in powers of B in κ. To this end, we adopt the procedure of the perturbation expansion in the field theory and define U (k, t) throughIt can be written explicitly aswhere T is the time-ordering operator andWe perform the expansion of eq. (D·5) up to the second order in B. As noted by Roth, 5 the term H ′(1) of eq. (D·6) does not contribute since it necessarily yields h αβ (∇ α k H)(∇ β k H) = 0. Thus, the B dependence through κ yields no terms first-order in B. It hence follows that this dependence of the first category can be treated independently from the others in obtaining the expression for the zero-field susceptibility. Thus, the expansion of U (k, t) up to the second order is given byLet us substitute eqs. (D·4) and (D·7) into eq. (D·3). We then carry out a similarity transformation in the trace to express S in terms of E ℓ . We also use the identity (p = 0, 1, 2, · · · ):where G ℓ ≡ −E −1 ℓ is Green's function. We finally perform an integration by parts to the contribution from the second term in the square bracket of eq. (D·7) as
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| []
|
[
"THÉORIE D'IWASAWA POUR LES FORMES MODULAIRES DE POIDS 1",
"THÉORIE D'IWASAWA POUR LES FORMES MODULAIRES DE POIDS 1"
]
| [
"Alexandre Maksoud "
]
| []
| []
| La conjecture principale pour les formes modulaires ordinaires de poids supérieur ou égal à 2 prédit que l'idéal caractéristique du groupe de Selmer est engendré par une fonction L p-adique analytique. Nous étudions les propriétés du groupe de Selmer des formes modulaires de poids 1 sous des hypothèses générales sur la représentation galoisienne associée. Nous formulons une conjecture principale dont nous démontrons une divisibilité grâce au théorème des systèmes d'Euler de Kato. | null | [
"https://arxiv.org/pdf/1811.05368v1.pdf"
]
| 119,641,446 | 1811.05368 | a99fa28a47fe9e96cad1d871686171f66c304b05 |
THÉORIE D'IWASAWA POUR LES FORMES MODULAIRES DE POIDS 1
13 Nov 2018
Alexandre Maksoud
THÉORIE D'IWASAWA POUR LES FORMES MODULAIRES DE POIDS 1
13 Nov 2018
La conjecture principale pour les formes modulaires ordinaires de poids supérieur ou égal à 2 prédit que l'idéal caractéristique du groupe de Selmer est engendré par une fonction L p-adique analytique. Nous étudions les propriétés du groupe de Selmer des formes modulaires de poids 1 sous des hypothèses générales sur la représentation galoisienne associée. Nous formulons une conjecture principale dont nous démontrons une divisibilité grâce au théorème des systèmes d'Euler de Kato.
Soit f (z) = n≥1 a n q n une forme modulaire parabolique primitive de poids 1, de niveau Γ 1 (N) et Nebentypus ǫ. Soit ρ : Gal(Q/Q) −→ GL 2 (C) la représentation continue et irréductible attachée à f par Deligne et Serre ( [DS74]). Elle se factorise par le groupe de Galois G d'un corps de nombres H ⊂ Q de degré d et est réalisable sur le corps de nombres Q[ρ] engendré par les valeurs prises par Tr ρ. Fixons des plongements ι ∞ : Q → C et ι ℓ : Q → Q ℓ pour tout nombre premier ℓ.
Soit p un nombre premier impair. On exclura dans la suite un nombre fini de valeurs de p avec l'hypothèse :
(hyp) :
p ∤ N d
Via le plongement ι p on peut voir ρ comme une représentation p-adique à coefficients dans l'anneau des entiers O d'une extension non-ramifiée suffisamment grande de Q p . On note α et β les racines du p-ième polynôme de Hecke X 2 − a p X +ǫ(p) de f , ainsi que
f α (z) = f (z)−β f (pz) et f β (z) = f (z) − α f (pz) les p-stabilisations de f .
Comme la forme modulaire f α est p-ordinaire, on peut suivant [Gre91] lui attacher pour tout K ⊆ Q un groupe de Selmer Sel( f α , K ), de façon analogue à celui des formes modulaires de poids k ≥ 2. Nous nous intéressons tout particulièrement au cas où K est égal à Q n , le n-ième étage L'ingrédient clef de la preuve de ce théorème est un résultat de [BD16], dont la preuve repose sur le théorème de Baker-Brumer sur l'indépendance linéaire des logarithmes p-adiques de nombres algébriques.
Nous étudions ensuite les propriétés du groupe de Selmer dual Notons L alg p ( f α , T) ∈ Λ un générateur de l'idéal caractéristique de X ∞ ( f α ). Alors :
(i) pour tout ζ ∈ µ p ∞ avec ζ = 1, on a L alg p ( f α , ζ − 1) = 0. Une grande partie de la preuve du Théorème B repose sur le calcul du O-rang du module des co-invariants X ∞ ( f α ) Gal(Q ∞ /Q n ) . On montre que les "flèches de contrôle"
Sel n ( f α ) −→ Sel ∞ ( f α ) Gal(Q ∞ /Q n ) .
sont des isomorphismes pour tout entier naturel n lorsque α = 1, et sont injectives et de conoyau de co-rang égal à 1 pour tout n lorsque α = 1. La formule du terme constant de la série L alg p ( f α , T) nécessite en plus de calculer explicitement l'ordre de Sel 0 ( f α ). Enfin, la non-existence de sous-Λ-modules finis non-nuls de X ∞ ( f α ) résulte du Théorème principal de [Gre16]. Ce résultat est présenté dans la proposition II.3.1 sous une forme moins générale et utilise le fait que X ∞ ( f α ) est de torsion, mais est plus adapté à nos applications.
Afin de formuler une conjecture principale pour f α , nous avons besoin de l'hypothèse de régularité suivante de f en p :
(reg) : α = β Sous les hypothèses (hyp) et (reg), la représentation résiduelleρ de ρ est irréductible et pdistinguée, et d'après [Wil95], la composante locale Hρ de l'algèbre de Hecke ordinaire universelle de niveau modéré N est Gorenstein. On peut alors, selon [EPW06], construire une fonction L p-adique analytique L an p (ρ, T) ∈ Hρ[[T]] uniquement déterminée à H × ρ -près par certaines propriétés d'interpolation. D'autre part, d'après [Wil88], à la forme f α est attaché un morphisme de La conjecture A est un analogue de la conjecture principale d'Iwasawa-Greenberg formulée pour les formes de poids supérieur ou égal à 2, étudiée entre autres dans [Kat04], [SU14] et prédisant en particulier que pour toute forme modulaire classique g congrue à f , il existe une unité u g de Z p [[T]] telle que (⋆) L alg p (g, T) · u g = L an p (g, T)
Notre dernier résultat est le suivant. Nous donnons à présent une esquisse de la preuve du Théorème C. L'anneau H F est une algèbre finie et plate sur Z p [[X ]], où X est une variable formelle paramétrant le poids. On normalise le poids de sorte que f α soit de poids X = 0, et plus généralement, une spécialisation g de F de poids k ≥ 2 ait un poids X = ζ(1 + p) k−1 − 1, où ζ ∈ µ p ∞ . Soit (X ) ⊆ p ⊆ H F l'idéal premier de hauteur 1 correspondant à la spécialisation sur f α . D'après le résultat principal de [BD16], ( On peut alors produire une suite de formes modulaires primitives g n p-stabilisées de niveau N p dont les coefficients du q-développement convergent p-adiquement vers ceux de f α . Une fois cette construction en main, on aura schématiquement :
L alg p (g n , T) (c) n→+∞ (a) divise L an p (g n , T) (b) n→+∞ L alg p ( f α , T) divise (d) L an p ( f α , T)
Les limites sont calculées dans Λ et les divisibilités sont dans Λ[ 1 p ]. La preuve du (a) est une application d'un célèbre théorème de Kato donnant une divisibilité de la conjecture principale pour les formes modulaires qui sont des p-stabilisations ordinaires de formes de poids k ≥ 2 et de niveau N. possède une fonction L p-adique L alg p (X , T). Pour montrer qu'elle satisfait les propriétés d'interpolation évidentes, on doit d'abord montrer que les groupes de Selmer attachés à f α et aux g n sont isomorphes aux changements de base de X ∞ (F † ). Puis on montre que X ∞ (F † ) ne possède pas de sous-modules pseudo-nuls non-triviaux, de manière similaire à la preuve du Théorème B. Notons que cette dernière propriété nécessite que l'on travaille sur des anneaux intégralement clos, ce qui n'est pas nécessairement le cas de H F . Un résultat standard implique alors que la formation de la fonction L p-adique algébrique commute avec les changements de base, et la preuve du point (c) s'en déduit facilement.
Enfin, la preuve du point (d) repose sur les trois points précédents et sur le Lemme VII.1.1 justifiant un passage à la limite dans les divisibilités. Précisons qu'une hypothèse essentielle pour que ce lemme s'applique est que l'élément L alg p ( f α , T) soit non-nul, ce qui est contenu dans l'énoncé du théorème B. Remarquons que nos méthodes permettent de démontrer que la conjecture (⋆) implique la conjecture A.
L-représentation irréductible (π, V π ) de G l'idempotent e π = dimπ d g∈G Tr •π(g −1 )g ∈ L[G]M ∨ ≃ Hom O (M, O ⊗ Z p Q p /Z p ). Si M est de plus muni d'une structure de O[G]-module (à gauche), alors M ∨ aussi via la formule (g. f )(m) = f (g −1 .m) pour f ∈ M ∨ et m ∈ M. Pour toute représentation irréductible π de G, on vérifie aisément que (e π M) ∨ ≃ e π ∨ M ∨ .
où π ∨ désigne la représentation contragrédiente de π.
Caractères cyclotomiques et algèbres d'Iwasawa. Soitχ cyc : G Q −→ Z × p le caractère cyclotomique et soit ω : G Q −→ µ p−1 le caractère de Teichmuller.χ cyc induit un isomorphisme entreΓ = Gal(Q(µ p ∞ )/Q) et Z × p , et ω un isomorphisme entre Gal(Q(µ p )/Q) et µ p−1 . Le caractère χ cyc := χ cyc ω −1 se factorise via le groupe de Galois Γ de la Z p -extension extension cyclotomique, notée Q ∞ /Q, et réalise un isomorphisme Γ ≃ 1 + pZ p . On notera γ ∈ Γ l'image inverse du générateur 1+ p de 1+ pZ p . On a ainsi Γ ≃ γ Z p . L'algèbre d'Iwasawa est l'anneau de groupe complété Z p [[Γ]].
Elle est isomorphe à l'anneau des séries formelles en une variable Z p [[T]] après identification de γ avec 1 + T.
On note Q n le n-ème étage de la tour cyclotomique, c'est-à-dire le corps fixé par Γ p n , et on note Γ n = Gal(Q n /Q) ≃ Z/p n Z. Si ζ ∈ µ p ∞ (Q p ) est une racine de l'unité d'ordre p n , on définit (1) Pour tout p ∈ P A , le localisé A p est un anneau de valuation discrète.
χ ζ : G Q −→ Q × p le
(2) On a A = p∈P A A p
(3) Pour tout x = 0 dans A, il n'existe qu'un nombre fini d'idéaux p ∈ P A tels que x ∈ p.
On notera ord p la valuation essentielle attachée à p. Plus généralement, si N est un A-module de type fini et de torsion, on notera ord p (N) la longueur (finie) du A p -module localisé N p de N. II.1.2. Définition. Soit n ∈ N ∪ {∞}. On appelle groupe de Selmer de niveau n attaché à (T, T + ) le noyau de la flèche de restriction globale-locale :
0 / / C / / M / / M ′ / / 0 Comme A/p m i i [p] = 0 pour tout 1 ≤ i ≤ d, on a M ′ [p] = 0, d'0 / / T + / / T + / / T − / / 0 On considère aussi les A-modules co-libres D = T ⊗ A A ∨ et D ± = T ± ⊗ A A ∨ . Pour n ∈ N ∪ {∞}, on définit D n une déformation cyclotomique de D comme étant l'induite Ind G Q G Qn D, c'est-à-dire le A-module des fonctions localement constantes f : G Q −→ D telles que ∀g ∈ G Q , ∀h ∈ G Q n , f (hg) = h f (g). Il est muni de l'action de G Q donnée par la formule (γ. f )(g) = f (gγ). Il est canoniquement isomorphe au A-module D ⊗ A A[Γ n ] si n est fini (resp. à D ⊗ A A[[Γ]] si n = ∞) muni de l'action de G QSel n (T, T + ) := ker H 1 (Q Σ /Q, D n ) −→ H 1 (I p , D − n ) × ℓ∈Σ,ℓ = p H 1 (I ℓ , D n )
Le groupe de Selmer dual est défini comme étant
X n (T, T + ) = Sel n (T, T + ) ∨
Ce sont des modules sur l'anneau de groupe A[Γ n ] si n est fini (resp. sur l'anneau de groupe complété A[[Γ]] si n = ∞).
D'après les propriétés de base de la cohomologie des modules discrets, on a Sel ∞ (T,
T + ) = lim − − →n Sel n (T, T + ). Par ailleurs, H 1 (Q Σ /Q, D n ) ∨ étant de type fini sur A[Γ n ] pour n fini (resp. sur A[[Γ]] pour n = ∞), cela est encore vrai pour X n (T, T + ).
II.1.3. Lemme de Shapiro. Soit D un A-module discret sur lequel un groupe profini G agit continûment, et soit H un sous-groupe ouvert normal de G. D'après le lemme de Shapiro, on a un isomorphisme canonique
H i (H, D) ≃ H i (G, Ind G H D) (II.1.3.1) II.1.4. Lemme. L'isomorphisme II.1.3.1 induit un isomorphisme : Sel n (T, T + ) = ker H 1 (Q Σ /Q n , D) −→ H 1 (I p (Q Σ /Q n ), D − ) × ℓ∈Σ,ℓ = p H 1 (I ℓ , D)
Démonstration. D'après le lemme de Shapiro pour G = G Q et H = G Q n , on a ainsi H 1 (Q n , D) ≃ H 1 (Q, D n ). Par ailleurs, on peut vérifier qu'un cocycle de H 1 (Q, D n ) est non-ramifié en ℓ si et seulement si son image dans H 1 (Q n , D) est non-ramifiée en au moins une place λ de Q n audessus de ℓ (auquel cas il sera non-ramifié en toute place au-dessus de ℓ). Enfin, la condition locale en p est aussi conservée sous cet isomorphisme. On a en effet I p (Q n /Q) = Γ n , et d'après
le lemme de Shapiro on a un isomorphisme H 1 (I p (Q/Q n ), D − ) ≃ H 1 (I p (Q/Q), D − n ) compatible à l'isomorphisme précédent. Cela prouve que Sel n (T, T + ) ≃ Sel 0 (T n , T + n ) pour tout entier naturel n. Comme D ∞ = lim − − →n
D n , le cas n = ∞ s'obtient par passage à la limite injective, qui est exact et qui commute avec la formation des groupes de cohomologie de modules discrets.
II.1.5. Remarque. Soit λ une place de Q ∞ au-dessus d'un nombre premier ℓ = p. Le groupe de Galois Gal(Q nr ℓ /Q ∞,λ ) = G Q ∞,λ /I ℓ est d'ordre premier à p, donc la suite exacte d'inflation- restriction montre que l'application de restriction H 1 (Q ∞,λ , D) −→ H 1 (I ℓ , D)
est injective. On a donc la description alternative suivante de Sel ∞ (T, T + ) :
Sel ∞ (T, T + ) ≃ ker H 1 (Q Σ /Q ∞ , D) −→ H 1 (I p (Q Σ /Q ∞ ), D − ) × λ|ℓ∈Σ,ℓ = p H 1 (Q ∞,λ , D)
Le lemme de Shapiro montre alors que l'on a aussi :
Sel ∞ (T, T + ) = ker H 1 (Q Σ /Q, D ∞ ) −→ H 1 (I p , D − ∞ ) × ℓ∈Σ,ℓ = p H 1 (Q ℓ , D ∞ )Sel ∞ (T/a, T + /a) −→ Sel ∞ (T, T + )[a].
Il s'agit d'un isomorphisme sous certaines conditions générales, comme le précise la proposition suivante, inspirée de [SU14, Proposition 3.7].
II.2.1. Proposition. Soit n ∈ N ∪ {∞}. Supposons que l'idéal a est principal, que I p (Q/Q n ) agit trivialement sur D − , et que les A-modules D G Qn et D I ℓ sont divisibles pour tout l ∈ Σ.
Alors on a Sel n (T/a,
T + /a) ≃ Sel n (T, T + )[a], et donc par dualité X n (T, T + )/aX n (T, T + ) ≃ X n (T/a, T + /a).
Démonstration. Soit x ∈ A un générateur de a. D'après le lemme II.1.3.1, on a
Sel n (T, T + ) = ker H 1 (Q Σ /Q n , D) −→ H 1 (I p (Q Σ /Q n ), D − ) × ℓ∈Σ,ℓ = p H 1 (I ℓ , D)
On montre d'abord que l'application naturelle τ :
H 1 (Q Σ /Q n , D[a]) −→ H 1 (Q Σ /Q n , D)[a] est un isomorphisme.
Comme a est engendré par x et que D est A-divisible, on a une suite exacte courte induite par la multiplication par x :
0 / / D[a] / / D / / D / / 0
Celle-ci induit une suite exacte longue associée en cohomologie :
D G Qn / / D G Qn / / H 1 (Q Σ /Q n , D[a]) τ / / H 1 (Q Σ /Q n , D) / / H 1 (Q Σ /Q n , D)
où la première et la dernière flèche sont induites par la multiplication par x. Par hypothèse la première flèche est surjective, et par définition le noyau de la dernière flèche est égal à
H 1 (Q Σ /Q n , D)[a]
. Donc τ est un isomorphisme.
Pour montrer que τ envoie Sel n (T/a, T + /a) sur Sel n (T, T + )[a], il suffit de prouver que les applica-
tions H 1 (I p (Q Σ /Q n ), D − [a]) −→ H 1 (I p (Q Σ /Q n ), D − )[a]
et, pour ℓ ∈ Σ,
H 1 (I ℓ , D[a]) −→ H 1 (I ℓ , D)[a])T * = Hom Z p (D, µ p ∞ ) et son dual D * = T * ⊗ A A ∨ .
On note encore les modules induits avec un indice. Le résultat principal de [Gre16] donne des conditions suffisantes pour qu'un groupe de Selmer dual, défini de manière générale, n'admettent pas de sous-modules pseudo-nuls différents de 0. Nous simplifions le critère dans le but de l'appliquer dans la suite aux groupes de Selmer attachés à une forme modulaire ordinaire ou à une famille de Hida.
II.3.1. Proposition. Supposons que :
(a) X ∞ (T, T + ) est de torsion sur A[[Γ]]. (b) H 2 (Q Σ /Q ∞ , D) ∨ est de torsion sur A[[Γ]]. (c) Rang A T Gal(C/R) = d + (d) L'inertie en p de Q ∞ agit trivialement sur D − . (e) En tant que F[G Q ]-représentation, D[m A ] n'a pas de quotient isomorphe à F ou à µ p .
Alors X ∞ (T, T + ) n'a pas de sous-modules pseudo-nuls non-triviaux.
L p = ker H 1 (Q p , D ∞ ) −→ H 1 (I p , D − ∞La condition RFX(D ∞ ) est vérifiée car D ∞ est réflexif, et LEO(D ∞ ) l'est aussi d'après notre hypothèse (b). Montrons que l'on a LOC (1) ℓ (D ∞ ) pour tout ℓ, c'est-à-dire que (T * ∞ ) G Q ℓ = 0. Il suffit de prouver que le rang sur A[[Γ]] de (T * ∞ ) G Q ℓ est nul. D'après [Gre06, Proposition 3.10], celui-ci est égal au corang de D * ∞ G Q ℓ = H 0 (Q ℓ , D * ∞ ). D'après le lemme de Shapiro, on a H 0 (Q ℓ , D * ∞ ) = λ|ℓ H 0 (Q ∞,λ , D *L p = ker H 1 (Q p , D ∞ ) a / / H 1 (Q p , D − ∞ ) b / / H 1 (I p , D − ∞ ) L'0 / / ker a / / L p / / ker b / / 0
Il suffit de montrer que (ker a) ∨ et (ker b) ∨ n'ont pas de sous-modules pseudo-nuls non-triviaux.
Comme ker a est un quotient de H 1 (Q p , D + ∞ ), il suffit de montrer la même propriété pour
H 1 (Q p , D + ∞ ) ∨ .H 2 (Q p , D + ∞ ) = 0. Décrivons maintenant ker b. D'après la suite exacte d'inflation-restriction, on a ker b ≃ H 1 (Ẑ, H 0 (I p , D − ∞ )) oùẐ est topologiquement engendré par Frob p . D'après le lemme de Shapiro, on a H 0 (I p , D − ∞ ) = H 0 (I p (Q/Q ∞ ), D − ) qui est égal à D − d'après notre hypothèse (d). On a donc ker b ≃ D − /(Frob p −1)D − et ainsi (ker b) ∨ est un sous-module de (D − ) ∨ . En tant que modules sur l'anneau A[[Γ]] ≃ A[[T]], on a (D − ) ∨ ≃ A d − = (A[[T]]/(T)) d − .Corang(H 1 (Q Σ /Q, D ∞ )) = Corang(Sel ∞ (T, T + )) + ℓ∈Σ−{∞} Corang(Q ℓ (T, T + )) où Q ℓ (T, T + ) = H 1 (Q ℓ , D ∞ )/L ℓ pour tout premier l ∈ Σ.(Q Σ /Q, D ∞ )) = Corang(D G Q ) + Corang(H 2 (Q Σ /Q ∞ , D)) + d − Corang(D Gal(C/R) ∞ )
D'après les hypothèses (b) et (e), les deux premiers termes de la somme sont nuls. D'autre part, on a Corang(D Gal(C/R)
∞ ) = Corang A (D Gal(C/R) ) = Rang A T Gal(C/R) = d + .
Cela montre que le corang de H 1 (Q Σ /Q, D ∞ ) est bien égal à d − . Calculons maintenant le terme de droite de l'égalité de CRK(D∞, L). On va montrer que le corang de Q ℓ (T, T + ) est 0 si l = p et qu'il est égal à d − pour ℓ = p. Cela terminera la vérification de CRK(D ∞ , L) vu que, d'après l'hypothèse (a), le
A[[Γ]]-module Sel ∞ (T, T + ) a un corang égal à 0. Si ℓ = p, on a Q ℓ (T, T + ) = H 1 (Q ℓ , D ∞ ). Son corang est égal à la somme des corangs de H 0 (Q ℓ , D ∞ ) et de H 0 (Q ℓ , T * ∞ ) ∨ d'Tr ρ f (Frob ℓ ) = a ℓ , et detρ f (Frob ℓ ) = ǫ(ℓ)
où Frob ℓ ∈ G désigne le Frobenius en ℓ. Comme ρ est impaire, son indice de Schur est égal à 1 et elle est réalisable sur le corps de nombres Q[ρ] engendré par les Tr(ρ(g)) quand g parcourt G. On a de plus Q[ρ] ⊆ Q(µ d ).
On suppose dans toute la suite que f vérifie l'hypothèse (hyp), c'est-à-dire p ∤ N d. On note v la place de Q au-dessus de p définie par ι p . Le complété H v de H est une extension non-ramifiée de Q p de degré f v et de groupe de Galois G v . Soit L ⊆ Q p une extension finie et non-ramifiée On notera encore ρ la représentation p-adique obtenue via ι p :
de Q contenant H v et Q p (µ d )ρ : G → GL 2 (O) On note simplement V = V ρ et T = T ρ . Posons D := V /T = T ⊗ Z p Q p /Z p = T ⊗ O L/O. C'est un
III.4. Groupes de Selmer auxiliaires.
III.4.1. Notation. Pour un entier naturel n et une place w de H n au-dessus de p, on note :
• U n,w = x ∈ O × H n,w / x − 1 ∉ O ×
H n,w le Z p -module des unités locales principales de H n,w .
• U n := w|p U n,w le produit des unités locales de H n au-dessus de p.
• E n := O × H n ⊗ Z Z p le complété p-adique des unités globales de H n .
• M ρ := e ρ (M ⊗ Z p O) la partie ρ-isotypique d'un Z p [G]-module M.
On enlèvera l'indice n lorsque n = 0. Notons que les suites (U n,w ) n , (U n ) n et (E n ) n forment des systèmes projectifs dont les flèches de transition sont les applications de norme.
III.4.2. Remarque.
• Les Z p -modules U n et E n sont libres. En effet, les complétés padiques du corps H n ne contiennent pas de racines p-ièmes de l'unité, car l'extension H/Q est non-ramifiée en p.
• On a la description alternative suivante pour U n . Soit (O H n ⊗ Z Z p ) × lib le Z p -module obtenu en tuant la torsion (d'ordre nécessairement premier à p) du groupe (O H n ⊗ Z Z p ) × . Les plongements de H n dans ses complétés p-adiques induisent un isomorphisme naturel de Z p [G n ]-modules
U n ≃ (O H n ⊗ Z Z p ) × lib .jectent respectivement dans Q p E ρ n := E ρ n ⊗ O Q p et dans Q p U ρ n := U ρ n ⊗ O Q p . Montrons que le mor- phisme de Q p [G n ]-modules Q p E ρ n −→ Q p U ρ n est injectif.
Comme G n est isomorphe au produit direct G × Γ n , toute représentation irréductible (sur Q p ) de G n est égale à un produit tensoriel π ⊗ χ où π est une représentation irréductible de G et χ un caractère de Γ n . En notant M ρ⊗χ la composante ρ ⊗χ-isotypique d'un Q p [G n ]-module M, on est ramené à montrer l'injectivité de la flèche (Q p E n ) ρ⊗χ −→ (Q p U n ) ρ⊗χ pour tout χ.
On peut décrire la structure des Q p [G n ]-modules Q p U n et Q p E n comme suit. Notons V π le Q p -espace vectoriel réalisant une représentation π de G n . Grâce aux logarithmes p-adiques on peut montrer que Q p U n = π V dimV π π où la somme est prise sur l'ensemble des représentations irréductibles de G n . D'autre part, la preuve de Minkowski du théorème des unités de Dirichlet montre que
Q p E n = π =1 V dimV + π π
où V ± π désigne le sous-espace propre de la conjugaison complexe associé à la valeur propre ±1.
Soit m π la multiplicité de π apparaissant dans coker[Q p E n −→ Q p U n ] = Q p X n . Le théorème de Baker-Brumer permet de montrer (comme dans la preuve de [EKW84, Theorem 1]) que l'on a m π < dimV π dès que π = 1 et V + π = 0. On peut l'appliquer à π = ρ ⊗ χ qui vérifie dimV ρ⊗χ = 2 et dimV + ρ⊗χ = 1. On obtient m ρ⊗χ ≤ 1, et finalement un comptage de dimensions donne
dimker (Q p E n ) ρ⊗χ −→ (Q p U n ) ρ⊗χ = 2 − 4 + 2m0 Sel n ( f α ) 0 / / Hom G (Cl p (H n ), D) / / Hom G (X n , D) / / Hom G (U n /ι(E n ), D) / / 0 Hom G v (I v (M n /H n ), D − ) ≃ / / Hom G v (U n,
IV. FINITUDE DES GROUPES DE SELMER
On prouve dans cette section le théorème A. Pour tout entier naturel n, le groupe des classes du corps de nombres H n est fini, donc il suffit de montrer que Sel # n est fini d'après la suite exacte III.4.6.1. On donne aussi une description explicite de Sel # 0 qui la formule du terme constant de la fonction L p-adique algébrique du théorème B. (1) Tout élément f ∈ Hom G (U n , V ) s'écrit de manière unique
IV.1. Traces et réseaux. On décrit dans la suite le O[Γ n ]-module Sel
f (x ⊗ c) = Tr g∈G log p ι p (g −1 (x))c ρ(g) z + z − où x ⊗ c ∈ (O H n ⊗ Z Z p ) × lib = U n (c.f. remarque III.4.2)
, et où z + , z − ∈ L n (la trace Tr est calculée coordonnée par coordonnée). De plus, la restriction de f à U n,v est égale à
f |U n,v = Tr f v −1 j=0 log p • Frob j v β − j · z + α − j · z − où f v est l'ordre de G v .
(2) L'application Φ n qui, au couple (z + , z − ) ∈ L 2 n associe f définie comme en (1) est un L-isomorphisme équivariant pour l'action naturelle de Γ n ≃ Gal(L n /L) sur L 2 n et sur Hom G (U n , V ).
Démonstration. On a Gal(H
n,v /Q p ) = G v × Γ n , ainsi qu'un isomorphisme G v × Γ n -équivariant Q p [G v × Γ n ] ≃ Hom Z p (O × H n,v , Q p ) (IV.1.4.1) g → log p •g −1 Soit f 0 = g∈G v ×Γ n z g log p •g −1 ∈ Hom(O × H n ,v , Q p ).
Si f 0 est à valeurs dans L n , alors pour tout h ∈ Gal(Q p /L n ), h • f 0 = f 0 . Comme les éléments log p •g −1 sont à valeurs dans L n , on en déduit que h(z g ) = z g pour h arbitraire, et donc z g ∈ L n pour tout g ∈ G v × Γ n . Supposons maintenant que f 0 ∈ Hom(O × H n,v , L). Pour tout γ ∈ Gal(L n /L) ≃ Γ n , on a γ • f 0 = f 0 , dont on déduit que γ(z gγ ) = z g pour tout g ∈ G v × Γ n . On a ainsi
f 0 = g∈G v γ∈Γ n z gγ log p •γ −1 g −1 = g∈G v γ∈Γ n γ −1 γ(z gγ ) log p •g −1 =
g∈G v γ∈Γ n γ −1 z g log p •g −1 = Tr g∈G v z g log p •g −1 De même, comme U n ≃ Ind G G v U n,v , un élément f 0 ∈ Hom((O H n ⊗ Z p ) × , L) s'écrit de manière unique f 0 = Tr g∈G z g log p •g −1 , et l'on a un isomorphisme similaire à l'isomorphisme IV.1.4.1.
Sous l'identification (O
H n ⊗ Z Z p ) × lib = U n (c.f. remarque III.4.2), on a f 0 (x ⊗ c) = Tr g∈G z g log p ι p • g −1 (x)c avec (z g ) g∈G ∈ L G . Considérons à présent f ∈ Hom G (U n , V ). Comme Hom G (U n , V ) = Hom Z p (O × H n , L) ⊗ L V G ,
on peut écrire f dans la base (t + , t − ) de V sous la forme
f (x ⊗ c) = Tr g∈G log p ι p • g −1 (x)c z + g z − g f étant G-équivariante, on a pour tout g ∈ G l'identité z + g z − g = ρ(g) z + 1 z − 1
, qui donne (en posant z + = z + 1 , z − = z − 1 ) la description du point (1). La définition de Φ n donnée par la formule Φ n (z + , z − ) = f est G-équivariante. En effet, si γ ∈ Γ n , alors γ commute avec g, ι p , log p et ρ(g) pour tout g ∈ G, d'où :
γ. f = f • γ −1 = Tr g∈G log p (ι p (g −1 (γ −1 (x)))c) ρ(g) z + z − = Tr γ −1 g∈G log p (ι p (g −1 (x))c) ρ(g) z + z − = Tr g∈G log p (ι p (g −1 (x))c) ρ(g) γ(z + ) γ(z − ) = Φ n (γ(z + ), γ(z − ))
IV.1.5. Soit π n une uniformisante de Z p,n . C'est aussi une uniformisante de O H n,v , de O n et on a par définition U n,v = 1 + π n O H n,v . Les propriétés usuelles du logarithme et de l'exponentielle p-adique impliquent que l'on a log p (1 + pO H n,v ) = pO H n,v , mais on n'a pas de description aussi explicite de l'image de U n,v . Néanmoins, pour 1 + π n a ∈ 1 + π n O H n,v , on a facilement v p log p (1 + π n a) = v p π n a + (π n a) 2 2 + · · · + (π n a) p n p n + · · · ≥ v p π p n n p n = 1 − n On a ainsi l'encadrement pO H n,v ⊆ log p (U n,v ) ⊆ p 1−n O H n,v .
IV.1.6. D'après la preuve de la proposition III.4.4, on a (avec les notations de la proposition)
Q p E ρ n ≃ (V ⊗ L Q p ) p n . Comme E n est sans torsion et ρ est résiduellement irréductible, on a donc E ρ n ≃ T p n .
IV.1.7. Notation.
• On définit R + n,loc ⊆ L n (resp. R − n,loc ⊆ L n ) comme étant l'image du sousmodule de U n,v ⊗ Z p O sur lequel Frob v agit par multiplication par β (resp. multiplication par α) par l'application composée
U n,v ⊗ Z p O log p ⊗1 / / H n,v ⊗ Z p O m / / L n
où m désigne la multiplication interne dans L n . On a donc, pour ζ = β ou ζ = α selon le choix du signe ±,
R ± n,loc = i a i f −1 j=0 ζ − j log p (F j (x i )), x i ∈ U n,v , a i ∈ O .
• On fixe un isomorphisme de G-modules E ρ n ≃ T ⊕ p n , et on définit de même R ± n,gl comme étant l'image du sous-O-module E ρ n ± := T ± ⊕ p n par la composée
E n ⊗ Z p O ι p ⊗1 / / U n,v ⊗ Z p O log p ⊗1 / / H n,v ⊗ Z p O m / / L n Autrement dit, on a R ± n,gl = i b i log p (ι p (x i )), x = i x i ⊗ b i ∈ E ρ n ±
On a clairement R ± n,gl ⊆ R ± n,loc ⊆ L n , car Frob v agit par multiplication par ζ sur les éléments de
IV.1.9. Lemme. Il existe x ∈ O H n,v tel que f −1 j=0 ζ − j Frob j v (x) soit une unité de O n .
Preuve du lemme. On réduit la somme modulo l'uniformisante π n de O n . Soit F n = O n /(π n ) le corps résiduel de L n . D'une part, le polynôme P(X ) =
f v −1 j=0 ζ − j X p j ∈ F n [X ] est de degré p f v −1 et d'
4). On a les équivalences
(
1) f ± (U n,v ) = 0 ⇐⇒ z ± = 0 et f ± (U n,v ) ⊆ T ± ⇐⇒ z ± ∈ R ± n,loc ⊥ (2) Si n = 0, alors f ± (U v ) ⊆ T ± ⇐⇒ pz ± ∈ O
Démonstration. D'après la description de f dans la proposition IV.1.4, on a f ± (U n,v ) = 0 (resp. f ± (U n,v ) ⊆ T ± ) si et seulement si Tr(xz ± ) = 0 (resp. Tr(xz ± ) ∈ O) pour tout x ∈ R ± n,loc . Par ailleurs, l'égalité Tr(xz ± ) = 0 est encore vraie pour tout x dans le L-espace engendré par R ± n,loc , qui est égal à L n . Donc f ± (U n,v ) = 0 équivaut à z ± = 0, ce qui prouve le point (1). Le point (2) se déduit du (1), car pour n = 0, on a R ± n,loc
⊥ = (pO) ⊥ = 1 p O.
IV.1.11. Notation. À l'aide de l'isomorphisme E ρ n ≃ T p n , on définit (x + i , x − i ) la préimage de la base (t + , t − ) de la i-ème copie de T. On notera aussi s + i et s − i les images de x + i et de x − i par l'application Λ p := m • (log p •ι p ⊗ 1) utilisée pour définir R ± n,gl . Notons que la famille s ± i 1≤i≤p n engendre R ± n,gl .
IV.1.12. Pour tout g ∈ G et pour tout 1 ≤ i ≤ p n , on a les relations g(
x + i ) = (x + i ) a(g) (x − i ) c(g) et g(x − i ) = (x + i ) b(g) (x − i ) d(g)
. On en déduit facilement une égalité de matrices lignes :
Λ p (g(x + i )) Λ p (g(x − i )) = s + i s − i ρ(g).
IV.1.13. Proposition. Soit f = Φ n (z + , z − ). Alors on a (avec les notations en IV.1.11) les équivalences suivantes :
f (E n ) = 0 ⇐⇒ ∀1 ≤ i ≤ p n , Tr(s + i z + + s − i z − ) = 0 f (E n ) ⊆ T ⇐⇒ ∀1 ≤ i ≤ p n , Tr(s + i z + + s − i z − ) ∈ O Démonstration. On a f (E n ) ⊆ T si et seulement si f (E ρ n ) ⊆ T si et seulement si f (x + i ), f (x − i ) ∈ T pour tout 1 ≤ i ≤ p n , c'est-à-dire si la matrice carrée M i := f (x + i ) f (x − i ) ∈ M 2 (L) est à coefficients dans O.
On peut réécrire cette matrice sous la forme :
M i = Tr g∈G ρ(g) z + z − Λ p (g −1 (x + i )) Λ p (g −1 (x − i )) = Tr g∈G ρ(g) z + z − s + i s − i ρ(g −1 ) d'n est isomorphe à f ∈ Hom G (U n , V ) / f − (U n,v ) ⊆ T − , f (E n ) ⊆ T f ∈ Hom G (U n , V ) / f (U n,v ) ⊆ T
En utilisant la description avec l'isomorphisme Φ n ainsi que le corollaire IV.1.10 et la proposition IV.1.13, ce dernier module est isomorphe à
(z + , z − ) ∈ L 2 n / z − ∈ R − n,loc ⊥ et ∀1 ≤ i ≤ p n , Tr(s + i z + + s − i z − ) ∈ O R + n,(z + , z − ) ∈ L 2 n / z − ∈ R − n,/ / Hom G (E n , T) × Hom G v (U n,v , T − ) Hom G (U n , V ) / / Hom G (E n , V ) × Hom G v (U n,v , V − ) 0 / / Sel # n / / Hom G (U n , D) / / Hom G (E n , D) × Hom G v (U n,v , D − ) 0 0
Les colonnes sont exactes par le même argument que celui pour la suite exacte IV.1.0.1. On va prouver que la flèche horizontale du milieu est un isomorphisme, grâce à la proposition suivante et à un simple comptage de dimensions. Cela suffira pour conclure que Sel # n est fini.
IV.2.1. Proposition. Soit
Sel n (V ) := ker
Hom G (U n /E n , V ) −→ Hom G v (U n,v , V − )
Alors Sel n (V ) = 0 pour tout entier naturel n.
IV.2.2. La représentation ρ est à coefficients dans un corps de nombres, donc on peut consi- V dim π π où la somme est prise sur toutes les représentations irréductibles π de G n avec π triviale ou bien telle que la conjugaison complexe Frob ∞ ∈ G n ne fixe aucun vecteur non-nul de V π .
dérer son Q[G]-module QV , ainsi que Q p V son Q p [G]-module associés. Comme L ⊆ Q p , on a une injection Sel n (V ) → Sel n (Q p V ) := ker Hom G (U n /E n , Q p V ) −→ Hom G v (U n,v , Q p V −
IV.2.6. Corollaire. On a
Hom(U n /E n , Q p ) Q ⊗ Q Q p V G = 0.
Démonstration. Soit W la représentation contragrédiente de Hom(U n /E n , Q p ) Q . Il suffit de montrer que Hom Q[G] (W, QV ) = 0. D'après le théorème de réciprocité de Frobenius, on a
Hom Q[G] (D n , QV ) = Hom Q[G n ] (W, QV ⊗ Q Q[Γ n ]).
Or, en tant que Q[G n ]-représentations, on a V ⊗ Q Q[Γ n ] = ζ p n =1 V ⊗ Q(ζ), où Q(ζ) désigne la droite Q munie de l'action de Γ n donnée par le caractère envoyant le générateur γ n ∈ Γ n sur ζ ∈ Q × . La conjugaison complexe Frob ∞ ∈ G n fixe l'extension Q ∞ /Q, donc on a Frob ∞ ∈ G ⊆ G n . Ainsi, l'action de Frob ∞ sur un V ⊗ Q(ζ) est donnée par l'action sur la première composante. Comme V est impaire, on a donc π Frob ∞ =1 = 0 pour π égale à la contragrédiente de V ⊗ Q(ζ). IV.2.9. Définition. On définit l'unité globale ǫ ρ comme étant un élément
i ǫ i ⊗α i ∈ O × H ⊗ Z Q[ρ] ρ tel que son logarithme p-adique log p (ǫ ρ ) = i ι p (α i ) log p (ι p (ǫ i )) engendre R + 0,Sel ♯ 0 ≃ O/ log p (ǫ ρ ) p O.
Démonstration. D'après le corollaire IV.1.14, on a Sel Par définition de l'action de Γ p n sur X ∞ = Gal(M ∞ /H ∞ ), le groupe (X ∞ ) Γ p n est le groupe de Galois de K /H ∞ , la sous-extension maximale de M n /H ∞ telle que K est abélienne sur H n . Il suffit de prouver que K = M n , ce que l'on montre par double inclusion. On a d'une part K ⊆ M n , par maximalité de H n /H n car l'extension K /H n est abélienne (par définition) non-ramifiée en que L p ( f α , 0) est nul si et seulement si δ α = 1, c'est-à-dire α = 1. Enfin, pour n quelconque, en retranchant (E 0 ) à (E n ) on voit que L p ( f α , T) est premier à ω n (T) T , dont les racines sont exactement les ζ − 1, où ζ parcourt µ p n − {1}. On a donc toujours L p ( f α , ζ − 1) = 0 pour tout ζ ∈ µ p n − {1}.
V.2. Absence de sous-modules pseudo-nuls et terme constant de L p ( f α ). L'hypothèse (c) est clairement vérifiée, car la représentation ρ est impaire. L'hypothèse (d) aussi, car ρ est non-ramifiée en p. Enfin, l'hypothèse (e) est vérifiée car ρ est résiduellement irréductible de dimension 2. On peut ainsi conclure que X ∞ ( f α ) n'a pas de sous-modules pseudo-nuls non-triviaux.
V.2.1. Proposition. Le Λ-module X ∞ ( f α ) n'g φ := φ(F) = n φ(a n (F))q n ∈ S k φ (N p r φ , ǫχ φ ω 1−k φ , O φ )
définit une forme parabolique ordinaire p-stabilisée de poids k φ et de niveau N p r φ . On dit que F est N-nouvelle si toutes ses spécialisations classiques le sont. Une famille de Hida primitive F est une forme parabolique ordinaire Z p [[X ]]-adique N-nouvelle qui est propre pour les opérateurs de Hecke U ℓ (resp. T ℓ , 〈ℓ〉) pour ℓ|N p (resp. ℓ ∤ N p).
gébrique g φ = φ(F) de F, l'image L p (g φ , T) ∈ O φ [[T]] de L p (F, T) par la projection naturelle H F [[T]] ։ O φ [[T]
] est la fonction L p-adique usuelle attachée à g φ . Autrement dit, pour tout 0 ≤ m ≤ k − 2 et ξ racine primitive p t−1 -ème de l'unité, on a la formule d'interpolation :
L p (g φ , ξ(1 + p) m − 1) = e g (ξ, m) p t ′ (m+1) m! a p (g) t ′ (−2iπ) m+1 G(ω m χ −1 ξ )Ω (−1) m g L(g φ , ω m χ −1 ξ , m + 1) où a p (g φ ) = φ(a p (F)), où χ ξ : G Q ։ Γ ։ µ p tρ F : G Q −→ GL H F (T F ),
non-ramifiée en-dehors de N p telle que Tr(ρ F (Frob ℓ )) = a ℓ (F) pour tout ℓ ∤ N p. On a en outre les propriétés suivantes :
• Pour toute spécialisation algébrique g φ = φ(F) de F, le O φ -module T φ := T F ⊗ H F ,φ O φ est
isomorphe à un réseau de la représentation de Deligne attachée à g φ . De même, on a
T ≃ T F ⊗ H F ,φ α O.
• Il existe T + F ⊆ T F un sous-H F -module libre de rang 1 qui est stable par G Q p vérifiant les propriétés suivantes. Le quotient T − F := T F /T + F est libre de rang 1, et l'action de G Q p sur T − F est donnée par le caractère non-ramifié δ tel que δ(Frob p ) = a p (F). De plus, pour toute spécialisation algébrique g φ de F, la filtration obtenue se spécialise sur la filtration ordinaire attachée à T φ . De même, on a T ± ≃ T ± ⊗ H F ,φ α O.
VI.4. Conjectures principales.
VI.4.1. Soit g φ une spécialisation classique de F. Le groupe de Selmer (resp. groupe de Selmer dual) attaché à g φ est par définition
le O φ [[T]]-module Sel ∞ (φ) := Sel ∞ (T φ , T + φ ), resp. X ∞ (φ) := Sel ∞ (φ) ∨ On pose L p (g φ , T) := L p (X ∞ (φ), T) lorsque X ∞ (φ) est de torsion sur O φ [[T]
]. La conjecture principale pour les formes primitives ordinaires de poids k ≥ 2 prédit l'égalité suivante.
VI.4.2. Conjecture. Le O φ [[T]]-module X ∞ (φ) est de torsion, et il existe une unité u φ de O φ [[T]]
telle que (1) Si a n divise b n dans A pour tout n, alors a divise b.
u φ · L p (g φ , T) = L p (g φ , T).− 1|k φ − 1 et χ φ = 1. Le O φ [[T]]-module X ∞ (φ) est de torsion, et L p (g φ , T) divise L p (g φ , T)) dans O φ [[T]]L p (g n , T) (c) n→+∞ (a) divise L p (g n , T) (b) n→+∞ L p ( f α , T) L p ( f α ,
(2) Soit π ∈ A un élément premier, régulier et topologiquement nilpotent. Si a n divise b n dans A[ 1 π ] pour tout n, et si a = 0, alors a divise b dans A[ 1 π ].
Démonstration. Commençons par le (1). Écrivons b n = a n c n avec c n ∈ A. Comme A est compact, on peut supposer, quitte à extraire une sous-suite, que (c n ) n converge. On a ainsi b = ac par passage à la limite, et donc (a) ⊇ (b).
Traitons (2). Soit n ∈ N. Il existe k(n) ∈ N et c n ∈ A tels que π k(n) b n = a n c n . L'élément π étant régulier, quitte à simplifier suffisamment de fois par π, on peut supposer que l'on a, ou bien k(n) = 0, ou bien π ∤ c n . Dans tous les cas, π étant premier, on a π k(n) |a n . Par hypothèse, a n converge vers a = 0 et π est topologiquement nilpotent, donc k(n) est borné avec n d'après (1). Ainsi, il existe un entier K suffisamment grand tel que (a n ) ⊇ (π K b n ) pour tout n, et donc (a) ⊇ (π K b) par (1), ce qu'on voulait démontrer.
On définit
F † (Y ) = m≥1 a m (F † ; Y )q m ∈ A[[q]] où a m (F † ; Y ) := Φ(a m (F))
Alors F † (Y ) paramètre F au voisinage de f α au sens suivant : on a F † (0) = f α , et pour tout entier k ≥ 2 tel que p re |k − 1, et pour tout choix y ∈ M d'une racine e-ième de u k−1 −1 p re , F † (y) est une spécialisation algébrique de F de poids k, de niveau N p et nebentypus ǫω 1−k .
Démonstration. On a en effet p re+1 |u k−1 −1, donc |y| p < 1 et il existe un unique morphisme Ψ y :
A −→ O M satisfaisant Ψ y (Y ) = y. On a de plus Ψ y (X ) = u k−1 − 1, donc la composée φ y = Ψ y • Φ : H F −→ O M est une spécialisation algébrique de H F de poids k et de caractère χ φ y = 1. La forme F † (y) = F(φ y ) définit bien une spécialisation algébrique de F avec les propriétés annoncées. VII.2.6. Notation. Quitte à agrandir M, on peut supposer que M contient toutes les extensions de Q p de degré ≤ e, et donc M contient toutes les racines e-ièmes d'éléments de Q × p . On fixe dans toute la suite une telle extension M.
Pour tout entier naturel n, on note k(n) = (p − 1)p n+re + 1. On fixe y n ∈ M une racine e-ième de u k(n)−1 −1 p re et on pose g n = F † (y n ). Comme p − 1|k(n) − 1, g n est la p-stabilisation d'une forme primitive classique de poids k(n), de niveau N,
de la Z p -extension cyclotomique Q ∞ de Q, et introduisons les O[Gal(Q n /Q)]-modules Sel n ( f α ) := Sel( f α , Q n ) (voir définition III.2.1). De même, on considère le groupe de Selmer Sel ∞ ( f α ) := Sel( f α , Q ∞ ) qui est un module sur l'anneau de groupe complété O[[Gal(Q ∞ /Q]]. Notre premier résultat est le suivant : Théorème A. Supposons (hyp) vérifiée. Alors, pour tout n ∈ N, le O-module Sel n ( f α ) est fini.
X
∞ ( f α ) := Hom ct (Sel ∞ ( f α ), Q p /Z p ). Rappelons que l'on a un isomorphisme entre O[[Gal(Q ∞ /Q)]] et l'algèbre d'Iwasawa Λ = O[[T]], envoyant un générateur topologique γ de Gal(Q ∞ /Q) sur 1 + T. Théorème B. Supposons (hyp) vérifiée. Alors le Λ-module X ∞ ( f α ) est de type fini et de torsion, et de plus il ne possède pas de sous-modules finis non-nuls.
p
( f α , 0) = 0 ⇐⇒ α = 1 De plus, lorsque α = 1, on a l'égalité suivante (définie à une unité p-adique près) :L alg p ( f α , 0) = log p (ǫ ρ ) p # Cl p (H) ρ oùCl p (H) ρ est la composante ρ-isotypique de la p-partie du groupe des classes de H, où log p est le logarithme p-adique et où ǫ ρ ∈ O × H ⊗ Z Q[ρ] ρ est une unité globale de H, vue dans O × grâce à ι p . (c.f. IV.2.10 pour une définition précise).
spécialisation
Hρ −→ O. On définit L an p ( f α , T), la fonction L p-adique analytique de f α , comme étant l'image de L an p (ρ, T) par le morphisme induit Hρ[[T]] −→ Λ. Conjecture A. Supposons que les hypothèses (hyp) et (reg) sont vérifiées. Il existe une unité u de Λ telle que L alg p ( f α , T) · u = L an p ( f α , T)
Théorème C .
.Supposons que les hypothèses (hyp) et (reg) sont vérifiées. AlorsL alg p ( f α , T) divise L an p ( f α , T) dans Λ[ 1 p ].La preuve du Théorème C utilise la théorie des familles de Hida. SoitF une famille de Hida primitive de niveau N se spécialisant sur f α . Elle correspond à un idéal premier minimal a du quotient new H new ρ de Hρ. D'après [BD16], F est par ailleurs unique à conjugaison près (c.f. [Dim14, Corollary 7.7]). On peut voir F à coefficients dans H F := H new ρ /a. L'image L an p (F, T) ∈ H F [[T]] de L an p (ρ, T) par la projection Hρ[[T]] ։ H F [[T]], satisfait la propriété suivante : pour toute spécialisation g classique de poids k ≥ 2 de F, correspondant à un morphisme H F −→ O g ⊆ Z p , l'image de L an p (F, T) dans O g [[T]] coïncide avec la fonction L p-adique analytique usuelle L an p (g, T) de g.
L'idée de la preuve des points (b) et (c) repose sur la construction d'une fonction L p-adique(algébrique et analytique) à deux variables (poids X et cyclotomique T) attachée à F † (X ) qui interpole (en X ) les fonctions L p-adiques des g n et de f α . La construction analytique est formelle et la preuve du point (b) est un simple passage à la limite dans les séries formelles. Du côté algébrique, on commence par définir un O[[X /p r , T]]-module de type fini et de torsion X ∞ (F † ). Comme l'anneau O[[X /p r , T]] est un anneau de séries formelles, il est factoriel, et donc X ∞ (F † )
On a ainsi, pour tout x = 0 dans A, v p (x) = ord p (A/(x)). D'après le point (3) du Théorème I.1.1, on en déduit que le uplet d'entiers (ord p (N)) p est presque nul. I.1.2. Définition (Idéal divisoriel). Soit a un idéal de A. On dit que a est un idéal divisoriel s'il est égal à l'intersection des idéaux principaux le contenant. D'après le point (2) du théorème I.1.1, pour tout x ∈ A et pour tout idéal entier divisoriel a, on a x ∈ a si et seulement si ord p (x) ≥ ord p (A/a) pour tout p ∈ P A . Réciproquement, si (m p ) p est un uplet d'entiers presque tous nuls, alors l'ensemble a = {x ∈ A : ord p (x) ≥ m p , ∀p ∈ P A } est un idéal divisoriel de A, et l'on a m p = ord p (a). En particulier, si a est un idéal entier divisoriel et p 1 , . . ., p d sont les idéaux premiers de hauteur 1 contenant a, alors, en posant m i = ord p i (A/a), on a a = p m 1 1 . . . p m d d . Ainsi, pour deux idéaux divisoriels a et b, on a a ⊆ b si et seulement si ord p (a) ≥ ord p (b) pour tout idéal p ∈ P A . I.1.3. Définition (Idéal caractéristique). Soit M un A-module de type fini et de torsion. On appelle idéal caractéristique de M, et l'on note car A (M), l'idéal entier divisorielcar A (M) = {x ∈ A : ord p (x) ≥ ord p (M), ∀p ∈ P A }.Comme l'application qui à un module de longueur finie associe sa longueur est additive sur les suites exactes, on en déduit le lemme suivant :I.1.4. Lemme. L'application qui, à un A-module de type fini et de torsion, associe son idéal caractéristique est multiplicative sur les suites exactes. Autrement dit, si l'on a une suite exacte0 / / N / / M / / P / / 0 alors car A (M) = car A (N)car A (P). I.1.5. Définition. Soit M un A-module de type fini. M est dit pseudo-nul s'il vérifie l'une des deux conditions équivalentes suivantes : (1) Pour tout idéal p ∈ P A ∪ {(0)}, on a M p = 0 (2) L'annulateur de M n'est contenu dans aucun idéal premier de hauteur 1. Un module pseudo-nul M est nécessairement de torsion, car M (0) = M ⊗ A K = 0. De plus, un module M de type fini et de torsion est pseudo-nul si et seulement si car A (M) = A. I.1.6. Définition. Soit φ une application linéaire entre A-modules de type fini M −→ N. On dit que φ est un peudo-isomorphisme si le noyau et le conoyau de φ sont pseudo-nuls. Si un tel φ existe, on dit que M est pseudo-isomorphe à N. De manière équivalente, φ est un pseudo-isomorphisme si et seulement si les applications localisées φ p : M p −→ N p sont des isomorphismes pour tout idéal p ∈ P A . Si M est pseudo-isomorphe à N, alors M est de torsion si et seulement si l'est, auquel cas on a car A (M) = car A (N) d'après le lemme I.1.4. Par ailleurs, la relation "être pseudo-isomorphe à" est symétrique si l'on se restreint à la catégorie des modules de type fini et de torsion sur A. I.1.7. Théorème (Théorème de structure). Soit M un A-module de type fini et de torsion. Il existe p 1 , . . ., p d ∈ P A , des entiers m i > 0 et un pseudo-isomorphisme de M vers i A, D sont des modules pseudo-nuls. Les idéaux p i et les entiers m i sont uniquement déterminés. On a en effet car A (M) = p m i = ord p i (M) pour tout 1 ≤ i ≤ d. I.2. Idéaux caractéristiques et changement de bases. I.2.1. Lemme. Soit M un A-module de type fini et de torsion, soit p ∈ P A . (1) Si M est pseudo-nul, alors M[p] et M/pM sont de torsion sur A/p. (2) Si M/pM est de torsion sur A/p, alors p ne divise pas car A (M).Démonstration. Supposons M pseudo-nul. Alors il existe un élément x ∈ A − p tel que xM = 0. Donc l'image de x dans A/p est un élément non-nul tuant M[p] et M/pM, ce qui prouve que ces deux modules sont de torsion sur A/p. Montrons le point (2) par contraposée et supposons que p divise l'idéal caractéristique de M. Alors d'après le théorème I.1.7, il existe une application A-linéaire M −→ A/p de conoyau pseudo-nul. En tensorisant par A/p au-dessus de A, on obtient un morphisme de A/p-modules M/pM −→ A/p dont le conoyau est de torsion d'après le point (1). Donc M/pM n'est pas de torsion sur A/p. I.2.2. Proposition. Soient M un A-module de type fini et de torsion, et p ∈ P A . Supposons que p est principal et que A/p est encore un anneau intégralement clos.(1) Si M est pseudo-nul, alors on a car A/p (M[p]) = car A/p (M/pM).
( 2 )
2Si M n'a pas de sous-A-modules pseudo-nuls différents de {0} et si M/pM est de torsion sur A/p, alors car A/p (M/pM) = car A (M)/p car A (M).
( 3 )
3Plus généralement, si l'on suppose uniquement que M/pM est de torsion sur A/p, alors M[p] est aussi de torsion sur A/p et on a la relation car A/p (M/pM) = car A/p (M[p]). (car A (M)/p car A (M)) . Démonstration. Sans perte de généralité, on peut supposer que A est de dimension de Krull au moins égale à 2. Soit x ∈ A un générateur de p. Supposons d'abord M pseudo-nul. D'après le lemme I.2.1, les A/p-modules M[p] et M/pM sont de torsion. Comme A/p est supposé intégralement clos, il a un sens de considérer leurs idéaux caractéristiques. La multiplication par x sur M induit une suite exacte de A-modules : 0 / / M[p] / / M / / M / / M/pM / / 0 Fixons maintenant un idéal q ∈ P A/p , et montrons que ord q (M[p]) = ord q (M/pM). Notons Q ⊆ A l'image inverse de q par la projection A ։ A/p. C'est un idéal premier de A de hauteur 2 contenant p. En localisant en Q la suite exacte précédente, on obtient une suite exacte : 0 / / M[p] Q / / M Q / / M Q / / (M/pM) Q / / 0 Comme M est pseudo-nul, le localisé M Q est pseudo-nul sur A Q , qui est un anneau noethérien intégralement clos de dimension de Krull égale à 2. L'annulateur de M Q contient donc une puissance de l'idéal maximal QA Q , et ainsi M Q est de longueur finie. On en déduit que ord Q M[p] = ord Q (M/pM). Comme ord q M[p] = ord Q M[p] et ord q (M/pM) = ord Q (M/pM), cela montre que les A/p-modules M[p] et M/pM ont la même q-longueur, ce qui termine la preuve du point (1). Supposons maintenant que M n'a pas de sous-modules pseudo-nuls non-triviaux, et que M/pM est de torsion sur A/p. Alors d'après le lemme I.2.1, on sait que p n'apparaît pas dans la suite des p 1 , . . ., p d ∈ P A divisant car A M. Par ailleurs, le pseudo-isomorphisme donné par le théorème I.1.7 est injectif, et donc définit une suite exacte m i sont des entiers positifs et D est un A-module pseudo-nul. En considérant la multiplication par x, on en déduit une suite exacte longue i A/p m i i [p] / / D[p] / / M/pM / / i A/p m i i ⊗ A A/p / / D/pD / / 0 Comme x n'appartient pas aux p i , le premier terme de cette suite exacte est nul. D'après le lemme I.1.4 et le premier point, on a donc car A/p (M/pM) = car Ale point (3). Supposons uniquement que M/pM est de torsion sur A/p. D'après le théorème I.1.7, il existe un pseudo-isomorphisme φ définissant une suite , D sont des A-modules pseudo-nuls. D'après le lemme I.2.1, p est distinct des p i p] = 0. On en déduit facilement que C[p] = M[p]. Comme C est pseudo-nul, C[p] est de torsion sur A/p d'après le lemme I.2.1, et donc M[p] aussi. On définit le A-module M ′ comme étant l'image de M par φ. Alors M ′ est pseudo-isomorphe à M et donc a le même idéal caractéristique que M. De plus, c'est un sous-module de i Adonc il n'a pas de sous-modules pseudo-nuls non-triviaux. Enfin, M ′ /pM ′ est un quotient de M/pM, donc il est de torsion sur A/p. D'après le point (2) précédemment démontré, on a donc car A/p (M ′ /pM ′ ) = car A (M)/p car A (M) Il reste à montrer que car A/p (M/pM) = car A/p (M[p]). car A/p (M ′ /pM ′ ). On a une suite exacte courte
où une suite exacte courte de A/pmodules de torsion induite par la multiplication par x : 0 / / C/pC / / M/pM / / M ′ /pM ′ / / 0 D'après le lemme I.1.4, on a donc car A/p (M/pM) = car A/p (C/pC). car A/p (M ′ /pM ′). Or, d'après le lemme I.2.1, on a car A/p (C/pC) = car A/p C[p], et l'on a vu précédemment que C[p] = M[p]. On a donc montré que car A/p (M/pM) = car A/p (M[p]). car A/p (M ′ /pM ′ ) = car A/p (M[p]). car A (M)/p car A (M), ce qui termine la preuve de la proposition. I.3. Fonctions L p-adiques algébriques. I.3.1. Définition. Soit M un module de type fini et de torsion sur A. La fonction L p-adique algébrique de M, notée L p (M), est un générateur de l'idéal caractéristique de M. Lorsqu'elle existe, elle est unique à multiplication par une unité de A près. Lorsque A est factoriel, ses idéaux premiers de hauteur 1 sont tous principaux, et donc la fonction L p-adique algébrique existe toujours. C'est le cas par exemple quand A est régulier d'après le théorème de Auslander-Buchsbaum, en particulier c'est vrai lorsque A est une algèbre de séries formelles.Considérons le cas particulier où A = A 0 [[T]], où A 0 est l'anneau des entiers d'une extension finie de Q p , d'uniformisante ̟. Rappelons que d'après le théorème de préparation de Weierstrass, une série formelle non-nulle f (T) ∈ A se décompose de manière unique en un produit f (T) = ̟ µ U(T)P(T), où µ ∈ N, U(T) ∈ A × et P(T) est un polynôme distingué. On s'intéresse en théorie d'Iwasawa aux invariants suivants : I.3.2. Définition. Supposons que A = A 0 [[T]], et soit M un A-module de type fini et de torsion. Soit f (T) la fonction L p-adique algébrique de M, que l'on décompose en f (T) = ̟ µ U(T)P(T). Les λ et µ invariants de M, notés λ(M) et µ(M), sont respectivement définis comme étant λ(M) = deg T P(T) et µ(M) = µ. Nous terminons cette section avec un lemme utile interprétant le terme constant de la fonction L p-adique algébrique comme étant un quotient de Herbrand : I.3.3. Lemme. Supposons que A = A 0 [[T]] et soit M un A-module de type fini et de torsion tel que M/T M est fini. Alors M[T] est aussi fini, et l'l'égalité à une unité de A 0 près. Démonstration. Soit p l'idéal premier de hauteur 1 de A engendré par T. On a A/p = A 0 , et L p (M; 0) est un générateur de car A (M)/p car A (M). Par ailleurs, un A/p module N de type fini est de torsion si et seulement si il est fini, auquel cas car A/p N est engendré par #N. Par hypothèse, M/pM est de torsion sur A/p, donc d'après le point (3) de la proposition I.2.2, M[p] est fini, et l'on a une égalité entre idéaux principaux (#M[p]).(L p (M; 0)) = (#(M/pM)), d'où l'égalité L p (M; 0)= #(M/T M) #M[T] . II. GROUPE DE SELMER D'UNE REPRÉSENTATION ORDINAIRE II.1. Groupe de Greenberg-Selmer. II.1.1. Soient A une Z p -algèbre profinie, et T un A-module libre de rang fini muni d'une action continue de Gal(Q Σ /Q), où Σ est un ensemble fini de places de Q contenant p et ∞, et Q Σ est la plus grande extension algébrique de Q non-ramifiée en dehors de Σ. On se donne une suite exacte courte de G Q p -modules libres sur A
sur les deux facteurs du produit tensoriel. On peut voir D n (resp. D ∞ ) comme un module colibre de rang d sur l'anneau A[Γ n ] (resp. sur l'anneau A[[Γ]]). On définit similairement D ± n , et on a aussi une suite exacte courte 0 / / D + n / / D n / / D − n / / 0 La définition suivante est tirée de la définition de [Gre91] du groupe de Selmer d'un motif vérifiant une condition d'ordinarité (voir aussi [Gre89]).
II. 2 .
2Changement de bases. On conserve les notations précédentes. Pour tout idéal a ⊆ A, on a un isomorphisme naturel (A/a) ∨ = A ∨ [a], et donc on a T/a ⊗ A/a (A/a) ∨ = D[a] (et similairement pour T + ). En particulier, on a une application naturelle
Démonstration.
On va montrer que l'on peut appliquer [Gre16, Proposition 4.1.1] au A[[Γ]]module D ∞ . Avec les notations de loc. cit., on pose L ℓ = 0 pour l ∈ Σ, l = p et enfin
∞
application a est surjective. En effet, coker a s'injecte dans H 2 (Q p , D + ∞ ). D'après la dualité locale de Tate, on a H 2 (Q p , ) G Qp = 0 par le même argument que précédemment. On a ainsi une suite exacte courte :
Ce dernier module n'a pas de sous-modules pseudo-nuls non-triviaux, donc (ker b) ∨ non plus et donc L ∨ p non plus. On vérifie maintenant l'hypothèse CRK(D ∞ , L), c'est-à-dire l'égalité des corangs sur A[[Γ]] :
après la proposition 4.2(b) de loc. cit. et d'après la dualité locale de Tate. Or ces deux modules sont de co-torsion sur A[[Γ]] d'après le même argument utilisant le lemme de Shapiro que précédemment, donc de corang nul. Enfin, pour ℓ = p, on a vu que L p et ker a ont le même corang. Comme a est surjectif, Q p (T, T + ) et H 1 (Q p , D − ∞ ) ont le même corang. D'après la proposition 4.2(a) de loc. cit. et un argument identique à celui pour ℓ = p, ce dernier a un corang égal à d − . Ceci termine la vérification de l'hypothèse CRK(D ∞ , L). Pour appliquer [Gre16, Proposition 4.1.1], il suffit maintenant de vérifier l'une des trois hypothèses supplémentaires. La condition (b) stipulant que D ∞ est co-libre et que la G Q -représentation résiduelle de D ∞ n'a pas de quotient isomorphe à µ p est clairement vérifiée, car cette dernière est isomorphe à D[m A ] et qu'elle vérifie la même condition d'après notre hypothèse (e). III. GROUPE DE SELMER D'UNE FORME MODULAIRE DE POIDS 1 III.1. Préliminaires. On reprend les notations du début de l'article. La représentation de Deligne-Serre ρ de f est une représentation complexe irréductible d'image finie, impaire de dimension 2, non-ramifiée en-dehors des places divisant N et vérifiant ∀ℓ ∤ N,
. On note O son anneau des entiers et F = O/pO son corps résiduel.
O[G]-module qui est isomorphe à (L/O) 2 en tant que O-module discret. On a par ailleurs un isomorphisme (canonique) O ∨ = L/O, et donc D ≃ T ⊗ O O ∨ . On utilisera plusieurs fois le lemme élémentaire suivant : III.1.1. Lemme. Soit M un O[G]-module.
( 1 )
1La partie ρ-isotypique e ρ M de M est (non-canoniquement) isomorphe en tant que Omodule à Hom G (T, M) ⊕2 . (2) Si M est de type fini, alors on a Hom G (T, M) ∨ ≃ Hom G (M, D). Démonstration. Montrons le point (1). Comme O[G] = χ T dimπ π , on a e ρ O[G] = T ⊕2 , et donc (par orthogonalité des idempotents) on a : e ρ M = Hom G (O[G], e ρ M) = Hom G (T ⊕2 , e ρ M) = Hom G (T, M) ⊕2 . Montrons le point (2). Notons d'abord que Hom(T, D) = coker (Hom(T, T) −→ Hom(T, V )) car T est libre, et donc Hom G (T, D) = coker (Hom G (T, T) −→ Hom G (T, V )) car G est d'ordre premier à d. Comme les réseaux G-stables de V sont tous L-homothétiques, Hom G (T, D) s'identifie naturellement à L/O. On a donc l'isomorphisme recherché pour M = T. Soit M un O[G]-module de type fini. On peut supposer que M = e ρ M. On a une présentation finie O[G] m → O[G] n → M → 0 qui, en appliquant l'idempotent e ρ , donne la présentation T 2m r / / T 2n s / / M / / 0 de O[G]-modules. On a Hom G (M, D) = ker r * et Hom G (T, M) = coker r * où r * : Hom G (T 2n , D) → Hom G (T 2m , D) et r * : Hom G (T, T 2m ) → Hom G (T, T 2n ) sont respectivement la pré-composition et la post-composition par r. En utilisant l'isomorphisme construit précédemment et l'exactitude du foncteur de dualité, on a aisément Hom G (T, M) ∨ = (coker r * ) ∨ ≃ ker r * = Hom G (M, D).Comme p est non-ramifié dans H, le sous-groupe G v de G est cyclique et engendré par le Frobenius arithmétique Frob v . De plus, les valeurs propresde ρ(Frob v ) sont α et β, donc il existe une O-base de T dans laquelle ρ f (Frob v ) = β 0 0 α . On note T = T + T − la décomposition correspondante. On a similairement des décompositions en sommes directes de G v -modules V = V + V − et D = D + D − .III.2. Définition de Sel n ( f α ). On définit les groupes de Selmer attachés à f α à l'aide des défi-nitions de la section précédente, avec les données A = O, T = T et T + = T + . III.2.1. Définition. Soit n ∈ N {∞}. Le groupe de Selmer de niveau n attaché à f α est le Omodule Sel n ( f α ) = Sel n (T, T + ) On note X n ( f α ) := Sel n ( f α ) ∨ le groupe Selmer dual. Comme T ⊗ O O ∨ ≃ D, on a donc, d'après le lemme II.1.4 : Sel n ( f α ) = ker H 1 (Q n , D) −→ H 1 (I p , D − ) × ℓ = p H 1 (I ℓ , D) . III.3. Isomorphismes de restriction. La représentation ρ étant d'image finie, la suite d'inflationrestriction en cohomologie galoisienne va nous permettre ici de décrire Sel n ( f α ) faisant intervenir l'arithmétique du corps de nombres H. Soit H ∞ = ∪ n H n = ∪ n HQ n l'extension cyclotomique de H. Soit n un entier naturel. Comme p ∤ d, les extensions H/Q et Q n /Q sont linéairement disjointes, et l'on a G n := Gal(H n /Q) ≃ G × Γ n Le groupe de Galois Gal(H ∞ /H) s'identifie à Γ, et l'algèbre de groupe complétée O[[Gal(H ∞ /H)]] à l'algèbre d'Iwasawa Λ = O[[T]].Soit M n /H n la pro-p-extension abélienne maximale de H n non-ramifiée en-dehors des places de H n divisant p et ∞. On note X n son groupe de Galois. La suite d'inflation-restriction permet de réécrire le groupe de Selmer de f α sous la forme d'un Hom, grâce à la condition p ∤ d.III.3.1. Lemme. La flèche de restriction de G Q n à G H n induit un isomorphisme : ker H 1 (Q n , D) −→ ℓ = p H 1 (I ℓ , D) ≃ Hom G (X n , D) où l'action de G ≃ Gal(H n /Q n ) sur X n est donnée par la conjugaison. Démonstration. La suite d'inflation-restriction fournit une suite exacte : 0 / / H 1 (G, D) / / H 1 (Q n , D) res / / Hom G (G H n , D) / / H 2 (G, D) Comme p ∤ d, les groupes de cohomologie H i (G, D) = 0 pour i > 0, et donc l'application de restriction réalise un isomorphisme H 1 (Q n , D) ≃ Hom G (G H n , D). Montrons à présent que les conditions de trivialité locale sont conservées après restriction. Soit c ∈ H 1 (Q n , D). Alors pour tout nombre premier ℓ, c |I ℓ (Q/Q n ) est trivial si et seulement si c |I ℓ (Q/H n ) l'est. En effet, la flèche H 1 (I ℓ (Q/Q n ), D) res / / H 1 (I ℓ (Q/H n ), D) est injective, car son noyau H 1 (I ℓ (H n /Q n ), D) est nul d'après le même argument que précédemment. On a donc : ker H 1 (Q n , D) −→ ℓ = p H 1 (I ℓ , D) ≃ ker Hom G (G H n , D) −→ ℓ = p Hom(I ℓ (Q/H n ), D) ≃ Hom G (X n , D) III.3.2. Corollaire. La flèche de restriction de G Q n à G H n induit un isomorphisme : Sel n ≃ ker Hom G (X n , D) −→ Hom(I p (M n /H n ), D − ) Démonstration. Comme p est non-ramifié dans H n /Q n , on a I p (M n /Q n ) = I p (M n /H n ) et donc la condition de trivialité en p pour un cocycle de ker H 1 (Q n , D) −→ ℓ = p H 1 (I ℓ , D) est la même que celle pour son image dans Hom G (X n , D).
III. 4 /
4.3. L'application de réciprocité d'Artin de la théorie des corps de classes induit une suite exacte de Z p [G n ]-modulesE n ι / / U n rec / / X n / / Cl p (H n ) / / 0 où Cl p (H n ) est la p-partie du groupe de classes de H n . La conjecture de Leopoldt prédit que ι est injective. Le théorème de Baker-Brumer implique que la restriction de ι à la ρ-composante E ρ n est injective. Autrement dit, on a la proposition suivante. III.4.4. Proposition. On a une suite exacte de O[G n ]/ Cl p (H n ) ρ / / 0 Démonstration. On applique à la suite exacte précédente le foncteur exact M → M ρ et l'on doit justifier l'injectivité de la première flèche. Comme les modules E ρ n et U ρ n sont O-libres, ils s'in-
ρ⊗χ ≤ 0 ce
0qui termine la preuve de l'injectivité et de la proposition. III.4.5. Pour tout Z p -module M, on a clairement Hom G (M, D) = Hom G (M ρ , D). En appliquant le foncteur (exact) contravariant Hom G (−, D) à la suite exacte de O[G]-modules de la proposition III.4.4, on obtient donc une suite exacte courte de O[Γ n ]-modules : 0 / / Hom G (Cl p (H n ), D) / / Hom G (X n , D) / / Hom G (U n /ι(E n ), D) / / 0 D'après la théorie des corps de classes locale, l'application de reciprocité locale envoie bijectivement le groupe d'inertie I v (M n /H n ) sur U n,v . On peut compléter la suite exacte précédente en un diagramme commutatif dont la ligne et la colonne centrale sont exactes :
v , D − ) III.4.6. Définition. Pour tout n ∈ N {∞}, on définit : Sel Iw n := Hom G (Cl p (H n ), D)resp.Hom G (U n /ι(E n ), D) −→ Hom G v (U n,v , D − ) resp. après le diagramme commutatif précédent, on a pour tout entier naturel n,0 −→ Sel Iw n −→ Sel n ( f α ) −→ Sel # n −→ 0 (III.4.6.1)une suite exacte courte de O[Γ n ]-modules. Elle reste valable pour n = ∞, car le passage à la limite directe est une opération exacte sur les modules discrets.
♯ n et d'autres sous-modules de Hom G (U n , D). Comme U n est libre sur Z p et comme le fait de prendre les G-invariants est exact, on a une suite exacte courte(IV.1.0.1) 0 −→ Hom G (U n , T) −→ Hom G (U n , V ) −→ Hom G (U n , D) −→ 0 IV.1.1. Définition. Pour tout n ∈ N, notons Q p,n ⊆ Q p le n-ième étage de la Z p -extension cyclotomique de Q p et L n = L · Q p,n , et Z p,n et O n leurs anneaux des entiers respectifs. Le choix de L non-ramifiée et contenant H v entraîne que L n ⊇ H n,v et aussi L ∩ Q p,n = Q p . Pour n = ∞, on posera Q p,∞ = n Q p,n et L ∞ = n L n . On définit enfin sur l'extension L ∞ /L un opérateur trace (normalisé) : Tr : L ∞ −→ L x → 1 p n Tr L n /L (x) si x ∈ L n Notons que Tr est bien définie, car si x ∈ L n et si m ≥ n, alors 1 p n Tr L n /L (x) = 1 p m Tr L m /L (x). IV.1.2. Soit n ∈ N. L'application (x, y) −→ Tr(x y) est une forme L-bilinéaire sur L n qui est symétrique et non-dégénérée. Pour R un sous-Omodule de L n , on définit R ⊥ := {y ∈ L n / ∀x ∈ R, Tr(x y) ∈ O}. L'application R → R ⊥ est décroissante, et l'on a (aR) ⊥ = a −1 R ⊥ pour tout a ∈ L × n . Si 0 = R ⊆ L n est un O n -module (nécessairement libre de rang 1) alors R ⊥ en est aussi un. De même, si R est un O-réseau de L n (i.e. R est un sous-O-module de rang p n ), alors R ⊥ aussi. IV.1.3. Notation. Dans toute la suite, on fixe une O-base (t + , t − ) de T (et donc aussi une Lbase de V ) adaptée à la décomposition T = T + T − . La matrice de ρ(g) dans cette base sera notée a g b g c g d g ∈ GL 2 (O). De plus, on notera ( f + , f − ) les coordonnées relatives à cette base d'un morphisme f à valeurs dans V . IV.1.4. Proposition.
.
Proposition. R ± n,loc est un réseau de L n , et pour n = 0, on a R ± 0,loc = pO.Démonstration. Posons ζ = β ou ζ = α selon le choix du signe ±. On a clairement R ± n,loc ⊆ p 1−n O H n,v · O n = p 1−n O n d'après le paragraphe IV.1.5. Montrons que R ± n,loc contient un réseau de L n . L'application définissant R ± n,loc est G v -équivariante, donc d'après le même paragraphe, R ± n,loc contient le O-module S = m (pO H n,v ⊗ O) Frob v =ζ . On a le lemme :
autre part, le corps résiduel de H n,v est F p fv ⊆ F n . Donc P n'est pas identiquement nul sur F p f , et donc il existe x ∈ O H n,v tel que P(x mod π n ) = 0, et un tel x convient.Soit x 0 comme dans le lemme. Posons y 0 = f −1 j=0 ζ − j F j (x). On a O · p y 0 ⊆ S ainsi que tp y 0 = p f −1 j=0 ζ − j F j (t·x 0 ) ∈ S pour tout t ∈ Z p,n .On a donc O·Z p,n · p y 0 ⊆ S. Comme L et Q p,n sont arithmétiquement disjoints, on a O · Z p,n = O n , et comme y 0 est une unité de O n , on a O.Z p,n .p y 0 = pO n . On a ainsi montré que pO n ⊆ S ⊆ R ± n,loc ⊆ p 1−n O n . Donc R ± n,loc est un réseau de L n , et pour n = 0, on a bien R ± 0,loc = pO.IV.1.10. Corollaire. Soit f = Φ n (z + , z − ) ∈ Hom G (U n , V ) (c.f. proposition IV.1.
après IV.1.12 où Tr désigne l'opérateur trace (défini en IV.1.1) appliqué à chaque entrée de la matrice. On observe que M i est une matrice commutant avec ρ(G) qui est absolument irréductible. Donc M i est scalaire, disons M i = λ i I 2 . En prenant la trace matricielle, on trouve λ i = d 2 Tr(s + i z + + s − i z − ). Comme p ∤ 2d, on obtient que f (E n ) ⊆ T si et seulement si Tr(s + i z + + s − i z − ) ∈ O pour tout 1 ≤ i ≤ p n , d'où la première équivalence. La deuxième équivalence est similaire : on a f (E n ) = 0 si et seulement si M i = 0 pour tout 1 ≤ i ≤ p n , c'est-à-dire si Tr(s + i z + + s − i z − ) = 0. IV.1.14. Corollaire. On a un isomorphisme de O[Γ n ]Grâce à la suite exacte IV.1.0.1, on voit que le O[Γ n ]-module Sel #
⊥
Enfin, pour z − ∈ R − n,loc ⊥ , on a z − ∈ R + n,gl ⊥ donc Tr(s − i z − ) ∈ O pour tout 1 ≤ i ≤ p n . Donc Sel # n est isomorphe à :
D'après le lemme de Schur et le théorème IV.2.5, on en déduit que Hom Q[G n ] (W, QV ⊗ Q Q(ζ)) = 0 quelque soit ζ ∈ µ p n . Ainsi, on a bien Hom Q[G] (W, QV ) = 0. Démonstration de la proposition IV.2.1. Soit f ∈ Sel n (Q p V ). D'après la preuve de la proposition IV.1.4, f peut s'écrire commef (x ⊗ c) = g∈G log p (ι p (g −1 (x))c) ρ(g) z + z −avec z + , z − ∈ Q p . D'après le corollaire IV.1.10, la condition f − (U n,v ) = 0 entraîne z − = 0. Quitte à normaliser f , on peut supposer que z + ∈Q. Comme les coefficientsde ρ(h) (pour h ∈ G) sont dansQ, on en déduit que f ∈ Hom(U n /E n ,Q p )Q ⊗ Q Q p V G . D'après le corollaire IV.2.6, on a ainsi f = 0. IV.2.7. Corollaire (Théorème A). Pour tout n ∈ N, Sel n ( f α ) et Sel ♯ n sont finis, et R + n,gl est un réseau de L n .Démonstration. Soit n ∈ N. D'après la preuve de la proposition III.4.4, les L-espaces vectorielsHom G (U n , V ) et Hom G (E n , V )×Hom G v (U n,v , V − ) onttous deux une dimension égale à 2p n . Donc d'après la proposition IV.2.1, la flèche verticale au milieu du diagramme commutatif IV.2.0.1 est un isomorphisme. Ainsi, Sel # n est fini. D'après IV.1.14, R + n,gl ⊥ est un réseau de L n , et donc R + n,gl aussi. Enfin, Sel n ( f α ) est fini d'après la suite exacte III.4.6.1 et ce qui a été montré précédemment. IV.2.8. Pour n = 0, le corollaire précédent dit que le générateur s unité multiplicative de O près) est non-nul. Cet élément est le logarithme p-adique d'une unité globale dont on rappelle la construction. On fixe un isomorphisme entre E ρ = e ρ (O × H ⊗ Z O) et T, et on choisit x + 0 ∈ E ρ la préimage d'un générateur du sous-O-module T + de T. Avec les notations IV.1.7, on a alors s + 0 = m • (log p •ι p ⊗ 1)(x + 0 ) ∈ pO. On écrit simplement s + 0 = log p (x + 0 ). Comme ρ est à coefficients dans Q[ρ] ≃ ι p (Q[ρ]) ⊆ L, on peut trouver un O × -multiple de s + 0 qui soit le logarithme d'un élément de (O × H ⊗ Z Q[ρ]) ρ .
.
p (ǫ ρ )O ⊥ = log p (ǫ ρ ) −1 O d'une part, et R + Le groupe de Selmer est de torsion. Rappelons que l'on a un isomorphisme entre l'algèbre de groupe complétée O[[Γ]] et l'algèbre d'Iwasawa Λ = O[[T]], envoyant le générateur topologique γ sur 1 + T. Nous prouvons dans ce paragraphe la proposition suivante. V.1.1. Proposition. Le Λ-module X ∞ ( f α ) est de torsion. Soit L p ( f α , T) sa fonction L p-adique. Alors L p ( f α , 0) = 0 si et seulement si α = 1, et l'on a toujours L p (ζ − 1) = 0 pour ζ ∈ µ p ∞ − {1}. D'après le corollaire III.3.2, on a pour tout n ∈ N ∪ {∞} : Sel n ( f α ) ≃ ker Hom G (X n , D) r n / / Hom G v (I p (M n /H n ), D − ) On a une application naturelle Sel n ( f α ) −→ Sel ∞ ( f α ) Γ p n , dont on va décrire le noyau et le conoyau. V.1.2. Lemme. La flèche de restriction X ∞ −→ X n induit un diagramme commutatif de Z p [G n ]-/ I p (M n /H ∞ ) / / ? I p (M n /H n ) / / ? I p (H ∞ /H n ) / / 0 Démonstration. On a clairement M n ⊇ H ∞ , et donc on a une tour d'extensions M ∞ /M n /H ∞ /H n .
a pas de sous-modules pseudo-nuls non-triviaux. Démonstration. On va vérifier les hypothèses de la proposition II.3.1, pour A = O et M = T.Le point (a) a été vérifié dans la proposition V.1.1. Notons Σ l'ensemble des diviseurs premiers de N p et montrons que H 2 (Q Σ /Q ∞ , D) est de cotorsion sur Λ. D'après la suite spectrale de Hochschild-Serre, le noyau de l'application de restrictionH 2 (Q Σ /Q ∞ , D) −→ H 2 (Q Σ /H ∞ , D) est contrôlé par la cohomologie du groupe Gal(H ∞ /Q ∞ ) ≃ G à valeur dans certains Z p -modules discrets, qui est triviale car p ∤ d. Il suffit donc de montrer que H 2 (Q Σ /H ∞ , D) est de cotorsion. Celui-ci est isomorphe à H 2 (Q Σ /H, D ∞ ) d'après le lemme de Shapiro. Comme Q Σ = H Σ , on peut utiliser la formule de caractéristique d'Euler-Poincaré de [Gre06, Proposition 4.1] donnant : Corang(H 2 (Q Σ /H ∞ , D)) = Corang(H 1 (Q Σ /H ∞ , D)) − Corang(H 0 (Q Σ /H ∞ , D)) − 2r 2 (H) où r 2 (H) est le nombre de places complexes de H. Le Λ-corang du H 0 est 0, car il est égal à D ≃ (L/O) 2 , qui est de co-torsion sur Λ. Il faut donc montrer que le corang du H 1 est égal à 2r 2 (H). Comme Q Σ est aussi la plus grande extension de H ∞ non-ramifiée en dehors de Σ, et que l'action de G H ∞ sur D est triviale, on a H 1 (Q Σ /H ∞ , D) = Hom(Gal(M ∞,Σ /H ∞ ), D) où M ∞,Σ est la plus grande pro-p extension abélienne de H ∞ non-ramifiée en dehors de Σ. Son groupe de Galois X ∞,Σ a le même rang sur Λ que X ∞ , qui est égal à r 2 (H) d'après [Iwa73, Theorem 17]. Comme Hom(X ∞,Σ , D) ∨ = Hom(T, X ∞,Σ ) ≃ X ⊕2 ∞,Σ d'après le lemme III.1.1, ce dernier Λ-module a pour rang 2r 2 (H), ce qui termine la vérification de l'hypothèse (b).
V. 2 . 2 .
22Proposition. Si α = 1, alors le terme constant de L p ( f α , T) est égal (à une unité de O près) à log p (ǫ ρ ) p # Cl p (H) ρ , où ǫ ρ est l'élément de la définition IV.2.10.Démonstration. D'après la proposition V.1.3, on sait queX ∞ ( f α )/T X ∞ ( f α ) est fini et à le même ordre que X 0 ( f α ), c'est-à-dire que Sel 0 ( f α). D'après le lemme I.3.3, le sous-module X ∞ ( f α )[T] est donc fini, et d'après la proposition V.2.1, il est nécessairement trivial. Ainsi, toujours d'après le lemme I.3.3, L p ( f α , 0) est égal (à une unité de O près) à l'ordre de Sel 0 ( f α ). Ce dernier module est une extension de Hom G (Cl p (H), D) par Sel # 0 . Le groupe Hom G (Cl p (H), D) a même ordre que son dual Hom G (T, Cl p (H)) qui est égal à # Cl p (H) ρ d'après le lemme III.1.1. Le groupe Sel # 0 a quant à lui un O-ordre égal à log p (ǫ ρ ) p d'après le corollaire IV.2.10. Cela termine la preuve de la formule annoncée. VI. THÉORIE D'IWASAWA ET FAMILLES DE HIDA VI.1. Familles de Hida. VI.1.1. Spécialisations. On pose u = 1 + p un générateur du groupe multiplicatif 1 + pZ p . Soit H une extension finie de Z p [[X ]]. On dit qu'un morphisme d'anneaux φ : H −→ Q p est une spécialisation classique de H s'il existe un entier k ≥ 1 et une racine de l'unité ζ ∈ µ p ∞ primitive d'ordre p r−1 (où r > 0) tels que φ(X ) = ζu k−1 − 1. On dit que φ est une spécialisation algébrique si, de plus, k ≥ 2. On pose k φ := k, r φ := r, et aussi χ φ = χ ζ . La spécialisation φ définit par ailleurs un idéal p φ := ker φ de H, qui est premier de hauteur 1, et un anneau O φ := im φ qui est une extension finie de Z p .VI.1.2. Une forme parabolique ordinaire Z p [[X ]]-adique de niveau modéré N et de caractère ǫ est un q-développement formel F = n≥0 a n (F)q n à coefficients dans une extension finie H de Z p [[X ]] telle que, pour toute spécialisation algébrique φ de H, le q-développement
VI.1.3. Soit H N l'algèbre de Hecke universelle ordinaire de niveau modéré N. C'est une Z p [[X ]]algèbre générée par les opérateurs de Hecke agissant sur l'espace des formes paraboliques ordinaires. Elle est libre de rang fini d'après [Hid86b, Theorem 3.1], et ses spécialisations algébriques correspondent aux formes paraboliques propres de niveau modéré N. Le quotient H new N de H N agissant fidèlement sur l'espace des formes N-nouvelles est de même une Z p [[X ]]algèbre finie réduite et sans torsion. Ses spécialisations algébriques sont de plus en bijection avec les (orbites galoisiennes de) formes propres classiques de poids k ≥ 2 et niveau modéré N qui sont nouvelles en N. VI.2. Fonctions L p-adiques analytiques en familles et en poids 1. Pour attacher une fonction L p-adique à f α , on commence par déformer f α p-adiquement, ce qui est possible grâce à [Wil88, Theorem 3], donnant le résultat suivant. VI.2.1. Théorème (Wiles). Il existe une famille de Hida primitive F se spécialisant en f α . D'après le théorème principal de [BD16], on sait même que F est unique à conjugaison près sous l'hypothèse (reg) (c.f. [Dim14, Corollary 7.7]). La famille F définit un morphisme d'anneaux H new N −→ Frac(Z p [[X ]]), dont le noyau a est un idéal premier minimal de H new N . On peut alors voir F à coefficients dans H F := H new N /a. On note φ α : H F −→ Q p la spécialisation classique de poids 1 associée à f α , et on note simplement p α = ker φ α l'idéal premier de hauteur 1 associé à f α . Les coefficients de Fourier de f α sont dans O, donc φ α est à valeurs dans O. VI.2.2. Remarque. Soit φ une spécialisation algébrique de poids k de H F telle que p − 1|k − 1 et χ φ = 1. Alors, par définition, g φ := F(φ) est de niveau N p et son nebentypus ǫ est de niveau N. Comme p ∤ N, g φ est nécessairement p-old d'après [Miy06, Theorem 4.6.17/2]. Ainsi, g φ est la p-stabilisation d'une forme primitive de niveau N. VI.2.3. La Z p [[X ]]-algèbre H N est finie donc semi-locale, elle est isomorphe au produit fini de ses localisés aux idéaux maximaux m (H N ) m . On note Hρ le localisé de H N en l'idéal maximal correspondant àρ. Lorsqueρ est absolument irréductible et p-distinguée, l'anneau Hρ est Gorenstein. [EPW06] construit alors un élément de H m [[T]] à l'aide de symboles modulaires qui interpole les fonctions L p-adiques usuelles des spécialisations algébriques de Hρ, c'est-àdire des formes paraboliques propres p-ordinaires de niveau modéré N dont la représentation galoisienne associée est résiduellement isomorphe àρ. VI.2.4. Proposition. Supposons (hyp) et (reg) vérifiées. Il existe un élément L p (F, T) ∈ H F [[T]], défini à une unité de H F près, satisfaisant la propriété suivante : pour toute spécialisation al-
−1 est le caractère envoyant γ sur ξ, avec e g (ξ, m) = 1 − a p (g φ ) −1 p m et t ′ = 0 lorsque ξ = 1 et p − 1|m, et avec e g (ξ, m) = 1 et t ′ = t sinon, où G(−) désigne la somme de Gauss usuelle, et où Ω ± g est une période canonique de g, définie à une unité de O φ près. Le choix des périodes peut par ailleurs être effectué simultanément pour toutes les spécialisations g φ . Démonstration. Sous les hypothèses (hyp) et (reg),ρ est irréductible et p-distinguée. On peut alors considérer l'élément L(m, N, ω 0 ) ∈ Hρ[[T]] défini (à une unité de Hρ près) dans [EPW06, Section 3.4], et on définit L p (F, T) comme étant son image par la projection naturelle Hρ[[T]] ։ H F [[T]]. La formule est la conséquence directe de la proposition 3.4.3 de loc. cit. VI.2.5. Définition. Supposons (hyp) et (reg) vérifiées. On définit L p ( f α , T) ∈ O[[T]] la fonction L p-adique analytique de f α comme étant l'image de L p (F, T) par l'application H F [[T]] −→ O[[T]] induite par φ α . Elle est bien définie à une unité multiplicative de O près.
T) Toutes les fonctions L p-adiques sont des éléments de l'anneau topologique O ′ [[T]], et les divisibilités sont dans O ′ [[T]][ 1 p ]. Le point (a) est une application du théorème de Kato (Théorème VI.4.3). Les points (b) et (c) sont respectivement démontrés dans le lemme VII.3.1 et la proposition VII.4.2, après avoir construit une fonction L p-adique analytique et algébrique au voisinage de f α dans les paragraphes VII.3 et VII.4. Une fois les points (a), (b) et (c) prouvés, la preuve du théorème C découle immédiatement du lemme suivant, pour A = O ′ [[T]], π une uniformisante de O ′ , a n = L p (g n , T), a = L p ( f α , T), b n = L p (g n , T), et b = L p ( f α , T). VII.1.1. Lemme. Soit A un anneau topologique compact. Soient (a n ) n et (b n ) n deux suites d'éléments de A convergeant respectivement vers a et b.
de nebentypus ǫ et à coefficients dans O M d'après le corollaire VII.2.5 la remarque VI.2.2. On note φ n : H F −→ O M le morphisme de spécialisation correspondant à g n et p n = p φ n ⊆ H F . Notons que dans O M , coefficients du q-développement de g n convergent vers ceux de f α lorsque n → ∞. VII.3. Fonction L p-adique analytique au voisinage de f α . On définit une fonction L padique analytique au voisinage de f α comme étant l'élément L † p (Y , T) ∈ A[[T]] obtenu en prenant l'image de L p (F, T) par le morphisme H F [[T]] → A[[T]] induit par Φ. Ainsi, d'après le corollaire VII.2.5 et avec les notations VII.2.6, on a :L † p (0, T) = L p ( f α , T) et L † p (y n , T) = L p (g n , T)pour tout entier naturel n.VII.3.1. Lemme. La suite (L p (g n , T)) n converge vers L p ( f α , T) dans O M [[T]].Démonstration. Il est clair que toute série formelleF(Y , T) ∈ O M [[Y , T]] définit une application continue y −→ F(y, T) sur O M − O × M à valeurs dans O M [[T]]. En considérant F(Y , T) = L † p (Y , T), on a donc lim n→+∞ L p (g n , T) = lim n→+∞ L † p (y n , T) = L † p (0, T) = L p ( f α , T) dans O M [[T]].VII.4. Fonction L p-adique algébrique au voisinage de f α . Il n'existe pas a priori de fonction L p-adique algébrique associée au groupe de Selmer X ∞ (T F , T + F ), ni même d'idéal caractéristique, car l'anneau H F [[T]] n'est pas nécessairement factoriel, ni même intégralement clos (c.f. définitions I.1.3 et I.3.1). Notre paramétrage local de la famille de Hida nous permet de surmonter ce problème et de travailler sur un anneau de séries formelles A[[T]] ≃ O M [[Y , T]], qui est factoriel et sur lequel on peut en outre appliquer les résultats des sections I et II.VII.4.1. On notera simplement lesA-modules T = T F ⊗ H F ,Φ A et D = T ⊗ A A ∨ . On définit similairement T ± et D ± . On définit aussi les A[[T]]-modules Sel ∞ (F † ) = Sel ∞ (T, T + ), resp. X ∞ (F † ) = Sel ∞ (F † ) ∨Rappelons que pour tout entier naturel n, le groupe de Selmer Sel ∞ (φ n ) attaché à g n est de torsion d'après le théorème VI.4.3. Nous démontrons dans la suite la proposition suivante.VII.4.2. Proposition. La suite (L p (g n , T)) n converge vers L p ( f α , T) dans O M [[T]]. VII.4.3. Lemme. On a des isomorphismes de O M [[T]]-modules : X ∞ (F † )/ (Y − y n ) · X ∞ (F † ) ≃ X ∞ (φ n ) ⊗ O φn O M , et X ∞ (F † )/Y · X ∞ (F † ) ≃ X ∞ ( f α ) ⊗ O O M(VII.4.3.1) pour tout entier naturel n. En particulier, X ∞ (F † ) est de torsion sur A[[T]]. Démonstration. Cela résulte d'une application directe de la proposition II.2.1. Avec les notations de la proposition, on pose A = A, T = T, T ± = T ± , et a est l'idéal principal de A engendré par Y − y n ou bien par Y . Montrons que ses hypothèses sont vérifiées. L'idéal a est principal par définition. Le groupe d'inertie en p agit trivialement sur D − parce que ceci est déjà le cas pour T − F . Il reste à montrer que D G Q∞ et D I ℓ (pour l|N) sont A-divisibles. Il suffit de démontrer qu'ils sont A-colibres. On note D g n = V g n /T g n le O M -module discret usuel construit à partir de la représentation de Deligne de g n , et on note encore D le O M -module D ⊗ O O M . Les propriétés de spécialisation énoncées dans le théorème VI.3.1 entraînent les identifications D g n ≃ D[Y − y n ] et D ≃ D[Y ]. Soit ̟ une uniformisante de O M , soit F M = O M /̟ et notons M = (̟, Y ) l'idéal maximal de A. La représentation résiduelle D de ρ F vérifie D ≃ D g n [̟] ≃ D[̟] en tant que F M [G Q ]-modules. On a déjà D G Q∞ = 0. En effet, comme l'action galoisienne est A-linéaire, on a D G Q∞ [Y ] = D G Q∞ . Comme H ∩Q ∞ = Q et ρ est irréductible, on a D G Q∞ = D G Q = 0. D'après le lemme de Nakayama, on a bien D G Q∞ = 0. Soit ℓ|N, et montrons que D I ℓ est colibre sur A. Soit P = D I ℓ ∨ le dual de Pontriyagin de D I ℓ . Alors on a P/MP = D I ℓ ∨ = D I ℓ . C'est un F M -espace vectoriel, de dimension 0 si a ℓ (F † ; Y ) ∈ M, et de dimension 1 si a ℓ (F † ; Y ) est une unité de A. Dans le premier cas, on a P = 0 par le lemme de Nakayama, donc P est libre. Supposons que P/MP est de dimension 1. Alors P est monogène, et donc P ≃ A/(f (Y )) pour un certain f (Y ) ∈ A. Par ailleurs, on a a ℓ (g n ) ∈ O × M pour tout entier naturel n, donc P/ (Y − y n ) P ≃ T g n I ℓ est de O M -rang égal à 1. Cela implique que l'élément f (Y ) s'annule en toutes les valeurs Y = y n , n ∈ N, et donc f (Y ) = 0 d'après le théorème de préparation de Weierstrass. Donc P est libre (de rang 1) comme annoncé, et cela termine la vérification. D'après le théorème B, X ∞ ( f α ) est de O[[T]]-torsion, donc X ∞ (F † )/Y · X ∞ (F † ) est de torsion sur O M [[T]]. L'assertion élémentaire suivante montre alors que X ∞ (F † ) est de torsion sur A[[T]]. Soit M un module de type fini sur un anneau B. Supposons qu'il existe un idéal premier Q ⊆ B tel que M/QM est de torsion sur B/Q. Alors il existe s ∈ B − Q qui tue M, et en particulier M est de torsion. VII.4.4. Lemme. Le A[[T]]-module X ∞ (F † ) n'a pas de sous-modules pseudo-nuls non-triviaux. Démonstration. On va montrer que l'on peut appliquer la proposition II.3.1 aux données A = A, T = T et T + = T + . L'hypothèse (a) est vérifiée d'après le lemme VII.4.3. Montrons que (b) est satisfaite, à savoir que le module H := H 2 (Q Σ /Q ∞ , D) ∨ est de torsion sur A[[T]]. On a montré dans la preuve de la proposition V.2.1 que H 2 (Q Σ /Q ∞ , D) ∨ est de type fini et de torsion sur O M [[T]]. La multiplication par Y définit une suite exacte application surjective H 2 (Q Σ /Q ∞ , D) ։ H 2 (Q Σ /Q ∞ , D)[Y ]. Donc le O M [[T]]-module H/Y · H est un sous-module d'un module de type fini et de torsion. Il est donc de type fini et de torsion, et il en est de même pour le A[[T]]-module H. L'hypothèse (c) est clairement vérifiée, car la représentation ρ F est impaire. L'hypothèse (d) aussi, car I p agit trivialement sur D − . Enfin, l'hypothèse (e) est vérifiée car ρ F est résiduellement irréductible de dimension 2. Cela termine la vérification des hypothèses, et donc la preuve du lemme. Une variante du lemme précédent pour X ∞ (F) = X ∞ (T F , T + F ) est montrée dans [Och06, Proposition 8.1] sous l'hypothèse que H F est régulier. Preuve de la proposition VII.4.2. Le groupe de Selmer X ∞ (F † ) est de type fini et de torsion sur l'anneau A[[T]] qui est factoriel, donc possède une fonction L p-adique algébrique L † Sel ∞ (φ n ) et de Sel ∞ ( f α ) sont engendrés respectivement par L † p (y n , T) et L † p (0, T). Autrement dit, on a : L p (g n , T) = L † p (y n , T) resp. L p ( f α , T) = L † p (0, T) L'argument de la preuve du lemme VII.3.1 montre alors que lim n→+∞ L p (g n , T) = L p ( f α , T) dans O M [[T]], ce qui termine la preuve de la proposition VII.4.2. RÉFÉRENCES
TABLE DES MATIÈRES
DESIdéaux caractéristiques et fonctions L p-adiques algébriques 6 I.1. Idéaux premiers de hauteur 1 et idéaux divisoriels 6 I.2. Idéaux caractéristiques et changement de bases 7 I.3. Fonctions L p-adiques algébriques 9 II. Groupe de Selmer d'une représentation ordinaire Lissité et paramétrage local de la famille de Hida 31 VII.3. Fonction L p-adique analytique au voisinage de f α 33 VII.4. Fonction L p-adique algébrique au voisinage de f αIntroduction
2
Quelques notations
5
I. 10
II.1. Groupe de Greenberg-Selmer
10
II.2. Changement de bases
12
II.3. Structure des groupes de Selmer
12
III. Groupe de Selmer d'une forme modulaire de poids 1
14
III.1. Préliminaires
14
III.2. Définition de Sel n ( f α )
15
III.3. Isomorphismes de restriction
15
III.4. Groupes de Selmer auxiliaires
16
IV. Finitude des groupes de Selmer
18
IV.1. Traces et réseaux
18
IV.2. Fin de la preuve du théorème A
23
V. Preuve du Théorème B
25
V.1. Le groupe de Selmer est de torsion
25
V.2. Absence de sous-modules pseudo-nuls et terme constant de L p ( f α )
27
VI. Théorie d'Iwasawa et familles de Hida
27
VI.1. Familles de Hida
27
VI.2. Fonctions L p-adiques analytiques en familles et en poids 1
28
VI.3. Groupe de Selmer en famille
29
VI.4. Conjectures principales
30
VII. Preuve du théorème C
30
VII.1. Articulation de la preuve
30
VII.2. 33
Références
35
INTRODUCTION
il existe un morphisme injectif de Z p [[X ]]-algèbres H F → O[[X /p r ]], fournissant un paramétrage local F † (X ) de F au voisinage de f α par le poids, dans un disque de centre X = 0 et de rayon p r .H F ) p
est local régulier de dimension 1, i.e. c'est un anneau valuation discrète (c.f.[Dim14, Proposi-
tion 7.4]). Supposons par commodité que (H F ) p est non-ramifié au-dessus de Z p [[X ]] (X ) . Alors,
La section I constitue des rappels d'algèbre commutative sur les idéaux caractéristiques, et redémontre des résultats standard relatifs aux changements de bases. La section II rappelle la définition du groupe de Selmer attaché à une représentation ordinaire. On étudie les changements de bases, et donnons des conditions suffisantes pour qu'ils soient sans sous-modules pseudo-nuls non-triviaux. La section III définit le groupe de Selmer de f α . Les sections IV et V sont dédiées aux preuves des théorèmes A et B. La section VI prépare la preuve du théorème C, qui est donnée dans la dernière section.Après complétion de ce travail nous avons pris connaissance de la prépublication de Greenberg et Vatsal ([GV18]) qui contient des résultats similaires aux théorèmes A et B.QUELQUES NOTATIONSPour E ⊆ F ⊆ Q on note I ℓ (F/E) le groupe d'inertie de la place définie par le plongement ι ℓ : Q →Q ℓ dans l'extension F/E. Représentation et caractères. Soit G un groupe fini d'ordre d premier à p, et soit O l'anneau des entiers d'une extension finie et non-ramifiée L de Q p contenant µ d . On peut attacher à toute
Comme d est inversible dans O, on a même e π ∈ O[G]. L'application de réduction GL n (O) −→ GL n (O/p) modulo p a pour noyau un pro-p groupe, donc toute représentation irréductible π de G à valeurs dans O est résiduellement irréductible. Le choix d'un réseau G-stable T π ⊆ V π est donc unique à homothétie près. Dans l'algèbre de groupe O[G], on a π e π = 1, et e π e π ′ = 0 pour toutes représentations irréductibles π = π ′ de G. Pour tout O[G]-module M, on a ainsi une décomposition en somme directe M = π e π M et pour M = O[G], on obtient Dualité de Pontryagin. Pour un Z p -modules localement compact M, on note M ∨ = Hom ct (M, Q p /Z p ) le dual de Pontryagin de M. Le foncteur M → M ∨ induit une équivalence de catégories entre la catégorie des Z p -modules discrets de torsion et les Z p -modules compacts. Un Z p -module est autodual si et seulement si il est fini. L'application de trace Tr L/Q p réalise un accouplement parfait O × O −→ Z p , donc on a un Oisomorphisme Hom ct (O, Z p ) ≃ O, et donc O ∨ ≃ O ⊗ Z p Q p /Z p . Plus généralement, pour un Omodule M discret de torsion ou bien compact, on aO[G] =
π
e π O[G] ≃
π
T dimπ
π
caractère galoisien se factorisant par Γ n et envoyant γ sur ζ. Le caractère χ ζ est d'ordre p n et de conducteur p n+1 . I. IDÉAUX CARACTÉRISTIQUES ET FONCTIONS L p-ADIQUES ALGÉBRIQUES Nous rappelons dans cette section comment attacher un idéal caractéristique a à un module M sur un anneau A. Il faut pour cela supposer que A est un anneau de Krull, et que M est de type fini et de torsion sur A. Notre référence principale est [Bou07, Chapitre VII]. Comme tous nos anneaux sont noethériens, il faut et il suffit de supposer que A est intégralement clos. Si de plus A est un anneau factoriel, alors a est principal et on définira la fonction L p-adique algébrique de M comme étant un générateur de a. Dans les applications nous considérerons typiquement des anneaux de séries formelles à coefficients dans O ou dans Z p . Nous étudions dans un deuxième temps le lien entre idéaux caractéristiques et changements de bases, en vue de la preuve du théorème C. Dans toute la suite, A désigne un anneau noethérien intégralement clos, de corps des fractions K . I.1. Idéaux premiers de hauteur 1 et idéaux divisoriels. Soit P A l'ensemble des idéaux premiers de hauteur 1 de A, c'est-à-dire l'ensemble des idéaux premiers non-nuls p ne contenant pas d'autre idéal premier que (0). On a le théorème classique suivant ([Bou07, Chap. VII, Théorème 1.7.4]). I.1.1. Théorème.
sont injectives. La première l'est clairement, car par hypothèse les H 1 sont des Hom, et la deuxième est injective par le même argument que précédemment (et utilisant le fait que D I ℓ est A-divisible). II.3. Structure des groupes de Selmer. Jusqu'à la fin de cette section, nous supposons que A est un anneau de séries formelles à coefficients dans une extension finie O de Z p , de corps résiduel F. On a ainsi A = O[[X 1 , . . ., X m ]], m ≥ 1 ou bien A = O. Notons m A son idéal maximal.On conserve les notations des paragraphes précédents, et on note d (resp. d ± ) le rang de T (resp. de T ∨ ). On introduit de plus le A-module libre (de rang d)
) .
)La remarque II.1.5 montre que le groupe de Selmer défini dans loc. cit. est bien égal à Sel ∞ (T, T + ), et par ailleurs, la conclusion de la proposition 4.1.1 de loc. cit. est équivalente à ce que X ∞ (T, T + ) n'a pas de sous-modules pseudo-nuls non-triviaux d'après [Gre06, Proposition 2.4].
). Ce dernier module est de corang fini sur A, en particulier son corang sur A[[Γ]] est nul, ce qu'on voulait démontrer. Montrons maintenant que L ∨ p n'a pas de sous-A[[Γ]]-modules pseudo-nuls non-triviaux. On a
Or ceci est vrai d'après [Gre06, Proposition 3.7], qui s'applique car on sait que
On va montrer que les deux termes de l'égalité sont égaux à d − sous les hypothèses de la proposition. D'après [Gre06, Proposition 4.1],
on a
Corang(H 1
loc⊥ et ∀1 ≤ i ≤ p n , Tr(s +
i z + ) ∈ O
R +
n,loc
⊥
× R −
n,loc
⊥
=
R +
n,gl
⊥
× R −
n,loc
⊥
R +
n,loc
⊥
× R −
n,loc
⊥ =
R +
n,gl
⊥
R +
n,loc
⊥ .
IV.2. Fin de la preuve du théorème A. Considérons le diagramme commutatif
(IV.2.0.1)
0
0
Hom G (U n , T)
Définition. On définit Hom(U n /E n , Q p ) Q comme étant l'intersection de Hom(U n /E n , Q p ) avec Hom(U n , Q p ) Q dans Hom(U n , Q p ).Hom(U n /E n , Q p ) Q ≃) . On va
montrer que Sel n (Q p V ) est nul. Suivant [BD16], on introduit les définitions suivantes.
IV.2.3. Définition. On note Hom(U n , Q p ) Q le sous-Q-espace vectoriel de Hom(U n , Q p ) engen-
dré par les log p (ι p • g −1 ⊗ 1) quand g décrit G, c'est-à-dire l'image de Q[G] via l'isomorphisme
IV.1.4.1.
IV.2.4. Le théorème de Baker-Brumer sur la Q-indépendance de logarithmes p-adiques de nombres al-
gébriques permet à Bellaiche et Dimitrov de déterminer la structure de Q[G n ]-module à gauche
de Hom(U n /E n , Q p ) Q .
IV.2.5. Théorème ([BD16], Theorem 3.5). En tant que Q-représentations de G n , on a
π=1 ou π Frob∞=1 =0
Kato a prouvé une divisibilité partielle dans le cas où g φ est la p-stabilisation d'une forme primitive ordinaire de niveau premier à p ([Kat04, Theorem 17.4]). D'après la discussion du paragraphe VI.2.2, ceci est valable précisément lorsque p − 1|k φ − 1 et χ φ = 1. On a donc le théorème suivant. VI.4.3. Théorème (Kato). Supposons que p
[ 1 p ] .
1]Nous formulons la conjecture suivante : VI.4.4. Conjecture (=Conjecture A). Supposons les hypothèses (hyp) et (reg) vérifiées. Il existe une unité u de O[[T]] telle que u · L p ( f α , T) = L p ( f α , T) Les arguments de passage à la limite utilisé dans la preuve du théorème C montreront (sous les hypothèses du théorème C) que la conjecture VI.4.2 implique la conjecture VI.4.4. Notre résultat est le suivant : VI.4.5. Théorème (= Théorème C). Supposons les hypothèses (hyp) et (reg) vérifiées. AlorsL p ( f α , T) divise L p ( f α , T) dans O[[T]][ 1 p ]. Deplus, si la conjecture VI.4.2 est vraie pour toute spécialisation g φ de poids k φ suffisamment proche p-adiquement de 1 et de niveau N p, alors la conjecture VI.4.4 est vraie. VII. PREUVE DU THÉORÈME C VII.1. Articulation de la preuve. Nous supposons dans cette section que les hypothèses (hyp) et (reg) sont vérifiées. Comme déjà noté dans l'introduction, on sait alors que (H F ) p α est un anneau de valuation discrète d'après le résultat principal de [BD16] (voir aussi [Dim14, Proposition 7.4]). Dans le paragraphe VII.2, on construit un paramétrage de F au voisinage de f α , donnant une suite de formes modulaires primitives p-stabilisées g n (c.f. notation VII.2.6) dont les coefficients de Fourier vivent dans une extension finie O ′ de O et convergent vers ceux de f α . Les étapes de la preuve du théorème C sont rassemblées dans le schéma suivant :
VII.2. Lissité et paramétrage local de la famille de Hida.VII.2.1. Soit F(W) = n c n W n ∈ Q p [[W]] une série formelle de rayon de convergence positif. Alors il existe un entier r tel que F(W) converge sur le disque de rayon p −r , c'est-à-dire que la suite (c n p nr ) n est bornée. On peut donc voir F(Y ) comme un élément du sous-anneauZ p [[W/p r ]][ 1 p ] de Q p [[W]].VII.2.2. Lemme. Soit F(W) ∈ Q p [[W]]. Supposons que F(W) est entier sur Z p [[W]]. Alors il existe une extension finie M de Q p et un entier r tel que F(W) ∈ O M [[W/p r ]]. Démonstration. Soit Q(W, Z) ∈ Z p [[W]][Z] un polynôme (en la variable Z) unitaire de degré d Q qui s'annule en F(W). Comme Q p [[W]] est intègre, Q(W, Z) a un nombre fini de racines dans Q p [[W]]. Le théorème d'approximation d'Artin [Art68, Theorem 1.2] implique que, pour tout entier c ≥ 0 et pour toute racine G(W) de Q(W, Z), il existe une série formelleĜ(W) ∈ Q p [[W]] de rayon de convergence positif qui est une racine de Q(W, Z) et qui satisfait la congruence G(W) ≡Ĝ(W) mod W c . En choisissant c suffisamment grand, on voit que l'on a G(W) =Ĝ(W), et donc toute racine du polynôme Q(W, Z) converge au voisinage de 0. En particulier, il existe un entier r tel que F(W) ∈ Z p [[W/p r ]][ 1 p ]. Soit M le compositum de toutes les extensions de Q p de degré inférieur ou égal à d Q . Pour tout w ∈ p r+1 Z p , l'équation Q(w, F(w)) = 0 implique que F(w) satisfait une équation de degré d Q , et donc f (w) ∈ M. On en déduit que F(W) est à coefficients dans M, et donc F(W) ∈ O M [[W/p r ]][ 1 p ]. Enfin, comme F(W) est entier, il est p-entier et donc F(W) ∈ O M [[W/p r ]] comme annoncé. VII.2.3. Soient r ≥ 0 et e ≥ 1 des entiers et M une extension finie de L. On notera O M [[X 1/e /p r ]] la O M [[X ]]-algèbre O M [[X , Y ]]/(p re Y e − X ). Un morphisme de spécialisation X = a pour a ∈ m C p s'étend à O M [[X 1/e /p r ]] dès que p −re a ∈ m C p , et dépend du choix d'une racine e-ième de p −re a dans C p . VII.2.4. Proposition. Il existe des entiers r et e ≥ 1, une extension finie M/L suffisamment grande et un morphisme (injectif) de Z p [[X ]]-algèbres Φ tel qu'on ait un diagramme commutatif : / / O M [[X 1/e /p r ]] Démonstration. Comme (H F ) p α est un anneau de valuation discrète d'égale caractéristique, son complété est un anneau de séries formelles en une variables sur sur son corps résiduel. Donc il existe un morphisme injectif d'anneaux topologiquesΦ : (H F ) p α → Q p [[Y 0 ]] où Y 0 est une variable formelle. L'élément X ∈ (H F ) p α satisfait lim n→+∞ X n = 0, donc comme élément de Q p [[Y 0 ]],X est une série formelle sans terme constant X = aY e 0 + bY e+1 0 + . . . où e ≥ 1 et a = 0. Soit H(Y 0 ) ∈ Q p [[Y 0 ]] une racine e-ième de la série a + bY 0 + . . ., de sorte que X = (Y 0 H(Y 0 )) e . Comme H(0) = 0, on voit que Y = Y 0 H(Y 0 ) définit une nouvellevariable formelle, i.e. on a un isomorphisme d'anneaux topologiques Q p [[Y 0 ]] ≃ Q p [[Y ]] ≃ Q p [[X 1/e ]]. La restriction de Φ à H F donne un morphisme de Z p [[X ]]-algèbres H F → Q p [[X 1/e ]]. Comme H F est finie sur Z p [[X ]], l'image de ce morphisme est inclue dans un anneau de la forme O M [[X 1/e /p r ]] d'après le lemme VII.2.2, où M est une extension finie de Q p . On note encore Φ le morphisme de Z p [[X ]]algèbres obtenu : Φ : H F → O M [[X 1/e /p r ]] Il vérifie par construction Φ −1 (X 1/e /p r ) = p α , ce qui rend le diagramme VII.2.4.1 commutatif. VII.2.5. Corollaire. Soit Y la variable formelle X 1/e /p r et soit A := O M [[X 1/e /p r ]] ≃ O M [[Y ]].(VII.2.4.1)
H F
φ α
/ /
Φ
' '
O M
Z p [[X ]]
O O
X =0
O O
Démonstration. Comme G agit trivialement sur Γ p n ≃ Z p , la composante ρ-isotypique de ce dernier module est triviale. D'après le lemme précédent, on a donc Hom G (X n , D) ≃ Hom G (X ∞ , D) Γ p n , ainsi qu'une suite exacte : Preuve de la proposition V.1.1. Notons ω n (T) = (1 + T) p n − 1. Avec l'identification γ = 1 + T, le groupe Γ p n est généré par ω n (T) + 1. Ainsi, pour un Λ-module Z, on a Z Γ p n = Z/ω n (T)Z.Notons δ α = 1 si α = 1 et δ α = 0 si α = 1. D'après la proposition V.1.3, le O-rang de X ∞ ( f α ) Γ p n est égal à δ α pour tout entier naturel n. Soit r le Λ-rang de X ∞ ( f α ). D'après le théorème de structure des Λ-modules de type fini, il existe des polynômes irréductibles P i (T) de Λ, des entiers naturels m i et un pseudo-isomorphismeOn a deux suites exactes courtesLa multiplication par ω n (T) induit deux suites exactes longues O-modules de type fini. Comme C et D sont finis, elles impliquent les deux égalités de rangs suivantes :On en déduit que pour tout entier naturel n, on a :(E n ) : δ α = r p n + i deg T pgcd(P i (T) m i , ω n (T)).En particulier, on obtient r = 0, c'est-à-dire X ∞ ( f α ) est de torsion sur Λ. On peut donc définir L p ( f α , T) comme étant égale au produit des P i (T) m i . Comme ω 0 (T) = T, l'équation (E 0 ) montre
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| []
|
[
"Information Rate Optimization for Non-Regenerative MIMO Relay Networks with a Direct Link",
"Information Rate Optimization for Non-Regenerative MIMO Relay Networks with a Direct Link"
]
| [
"Giorgio Taricco "
]
| []
| []
| We consider the optimization of a two-hop relay network based on an amplify-and-forward Multiple-Input Multiple-Output (MIMO) relay. The relay is assumed to derive the output signal by a Relay Transform Matrix (RTM) applied to the input signal. Assuming perfect channel state information about the network at the relay, the RTM is optimized according to two different criteria: i) network capacity; ii) network capacity based on Orthogonal Space-Time Block Codes. The two assumptions have been addressed in part in the literature. The optimization problem is reduced to a manageable convex form, whose KKT equations are explicitly solved. Then, a parametric solution is given, which yields the power constraint and the capacity achieved with uncorrelated transmitted data as functions of a positive indeterminate. The solution for a given average power constraint at the relay is amenable to a water-fillinglike algorithm, and extends earlier literature results addressing the case without the direct link. Simulation results are reported concerning a Rayleigh relay network and the role of the direct link SNR is precisely assessed. | null | [
"https://arxiv.org/pdf/2104.05236v2.pdf"
]
| 238,253,478 | 2104.05236 | d14acc7bd95fe85508d4e4ce1ef214f804762610 |
Information Rate Optimization for Non-Regenerative MIMO Relay Networks with a Direct Link
Giorgio Taricco
Information Rate Optimization for Non-Regenerative MIMO Relay Networks with a Direct Link
1Index Terms-Relay networksInformation rateRelay Trans- form MatrixConvex OptimizationWater-filling
We consider the optimization of a two-hop relay network based on an amplify-and-forward Multiple-Input Multiple-Output (MIMO) relay. The relay is assumed to derive the output signal by a Relay Transform Matrix (RTM) applied to the input signal. Assuming perfect channel state information about the network at the relay, the RTM is optimized according to two different criteria: i) network capacity; ii) network capacity based on Orthogonal Space-Time Block Codes. The two assumptions have been addressed in part in the literature. The optimization problem is reduced to a manageable convex form, whose KKT equations are explicitly solved. Then, a parametric solution is given, which yields the power constraint and the capacity achieved with uncorrelated transmitted data as functions of a positive indeterminate. The solution for a given average power constraint at the relay is amenable to a water-fillinglike algorithm, and extends earlier literature results addressing the case without the direct link. Simulation results are reported concerning a Rayleigh relay network and the role of the direct link SNR is precisely assessed.
I. INTRODUCTION
Wireless communication systems have been using relaying techniques for several decades in order to extend the range coverage of radio networks. Among the benefits, relays help to combat shadowing and fading effects which may limit the signal propagation in wireless environments. Back in the day, the concept of relaying has been rationalized, from an information theoretical point of view, by the introduction of the three-terminal channel model (source-relay-destination) in the seminal papers by Van Der Meulen [1] and Cover and El Gamal [2], which determined the achievable rate under several operating conditions. During the last two decades, several results emerged in the framework of single and multiple antenna systems. As far as single-antenna systems are concerned, Sendonaris et al. studied the effects of relaying as a user cooperation diversity technique to increase the cellular coverage of third-generation systems based on CDMA [3]; Nabar et al. investigated different time-division multiple-access-based cooperative protocols with relay terminals operating in either amplify-and-forward or decode-and-forward modes by using the achievable rate as the metric of interest [4]; Laneman et al. proposed low-complexity cooperative diversity protocols to combat multipath fading in wireless networks by using Giorgio Taricco ([email protected]) is with Politecnico di Torino (DET). A limited part of these results have been presented in IEEE ISIT 2021 [27]. several strategies including fixed relaying schemes [Amplifyand-Forward (AF) and Decode-and-Forward (DF)], adaptive relaying schemes, and incremental relaying schemes based upon limited feedback from the destination terminals [5]; Host-Madsen et al. provided lower and upper bounds to the outage and ergodic capacity of a three-terminal wireless relay channel in Rayleigh fading while taking into account practical constraints at the relay node and the impact of power allocation [6].
Relaying based on Multiple-Input Multiple-Output (MIMO) wireless terminals has been studied in [7], [8]. Specifically, joint transmission and reception at the relay was addressed in the paper by Wang et al. [7] but, as pointed out by Tang and Hua, [8], it may entail unwanted side effects since, typically, the transmitted signal power at the relay overshadows the power of the received signal. As a result, a more practical approach consists of keeping the reception and transmission processes at the relay orthogonal with respect to each other. Orthogonality can be implemented by operating the system in a two-hop time division or by frequency-domain division multiple access scheme.
Focusing on the relayed signal, two basic approaches have been considered in the literature, which can be classified as regenerative or non-regenerative. The regenerative approach consists of rebuilding the transmitted signal after decoding the received signal, and is commonly referred to as decode-andforward (DF). The non-regenerative consists of forwarding the received signal after amplification, thereby including the received noise. This latter approach is commonly referred to as amplify-and-forward (AF). For single-antenna systems, it has been observed that AF schemes are advantageous in terms of achievable diversity order with respect to DF schemes while the situation is not clearly understood as far as concerns capacity. Nevertheless, AF schemes offer a number of benefits making them preferable to DF schemes [8]. More recently, it has been pointed out that AF schemes enable to retain the soft information of the transmitted signal and guarantee a limited signal delay at the same time [9]- [11].
In the framework of MIMO-AF relay schemes, the transmitted signal is obtained by the joint amplification of the different received signal components so that it can be characterized by a Relay Transform Matrix (RTM), which derives the transmitted signal vector through multiplication by the received signal vector. Following the classification introduced by Tang and Hua [8], we consider three operating schemes for the MIMO relay system considered: i) Direct Link without Relay; ii) Relay without Direct Link; and iii) Relay with Direct Link. A key contribution from [8] is the derivation of the information-theoretically optimum RTM (i.e., maximizing the system capacity) for the second operating scheme (Relay without Direct Link). The authors also considered the more general third operating scheme (Relay with Direct Link) but didn't find the optimum RTM in this case and claimed this case to be an open problem [8, p.1400]. This operating scheme was also considered by Shariat and Gazor [9], who focused on the optimization of the capacity constrained to the use of Orthogonal Space-Time Block Codes (OSTBC). Their approach is also equivalent to the maximization of the overall Signal-to-Noise Ratio (SNR).
Several works considered the Relay with Direct Link scheme (also in the presence of a precoder at the source) with the goal of obtaining lower or/and upper bounds to the achievable rate [12]- [15]. More recently, some work considered the impact of imperfect channel state information, known according to its statistic distribution, and obtained upper bounds to the ergodic capacity [16].
The RTM optimization has also been studied under a different optimization criterion, namely, the Minimum Mean-Square Error (MMSE) minimization. Several works adopted this approach, such as [17]- [19]. In general, MMSE optimization is not equivalent to capacity optimization but this approach lends itself to a simpler solution of the optimization problem.
In this work we present an algorithm to derive the RTM optimizing the capacity of a two-hop relay network. The case considered here is more general than [8], [9] for many reasons. First of all, we allow the number of transmit and receive antennas at the relay to be arbitrarily different, as well as the channel matrix ranks. On the contrary, it was assumed in [8] that the number of transmit and receive antennas at the relay was the same and [9] assumed that the rank of the source to destination channel matrix (H 1 in this paper) was equal to the number of receive antennas, so that the case t < r (see Fig. 1 for the definitions) was not included. Additionally, the solution presented here applies to the joint direct link and relay transmission case (labeled as "Case (C) Relay With Direct Link" in [8]), recognized as an open problem by the authors of [8]. Our solution can be obtained, for a specific relay power constraint, by resorting to a water-filling-like algorithm, bearing some similarity with [8, Sec.IV] (which is nevertheless not applicable to this case). For validation purposes, we report numerical simulation results coherent with those from [8, (B) Relay Without Direct Link] by forcing the direct link channel matrix to zero. Then, we extend the analysis by considering also the case of a full relay network, including the direct link, first with an overall constant number of antennas in the relay network, next, with different number of antennas. A distinguishing feature of this work with respect to earlier literature results (e.g., [12]- [16]) is the consideration of the actual capacity instead of some lower or upper bound (which nevertheless provide valuable contributions).
Summarizing, the paper organization is as follow. Section II introduces the system model for the MIMO relay network with all relevant parameters which characterize it completely. Then, Section III solves the optimization problems corresponding to capacity and OSTBC-capacity maximization in the fully general case of arbitrary channel matrix ranks and dimensions. Section III-A addresses capacity optimization and extends the work of [8]. Section III-B proposes the relevant parametric solution. Section III-C addresses OSTBC-capacity optimization and extends partly the work of [9] and Section III-D proposes the relevant parametric solution. Section IV collects three types of relay network scenarios to illustrate the application of the theoretical results of the previous section. The first scenario consists of a relay network without the direct link and is considered for validation and comparison with the results of [8]. The other scenarios consider a full relay network with constant number of antennas (where [9] is applicable as far as OSTBC-capacity is concerned) an a second scenario with different numbers of antennas (where [9] is not applicable even in the case of OSTBC-capacity). Concluding remarks are collected in Section V.
II. SYSTEM MODEL
We consider a MIMO relay network consisting of three nodes: the source (S) equipped with t transmit antennas; the destination (D), equipped with r receive antenna; and the relay (R), equipped with u transmit and s receive antennas. The channel matrices corresponding to the three different links of interest are labeled as H 0 (S→D), H 1 (S→R), and H 2 (R→D). The system operates in two-hop relaying mode: the source transmits during the first hop and the relay during the second hop. The average power transmitted by the source and the relay are upper bounded by P 1 and P 2 , respectively. We assume that the relay applies a u × s Relay Transformation Matrix (RTM) X to the received signal before forwarding it to the destination in the second hop. The resulting channel equations are given as follows:
y 0 = H 0 x +z 0 (Hop 1, S→D) y 1 = H 1 x +z 1 (Hop 1, S→R) y 2 = H 2 Xy 1 +z 2 (Hop 2, R→D) = H 2 XH 1 x +H 2 Xz 1 + z 2(1)
We assume, w.l.o.g., that the received noise components are iid (otherwise, we can pre-multiply the received vectors and channel matrices by the inverse matrix square roots of the corresponding noise correlation matrices). Then, 1
z 0 , z 2 ∼ CN (0, I r ), z 1 ∼ CN (0, I s )(2)
The equivalent channel equation becomes wherez ∼ CN (0, I 2r ). This channel equation leads directly to the equation representing the capacity of the relay network reported in the following eq. (5).
y = H 0 H 2 XH 1 x + z 0 H 2 Xz 1 + z 2(3)
III. RTM OPTIMIZATION
In this section we address the calculation of the optimum RTM based on the assumption that the relay knows all the channel matrices involved in eq. (1). Specifically, we look for the RTM which maximizes the two-hop relay channel capacity.
A. Optimum RTM
In the absence of Channel State Information at the Transmitter (CSIT), the capacity is achieved when x ∼ CN (0, P1 t I t ) and is given by
C = log 2 det I t + P 1 t H H 0 H 0 + H H 1 X H H H 2 (I r + H 2 XX H H H 2 ) −1 H 2 XH 1(5)
The average power constraint at the relay can be expressed in terms of the RTM X and the channel matrices as follows:
tr XX H + P 1 t XH 1 H H 1 X H = tr X I s + P 1 t H 1 H H 1 X H ≤ P 2 . (6)
The optimum RTM (maximizing the capacity (5) under the constraint (6)) is given by the following Theorem.
Theorem 1 Given the two-hop MIMO relay network described by eqs. (1) with average source and relay power constraints P 1 and P 2 , the optimum (capacity-maximizing) RTM X is given by
X = U B Λ −1/2 B Λ 1/2 U H A ,(7)
where the matrices U B , Λ B , U A are obtained by the "thin" unitary diagonalizations (UD's) [20, Th. 7.3.2]: 2
A = U A Λ A U H A , B = U B Λ B U H B .(8)
2 A "thin" UD U ΛU H of an n × n matrix is characterized by an m × m diagonal matrix Λ whose diagonal entries are sorted in nonincreasing order,
i.e., (Λ) i,i ≥ (Λ) i+1,i+1 for i = 1, . . . , m − 1 and a semi-unitary n × m matrix U with the property that U H U = Im. where A H 1 t P 1 I t + H H 0 H 0 + H H 1 H 1 −1 H H 1 B H H 2 H 2 , C I s + P 1 t H 1 H H 1 (9)
We also have
Λ diag(x 1 , . . . , x ρ , 0, . . . , 0 ρ B −ρ )(10)
where ρ min(s, ρ B ) and ρ B rank(B) = rank(H 2 ) ≤ min(u, r). The diagonal matrix Λ A is possibly extended by zero padding to the size ρ B × ρ B . The matrix Λ has ρ ≤ ρ B possibly positive eigenvalues, obtained by solving the convex optimization problem
min x≥0 − ρ i=1 ln 1 − α i 1 + x i s.t. ρ i=1 β i x i ≤ P 2 , x i ≥ 0, i = 1, . . . , ρ(11)
where, for i = 1, . . . , ρ,
α i ( Λ A ) i,i , β i ( U H A C U A ) i,i ( Λ B ) i,i(12)
Proof: See App. A.
B. Parametric Water-Filling solution
We can get a closed-form parametric solution of the optimization problem (11) in Theorem 1 based on a single ξ > 0. To this end, we define 3
ϕ i (ξ) α i 2 − 1 + α 2 i 4 + α i β i ξ + , i = 1, . . . , ρ. (13)
These functions provide the components of the vector x, solution of the optimization problem (11) in Theorem 1, as
x i = ϕ i (ξ)
. Accordingly, we obtain two parametric equations:
P 2 = ρ i=1 β i ϕ i (ξ)(14)C = log 2 det I t + P 1 t (H H 0 H 0 + H H 1 H 1 ) + ρ i=1 log 2 1 − α i 1 + ϕ i (ξ)(15)
These expressions are obtained by solving the KKT equations corresponding to the optimization problem (11) and are derived in detail in App. B. The uniqueness of the solution of (14) stems from the fact that the functions ϕ i (ξ) are monotonically increasing for ξ > ξ i (1 − α i )β i . Since ϕ i (ξ) = 0 for ξ ≤ ξ i , we can find solve (14) numerically by dividing the real positive line {ξ : ξ > 0} through the sorted thresholds ξ i and considering over each interval so determined only the positive functions. This remains nevertheless a nonlinear equation. The approach recalls the solution of the water-filling equation arising in the case of independent additive Gaussian channels with an overall average power constraint [23].
C. RTM optimization based on OSTBC capacity
Instead of considering the optimization of the RTM to maximize the capacity, one may consider the maximization of the Orthogonal Space-Time Block Coding (OSTBC) capacity, as defined in [21]. This approach has been followed in [9]. Unfortunately, it was assumed in [9] that the matrix H 1 H H 1 has always full rank s, which may not be true, for example, if t < s, and limits the generality of the result. For this reason we provide here a derivation of the optimum RTM achieving the OSTBC capacity in the general case.
The OSTBC capacity with symbol rate R of the MIMO relay channel is given by [21]:
C OSTBC = R log 2 1 + P 1 tR tr H H 0 H 0 + H H 1 X H H H 2 (I r + H 2 XX H H H 2 ) −1 H 2 XH 1(16)
The optimum RTM (maximizing the above capacity for every R) is given in the following Theorem.
Theorem 2 Given the two-hop MIMO relay network described by eqs.
(1) with average source and relay power constraints P 1 and P 2 , the optimum (OSTBC capacitymaximizing) RTM X is given by
X = U B Λ −1/2 B Λ 1/2 U H A ,(17)
where the matrices U B , Λ B , U A are obtained by the "thin" unitary UD'š
A = U A Λ A U H A ,B = U B Λ B U H B .(18)
wherě
A H 1 H H 1 ,B H H 2 H 2 , C I s + P 1 t H 1 H H 1(19)
We define Λ as in (10), ρ min(s, ρ B ) and ρ B rank(B) = rank(H 2 ) ≤ min(u, r). Λ A is possibly extended by zero padding to the size ρ B ×ρ B . The x i , i = 1, . . . , ρ are obtained by solving the optimization problem
min x≥0 ρ i=1 α i 1 + x i s.t. ρ i=1 β i x i ≤ P 2 , x i ≥ 0, i = 1, . . . , ρ(20)
where, for i = 1, . . . , ρ,
α i ( Λ A ) i,i , β i ( U H A C U A ) i,i ( Λ B ) i,i(21)
Proof: See App. C.
D. Parametric Water-Filling solution
Here we provide a closed-form parametric solution to the optimization problem considered in Theorem 2, based on an independent positive variable ξ. Using the definitions of Theorem 2, we define
ψ i (ξ) ξ α i β i − 1 +(22)
Accordingly, we obtain these two parametric equations:
P 2 = ρ i=1 β i ψ i (ξ)(23)C = log 2 1 + P 1 t H H 0 H 0 + H H 1 H 1 − ρ A i=1 (Λ A ) i,i 1 + ψ i (ξ)(24)
These expressions are derived in detail in App. D.
IV. NUMERICAL RESULTS
The numerical results in this section are presented to validate the algorithms derived in cases already handled in the literature and to show their applicability to cases where the literature algorithms are not applicable.
A. Validation of the results
Here, we compare our algorithms with the results presented by Tang and Hua in their paper, specifically [8,Figs. 3 and 4]. In that use case, the authors assumed that all the antenna numbers are the same, i.e., t = r = u = s = M = 4, and considered the ergodic capacity corresponding to a relay system whose channel matrices have all iid Rayleigh distributed fading gains with unit variance without a direct link from source to destination (more precisely, the entries of H 1 , H 2 are iid CN (0, 1) and H 0 ≡ 0). We resort to the following definitions of SNR's:
ρ 1 P 1 M σ 2 1 , ρ 2 P 2 M σ 2 2 .(25)
According to the previous assumptions, we can simplify the expression of the relay network capacity (5) as
C = max X:tr{X(I M +ρ1H1H H 1 )X H }=M ρ1 log 2 det I M + ρ 2 H H 1X H H H 2 I M + ρ 2 ρ 1 H 2XX H H H 2 −1 H 2X H 1(26)
Similarly, the OSTBC relay network capacity (16) becomes
C OSTBC = max X:tr{X(I M +ρ1H1H H 1 )X H }=M ρ1 log 2 1 + tr ρ 2 H H 1X H H H 2 I M + ρ 2 ρ 1 H 2XX H H H 2 −1 H 2X H 1(27)
In both cases, we setX P 2 /P 1 X so that the capacity expressions are independent of P 1 , P 2 and depend only on ρ 1 , ρ 2 .
For this scenario, Fig. 2 illustrates the ergodic capacity vs. ρ 1 at fixed ρ 2 = 10 dB and Fig. 3 illustrates the ergodic capacity vs. ρ 2 at fixed ρ 1 = 10 dB. Each figure reports the six curves with a label composed of two tags: the first tag denotes the type of RTM used (OPT1,OPT2,NAF) and the second tag denotes the type of capacity plotted (ergodic capacity (26) or ergodic OSTBC capacity (27)). The types of RTM's considered are: i) OPT1: RTM maximizing the relay network capacity (5); ii) OPT2: RTM maximizing the OSTBC capacity (16); iii) NAF: Naive Amplify and Forward, where the RTM is a scaled identity matrix. The results agree exactly with those reported in [8,Figs. 3 and 4].
The performances illustrated in Figs. 2 and 3 agree with the basic expectations. The OPT1 RTM maximizes the ergodic capacity and is suboptimal for the OSTBC capacity. The OPT2 RTM maximizes the ergodic OSTBC capacity and is suboptimal for the capacity. The NAF RTM has the worst performance in all cases. It is quite noticeable from Fig. 2 that the OPT2 capacity curve is strongly degraded at large ρ 1 , as a consequence of the profound mismatch between the capacity and the OSTBC capacity.
As far the asymptotic behavior of the capacity curves in Figs. 2 and 3 is concerned with, we notice a key difference. In Fig. 2, when ρ 1 → ∞, all curves converge to different limits, which depend on the fixed value of ρ 2 . On the contrary, in Fig. 3, when ρ 2 → ∞, all the capacity and OSTBC capacity curves converge to the same limits, respectively, which depend on the fixed value of ρ 1 . The difference is consistent with the following information theoretical interpretation. The relay network is the cascade of two channels, channel 1 (source to relay) and channel 2 (relay to destination). As such, the dataprocessing inequality [23] must be satisfied and the overall capacity is upper bounded by the capacity of each channel. In the case illustrated in Fig. 2, when ρ 1 → ∞, channel 1's capacity increases without bound so that the relay network capacity coincides with that of channel 2, and is affected by the RTM. Hence, the different limits. As far as OSTBC-capacity maximization is concerned, when ρ 1 → ∞, we can see that the RTM rank tends to 1, so that the capacity and the OSTBCcapacity have the following limiting behavior:
C → max X log 2 det{I M + ρ 2 H H 1X H H H 2 H 2X H 1 } (28) C OSTBC → max X log 2 {1 + tr(ρ 2 H H 1X H H H 2 H 2X H 1 )} (29)
Thus, under the limit power constraint tr{XH 1 H H
1X
H } = M , they tend to the same limit.
On the contrary, in the case illustrated in Fig. 3, when ρ 2 → ∞, channel 2's capacity goes to infinity so that the relay network capacity coincides with that of channel 1, which is independent of the RTM. Hence, the coincidence of the limits. Moreover, the upper ergodic capacity limits in both figures coincide. Finally, we notice that the ergodic OSTBC capacity always entails a major loss (even in the full-rate case) with respect to the ergodic capacity.
B. Full Relay Network -Equal Number of Antennas
In this case we consider a relay network where also the direct link is present, contrary to the scenario considered in Section IV-A. We still assume that all antenna arrays have the same number of antennas, t = r = u = s = M = 4, and the channel matrices are iid Rayleigh as before (i.e., all entries of H 0 , H 1 , H 2 are uncorrelated CN (0, 1) distributed). Finally, and we define the SNR's as
ρ 0 P 1 M σ 2 2 , ρ 1 P 1 M σ 2 1 , ρ 2 P 2 M σ 2 2 .(30)
Thus, we can simplify the relay network capacity (5) as:
C = max X:tr{X(I M +ρ1H1H H 1 )X H }=M ρ1 log 2 det I M + ρ 0 H H 0 H 0 + ρ 2 H H 1X H H H 2 I M + ρ 2 ρ 1 H 2XX H H H 2 −1 H 2X H 1(31)
Similarly, the OSTBC relay network capacity (16) becomes
C OSTBC = max X:tr{X(I M +ρ1H1H H 1 )X H }=M ρ1 log 2 1 + tr ρ 0 H H 0 H 0 + ρ 2 H H 1X H H H 2 I M + ρ 2 ρ 1 H 2XX H H H 2 −1 H 2X H 1(32)
The two relay network capacity expressions are independent of P 1 , P 2 for given ρ 0 , ρ 1 , ρ 2 . Fig. 4 illustrates the ergodic capacity behavior vs. ρ 2 with ρ 1 = 10 dB and two values of ρ 0 = −10, 10 dB. Again, the optimum for capacity (OPT1), for OSTBC-capacity (OPT2) and naive amplify and forward (NAF) RTM's are considered. Interestingly, we note that, for low ρ 0 , e.g., −10 dB, the results are very close to those reported in Fig. 3. This condition is close to having no direct link because most power passes through the relay. Increasing ρ 0 impacts drastically on the performance results, as illustrated. These results allow to assess the trade-offs implied by the presence of the direct link, which is a key contribution of this work.
In a similar way, Fig. 5 plots the ergodic capacity vs. ρ 0 with fixed ρ 1 = 10 dB and several values of ρ 2 : 0, 10, 20, 30 dB. In this case, the curves increase monotonically with respect to the link SNR ρ 0 and reach a limit as ρ 2 → ∞.
By these results we can see when the RTM optimization is worth the effort or rather naive amplify and forward is sufficient for a given scenario. For example, we can see from Fig. 5 a clear advantage when ρ 2 = 10 dB, which decreases progressively by increasing ρ 2 , until it becomes very small for ρ 2 = 30 dB. Then, if the relay-to-destination SNR ρ 2 is very large, there is little gain available from RTM optimization, while the gain is substantial in the range of moderate values as 10 dB.
C. Full Relay Network -Different Number of Antennas
To conclude this selection of simulation scenarios we consider the case when the number of antennas in the relay network is variable so that the results of the literature are not applicable both for the capacity [8] and for the OSTBCcapacity [9]. In particular, we consider the scenario where the number of transmit and receive antennas of the source and destination are t = r = 2 and the number of transmit and receive antennas at the relay are s = u = 2 or 4 or 8. Here, we define Ergodic capacity (bit/s/Hz) Ergodic capacity (bit/s/Hz) We can see that increasing the number of relay antennas is quite beneficial to the relay network. In fact, the limit ergodic capacity with ρ 0 = ρ 1 = 10 dB and ρ 2 → ∞ increases from 9.9 to 11.4 and 13.0 bit/s/Hz as the number of relay antennas increases from s = u = 2 to 4 and 8, respectively, while the number of transmit and receive antennas at the source and destination remain fixed and equal to 2. Comparatively, the capacity of the direct link without the relay for ρ 0 = 10 dB is 7.14 bit/s/Hz [26]. These results show the effectiveness of a MIMO relay with different numbers of antennas on the capacity.
ρ 0 P 1 tσ 2 2 , ρ 1 P 1 tσ 2 1 , ρ 2 P 2 uσ 2 2 .(33)OPT1-CAP-SNR0= -10dB OPT1-OSTBC-SNR0= -10dB OPT2-CAP-SNR0= -10dB OPT2-OSTBC-SNR0= -10dB NAF-CAP-SNR0= -10dB NAF-OSTBC-SNR0= -10dB OPT1-CAP-SNR0= 10dB OPT1-OSTBC-SNR0= 10dB OPT2-CAP-SNR0= 10dB OPT2-OSTBC-SNR0= 10dB NAF-CAP-SNR0= 10dB NAF-OSTBC-SNR0= 10dBOPT1-CAP-SNR2= 10dB OPT1-OSTBC-SNR2= 10dB OPT2-CAP-SNR2= 10dB OPT2-OSTBC-SNR2= 10dB NAF-CAP-SNR2= 10dB NAF-OSTBC-SNR2= 10dB OPT1-CAP-SNR2= 20dB OPT1-OSTBC-SNR2= 20dB OPT2-CAP-SNR2= 20dB OPT2-OSTBC-SNR2= 20dB NAF-CAP-SNR2= 20dB NAF-OSTBC-SNR2= 20dB OPT1-CAP-SNR2= 30dB OPT1-OSTBC-SNR2= 30dB OPT2-CAP-SNR2= 30dB OPT2-OSTBC-SNR2= 30dB NAF-CAP-SNR2= 30dB NAF-OSTBC-SNR2= 30dB
V. CONCLUSIONS
The work focuses on the optimization of the Relay Transformation Matrix (RTM) in a two-hop amplify-and-forward Ergodic capacity (bit/s/Hz) relay network. The contributions extend nontrivially earlier results from the literature. The seminal work by Tang and Hua [8] provided the solution of the optimization problem with the capacity as objective function for a pure relay network (without the direct link). The authors emphasized that the relay network with a direct link case was an open problem at the time and to the author's knowledge it remained so until now. The work by Shariat and Gazor [9] established the interest in the full relay network but focused on the OSTBCcapacity only. Though the OSTBC-optimized RTM provides good results in terms of capacity in many cases, it remains a suboptimal approach and may lead sometimes to considerable performance degradation (see Fig. 2). Moreover, reference [9] imposed some conditions on the number of antennas of the relay network limiting the generality of the results. These limitations are overcome in this work which does not assume any conditions on the channel matrices' ranks and on the number of antennas.
OPT1-CAP-SNR0= -10dB OPT1-OSTBC-SNR0= -10dB OPT2-CAP-SNR0= -10dB OPT2-OSTBC-SNR0= -10dB NAF-CAP-SNR0= -10dB NAF-OSTBC-SNR0= -10dB OPT1-CAP-SNR0= 10dB OPT1-OSTBC-SNR0= 10dB OPT2-CAP-SNR0= 10dB OPT2-OSTBC-SNR0= 10dB NAF-CAP-SNR0= 10dB NAF-OSTBC-SNR0= 10dB
The optimum RTM has been derived in Theorems 1 and 2 for the capacity and OSTBC-capacity, respectively, by different simplified convex optimization problems, whose parametric solutions have been derived in Sections III-B and III-D, respectively. The KKT equations corresponding to the relevant optimization problems have been solved and used to provide parametric expressions of the average power constraint and the capacity as depending only on a single parameter. The solution recalls the structure of water-filling equations.
Simulation results have been presented to compare the capacity achieved by the optimum RTM and by naive amplifyand-forward. It is shown that the capacity advantage due to RTM optimization decreases as the SNR increases but it is still sizable for practical SNR values. To assess the effectiveness of a MIMO relay on an existing 2×2 MIMO link we compared different simulation scenarios corresponding to increasing numbers of relay antennas. For example, we showed in Section IV-C that capacity increases from 7.1 bit/s/Hz (w/o relay) to 9.9, 11.4, and 13.0 bit/s/Hz, by using a relay with 2, 4, and 8 antennas, respectively.
APPENDIX A PROOF OF THEOREM 1
Proof: In order to prove the statement of Theorem 1, we begin with the following elementary linear algebra identity:
K H (I + KK H ) −1 K = K H K(I + K H K) −1 = I − (I + K H K) −1 .(34)
Setting K = H 2 X in (34), we can rewrite the capacity (5) as
C = log 2 det I t + P 1 t (H H 0 H 0 + H H 1 H 1 ) − P 1 t H H 1 (I s + X H H H 2 H 2 X) −1 H 1(35)
According to the definition given in Section III, the optimum RTM is the matrix X that maximizes the capacity reported in eq. (5), under the constraint given by the previous eq. (6).
Subtracting from eq. (35) the constant term (with respect to X)
log 2 det I t + P 1 t (H H 0 H 0 + H H 1 H 1 ) ,(36)
we can see that the optimum RTM is found by solving the following optimization problem:
max X det[I t − A H (I s + X H BX) −1 A] (37) s.t. tr(XCX H ) ≤ P 2(38)
where we defined the matrices A, B, C as in (9). Now, consider the following UD's:
A = U A Λ A U H A (39) X H BX = U ΛU H(40)
The objective function can be upper bounded as follows:
det[I t − A H (I s + X H BX) −1 A] = det[I s − U A Λ A U H A (I s + U ΛU H ) −1 ] = det[I s − U H U A Λ A U H A U (I s + Λ) −1 ] = det[I s − QΛ A Q H (I s + Λ) −1 ] = det(I s + Λ − QΛ A Q H ) det(I s + Λ) ≤ s i=1 1 − (Λ A ) i,i 1 + (Λ) i,i .(41)
Here, we set Q U H U A (i.e., a unitary matrix) and then we applied [24, eq.(2)] after noticing that both Λ and I s − To find an expression of the RTM X, we notice that both sides of (40) have the same rank:
ρ rank(X H BX) = rank(U A ΛU H A ) ≤ min(s, u, r). (42)
If ρ B rank(B), then ρ B ≥ ρ, and we have the following "thin" UD's:
B = U B Λ B U H B u×ρ B ×ρ B ×u U A ΛU H A = U A Λ U H A s×ρ B ×ρ B ×s (43) where U H B U B = U H A U A = I ρ B , Λ B
is the diagonal submatrix of Λ B with the positive elements, and Λ is the unknown diagonal submatrix of Λ with nonnegative elements while the other elements of Λ (if any) are all equal to 0. Thus, eq. (40) is satisfied by setting
X = U B Λ −1/2 B Λ 1/2 U H A .(44)
Remark A.1 Notice that the maximum in (41) is attained regardless of any constraint by the matrix X with the structure given in (44). For every pair of Hermitian positive semidefinite matrices A and B, the matrix defined in (44) maximizes
det[I t − A H (I s + X H BX) −1 A],(45)
and thus the capacity (35). The structure (44) contains ρ free parameters as the diagonal elements of Λ. The relay power constraint is introduced in the following optimization problem. Now, we have to choose Λ in order to i) maximize the upper bound in (41), namely,
s i=1 1 − (Λ A ) i,i 1 + (Λ) i,i ,
and ii) satisfy the relay power constraint (6), which can be written as
ρ i=1 ( U H C U ) i,i ( Λ B ) i,i (Λ) i,i = P 2 .(46)
Notice that the inequality in (6) is turned into an equality since a possibly optimum solution Λ 0 such that
ρ i=1 ( U H C U ) i,i ( Λ B ) i,i (Λ 0 ) i,i = ρP 2 < P 2 ,(47)
for some 0 < ρ < 1, cannot be optimum since ρ −1 Λ 0 would increase all the factors in the upper bound in (41) since
1 − (Λ A ) i,i 1 + ρ −1 (Λ 0 ) i,i > 1 − (Λ A ) i,i 1 + (Λ 0 ) i,i .(48)
The detailed solution of this optimization problem is reported in the following App. B and completes the proof of Theorem 1. (11) From the statement of the optimization problem reported in eq. (11) of Theorem 1, we derive the Lagrangian function of the problem as follows:
APPENDIX B PARAMETRIC SOLUTION OF OPTIMIZATION PROBLEM
L(x,λ 0 , λ 1 , . . . , λ ρ ) = − ρ i=1 ln 1 − α i 1 + x i + λ 0 (β T x − P 2 ) − ρ i=1 λ i x i .(49)
where 0 < α i < 1, β i > 0 and λ i , i = 0, . . . , ρ, are the Lagrange multipliers [22]. Here, we did not consider the constraints x 1 ≥ x 2 ≥ · · · ≥ x ρ since, by Lemma E.1, these constraint are automatically satisfied by any nonnegative solution (x i ≥ 0, i = 1, . . . , ρ). In fact, otherwise, a permutation of the variables would lead to a further decrease of the objective function. Thus, we can save the extra effort that would be required. The KKT equations are obtained according to [22,Sec. 5.5.3]. First, we take the partial derivatives of the Lagrangian function with respect to the variables x i , for i = 1, . . . , ρ:
∂L ∂x i = 1 1 + x i − 1 1 − α i + x i + λ 0 β i − λ i(50)
Then, we have the following KKT equations:
β T x − P 2 ≤ 0 λ 0 (β T x − P 2 ) = 0 λ 0 ≥ 0 −x i ≤ 0 i = 1, . . . , ρ λ i x i = 0 i = 1, . . . , ρ λ i ≥ 0 i = 1, . . . , ρ ∂L ∂x i = 0 i = 1, . . . , ρ(51)
We can see that the objective function
f (x) − ρ i=1 ln 1 − α i 1 + x i(52)
is convex for x ≥ 0 because
∂ 2 f ∂x 2 i = α i (2 − α i + 2x i ) (1 + x i ) 2 (1 − α i + x i ) 2 ≥ 0.(53)
The mixed derivatives ∂ 2 f /(∂x i ∂x j ) = 0 for all i = j. Therefore, we have a convex optimization problem. We can see that Slater's condition is satisfied, so that the KKT equations are sufficient for optimality. The constraint β T x−P 2 ≤ 0 is achieved with equality since f (x) is decreasing with every x i . Therefore, we have λ 0 ≥ 0.
Finally, we obtain from the gradient equations:
1 1 − α i + x i − 1 1 + x i = λ 0 β i − λ i , i = 1, . . . , ρ. (54)
For a given λ 0 ≥ 0, recalling that λ i ≥ 0, x i ≥ 0, λ i x i = 0, there are two possible cases • λ i = 0, which implies that the equation is equivalent to
x 2 i + (2 − α i )x i + 1 − α i − α i λ 0 β i = 0(55)
Since 0 < α i < 1, a solution x i > 0 exists only if
1 − α i − α i λ 0 β i < 0 =⇒ λ 0 < α i (1 − α i )β i(56)
and is given by
x i = α i 2 − 1 + α 2 i 4 + α i λ 0 β i .(57)
• λ i > 0, which implies that one root of (55) must be equal to 0 to satisfy the KKT condition λ i x i = 0. In turn, this implies that
1 − α i − α i λ 0 β i − λ i = 0(58)
and hence
λ 0 = α i (1 − α i )β i + λ i β i > α i (1 − α i )β i ,(59)
so that
α i 2 − 1 + α 2 i 4 + 1 λ 0 β i < α i 2 − 1 + α 2 i 4 + 1 − α i = 0
(60) Summarizing, we can write the solution in all cases as
x i = α i 2 − 1 + α 2 i 4 + α i λ 0 β i +(61)
Thus, the unknown λ 0 ≥ 0 can be found by solving the nonlinear equation
P 2 = ρ i=1 β i α i 2 − 1 + α 2 i 4 + α i λ 0 β i +(62)
A unique solution always exists because the rhs is a monotonically decreasing function of λ 0 , which is identically equal to 0 when λ 0 ≥ max 1≤i≤ρ αi (1−αi)βi . Setting ξ 1/λ 0 yields the parametric solution reported in eqs. (13) to (15).
with N ≤ s 2 −s+1, w ≥ 0, = 1, . . . , N , and N =1 w = 1. Thus,
miň Q:QQ H =I tr{(I +Λ) −1QΛ AQ H } = N =1 w λ T Π λ A = min 1≤ ≤Nλ T Π λ A = min 1≤ ≤N s i=1 (Λ A ) π (i),π (i) 1 + (Λ) i,i(74)
where π is the permutation associated to the permutation matrix Π defined by
(Π ) i,j = δ π (i),j(75)
where δ a,b = 1 if a = b and 0 otherwise (Kronecker delta function). The optimum permutation can be found by applying the lower bound of Lemma E.1 from Appendix E:
miň Q:QQ H =I tr{(I +Λ) −1QΛ AQ H } = ρ A i=1 (Λ A ) i,i 1 + (Λ) i,i(76)
where ρ A rank(Ǎ) = rank(H 1 ) ≤ min(t, s). We notice that the optimum solution found above corresponds to settinǧ Q = I s , which impliesǓ =Ǔ A . We also have to take into account the additional constraint stemming from the inequality ρ rank(Λ) ≤ min(ρ B , s)
where ρ B rank(B) = rank(H 2 ) ≤ min(r, u). Using the "thin" UDB
= U B Λ B U H B u×ρ B ×ρ B ×u ,(78)
eq. (40) is satisfied by setting
X = U B Λ −1/2 B Λ 1/2 U H A ,(79)
where U A is obtained by taking the first ρ columns of U A , and the relay power constraint (6) becomes
ρ i=1 ( U H A C U A ) i,i ( Λ B ) i,i (Λ) i,i ≤ P 2 ,(80)
which completes the proof of Theorem 2. (20) We proceed as in App. B with the Lagrangian L(x,λ 0 , λ 1 , . . . , λ ρ ) =
APPENDIX D PARAMETRIC SOLUTION OF OPTIMIZATION PROBLEM
ρ i=1 α i 1 + x i + λ 0 (β T x − P 2 ) − ρ i=1 λ i x i .(81)
The Lagrangian derivatives are
∂L ∂x i = − α i (1 + x i ) 2 + λ 0 β i − λ i(82)
The KKT equations remain the same as (51) from App. B.
The objective function is
f (x) ρ i=1 α i 1 + x i ,(83)
which is plainly convex for x ≥ 0 so that we have a convex optimization problem. Slater's condition is satisfied so that the KKT equations are sufficient for optimality. Again, β T x − P 2 ≤ 0 is achieved with equality since f (x) is decreasing with every x i , so that λ 0 ≥ 0. The gradient equations are:
α i (1 + x i ) 2 = λ 0 β i − λ i , i = 1, . . . , ρ.(84)
For a given λ 0 ≥ 0, recalling that λ i ≥ 0, x i ≥ 0, λ i x i = 0, we can get the solution:
x i = ξ α i β i − 1 +(85)
where ξ λ −1/2 0 and {·} + max(0, ·). Thus, the unknown ξ > 0 can be found by solving the nonlinear equation 4
P 2 = ρ i=1 (ξ α i β i − β i ) · 1 ξ> √ βi/αi
A unique solution always exists because the rhs is a monotonically increasing function of ξ for ξ ≥ min 1≤i≤ρ β i α i .
APPENDIX E SEQUENCE PRODUCT SUM LEMMA
Lemma E.1 Given any two real nonnegative nonincreasing sequences α i , β i , i = 1, . . . , n such that α i ≥ α i+1 and β i ≥ β i+1 , for i = 1, . . . , n − 1, we have, for every permutation π, the following inequality:
n i=1 α i β n+1−i ≤ n i=1 α i β π(i) ≤ n i=1 α i β i .(86)
Proof: Since every permutation π ∈ S n can be expressed as a product of disjoint cycles [25, Sec. III.70], we have to prove the inequalities only when π is a cycle and then apply it to any π ∈ S n after proper relabeling of the indexes. Let us assume, w.l.o.g., that π = (1, . . . , n), i.e., the permutation 1 → 2 → 3 → · · · → n → 1. For the upper bound, we have to show that α 1 (β 1 − β 2 ) + α 2 (β 2 − β 3 ) + · · · + α n (β n − β 1 ) ≥ 0.
The above inequality stems from the following: α 1 (β 1 − β 2 ) + α 2 (β 2 − β 3 ) + · · · + α n (β n − β 1 ) = (α 1 − α n )(β 1 − β 2 ) + · · · + (α n−1 − α n )(β n−1 − β n ) ≥ 0, since α i −α n ≥ 0 and β i −β i+1 ≥ 0 for every i = 1, . . . , n−1. 4 1 A = 1 when A is true and 0 otherwise. Similary, for the lower bound, we have to show that α 1 (β n − β n−1 ) + α 2 (β n−1 − β n−2 ) + · · · + α n (β 1 − β n ) ≤ 0.
The above inequality stems from the following: α 1 (β n − β n−1 ) + α 2 (β n−1 − β n−2 ) + · · · + α n (β 1 − β n ) = (α 1 − α n )(β n − β n−1 ) + · · · + (α n−1 − α n )(β 2 − β 1 ) ≤ 0, since α i −α n ≥ 0 and β i −β i−1 ≤ 0 for every i = 1, . . . , n−1.
Fig. 2 .Fig. 3 .
23Plot of the ergodic capacity vs. ρ 1 (denoted by SNR1) with ρ 2 = 10 dB, iid Rayleigh fading and three types of RTM. i) OPT1: optimum RTM for capacity. ii) OPT2: optimum RTM for full-rate OSTBC capacity. iii) NAF: Naive Amplify and Forward, the RTM is a scaled identity matrix. Plot of the ergodic capacity vs. ρ 2 (denoted by SNR2) with ρ 1 = 10 dB, iid Rayleigh fading and three types of RTM. i) OPT1: optimum RTM for capacity. ii) OPT2: optimum RTM for full-rate OSTBC capacity. iii) NAF: Naive Amplify and Forward, the RTM is a scaled identity matrix.
Figs. 6
6to 8 show the ergodic capacity of this relay network vs. ρ 2 with ρ 1 = 10 dB and two values of ρ 0 = −10, 10 dB.
Fig. 4 .
4Plot of the ergodic capacity vs. ρ 2 (denoted SNR2) with ρ 0 = −10, 10 dB, ρ 1 = 10 dB, iid Rayleigh fading and three types of RTM. i) OPT1: optimum RTM for capacity. ii) OPT2: optimum RTM for full-rate OSTBC capacity. iii) NAF: Naive Amplify and Forward, the RTM is a scaled identity matrix.
Fig. 5 .
5Plot of the ergodic capacity vs. ρ 0 (denoted SNR0) with ρ 2 = 0, 10, 20, 30 dB, ρ 1 = 10 dB, iid Rayleigh fading and three types of RTM. i) OPT1: optimum RTM for capacity. ii) OPT2: optimum RTM for full-rate OSTBC capacity. iii) NAF: Naive Amplify and Forward, the RTM is a scaled identity matrix.
Fig. 6 .Fig. 7 .Fig. 8 .
678Plot of the ergodic capacity vs. ρ 2 (denoted SNR2) with ρ 0 = −10, 10 dB, ρ 1 = 10 dB, iid Rayleigh fading, relay network with t = r = 2 and s = u = 2, and three types of RTM. i) OPT1: optimum RTM for capacity. ii) OPT2: optimum RTM for full-rate OSTBC capacity. iii) NAF: Naive Amplify and Forward, the RTM is a scaled identity matrix. Same asFig. 6but s = u = 4. -OSTBC-SNR0= -10dB OPT1-CAP-SNR0= 10dB OPT1-OSTBC-SNR0= 10dB OPT2-CAP-SNR0= 10dB OPT2-OSTBC-SNR0= 10dB NAF-CAP-SNR0= 10dB NAF-OSTBC-SNR0= 10dB Same asFig. 6but s = u = 8.
QΛ A Q H are Hermitian positive semidefinite matrices and the nondecreasingly ordered eigenvalues of I s − QΛ A Q H are 1 − (Λ A ) i,i , i = 1, . . . , s. The upper bound is attained by setting Q = I s . Hence, U = U A .
The notation z ∼ CN (µ, Σ) is associated to the circularly-symmetric complex Gaussian distribution of the random vector z and the corresponding pdf is defined by fz(z) = det(πΣ) −1 exp[−(z − µ) H Σ −1 (z − µ)].
Hereafter, {·} + max(0, ·).
APPENDIX C PROOF OF THEOREM 2Proof: We proceed, as in the proof of Theorem 1 of Appendix A, to apply the identity (34) to the trace argument of (16). We obtain:After defining the matricesǍ,B, C as in(19), we get the following expression for the optimization problem to maximize the OSTBC capacity(16):Calculating the UD'šand defining the matrixQ Ǔ HǓ A , we can rewrite optimization problem (65) asThe objective function can be written aswhere we defined the column vectorsλ, λ A and the matrix Q bySince Q is a unitary matrix, Q is a doubly stochastic matrix and, by Birkhoff's theorem[20,Th.8.7.2], it can be written as the weighted sum of a certain number of permutation matrices:
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| []
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[
"Dynamical decoupling of a singlet-triplet qubit afflicted by a charge fluctuator",
"Dynamical decoupling of a singlet-triplet qubit afflicted by a charge fluctuator"
]
| [
"Guy Ramon \nDepartment of Physics\nSanta Clara University\n95053Santa ClaraCA\n"
]
| [
"Department of Physics\nSanta Clara University\n95053Santa ClaraCA"
]
| []
| The efficiency of dynamical decoupling pulse sequences in removing noise due to a nearby charge fluctuator is studied for a singlet-triplet spin qubit. We develop a numerical method to solve the dynamical equations for all three components of the Bloch vector under a general pulse protocol, where pulses are applied along an arbitrary rotation axis. The qubit is shown to undergo both dephasing and dissipative dynamics, pending on its working position. Analytical solutions are found for the limits of weakly and strongly coupled fluctuators, shedding light on the distinct dynamics in the different parameter regimes. Scaling of the qubit decay time with the number of control pulses is found to follow a power law over a wide range of parameters and qubit bias points. | 10.1103/physrevb.86.125317 | [
"https://arxiv.org/pdf/1206.0075v2.pdf"
]
| 118,527,939 | 1206.0075 | 283cabb84a0f928f51d5854c2c74570111e107cf |
Dynamical decoupling of a singlet-triplet qubit afflicted by a charge fluctuator
22 Sep 2012
Guy Ramon
Department of Physics
Santa Clara University
95053Santa ClaraCA
Dynamical decoupling of a singlet-triplet qubit afflicted by a charge fluctuator
22 Sep 2012numbers: 0367Lx0367Pp7321La7323Hk
The efficiency of dynamical decoupling pulse sequences in removing noise due to a nearby charge fluctuator is studied for a singlet-triplet spin qubit. We develop a numerical method to solve the dynamical equations for all three components of the Bloch vector under a general pulse protocol, where pulses are applied along an arbitrary rotation axis. The qubit is shown to undergo both dephasing and dissipative dynamics, pending on its working position. Analytical solutions are found for the limits of weakly and strongly coupled fluctuators, shedding light on the distinct dynamics in the different parameter regimes. Scaling of the qubit decay time with the number of control pulses is found to follow a power law over a wide range of parameters and qubit bias points.
I. INTRODUCTION
Considerable attention has been given in recent years to the proposal to encode the logical qubit states into spin singlet (S), and unpolarized triplet (T 0 ) states of two electrons localized in a double quantum dot (QD). [1][2][3] In a Bloch sphere representation of the S − T 0 qubit, |S = (|↑↓ − |↓↑ )/ √ 2 and |T 0 = (|↑↓ + |↓↑ )/ √ 2 lie in the north and south poles, and rotations around the z and x axes are performed by the exchange interaction, J, and a magnetic field gradient across the two dots, δh, respectively. Concerted efforts have resulted in a series of impressive advances in the initialization, readout, control, and coupling of these systems. Rapid electrostatic control over the interdot bias provides a highly tunable exchange interaction that enables singlet preparation and singleshot readout by utilizing Pauli spin blockade, as well as fast rotations around the z axis. 4 Indeed, aside from their robustness against uniform nuclear fluctuations, the main advantage of S − T 0 qubits over single spin qubits is their amenability to fast single-qubit operations.
To complete single-qubit control, nuclear polarization cycles, in which bias is swept across the S − T + degeneracy point, have been used to exchange spin polarization between the electrons and the nuclei, 5,6 and were shown to generate different Overhauser (hyperfine) fields in the two QDs. 7 Together with J, the resulting field gradient has provided an allelectric scheme to perform single qubit rotations around an arbitrary axis. 7 These nuclear pump cycles were later perfected by utilizing the hyperfine coupling in a feedback loop, which not only generated a stable nuclear field gradient of δh = 23 mT, but also produced a narrowed nuclear state distribution, leading to a prolonged dephasing time of T * 2 = 94 ns. 8 Other methods to generate local magnetic field gradients were also demonstrated, including inhomogeneous Zeeman fields generated by on-chip micromagnets. 9 Implementing two-qubit gates in a system of two doubledots has proven to be a challenging roadblock, and a first experimental demonstration of conditional operation on S − T 0 qubits was reported only last year. 10 Improved gate performance was recently demonstrated in a work that included a complete measurement of the system's density matrix using state tomography. 11 While both experiments used capacitive coupling to generate a CPHASE gate (that can be transformed into a CNOT gate with the addition of two Hadamard gates on the target qubit), the most notable new feature in the design of the latter experiment is the incorporation of nuclear state preparation that generated a stabilized field gradient. Shulman et al. then used this gradient (δh = 5 mT, Zeeman energy of 0.125µeV) to apply a spin echo (SE) control pulse along the x axis, mitigating charge noise and enabling them to work near the singlet avoided crossing, where sizable couplings between the two double dots can be obtained. At this bias, where J ≫ δh, noise due to nuclear fluctuations is suppressed and charge noise plays a significant role. The employed SE pulse extended the two qubits' coherence time and allowed for their entanglement with a Bell state fidelity of 0.72. 11 Evidently, dynamical decoupling (DD) pulse sequences have been applied to S − T 0 qubits prior to the above experiment with remarkable success, starting with a single pulse SE, 4 and following by more advanced control sequences, including Carr-Purcell-Meiboom-Gill (CPMG), Concatenated DD (CDD), and Uhrig DD (UDD) schemes. [12][13][14] The coherence time was extended to a record of more than 200µs using a 16 pulse CPMG sequence and pulse optimization techniques. 14 In these experiments, dephasing times were measured at a negative bias, where exchange is small, and decoherence is attributed mainly to the varying Overhauser fields. The π pulses used in these sequences were thus appropriately performed (roughly) along the z axis by pulsing the bias near the anticrossing where J ≫ δh. The discussion above suggests, however, that when the qubit resides closer to the anticrossing (e.g., during two-qubit operations), nuclear fluctuations (causing x rotations) will be suppressed and charge-noise-induced fluctuations (causing z rotations) will become more prominent, thereby requiring π x pulses to effectively extend the qubit coherence.
The effectiveness of various decoupling schemes at mitigating nuclear-induced dephasing has been extensively addressed theoretically for single-spin qubits [15][16][17][18] and for twospin qubits under SE. 19 In contrast, their effectiveness in handling charge noise is less clear. In the current work we address this question by extending numerical and analytical stochastic methods that were developed in the context of superconducting qubits. We shall consider a qubit coupled to a single twolevel-fluctuator (TLF), treating the latter as a classical source of random telegraph noise. While quantum telegraph noise was considered before (see, e.g., Ref. 20), our classical treatment of the TLF seems reasonable, given the fact that the area surrounding the qubit is likely to be depleted of charge traps.
The qubit Hamiltonian reads H q = B · σ, where B = 1 2 (δh, 0, J), and σ is the vector of Pauli spin matrices for the pseudospin states S and T 0 . Unless otherwise noted, we take δh = 0.125µeV, and the qubit working position, ϕ = arctan(δh/J), is determined by the interdot bias, ε, that controls J. Previous studies were mostly focused on the ϕ = 0 point, where pure dephasing is expected, and the performance of the pulse sequence depends only on the TLF characteristics. [21][22][23] Bergli and Faoro considered periodic DD (PDD) at the ϕ = π/2 point, where both dephasing and dissipative dynamics take place. 24 Here we extend the work reported in Ref. 24, by treating an arbitrary working position and a general pulse sequence. While we present results pertaining for two-spin qubits in gate-defined GaAs double dots, our work is relevant for a variety of systems, including superconducting qubits and other QD materials. In particular, S − T 0 qubits were recently implemented in a Si/SiGe double QD, where dephasing time of 360 ns was measured. 25 We expect that charge noise will play a dominant role in Si, where the hyperfine interaction strength is three orders of magnitude smaller, due to reduced coupling to-and number of nuclear spins, as compared with GaAs. DD has been successfully implemented in other systems such as electron spins in irradiated malonic acid single crystals. 26 Finally, very recently DD was incorporated with two-qubit gates in a hybrid system consisting of an electron spin and a nuclear spin in a single nitrogen-vacancy center in diamond. 27 The paper is organized as follows. In Sec. II we present our model and discuss the qubit-TLF couplings and their dependence on various system parameters. In Sec. III we detail the transfer matrix numerical approach to calculate qubit decoherence under a general pulse sequence. Sec. IV provides analytical results for the PDD and CPMG cases, shedding light on the effectiveness of these sequences at different parameter regimes. In Sec. V we use our formulation to estimate qubit coherence times for various scenarios, and in Sec. VI we provide a short summary of our work, and outline future research directions. Appendix A includes formulae for the special case of pure dephasing, and Appendix B explains how to use the results given in Sec. IV to explicitly find the dynamics of a qubit with an initial state along the equator of the Bloch sphere.
II. QUBIT-TLF COUPLINGS
A TLF residing near a double dot couples differently to the two-electron S and T 0 spin states due to their different charge distributions. 28 This results in fluctuations in the exchange interaction that lead to qubit decoherence and gate errors. 29,30 Various sources can contribute to charge noise in lateral gated devices, including donor centers near the gate electrodes, switching events in the doping layer, and charge traps near quantum point contacts. 31,32 Charge noise measurements in GaAs QDs revealed a linear temperature dependence characteristic of 1/f noise, 33 which was shown theoretically to emerge from a TLF ensemble with an exponentially broad distribution of switching rates. [34][35][36] These charge fluctuators behave classically and are characterized by switching rates, γ ± , and qubit coupling strength, v. In this picture we can write the qubit-TLF interaction as
H int = vξ(t)σ z ,(1)
where ξ(t) = ±1 is a classical noise representing a random telegraph process, switching between ±1 with rates γ ± . In a previous work we have developed a multipole expansion technique to calculate the Coulomb couplings between the S − T 0 qubit orbital states and the fluctuator. 30 Assuming the TLF is a two-site trap, sufficiently remote from the double dot, there is no qubit-TLF tunnel coupling, and the interaction Hamiltonian includes the terms:
−v β σ Q z − v γ σ Q z σ T z , where σ Q z (σ T z )
is the qubit (TLF) Pauli operator. We identify two possible scenarios for qubit-TLF couplings: (i) γcoupled, where the charge fluctuates between two sites in the trap, and (ii) β-coupled, where the charge jumps in and out of the trap. β-coupled TLFs require a nearby charge reservoir (such as the 2DEG layer or quantum point contacts), and are expected to be less abundant. In the classical limit considered here, the qubit-TLF Hamiltonian reads:
H = H q + H int = B(t) · σ,(2)
where B(t) is
B(t) = 1 2 (δh, 0, J − 2v γ ξ(t)), γ − coupling 1 2 (δh, 0, J + v β − v β ξ(t)), β − coupling(3)
It is stressed that in the case of classical TLF considered in the current work, the two couplings have the same qualitative effect on the qubit dynamics, and here we distinct them only for the purpose of evaluating their strength.
The v β and v γ couplings were calculated in Ref. 30 within a Hund-Mulliken orbital model using a multipole expansion to quadrupole-quadrupole order. It was shown that the leading term in the β-type (γ-type) coupling is dipole-charge (dipoledipole), where left and right entries correspond to the qubit and TLF, respectively. 37 Analytical approximations of these couplings are found by keeping only a subset of the orbital two-electron states: S(0, 2), S(1, 1), where (i, j) indicate the number of electrons in each dot. Within this simplified model we find the leading terms:
v 21 β (ε) = cã q R 2 J 2 (ε) 2J 2 (ε) + T 2 c sin θ cos φ (4) v 22 γ (ε) = cã qãt R 3 J 2 (ε) 2J 2 (ǫ) + T 2 c [sin θ T cos φ T − 3 sin θ cos φ × sin θ sin θ T cos(φ − φ T ) + cos θ cos θ T ] ,(5)
where the left (right) superscript denotes contribution from a particular multipole moment of the qubit (TLF): charge (1), dipole (2), etc. Here, couplings are normalized to QD confinement energy, ω 0 , c = (e 2 /κa B )/ ω 0 is the Coulomb to confinement energy ratio, a q (a t ) is the half interdot (TLF intersite) separation, R is the qubit-TLF distance, and tilde denotes length normalized to the QD Bohr radius a B . Furthermore, J(ε) is the bias-dependent exchange and T c is the Coulomb-assisted tunnel coupling between the two dots. The angular dependence of v β and v γ is specified by four angles (θ, φ, θ T , φ T ), where the first two define the orientation of the qubit-TLF axis, and the last two define the TLF intersite axis. 38 Fig . 1 shows the dependence of the qubit-TLF couplings on various parameters. Here and throughout the paper we have diagonalized the full Hund-Mulliken Hamiltonian, which includes all two-electron Coulomb matrix elements. Nevertheless, the analytic formulas in Eqs. (4), and (5) are instructive as they qualitatively capture many of the couplings features. We model the double dot with a quartic potential with dot confinement ω 0 = 3meV (a B ≈ 20 nm), and a q = 2.8, and use the dielectric constant for GaAs, κ = 13.1. In addition, unless otherwise noted, we take the TLF center radius D t = 5 nm, and half intersite distance a t = 20 nm, chosen to characterize δ-doped dopants in the insulator with a typical small radius and a large intersite separation. Lastly, unless otherwise noted, we consider an external magnetic field B = 0.7 T applied along the device plane, perpendicular to the interdot axis, providing a Zeeman triplet splitting of 17.5µeV. This field has been used in recent experiments that utilized nuclear state preparation to generate Overhauser field gradients, 8,11 and in experiments that studied control sequences to enhance qubit coherence. 12,13 In Figs. 1(a) and 1(b) we examine the couplings dependence on the polar and azimuthal orientations of the qubit-TLF axis, respectively. We fix the TLF polar angle θ T to either 0 (vertical traps) or π/2 (lateral traps) and look at the coupling dependence on θ, φ for the two θ T orientations, averaging out the remaining two angles. The blue lines in Figs. 1(a) and 1(b) clearly replicate the angular dependence of v 21 β , given in Eq. (4), with minimal dependence on θ T from higher order terms. For v 22 γ , averaging over the remaining two angles results inv
For θ T = π/2 we havē v 22 γ (θ) ∼ 1 − 3 2 1 + 2 π sin 2 θ v 22 γ (φ) ∼ |1 − 3(cos 2φ + sin 2φ)| .
Notice that at angles where v 22 γ changes sign, contributions from higher order terms in the multipole expansion become important. In addition, the relative strength of these higher order terms increases with reduced qubit-TLF distance R, resulting in a more complicated angular dependence of the couplings. Both v β and v γ depend only mildly on the TLF azimuthal angle φ T (not shown). angle of the TLF-qubit axis to θ = arccos(L z /R), where L z = 90 nm is the depth of the 2DEG plane below the surface. We see that v β ∼ R −2 for R 50 nm, and v γ ∼ R −3 for R 30 nm, corresponding to the leading contributions in the multipole expansion. When θ is fixed by R, the leading terms in both v β and v γ change sign as R approaches L z (θ → 0) and higher order terms become dominant. In the rest of the paper, having no a-priori knowledge of the TLF location and relative orientation, we calculate the couplings by performing averaging over all angles. 39 In Fig. 1(d) we show the couplings dependence on the TLF site radius D t , where the intersite half separation is varied accordingly as a t = 4D t . v γ shows a linear dependence in D t , matching the linear dependence of v 22 γ on a t . On the other hand, v β depends very mildly on D t as long as D t 40 nm, since its lowest order term that depends on D t and a t is dipole-quadrupole, which contributes only slightly to v β within the range of relevant trap sizes. Fig. 1(e) shows v γ , v β , and J dependence on the interdot bias point, ε, normalized to QD confinement, where detuning is measured from the S(1, 1) − S(0, 2) anticrossing point. The bias dependence is qualitatively approximated by the analytical formulas for the leading contributions, Eqs. (4), and (5), with 40
J(ε) ≈ 2T 2 c ε 2 + 4T 2 c − ε .
Finally, the magnetic field dependence of v γ , v β , and J, is shown in Fig. 1(f). A decrease of more than four orders of magnitude in the couplings is observed between B = 100 mT to B = 1.5 T. We note that the magnetic compression of the orbitals reduces the wave function overlap by a factor of ≈ 10 within this field range, thereby it alone cannot account for these results. In fact, the main reason for the high sensitivity of the couplings to the magnetic field is that varying B also shifts the bias position of the singlet anticrossing. In Fig. 1(f) we have fixed the bias at the anticrossing point found for B = 0.7 T. As B increases, the anticrossing shifts to a more positive bias, and as a result our fixed bias becomes negative, thereby reducing the couplings. In experiments the bias position is likely calibrated when B is varied and the high sensitivity of the exchange and qubit-TLF couplings would be greatly reduced.
We conclude that the qubit-TLF coupling strength may vary by several orders of magnitude and is particularly sensitive to the qubit working position (interdot bias) and the TLF distance. For concreteness, we consider in the rest of this paper only γ-type TLFs, given their relative abundance, and will henceforth drop the γ subscript in v γ to avoid clutter.
III. TRANSFER MATRIX METHOD
In this section we outline the formulation for qubit decoherence due to a single classical TLF. 24 We include asymmetric TLF switching, relevant in cases where the temperature is lower or comparable to the TLF level spacing.
We first write the qubit-TLF Hamiltonian, Eq. (2), in the basis of the qubit eigenstates:
H = ∆σ ′ z + vξ(t)(sin ϕσ ′ x − cos ϕσ ′ z ),(6)
where ∆ = 1 2 √ δh 2 + J 2 , ϕ = arctan(δh/J), and σ ′ x,z are the Pauli matrices in the rotated frame. In this frame, the qubit evolves under a static field in the z axis, with noise in both x and z axes. Since we are dealing with a bistable fluctuator, the Bloch vector representing the qubit state in the rotated frame precesses around either of the two effective fields B ± = (±v sin ϕ, 0, ∆ ∓ v cos ϕ).
We denote the two TLF states corresponding to B + and B − as up and down, respectively, and γ + (γ − ) is the switching rate from state up to down (down to up). Unlike the case of pure dephasing (ϕ = 0), the Bloch vector does not precess around the z axis alone but can reach any point on the Bloch sphere, resulting in dissipative dynamics. Denoting by p(r, t) the probability to reach point r = (x, y, z) on the Bloch sphere at time t, we introduce p + (r, t) and p − (r, t) as the probabilities to reach the point r when the TLF is in state up and down, respectively. The equations for p ± (r, t) are found to be
p ± (r, t + τ ) = (1 − γ ± τ )p ± (U −1 ± r, t) + γ ∓ τ p ∓ (U −1 ± r, t),(8)
where U ± = e τ B±·R rotate the qubit around the B ± axes. We consider an infinitesimal time step dt, such that γ ± dt are the switching probabilities for the two TLF states within time dt. Using the fields B ± in Eq. (7), and the rotation matrices 24 we find the master equations:
R ≡ (R x , R y , R z ),p ± = [(∆ ∓ v cos ϕ)(y∂ x −x∂ y ) ± v sin ϕ(z∂ y −y∂ z )]p ± +(γ ∓ p ∓ − γ ± p ± ).(9)
The expectation values of the Bloch vector components in the rotated frame evolving under the fields B ± are defined as:
r ± = drp ± (r, t)r,(10)
and their dynamical equations are found from Eqs. (9) to bė
x ± = (γ ∓ x ∓ − γ ± x ± ) − (∆ ∓ v cos ϕ) y ± y ± = (γ ∓ y ∓ − γ ± y ± ) + (∆ ∓ v cos ϕ) x ± ∓ v sin ϕz ± z ± = (γ ∓ z ∓ − γ ± z ± ) ± v sin ϕy ± .(11)
Finally we transform these coupled equations to a set of equations for the total expectation values, x = x + + x − , y = y + + y − , z = z + + z − , and their differences,
δx = x + − x − , δy = y + − y − , δz = z + − z − : x = −∆y + v cos ϕδẏ y = ∆x − v cos ϕδx − v sin ϕδż δz = −δγz − 2γδz + v sin ϕẏ δx = −δγx − 2γδx − ∆δy + v cos ϕẏ δy = −δγy − 2γδy + ∆δx − v cos ϕx − v sin ϕż z = v sin ϕδy,(12)
where we have defined the average switching rate, γ = (γ + + γ − )/2, and switching rate difference, δγ = γ + − γ − . It is convenient to write Eqs. (12) in a matrix forṁ
k = M 1 k,
where entries 1,2,6 of the vector
k = x y δz δx δy z (13)
define the Bloch vector components, and the matrix M 1 reads
M 1 = 0 −∆ 0 0 v cos ϕ 0 ∆ 0 −v sin ϕ −v cos ϕ 0 0 0 v sin ϕ −2γ 0 0 −δγ −δγ v cos ϕ 0 −2γ −∆ 0 −v cos ϕ −δγ 0 ∆ −2γ −v sin ϕ 0 0 0 0 v sin ϕ 0 .(14)
We now consider the dynamics of the Bloch vector subject to sequences of control π pulses along an arbitrary axis. The application of a π-pulse flips the components of the effec-tive fields in Eqs. (7) perpendicular to the pulse axis. While our dynamical equations are written in the rotated frame, we consider π-pulses along the original frame axes since these are more readily available in current experiments in S − T 0 qubits. 7 The resulting effective fields after the application of the control pulses read:
B πx ± = (∆ sin 2ϕ ∓ v sin ϕ, 0, −∆ cos 2ϕ ± v cos ϕ) B πy ± = − (±v sin ϕ, 0, ∆ ∓ v cos ϕ) B πz ± = (−∆ sin 2ϕ ± v sin ϕ, 0, ∆ cos 2ϕ ∓ v cos ϕ) (15)
These fields lead to qubit state evolution governed by matrices M j 2 , where j = x, y, z denotes the control pulse axis (in the original frame). M x 2 is found as
M x 2 = 0 ∆ cos 2ϕ 0 0 −v cos ϕ 0 −∆ cos 2ϕ 0 v sin ϕ v cos ϕ 0 −∆ sin 2ϕ 0 −v sin ϕ −2γ 0 ∆ sin 2ϕ −δγ −δγ −v cos ϕ 0 −2γ ∆ cos 2ϕ 0 v cos ϕ −δγ −∆ sin 2ϕ −∆ cos 2ϕ −2γ v sin ϕ 0 ∆ sin 2ϕ 0 0 −v sin ϕ 0 ,(16)M y 2 is found by substituting ∆ → −∆, v → −v in M 1 , and M z 2 is found by substituting ∆ → −∆, v → −v in M x 2 .
The evolution of the qubit state after a sequence of N π jpulses can be written as
k(t) = U j k(0)(17)
with
U j = 1 2 (N +1) i=1 e M j 2 τ2i e M1τ2i−1 , N ∈ odd e M1τN+1 N/2 i=1 e M j 2 τ2i e M1τ2i−1 , N ∈ even(18)
where τ i is the time interval between the (i − 1)th and ith pulses. When ϕ = π/2, and for symmetric switchings (δγ = 0), the matrices M 1 and M y 2 become block diagonal and the equations for (x, y, δz) decouple from the equations for (δx, δy, z). 24 Previously found results for pure dephasing (ϕ = 0) can be obtained more easily by solving a subset of Eqs. (12), as shown in Appendix A.
For all the results presented below we take the initial state of the qubit to lie along the rotated x axis, tilted by an angle ϕ in the x-z plane. Such initialization can be realized in two steps. First, the initial singlet state is transformed into the ground state of the nuclear field, (|S +|T 0 )/ √ 2, by lowering the exchange adiabatically with respect to the nuclear mixing time. 4 Second, a magnetic field gradient is ramped slowly so that the qubit state follows it, reaching the Hamiltonian eigenstate. We remark that this last step is challenging with current experimental techniques that generate nuclear field gradient, since the employed dynamical nuclear polarization cycles are slower than the electron spin dynamics. Our choice of initial state is motivated by the clean context it provides for the analysis of the qubit dynamics. In Appendix B, we provide formulas for qubit decay, when its initial state is set as (|S + |T 0 )/ √ 2, and discuss its relation to the results given in the main text.
The probabilities of finding the TLF state up or down are γ − /(γ + + γ − ) and γ + /(γ + + γ − ), respectively, therefore
p ± (r, t = 0) = γ ∓ γ + + γ − δ (3) (r −x),(19)
and the initial qubit's state reads
k(0) = 1, 0, 0, − δγ 2γ , 0, 0 .(20)
The ratio between the TLF switching rates is given by the Boltzmann factor
γ − γ + = e −∆Et/kB T ,(21)
where ∆E t is the TLF level splitting, taking into account intersite tunneling. 30 For the TLF parameter range considered in this work, ∆E t < 5µeV, thus for sample temperature of 100 mK, we always have δγ = 2γ tanh(∆E t /2k B T ) < γ. We now address the choice of rotation axis for the control pulses. First, since charge noise is induced along the original z axis, we expect that its decohering effects will not be corrected by π z pulses, which have been extensively employed to reduce nuclear induced noise. This is demonstrated in Fig. 2(a) where we plot qubit decay under both free induction (FID), and a single π z pulse (SE). 41 Here we take TLF switching time of 0.1
χ FID ≈ e −γt cos vt + γ v sin vt .(22)
At R = 300 nm, v = 62peV γ, and the strong coupling approximation is compromised [see dashed green lines in Fig. 1(a)]. In both cases the π z pulse does not extend the qubit coherence as compared with FID, resulting in dephasing times of 22µs, and 95µs, respectively. These relatively short dephasing times were likely circumvented in experiments by selecting relatively quiet samples.
The effectiveness of π x vs. π y control pulses in mitigating charge noise depends on the TLF parameters, as well as on the qubit working position. At pure dephasing (δh = 0), both pulse types eliminate noise from TLFs that are static during the time interval between pulses, τ , since the qubit state evolution depends only on the σ z operator [see Eq. (6), and Ref. 24 for more details]. At any other working position, π y pulses are always superior to π x pulses, where the relative improvement depends on the TLF parameters. The most pronounced improvement is expected at large negative bias where ϕ π/2, as verified in Fig. 2(b). In Fig. 2(c) we tune the in-terdot bias to the singlet anticrossing (zero detuning), resulting in J = 4.15µeV ≫ δh. Since ϕ ≈ 0 the performance of π x and π y are similar. However if we stay at the anticrossing and take δh = 0.5µeV, increased dephasing is exhibited by the π x pulse [see red dashed line in Fig. 2(d)]. Two-qubit gates that are performed near zero detuning (to increase coupling strength) will thus benefit greatly from π y pulses that will allow to increase δh, shortening gate times.
Finally we note that it is possible to effectively eliminate the exchange interaction so thatϕ = π/2 while retaining substantial qubit-TLF coupling. Normally, J and v decrease simultaneously with bias [see Fig. 1(e)], but, as was recently shown, at certain interdot bias points, the inter-qubit Coulomb couplings balance the internal exchange interaction, resulting in a zero effective exchange. 40 For a distance of 400 nm between the centers of the two DDs, this zero-exchange position is found at a moderately negative (dimensionless) bias of ε = −0.05. While this position is typically ideal from the perspective of charge noise, for sufficiently close TLFs, π x pulses will introduce substantial noise.
The above considerations suggest that π y pulses are more effective in mitigating charge noise in several important scenarios. Moreover, we expect π x pulses to be completely inefficient in correcting nuclear-induced noise, similarly to the inadequacy of π z pulses in correcting charge noise. This will play an important role at negative bias, where nuclear noise is dominant. While π y pulses are harder to implement and were previously generated with limited fidelity, 7 it is expected that high-fidelity π y pulses will be available in the near future. 42
IV. ANALYTIC RESULTS FOR PDD AND CPMG PULSE SEQUENCES
In this section we provide exact analytic solutions for the qubit dynamics due to its coupling with a single TLF, in the limits of weakly (v ≪ γ) and strongly (v ≫ γ) coupled fluctuators. As we show below, the qubit dynamics is different in these two regimes, and the asymptotic behavior in these limits proves valuable for the interpretation of our results. Our analysis follows Ref. 24, extending it in three respects: (i) we consider an arbitrary qubit working point ϕ, (ii) we include asymmetric TLF switching, and (iii) we find explicitly the weights of the different decay rates in the final solution, providing further insight into the qubit dynamics, and the enhanced performance of CPMG over PDD control pulses.
In the following we consider an N -pulse PDD protocol with a constant time interval between pulses, τ = t/(N +1), where t is the total time. (CPMG protocol will be considered in section IV C.) In light of the discussion ending the previous section, we consider only π y pulses. In order to calculate the qubit dynamics, we first note that after a π y pulse is applied, the matrix that governs the qubit evolution, M y 2 , can be written in terms of M 1 in Eq. (14) as:
M y 2 = LM 1 L,(23)
where L = diag(1, −1, 1, 1, −1, 1). Thus the qubit evolution under the full control sequence is found by
k(t) = T N +1 k(0),(24)
where (N is odd for PDD)
T = e M y 2 τ e M1τ = Le M1τ .(25)
The decay rates of the Bloch vector components are found in terms of the six eigenvalues, χ i , of the evolution operator T :
Γ i = − ln |χ i | τ ,(26)
and the general solution for the decay of the qubit state rotated components is
j(t) = 6 i=1 w j i e −Γit , j = x, y, z,(27)
where w j i is the weight of the ith eigenvalue in the solution of the jth component. It should be noted that the decay rates defined in Eq. (26) and calculated below, are time dependent, unlike the more commonly encountered decay coefficients obtained in Bloch-Redfield theory. Notice also that while we are only interested in the dynamics of the three rotated Bloch vector components, in the strong coupling case discussed below there are nonzero weights for all six eigenvalues of matrix T .
A. PDD: Weak Fluctuator
In order to find the eigenvalues χ i of T we need to exponentiate M 1 in Eq. (14). In the case of weak coupling we perform second order (degenerate) perturbation in v/γ. Note that while M 1 is non hermitian due to the unequal switching rates,
δγ = 0, we still have M 1 = M R 1 + iM I 1 ,
where M R 1 , M I 1 are both hermitian. In this case, the eigenstates of M 1 and M † 1 form a bi-orthogonal set and a perturbation theory is readily available. 43 The three relevant decay rates are found as:
Γ w 1 = 1− δγ 2 4γ 2 γv 2 ∆ 2 + 4γ 2 (1−A−B 1 ) sin 2 ϕ + C cos 2 ϕ Γ w 2,3 = 1− δγ 2 4γ 2 γv 2 ∆ 2 + 4γ 2 (2−B 1 −B 2 ) sin 2 ϕ − F ∓ F 2 + D 2 sin 2 2ϕ ,(28)
where the different functions of τ are given by:
A = ∆ 2 − 4γ 2 ∆ 2 + 4γ 2 sincτ B 1 = 8γ 2 ∆ 2 + 4γ 2 cos 2τ 2 tanh γτ γτ B 2 = 8γ 2 ∆ 2 + 4γ 2 sin 2τ 2 coth γτ γτ C = ∆ 2 + 4γ 2 2γ 2 1 − tanh γτ γτ D = cosτ 2 tanh γτ γτ − sincτ 2 F = 1 2 (1 − A − B 1 ) sin 2 ϕ − C cos 2 ϕ .(29)
Here we used sincτ ≡ sinτ /τ , whereτ ≡ ∆τ is the normalized time interval between pulses. We note that the effect of asymmetric TLF switching in this regime amounts to reducing all rates by a common factor that becomes appreciable for TLF level splitting above 1µeV . The three additional eigenvalues, associated with the dynamics of (δx, δy, δz), induce much faster decay rates, 2γ − O(v 2 /γ 2 ), but their weights vanish for all three Bloch vector components. Eqs. (28), (29) recover previous results obtained for ϕ = 0 and ϕ = π/2 with δγ = 0. 24 Specifically, for the case of pure dephasing (ϕ = 0), we have a single decay rate (see also Appendix A):
Γ w 1 (ϕ = 0) = Γ w 3 (ϕ = 0) = v 2 2γ 1 − tanh γτ γτ .(30)
For J = 0 (ϕ = π/2), the equations for x and y decouple from z, the latter having no dynamics when the qubit state is initially along the x axis, and we obtain two decay rates:
Γ w 1 (ϕ = π/2) = γv 2 ∆ 2 + 4γ 2 (1−A−B 1 ) Γ w 2 (ϕ = π/2) = γv 2 ∆ 2 + 4γ 2 (1+A−B 2 ).(31)
In order to explicitly find the time dependence of the Bloch vector components, we diagonalize the matrix T in Eq. (24), T = SW S † , where W is diagonal matrix, similar to T , whose ith element is e −Γiτ , and the columns of S are the eigenvectors of T . Up to second order in v/γ we can use the unperturbed eigenvectors, and by applying the resulting operator to the initial vector, Eq. (20), we find the following weights Notice that i w x i = 1, and i w y i = i w z i = 0. In Fig. 3 we illustrate the above results for a working position ϕ = π/4 (J = δh = 0.125µeV), obtained at negative bias ǫ = −0.054. In Fig. 3(a) we consider a fast TLF with average switching time of 0.1µs, and coupling strength v = 2.5 neV (R = 300 nm), such that ∆ ∼ γ ≫ v. All three decay rates in Eqs. (28) are comparable and contribute to the decay of all three Bloch vector components shown in Figs. 3(b) and (c), where we consider the qubit dynamics under an 11-pulse PDD protocol. In Figs. 3(d)-(f) we consider a slow TLF with an average switching time of 0.1 ms. To stay in the weak coupling regime, we take R = 2000 nm, resulting in v = 8.4 peV, so that ∆ ≫ γ ≫ v. In this case the function C in Eqs. (29) dominates and we obtain two distinct decay rates:
w x 1 = sin 2τ 2 w x 2,3 = 1 2 cos 2τ 2 1 ± F F 2 + D 2 sin 2 2ϕ w y 1 = 1 2 sinτ w y 2,3 = − 1 4 sinτ 1 ± F F 2 + D 2 sin 2 2ϕ w z 1 = 0 w z 2,3 = ± 1 2 cosτ 2 D sin 2ϕ F 2 + D 2 sin 2 2ϕ .(32)Γ w 1 ≈ Γ w 3 ≈ v 2 2γ (1 − tanh γτ γτ
) cos 2 ϕ, and Γ w 2 ≈ 2γv 2 ∆ 2 sin 2 ϕ, the latter being independent of τ . Whereas Γ w 1 , Γ w 3 ≫ Γ w 2 , the weight w x 2 is vanishingly small, and overall the qubit state decays with a single (faster) rate. We also observe that in this limit there is almost no dissipative dynamics since w y 2 ≈ 0, and w y 1 = −w y 3 for the y component, and w z 2,3 ∼ (γ/∆) 2 ≪ 1 for the z component. Finally, we note that the leading term in the faster decay rate, Γ w 1 , is eliminated at ϕ = π/2, and the qubit decay is governed by the much slower rate Γ w 2 , as well as by higher order corrections in γ/∆, and v/γ to Γ w 1 and Γ w 3 . This case is shown in Fig. 3(f), where we set J = 0 and obtain more than 5 orders-of-magnitude increase in coherence time.
B. PDD: Strong Fluctuator
In the limit of strong coupling, γ ≪ v, it is sufficient to perform first order perturbation in γ/v. Here the unperturbed eigenvalues of T form two degenerate subspaces of dim = 2, and dim = 4. The analytic results are complicated and we present here only the leading term (third order) in the expansion in v/∆, in the case of symmetric switching (δγ = 0). The results below are thus valid at negative detunings, where v ≪ ∆ (for a fixed δh = 0.125µeV ), but will be applicable to an extended bias regime if larger δh is considered. The three decay rates associated predominantly with the Bloch vector components are found as:
Γ s 1 = γv 2 ∆ 2 Ã sin 2 ϕ +τ 2 6 cos 2 ϕ Γ s 2,3 = γv 2 ∆ 2 B sin 2 ϕ −F ± F 2 +D 2 sin 2 2ϕ ,(33)
where the different functions are given bỹ
A = 1 − sincτ B = 2 1 − sinc 2τ 2 D = cosτ 2 − sincτ 2 F = 1 2 Ã sin 2 ϕ −τ 2 6 cos 2 ϕ .(34)
We consider strongly coupled TLF in two scenarios: ∆ ≫ v ≫ γ, and ∆ v ≫ γ, shown in Figs. 4 and 5, respectively. Examining the asymptotic behavior of the decay rates at τ ≪ 1, we haveà =B = −2D =τ 2 /6. As a result, at short time intervals, the qubit dynamics is governed by a single decay rate: Γ s 1 ≈ Γ s 2 ≈ γv 2 τ 2 /6, depicted by the solid blue and dotted red lines in Fig. 4(a). Γ s 3 scales as τ 4 and is considerably smaller. At the limit ofτ ≫ 1 we haveÃ,B,D ≪τ , thus Γ s 1 ≈ Γ s 2 ≈ γv 2 τ 2 cos 2 ϕ/6 ≫ Γ s 3 ≈ 2γv 2 sin ϕ 2 /∆ 2 . Note that for pure dephasing, both asymptotes coincide and Γ s (ϕ = 0) = γv 2 τ 2 /6 is the single solution to Eqs. (33). We conclude that when v ≪ ∆, unless ϕ = π/2, the qubit dynamics is governed at all times by a single decay rate that scales as τ 2 . In contrast, when J = 0, the cos 2 ϕ term is removed and Γ s 1,2 decrease substantially in the largeτ limit, becoming comparable to Γ s 3 . In this case the rates read:
Γ s 1 (ϕ = π/2) = γv 2 ∆ 2 (1 − sincτ ) Γ s 2 (ϕ = π/2) = γv 2 ∆ 2 1 − 2sinc 2τ 2 + sincτ ,(35)
both contribute to the qubit decay, and their magnitude in the largeτ limit is curbed at a value γv 2 /∆ 2 , as shown in Fig. 4(b). As a result, decay time is nearly three orders of magnitude longer, as compared with the J = δh case. This is shown in Figs. 4(c), and (d) for an 11 pulse PDD sequence. We note that in this case, the equations for (x, y, δz) decouple from those for (δx, δy, z), and Γ s 3 is associated solely with the z component, having no weight in the decay of our initial state.
In order to fully capture the qubit dynamics, we calculate the weights of the different rates in Eq. (33). As in the weak coupling case, the three additional eigenvalues, associated predominantly with the dynamics of (δx, δy, δz), induce much faster decay rates: 2γ − O(γv 2 /∆ 2 ), but in this case they have a small yet nonzero weight that contributes to the decay of the Bloch vector. We demonstrate this behavior for ϕ = π/2, where the analytic formulas are simpler due to the block diagonal form of M 1 in Eq. (14). Here we only need to consider the three eigenvalues of the block (x, y, δz), where two decay rates are given in Eqs. (35), and the third rate is Γ 3 ≈ 2γ, associated predominantly with δz. Considering the perturbed eigenvectors of T to second order in v/∆, the weights for the x and y components (z has no dynamics) are calculates to be:
w x 1 = sin 2τ 2 − (vτ ) 2 4∆ 2 Ã −B 2 w x 2 = cos 2τ 2 + v 2 ∆ 2 τ 2 4 Ã −B 2 −Ã 2 w x 3 = v 2 ∆ 2Ã 2(36)
and Fig. 4(f).
w y 1 = 1 2 sinτ + v 2τ 4∆ 2 Ã +τ 2 4 (B − 2) w y 2 = − 1 2 sinτ − v 2τ 4∆ 2 (B − 2) τ 2 4 −Ã +Ã w y 3 = − v 2τ 4∆ 2Ã (B − 2),(37)
Turning to the case of strongly coupled TLF with ∆ v ≫ γ, an analysis similar to the one above qualitatively captures the dynamics, although the analytic decay rates in Eqs. (33) are accurate only to third order in v/∆. To illustrate this regime we consider δh = J = 0.01µeV, and take J = δh or J = 0. The asymptotic behavior at the short and long τ limits remains as before, but the crossover from the τ 4 shorttime scaling to the constant long-time behavior of Γ s 3 occurs at a much larger τ , as seen in Fig. 5(a). Since the qubit dynamics are governed by a single decay rate: Γ s 1 = Γ s 2 , which is independent of ∆, the resulting qubit decay depicted by the solid blue line in Fig. 5(c) is largely unchanged. In contrast, for J = 0, the two contributing decay rates scale like 1/∆ 2 and are almost two-orders-of-magnitude larger than the rates plotted in Fig. 4(b). As a result, a dramatic decrease in coherence time is shown by the dashed red line in Fig. 5(c), as compared with the ∆ ≫ v ≫ γ case. Finally, the dissipative dynamics in the y and z components are much more pronounced for the ∆ v case, as seen in Fig. 5(d).
C. CPMG
Here we extend the above analysis to treat the CPMG protocol, where τ j = t/N for 2 ≤ j ≤ N and τ 1 = τ N +1 = t/2N . The evolution of the qubit state under N -pulse CPMG sequence can be written as:
k(t) = T 1/2 T N −1 T 1/2 , k(0), N ∈ odd LT 1/2 T N −1 T 1/2 k(0), N ∈ even,(38)
where T 1/2 is defined by Eq. (25) with τ → τ /2. Focusing on the x component of the Bloch vector, we consider first a weakly coupled TLF (v ≪ γ), in the case of J = 0. Here the z component is again decoupled from x and y, and we can consider only the 3 × 3 block in T corresponding to (x, y, δz). Furthermore, since there is no weight for the fast ∼ 2γ decay rate, predominantly associated with δz, in the weak coupling case, we can work within the 2 × 2 subspace of (x, y). The similarity transformation that diagonalizes our subset of T is:
S = sinτ 2 − cosτ 2 cosτ 2 sinτ 2 ,
and we can write Eqs. (38) for the (x, y) components as:
x(t) y(t) =LS χ N −1 1 0 0 χ N −1 2 S † 1 0 ,(39)
where χ i are the two eigenvalues of T that are associated with the decay rates in Eqs. (31) [recall Eq. (26)], andL = diag(1, ±1), with the upper (lower) sign corresponding to odd (even) number of pulses. Note that x(t) is unaffected bỹ L, and we obtain an 'even-odd effect' only for the y component. We find that x(t) decays with a single rate, Γ w 2 , given in Eqs. (31). The well known superior performance of the CPMG protocol over PDD can thus be explained as follows. For γ ≪ ∆, Eqs. (31) can be approximated by Γ w 1 ≈ γv 2 ∆ 2 (1−sincτ ), and Γ w 2 ≈ γv 2 ∆ 2 (1+sincτ −2sinc 2τ /2).
An efficient noise suppression is thus achieved when the time interval between pulses satisfiesτ ≪ 1. At these short time intervals, Γ w 1 ∼ γv 2 τ 2 ≫ Γ w 2 ∼ γv 2 ∆ 2 τ 4 . Since only Γ w 2 is present in the CPMG pulse sequence, it is more effective than PDD at the ϕ = π/2 point. Similar calculation can be carried for a general working position. In the general case, we need to include the z component that is now coupled to x and y. We find again that the x component decays with a single rate given by: (40) where A, B 2 , and C were given in Eqs. (29). The cos 2 ϕ term in Eq. (40) dominates at short τ , and the advantage of CPMG over PDD is largely eliminated at ϕ = π/2.
Γ w = γv 2 ∆ 2 + 4γ 2 (1 + A − B 2 ) sin 2 ϕ + C cos 2 ϕ ,
Turning to the case of a strongly coupled TLF (v ≫ γ), we include the small weight of the fast decaying (δx, δy, δz) components. Considering only the ϕ = π/2 case, we perform a calculation similar to the above within the subspace (x, y, δz), with Γ s 1,2 given in Eqs. (35) and the additional small weighted rate of 2γ. As in the weak coupling case, we find that the weight of Γ s 1 is vanished and the resulting qubit decay reads:
x(t) = 1 − v 2 ∆ 2Ã 2 e −Γ s 2 t + v 2 ∆ 2Ã 2 e −2γt ,(41)
whereà was given in Eqs. (34). While the weight of the second term is small, its fast decay rate induces a sizable effect on the qubit decay.
V. TLF-INDUCED DECOHERENCE UNDER GENERAL PULSE SEQUENCE: RESULTS
In this section we present qubit coherence times due to its coupling to a single TLF. With limited knowledge of the TLF characteristics, we wish to map a wide range of TLF switching rates and coupling strengths, at various qubit bias points. For brevity we focus only on the qubit decay time, T 2 , which we define as the time it takes the x component of the Bloch vector of a qubit initially prepared along the x axis to reach 50% of its initial value. We remark that it is often desirable to study the initial qubit dynamics, which may be different for two pulse protocols that have similar T 2 . In addition, T 2 does not capture the dissipative dynamics that leads to nonzero y and z components (see Figs. [3][4][5]. Nevertheless, the results presented below give good indication for the qubit dynamics in the various regimes.
As a benchmark case, we consider a 10-pulse CPMG sequence. Fig. 6 shows T 2 times vs. TLF average switching times for R = 300 nm, at three bias points. We identify three coupling strength regimes at which very different dynamics occur, as explained below: (i) very strong coupling, (ii) strong coupling, and (iii) weak coupling, marked by black, green, and red dashed lines, respectively. Note that by weak and strong we refer only to the ratio v/γ. In Fig. 6(a) we consider zero detuning, where J = 4.15µeV ≫ δh (ϕ ≈ 0), v = 1.4µeV, and the qubit is most susceptible to charge noise.
Here we are in the very strong coupling regime for the entire range of TLF switching rates. Since v/∆ is not small, it is more straightforward to consult the results for pure dephasing given in Appendix A. In this quasi static regime, for our case of 10 pulses, the τ values relevant for the qubit T 2 times satisfy τ ∼ 1/γ ≫ 1/v for almost the entire range. The large τ limit of Eq. (A15) results in Γ s ± ≈ γ for both rates, thus the qubit approximately decays with a single rate, γ, and T 2 = ln 2/γ.
Turning to the case of large negative detuning, where J = 0.02µeV ≪ δh (ϕ π/2) and v = 62 peV, Fig. 6(c) includes all three coupling strength regimes. Focusing first on the weak coupling (left side of the vertical dotted line), the qubit decays with a single rate given by Eq. (40). For ∆ ≫ γ, satisfied here, the cos 2 ϕ term dominates, as long as we are not too close to the ϕ = π/2 point [see Eqs. (29)], and the decay rate reads:
Γ w = v 2 2γ 1 − tanh γτ γτ cos 2 ϕ.(42)
The long-and short-time asymptotes of Eq. (42) yield the fol-lowing dephasing times:
T lt 2 = 2γ ln 2 v 2 cos 2 ϕ , γτ ≫ 1,(43)T st 2 = 6N 2 ln 2 γv 2 cos 2 ϕ 1/3 , γτ ≪ 1(44)
where N is the number of pulses. Observing the solid magenta line in Fig. 6(c) that depicts Eq. (42), and the dashed red and green lines, corresponding to its asymptotes, we notice that Eq. (42) works well into the strong coupling regime. This can be explained by taking the large τ limit of Eqs. (33), where we have found a single decay rate: Γ s = γv 2 τ 2 cos 2 ϕ/6, which is identical to the large τ limit of Eq. (42). For very weak coupling [left end of Fig. 6(c)], the approximate result of Eq. (42) breaks down since ∆ ≫ γ no longer holds. We note that although J ≪ δh, the dynamics shown here is very different from the case of J = 0. In the latter case, T 2 times increase substantially, due to much smaller rates given by Eq. (40). To realize this limit one needs to work at a very large negative bias, or to utilize inter-qubit couplings that can turn off the exchange completely. Fig. 6(b) shows decay times for moderate negative bias where J = δh, exhibiting the complex dynamics discussed above.
In Fig. 7 we show T 2 times vs. qubit-TLF distance, for a TLF average switching time of 0.1 ms, at the same three bias points. At zero detuning we are again at the very strong coupling regime throughout the considered range, resulting in a constant T 2 = ln 2/γ = 69.3µs, as shown in Fig. 7(a). At large negative detuning, Fig. 7(c) exhibits the weak and strong coupling regimes captured by Eqs. (43), and (44). Here, the relevant τ values scale as 1/v. To the left of the vertical dotted line, marking v = γ, we have strong coupling, thus γτ ≪ 1, and Eq. (44) applies. Since the dominant term in the qubit-TLF coupling scales as R −3 [see Fig. 1(c) and discussion therein], T 2 scales like R 2 in this regime. To the right of the v = γ line we have γτ ≫ 1, and Eq. (43) applies, thus T 2 scales as R 6 . The T 2 times in the intermediate bias position shown in Fig. 7(b), are understood by the same arguments all the way down to R ≈ 250 nm, below which, the very strong coupling, quasi-static regime applies, and T 2 is roughly constant. The increase in T 2 times below R = 100 nm is not explained within our analytic results, which do not apply to this v ≫ ∆ regime.
The discussion above suggests that prolonging the qubit coherence time by reducing the time interval between pulses (or equivalently increasing the number of pulses) is effective only when Eq. (44) holds. Within this regime, marked by the overlap of the dashed green lines with the full calculation in Figs. 6, and 7, decay time scales with the number of pulses as N 2/3 . This power law was predicted for pure dephasing due to random telegraph noise in Ref. 23. There it was explained by observing that as N increases, the non-Gaussian noise attributes are suppressed by the DD pulses. The N 2/3 scaling was then found by considering Gaussian noise with spectral density having a soft cutoff (e.g. ω −2 ). 23 double-axis (XY) DD pulse sequences were applied. 44 In this system, the dephasing of the central spin is induced by its coupling to a bath of spins with dipolar intra-bath coupling. By identifying the noise as a Gaussian and Markovian Ornstein-Uhlenbeck process, the authors were able to explain the N 2/3 scaling using the arguments given in Ref. 23. We find estimates for the number of pulses required to crossover into the large-N regime from either side. At the very strong coupling regime (dashed black lines), T 2 = ln 2/γ, therefore to avoid the large τ limit of Eq. (A15), we require vτ ≈ (v/γ) ln 2/N 1, or N (v/γ) ln 2. At the weak to strong coupling crossover we have T 2 = ln 2/Γ st , given by Eq. (44), and to satisfy the condition γτ 1, we need N 6 ln 2(γ/v cos ϕ) 2 . To summarize, the minimum number of pulses needed to enter the regime at which T 2 scales as N 2/3 is estimated by:
N max v ln 2 γ , 6 ln 2 cos 2 ϕ γ 2 v 2 .(45)
These considerations are demonstrated in Fig. 8, where we plot T 2 times vs. the number of CPMG pulses for a qubit-TLF distance of R = 300 nm, at the above three bias points. The dashed lines correspond to the power law: T 2 ∝ N 2/3 . The vertical dotted lines mark the minimum number of pulses It is interesting to note that the 2/3 power law is close to the value 0.72, recently found for even number of CPMG pulses in a setting that was likely dominated by nuclear-induced dephasing. 13 Finally, we comment that we have assumed that our control pulses are ideal, in that they are zero-width π pulses applied exactly along the y axis. In reality, control pulses have errors in both their rotation angle and axis, which may introduce more noise than they can effectively remove. 24,45 This is particularly relevant for large-N sequences, thus the estimates in Fig. 8 should not be taken too seriously above N ≈ 50.
Next we compare T 2 times under several widely used DD protocols. The concatenated DD sequence (CDD) is defined recursively, 46 with the lth order of concatenation given by:
CCC l (t) = CDD l−1 (t/2) − π − CDD l−1 (t/2),(46)
where CDD 0 is free induction, and the middle π pulse is included only for odd l. The number of pulses in the lth level concatenation is found as: The Uhrig DD sequence (UDD) is defined by pulse times:
N l = 1 3 2 l+1 − 1 , l ∈ odd 1 3 2 l+1 − 2 , l ∈ even.(47)t i = t sin 2 iπ 2(N + 1) , 1 ≤ i ≤ N(48)
where the time interval between the (i − 1)th and ith pulses is 47 We have not attempted to obtain analytical results for the decay rates of pulse sequences with unequal time intervals between pulses. In Fig. 9 we compare the performance of PDD, CPMG, CDD, and UDD sequences at the above three bias points. At zero detuning, Fig. 9(a) shows no advantage in increasing the number of control pulses up to N = 50, for any of the protocols, even though we are considering a relatively fast TLF (γ −1 = 1µs). At this strong coupling limit, the qubit state decay is characterized by plateaus (see, e.g., Fig. 10) that result in the observed oscillations in the dependence of T 2 on N . A better performance measure may be decay time to 95% of initial value, plotted in Fig. 9(d), where coherence times are consistently improved with the number of pulses. Decay times at moderate-and large-negative bias points are shown in Figs. 9(b) and (c), respectively, with their respective 5% drop times given in Figs. 9(e) and (f). Throughout the wide TLF parameter range considered, and at all qubit bias points, the CPMG protocol exhibits superior performance in mitigating charge noise. These results agree and complement previous observations of the superiority of CPMG over UDD in fighting noise with power-law high-frequency tail. 23,48 We also note that all pulse protocols roughly follow the N 2/3 power law over a wide range of TLF parameters and qubit working positions.
τ i = t i − t i−1 (t 0 = 0, t N +1 = t).
VI. CONCLUSION
In this work we have studied the dynamics of a singlettriplet spin qubit afflicted by a charge fluctuator, under dynamical decoupling control pulses. We have presented a theory that predicts rich dynamics governed by the qubit-TLF coupling strength and the TLF switching rates, as well as by the qubit working position. For a relatively small fixed magnetic field gradient, the exchange interaction at zero detuning results in pure dephasing that is predominantly dependant on the TLF parameters. In contrast, at negative bias points the qubit dynamics becomes dissipative and the effectiveness of the DD pulses depends on both TLF and qubit characteristics.
We have demonstrated that π y pulses are preferable over π x pulses in several scenarios, and moreover we expect that π y pulses will be superior in eliminating nuclear fluctuations. Finding analytical formulas for the qubit decay rates in the limits of weak and strong coupling for PDD and CPMG sequences, enabled us to explain our results for qubit coherence times as functions of TLF distance and switching rate at various bias points. Over a large range of system parameters, coherence times follow a power law T 2 ∝ N 2/3 , where N is the number of pulses. In addition, comparing the performance of several pulse protocols, we found that CPMG is the most effective protocol to eliminate charge noise. While we have presented specific results for two-spin qubits in gate-defined GaAs QDs, the formulation presented in this paper should be relevant to a wide variety of systems afflicted by charge fluctuators. A natural extension of our work will include pulseerror analysis. 24,45 In particular, the tolerance of various pulse sequences to accumulation of systematic and random pulse errors should be evaluated in the context of charge noise.
As current experiments move to more complicated QD structures, such as two coupled double QDs, 10,11 and threespin qubits, 49,50 we expect that charge noise will play an increasing role in the system decoherence. There are two main aspects needed to be considered when analyzing charge noise in larger devices. First, as the sample size increases, larger number of active charge traps can couple to the qubit(s). It is well known that a collection of weakly-coupled TLFs with a broad distribution of switching rates can lead to 1/f noise spectrum. On the other hand, qubit decoherence due to small mesoscopic ensembles can be dominated by the strongest fluctuator(s), and the generated noise is non-Gaussian with large variability between samples. The system dynamics can thus be quite different when crossing over from small to larger devices. A second point is that coupled qubits may provide access to working points that are unavailable for a single qubit. As the interdot bias controls simultaneously the exchange interaction and TLF coupling strength, the results of section V were restricted to the available subset of system parameters. With two or more qubits, inter-qubit couplings provide an additional handle over J, which is unrelated to the TLF coupling. In particular, optimal bias points at which J = 0, become available, where a very different dynamics is predicted.
Finally, experimental measurement of the qubit coherence time under DD sequences can provide insight into the spectral characteristics of the dominant noise processes. 51,52 The distinctive dynamics found for weak and strong fluctuators under DD, can therefore facilitate the characterization of charge fluctuators in solid-state devices. Moreover, by comparing our calculated decay rates for a general working point, where dissipative dynamics occur, with the results of a theory based on a Gaussian and Markovian noise, one can find both dephasing and relaxation functions, and determine the validity range of a Gaussian noise theory for qubit-TLF coupling under dynamical decoupling. v). This case is realized when δh = 0 and has been studied extensively. We outline its solution here to provide context to the general case studied in the main text. The controlled z rotation generated by J is eliminated in all cases except for free induction, thus we disregard J with the understanding that a qubit initially prepared along the x axis acquires only a random phase. Since we need to consider only the x component dynamics, it is more straightforward to evaluate the signal decay by:
χ(t) = dφp(φ, t)e iφ .(A1)
Dividing the probability distribution to p ± (φ, t) to accumulate phase φ while the TLF is in the up or down state, we can write a set of coupled equations for χ ± , analogous to Eqs. (9):
χ ± = −γ ± χ ± + γ ∓ χ ∓ ∓ ivχ ± .(A2)
These can be converted into equations for χ = χ + + χ − , and δχ = χ + − χ − :χ
= −ivδχ δχ = −δγχ − 2γδχ − ivχ,(A3)
which, in turn, are transformed into a second order equation for χ:χ
+ 2γχ + v 2 χ − ivδγχ = 0,(A4)
with initial conditions χ(t = 0) = 1, andχ(t = 0) = iv tanh(∆E t /2k B T ). The general solution to Eq. (A4) is 21,22,24
χ(t) = a + e −γ(1−µ)t + a − e −γ(1+µ)t ,(A5)
where
µ = 1 − v γ 2 + 2iv γ tanh ∆E T 2k B T a ± = (µ R ± 1)(1 ± iµ I ) 2µ ,(A6)
and µ R , µ I are the real and imaginary parts of µ. The same result is obtained from the more general formulation presented in section III. For pure dephasing with qubit initially along the x axis, z ± (t) = 0, and the remaining four equations in Eqs. (12) decouple into two sets (x, δy), and (δx, y), with a solution given by appropriately rotating Eqs. (A5)-(A6). In order to consider the effects of a sequence of π pulses we first note that in the case of pure dephasing, π x and π y pulses flip the qubit state in the same way, since the qubit evolution involves only the σ z operator (δh = 0). Repeated applications of π pulses alternately change the sign of the last term in Eq. (A4), resulting in a solution similar to Eq. (A5) with µ → µ * . A solution for the qubit signal decay after the application of a general sequence of π pulses can be found by stitching the solutions between pulses. This is done using a transfer matrix approach analogous to Eqs. (17)- (18). For a sequence of N pulses we find:
a j+1 + a j+1 − = Λ o (τ j ) a j + a j − , j ∈ odd (A7) and a j+1 + a j+1 − = Λ e (τ j ) a j + a j − , j ∈ even (A8) where Λ o (τ j ) = e −γ(µ+1)τj µ * e 2γµτj (1 − iµ I ) 1 + µ R e 2γµτj (µ R − 1) −(1 + iµ I ) ,(A9)
τ j is the time interval between pulses j and j + 1, Λ e = Λ * o , and a 1 ± are given by Eq. (A6). The final solution for the qubit signal decay after N control pulses reads
χ dd (t) = a N +1 + e −γ(1−µ)t + a N +1 − e −γ(1+µ)t ,(A10)
for even N , and a similar solution with µ → µ * for odd N .
As an example we retrieve below the known result for SE, as well as the signal decay in the case of two-pulse CPMG χ SE (t) = e −γt 2|µ| 2 (µ 2 I + 1)
± (1 ± µ R )e ±γµRt + (µ 2 R − 1) ± (1 ± iµ I )e ±iγµI t ,(A11)
χ CP2 (t) = e −γt 2µ|µ| 2 4(µ 2 R − 1)(µ 2 I + 1) sinh
γµ * t 2 + ± (µ R ± 1)(1 ± iµ I ) (µ 2 R − 1)e ±iγµI t + (1 + µ 2 I )e ±γµRt .(A12)
For N pulse PDD or CPMG sequences, analytical results are presented in section IV. For PDD with symmetric TLF switching, it can be shown that the qubit dephasing is given by two exponentials, similarly to Eq. (A10). These exponentials are the eigenvalues of the matrix Λ e (τ )Λ o (τ ), where τ = t/(N + 1). They are found as: 24
χ ± = e −γτ µ sinh γµτ ± cosh 2 γµτ − v 2 γ 2 . (A13)
In the limit of weak coupling, v ≪ γ, the χ + eigenvalue dominates, and we obtain a single decay rate by using Eq. (26):
Γ w = v 2 2γ 1 − tanh γτ γτ . (A14)
This result was found in the main text, by taking ϕ = 0 in Eqs. (28). and (29).
In the limit of strong coupling, v ≫ γ, Eq. (A13) results in two oscillating rates:
Γ s ± = γ (1 ± sincvτ ) ,(A15)
which contribute to the qubit dephasing with corresponding weights:
w ± = 1 2 (1 ± cos vτ ).(A16)
The short τ limit of Eq. (A15) can be retrieved from Eqs. (33) and (34) in the main text, where we presented the strong coupling results for the case of v ≪ ∆. Fig. 10 shows signal decay for several DD sequences, calculated for a TLF with an average switching time of 1 ms, and coupling strength of v = 0.1 neV. The solid blue and red lines correspond to SE and 2-pulse CPMG given by Eqs. (A11) and (A12), respectively. The green line, depicting 10-pulse CPMG, was generated using the transfer-matrix method, Eqs. (A7)-(A10). To verify our analytical results, we performed a numerical simulation of the TLF random switching, using a Poisson process with time constants γ ± . The symbols in Fig. 10 depict an average over 1000 realizations of the random switching, with appropriately weighted initial TLF states. We have carried similar simulations to verify our results for other DD sequences at general working positions, presented in the main text.
APPENDIX B: Effects of frame rotation on qubit dynamics
The analysis presented in the main text was applied to find the dynamics of the qubit state components in the rotated frame. The initial qubit state defined in Eq. (20) is thus appropriately set to lie along the rotated x axis. Current experimental techniques in QD S − T 0 spin qubits have limited accessibility to this state. The purpose of this appendix is to provide a more direct connection to currently investigated S − T 0 qubits by including the effects of the frame rotation. In this appendix only the Bloch components are defined in the original frame, thus the x axis refers to the state (|S + |T 0 )/ √ 2. Rotated quantities stated in the main text are denoted here with a prime. We note that for pure dephasing (ϕ = 0) the two frames coincide and all our previous results are intact. The qubit dynamics at the anticrossing point, where J ≫ δh and charge noise dominates, is thus adequately captured in the main text.
Eq. (6), from which our dynamical equations are derived is defined in a frame rotated by an angle ϕ in the x − z plane. Vectors and matrices are transformed to this frame by: To find the qubit's evolution under N -pulse PDD control sequence we rotate Eq. (24) back to the original frame:
k ′ = Ok (B1) M ′ = OM O −1 ,(B2)k(t) = O −1 T N +1 Ok(0),(B5)
where T is defined in Eq. (25). The decay rates, Γ i , found in Sec. IV from the eigenvalues of T , are unaffected by the above rotation, and only their weights in the solution for the Bloch components are modified. In order to find the transformed weights we use the eigenvectors of T , as described in Sec. IVA. For the case of weak coupling the weights of the three decay rates, Eqs. (28), in the three (non-rotated) Bloch components are found as:
whereτ , D, and F are defined in Eqs. (29), and i w x i = 1, and i w y i = i w z i = 0. Note that the rotation matrix mixes x and z components only when ϕ = 0, π/2, thus dissipative dynamics is minimal when δh ≫ J (large negative detuning) or J ≫ δh (at the singlet anticrossing).
For strong coupling there is a nonzero weight for the fast 2γ decay rate predominantly associated with the dynamics of (δx, δy, δz), in addition to the contributions from the three rates given in Eqs. (33). For ϕ = π/2 the transformation in and out of the rotated frame eliminates the weights of the two rates given in Eqs. (35), and instead the dynamics is governed by the rate:
Γ ′s 1 (ϕ = π/2) = 2γv 2 ∆ 2 1 − sinc 2τ 2 ,(B7)
and an additional non-zero contribution from Γ ′s 2 = 2γ. The weights of these two rates in the solution for the x component are found as:
w x 1 = 1 − 2v 2 ∆ 2 Ã −B 4 w x 2 = 2v 2 ∆ 2 Ã −B 4 ,(B8)
whereà andB are defined in Eqs. (34). Note that at ϕ = π/2 there is no dynamics in the y and z axes for our initial state. Fig. 11 shows the time evolution of the Bloch vector components in the non-rotated frame under 11-pulse PDD sequence. Fig. 11(a) captures a weakly-coupled fast TLF with parameters as in Figs. 3(b) and (c), while Fig. 11(b) shows a weakly-coupled slow TLF with parameters as in Fig. 3(e). The dashed red lines depict the transfer matrix calculation while the solid blue lines correspond to the analytic solution given by Eqs. (27), (28), and (B6). In both cases the most notable difference with respect to the results of Fig. 3 is the prominent dissipative dynamics associated with the z component. The y component (not shown) is an order-of-magnitude larger than its counterpart in Fig. 3, but is still considerably smaller.
Similarly, Figs. 11(c), and (d) show qubit dynamics under 11-pulse PDD sequence for a strongly-coupled TLF, with parameters corresponding to Figs. 4 and 5, respectively. The mixing between the x and z components, caused by the rotation, is apparent in the strong case as well. The y component shows appreciable dynamics only when ∆ v, as seen by the dotted red line in Fig. 11(d). For finite δh and zero exchange (ϕ = π/2) the x component in all cases shows comparable (slightly faster) decay to those given in Figs. 3(f), 4(d), and 5(c). As seen from Eqs. (B6) there is no relaxation in this case (y(t) = z(t) = 0).
22 γ
22∼ sin 2θ cos φ, for the case of θ T = 0, in close agreement with the solid red lines inFigs. 1(a), and 1(b).
Fig. 1 (FIG. 1 :
11c) shows the R dependence of the couplings at zero detuning. Solid lines depict angular-averaged results, whereas dashed lines show couplings calculated by fixing the polar (color online) qubit-TLF couplings vs. (a) polar angle θ; (b) azimuthal angle φ; (c) qubit-TLF distance R; (d) TLF Bohr radius DT ; (e) interdot bias normalized to QD confinement and measured from the singlet anticrossing point; (f) external magnetic field. In all plots blue (red) lines correspond to v β (vγ ). Left (right) y axis in panels (a) and (b) correspond to v β (vγ ). Green lines in panels (e) and (f) depict the exchange interaction. Unless otherwise noted we consider the singlet anticrossing point (zero detuning), and qubit-TLF distance R = 300 nm. Other system parameters are given in the main text.
FIG. 2 :
2(color online) Qubit decay under a SE control pulse with different rotation axes. (a) πz pulse (symbols) and FID (solid lines) decays for TLF distances of R = 200 nm and R = 300 nm. The qubit is at large negative bias, J = 0.02µeV, and δh = 0.125µeV (ϕ π/2). Dashed green lines correspond to the strong coupling approximation, Eq. (22); (b) qubit decay under πx and πy pulses for same parameters as in panel (a); (c) qubit decay under πx and πy at zero detuning, where J = 4.15µeV ≫ δh = 0.125µeV (ϕ ≈ 0), for R = 300 nm (v = 1.4µeV); (d) Same as panel (c) with δh = 0.5µeV. In all plots the TLF average switching time is 0.1 ms. ms and consider a large negative bias such that J = 0.02µeV. For a qubit-TLF distance, R = 200 nm (v = 0.2 neV), we have v ≫ γ, where the FID decay is dominated by v:
FIG
. 3: (color online) Panels (a)-(c): Weakly-coupled fast TLF (1/γ = 0.1µs, v = 2.5 neV), ∆ ∼ γ ≫ v. (a) Decay rates vs. time interval between pulses, τ , from Eqs. (28) and (29); (b), (c) Bloch vector components under 11-pulse PDD sequence. Panels (d)-(f): Weakly-coupled slow TLF (1/γ = 0.1 ms, v = 8.4 peV), ∆ ≫ γ ≫ v. (d) Decay rates from Eqs. (28) and (29); (e) x component under 11-pulse PDD sequence (y and z components are almost unaffected by the TLF); (f) Same as panel (e) with J = 0. In panels (b), (c), (e), and (f), solid blue lines depict Eqs. (27) and (32), and dashed red lines show transfer matrix calculation. In all plots J = δh = 0.125µeV (ϕ = π/4), except for panel (f) where J = 0.
FIG
. 4: (color online) Strongly coupled TLF with v = 2.5 neV (R = 300 nm) and 1/γ = 0.1 ms, for δh = 0.125µeV, satisfying ∆ ≫ v ≫ γ. (a), (b) Decay rates vs. time interval between pulses, τ , from Eqs. (33) for J = δh (ϕ = π/4), and J = 0 (ϕ = π/2), respectively; (c), (d) qubit decay under 11-pulse PDD sequence for J = δh and J = 0, respectively; (e) Bloch vector y component dynamics under 11-pulse PDD sequence (solid blue line) and J = 0 (dashed red line); (f) Bloch vector z component dynamics under 11pulse PDD sequence for J = δh (no z dynamics for J = 0). The dotted green lines in panels (d) and (e) correspond to Eqs. (36) and (37). The insets in panels (e) and (f) depict the long time dynamics of the transverse Bloch vector components.
FIG. 5 :
5(color online) Strongly coupled TLF with v = 2.5 neV (R = 300 nm) and 1/γ = 0.1 ms, for δh = 0.01µeV, satisfying ∆ v ≫ γ. (a), (b) Decay rates vs. time interval between pulses, τ , from Eqs.(33) for J = δh (ϕ = π/4), and J = 0 (ϕ = π/2), respectively; (c) qubit decay under 11-pulse PDD sequence for J = δh and J = 0; (d) y and z components dynamics under 11-pulse PDD sequence (no z dynamics for J = 0). Notice that for comparison withFig. 4we kept the same coupling strength, although normally it would reduce with J. One can tune J without changing the bias (thus keeping v fixed) by introducing a second double dot.
satisfying i w x i = 1 ,
1and i w y i = 0. The dashed red lines in Figs. 4(d) and (e) are plotted from Eq. (27), using the weights in Eqs. (36), and (37), whereas the dynamics of the z component, present only for J = 0, is shown in
FIG. 6 :
6(color online) Qubit decay times vs. TLF average switching times under a 10-pulse CPMG sequence, for a qubit-TLF distance R = 300 nm. (a) Zero detuning (J = 4.15µeV, v = 1.4µeV); (b) moderate negative detuning (J = 0.125µeV, v = 2.5 neV); and (c) large negative detuning (J = 0.02µeV, v = 62 peV). In all plots δh = 0.125µeV. The black, green, and red dashed lines correspond to very strong, strong, and weak coupling regimes, respectively. The vertical dotted lines in panels (b) and (c) mark v = γ (in panel (a) it is not within range).
FIG. 7 :
7Furthermore, the same power law was observed in experiments on spin qubits in nitrogen vacancy centers in diamond, where both CPMG and (color online) Qubit decay times vs. qubit-TLF distance under a 10-pulse CPMG sequence, for a TLF average switching time of 0.1 ms. (a) Zero detuning (J = 4.15µeV); (b) moderate negative detuning (J = 0.125µeV); and (c) large negative detuning (J = 0.02µeV). In all plots δh = 0.125µeV. The black, green, and red dashed lines correspond to very strong, strong, and weak coupling regimes, respectively. The vertical dotted lines in panels (b) and (c) mark v = γ.
FIG. 8 :
8(color online) Qubit decay times vs. number of CPMG pulses for R = 300 nm. (a) Zero detuning (J = 4.15µeV); (b) moderate negative detuning (J = 0.125µeV); and (c) large negative detuning (J = 0.02µeV). In all plots δh = 0.125µeV. The dashed lines correspond to Eq. (44). The vertical dotted black (red) lines mark the crossover from very strong (weak) to strong coupling regime. necessary to enter the above regime, where black and red lines correspond to the left and right conditions in Eq. (45), respectively. While the N 2/3 power law is in effect only for N satisfying Eq. (45), considerable increase in coherence time is still obtained for smaller number of pulses, to the left of the vertical dotted lines.
FIG. 9 :
9(color online) Qubit decay times vs. number of pulses for R = 300 nm under various protocols. (a) Zero detuning (J = 4.15µeV), TLF average switching time of 1µs; (b) moderate negative detuning (J = 0.125µeV), TLF average switching time of 0.1 ms; and (c) large negative detuning (J = 0.02µeV), TLF average switching time of 1 ms. Panels (d), (e), and (f) show decay times to 95% of initial value, for same parameters as in panels (a), (b), and (c), respectively. In all plots δh = 0.125µeV. CDD2 and 2-pulse UDD are the same as 2-pulse CPMG, and PDD has only odd number of pulses.
FIG. 10: (color online) Signal decay for ϕ = 0 under SE (blue line, circles), 2-pulse CPMG (red line, triangles), and 10-pulse CPMG (green line, squares). Solid lines show analytic results and symbols depict averages and standard deviations of 1000 numerical realizations of the TLF random switching process. In all plots δh = 0, v = 0.1 neV, average TLF switching time is γ = 1 ms, and TLF level splitting is ∆ET = 0.45µeV.
following we consider a qubit initially prepared along the x axis: |ψ(t = 0) = (|S + |T 0 )/ √ 2. In the rotated frame the initial state replacing Eq.(20) reads: k ′ (0) = cos ϕ, 0, − sin ϕ δγ 2γ , − cos ϕ δγ 2γ , 0, sin ϕ . (B4)
F 2 +
2D 2 sin 2 2ϕ ,
ACKNOWLEDGEMENTSThe author thanks Joakim Bergli for helpful discussions and acknowledges funding from Research Corporation.APPENDIX A: PURE DEPHASINGFor pure dephasing (ϕ = 0), Eq. (7) reads B ± = (0, 0, J ∓
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"A New Interpolation Approach and Corresponding Instance-Based Learning",
"A New Interpolation Approach and Corresponding Instance-Based Learning"
]
| [
"Shiyou Lian [email protected] \nXi'an Shiyou University\nXi'anChina\n"
]
| [
"Xi'an Shiyou University\nXi'anChina"
]
| []
| Starting from finding approximate value of a function, introduces the measure of approximation-degree between two numerical values, proposes the concepts of "strict approximation" and "strict approximation region", then, derives the corresponding one-dimensional interpolation methods and formulas, and then presents a calculation model called "sum-times-difference formula" for high-dimensional interpolation, thus develops a new interpolation approach ADB interpolation. ADB interpolation is applied to the interpolation of actual functions with satisfactory results. Viewed from principle and effect, the interpolation approach is of novel idea, and has the advantages of simple calculation, stable accuracy, facilitating parallel processing, very suiting for high-dimensional interpolation, and easy to be extended to the interpolation of vector valued functions.Applying the approach to instance-based learning, a new instance-based learning method learning using ADB interpolation is obtained. The learning method is of unique technique, which has also the advantages of definite mathematical basis, implicit distance weights, avoiding misclassification, high efficiency, and wide range of applications, as well as being interpretable, etc. In principle, this method is a kind of learning by analogy, which and the deep learning that belongs to inductive learning can complement each other, and for some problems, the two can even have an effect of "different approaches but equal results" in big data and cloud computing environment. Thus, the learning using ADB interpolation can also be regarded as a kind of "wide learning" that is dual to deep learning. Definition 2-1 Let R be real number field, x 0 [a, b]R, and [ 0 , 0 ][a, b] be a neighborhood of x 0 , which is called the approximation region of x 0 . For x[a, b], say x is approximate to x 0 if and only if x[ 0 , 0 ]. | 10.36227/techrxiv.15956145.v1 | [
"https://arxiv.org/pdf/2108.11530v1.pdf"
]
| 237,303,845 | 2108.11530 | 87578d5054f8cb0880730add743dd82a5f95fb22 |
A New Interpolation Approach and Corresponding Instance-Based Learning
Shiyou Lian [email protected]
Xi'an Shiyou University
Xi'anChina
A New Interpolation Approach and Corresponding Instance-Based Learning
1Approximation-DegreeInterpolationStrict ApproximationSum-Times-Difference FormulaInstance-Based LearningWide Learning
Starting from finding approximate value of a function, introduces the measure of approximation-degree between two numerical values, proposes the concepts of "strict approximation" and "strict approximation region", then, derives the corresponding one-dimensional interpolation methods and formulas, and then presents a calculation model called "sum-times-difference formula" for high-dimensional interpolation, thus develops a new interpolation approach ADB interpolation. ADB interpolation is applied to the interpolation of actual functions with satisfactory results. Viewed from principle and effect, the interpolation approach is of novel idea, and has the advantages of simple calculation, stable accuracy, facilitating parallel processing, very suiting for high-dimensional interpolation, and easy to be extended to the interpolation of vector valued functions.Applying the approach to instance-based learning, a new instance-based learning method learning using ADB interpolation is obtained. The learning method is of unique technique, which has also the advantages of definite mathematical basis, implicit distance weights, avoiding misclassification, high efficiency, and wide range of applications, as well as being interpretable, etc. In principle, this method is a kind of learning by analogy, which and the deep learning that belongs to inductive learning can complement each other, and for some problems, the two can even have an effect of "different approaches but equal results" in big data and cloud computing environment. Thus, the learning using ADB interpolation can also be regarded as a kind of "wide learning" that is dual to deep learning. Definition 2-1 Let R be real number field, x 0 [a, b]R, and [ 0 , 0 ][a, b] be a neighborhood of x 0 , which is called the approximation region of x 0 . For x[a, b], say x is approximate to x 0 if and only if x[ 0 , 0 ].
Introduction
Instance-based learning [1,2] is also called nonparametric approach [3,4] . Instead of establishing a global model of sample data, the approach uses sample data to interpolate directly to achieve objective function approximation. Examples are the experience, and the specific manifestation of a general rule. From the cognitive perspective, the way of learning based on instances is closer to human learning. So, it makes sense to give machines this learning ability. Instance-based learning has been studied for a long time and many achievements have been made (such as k-nearest neighbor algorithm, distance weighted nearest neighbor algorithm, locally weighted regression algorithm etc.), but there are still some problems and shortcomings on which (such as high-dimensional interpolation and misclassification). Therefore, instance-based learning as well as corresponding interpolation technique still needs us to continue to research and develop. On the other hand, the current environments of big data and cloud computing undoubtedly provide strong support for instance-based learning and interpolation, and for which also open up a new place to display its prowess. Inspired by the approximate evaluation method of flexible linguistic functions [5,6] in reference [7], in this paper, we intend to introduce a measure of degree of approximation between numerical values to study the approximate evaluation of numerical functions, and then explores new interpolation approaches based on the degree of approximation and corresponding instance-based learning methods.
In contrast to the strict approximation region in Definition 2-2, we refer to the "circle" region centered on point x 0 as the ordinary approximation region of x 0 U. The relation between the strict approximation region and the ordinary approximation region of the same (2D) point x 0 is shown in Figure 2-1. The illustration also shows the relationship between the strict approximation and the ordinary approximation. In fact, the reference [8] has stated: the geometric meaning of "close to point (x 1 , x 2 ,…, x n ) " is different from that of "close to x 1 and close to x 2 and close to x n ". to be called the degree of approximation, shortening as approximation-degree, of x to x 0 .
Where x 0 0 =r l is called left approximation radius of x 0 , 0 x 0 =r r is called right approximation radius of x 0 . We call the function relation defined by the Equation (2-1) to be the approximation-degree function of x 0 . According to the definition of the approximation-degree function above, the approximation-degree function of x 0 is
Approximate Evaluation of Functions Based on Approximation-Degree
− 1 0 − 1 , x[a 1 , x 0 ] 0 ( )= (3-1) − 1 0 − 1 , x[x 0 , b 1 ]
And let the approximation region of y 0 is [c 1 , d 1 ][c, d], the approximation-degree function of y 0 is
− 1 0 − 1 , y[c 1 , y 0 ] 0 ( )= (3-2) − 1 0 − 1 , y[y 0 , d 1 ]
It can be seen that the range of approximation-degree function 0 ( ) is [0, 1] and which is also reversible. In fact, it's easy to obtain that
d y ( 0 − 1 )+c 1 , d y [0, 1] 0 ( ) 1 = (3-3) d y ( 0 − 1 )+d 1 , d y [0, 1]
where d y is the approximation-degree of y to y 0 . Now we find the approximation-degree 0 ′ , and then set 0 ( ′)= 0 ( ′) (that is, transmitting the approximation-degree of x' (to x 0 ) to y' (to y 0 )); Further, we derive the required approximate value of y' from approximation-degree 0 ( ′) and inverse function 0 ( ) 1 of approximation-degree function of 0 . It can be seen that the inverse function 0 ( ) 1 of 0 ( ) is a piecewise function, which has two parallel expressions. Thus, substituting approximation-degree 0 ( ′)=d into 0 ( ) 1 , we can get two y (y 1 and y 2 ). Then, which y should be chosen as the desired approximate value of function?
Obviously, the desired y is related to the position of x' relative to x 0 and the trend (i.e., being increasing, decreasing, or a constant) of f(x) near x 0 . Thus, we have the following ideas and techniques:
(1) consider whether the derivative f'(x 0 ) of the function f(x) at point x 0 knows. If the derivative f'(x 0 ) is known, we can estimate the trend of f(x) near x 0 according to the f'(x 0 ) being positive, negative or zero and then determine the choice of y's value.
(2) consider whether there is a point x* on the x' side near the point x 0 (which does not beyond the approximation region of x 0 ), whose corresponding value of function, f(x*) = y*, is known. If there is such a point x*, we can estimate the trend of f(x) between the x 0 and x* by utilizing the size relation between the corresponding y* and y 0 , and then determine the choice of y's value. For instance, when x*<x'<x 0 , if y*<y 0 , which then shows that the general trend of function f(x) is increasing on the sub interval (x*, x 0 ), thus the y 1 , i.e., that value less than y 0 , should be chosen; while if y*>y 0 , which then shows that the general trend of function f(x) is decreasing on the sub interval (x*, x 0 ), thus the y 2 , i.e., that value larger than y 0 , should be chosen.
(3) If the derivative f'(x 0 ) is unknown and there is no such reference point x*, take the average 1 + 2 2 or take the y 0 directly as the approximate value of f(x').
Due to the space limit, in the following, we only discuss the second method further, and use third method to classification problems. As for the first method, it will be introduced in another article.
Finding Approximate Value of a Multivariate Function Based on Approximation-Degree
Let's take the function of two variables as an example to discuss this problem.
Let z=f(x, y) be a function (relation) from [a 1 , b 1 ]×[a 2 , b 2 ] to [c, d]. Suppose a pair of corresponding values, ((x 0 , y 0 ), z 0 ) of function z=f(x, y) and point (x', y') approximate to point (x 0 , y 0 ) are known. In the case that expression of function f(x, y) is unknown or not used, find the approximate value of f(x', y').
By the definition of strict approximation, (x', y') approximate to (x 0 , y 0 ) is equivalent to x' approximate to x 0 and y' approximate to y 0 . Thus, we can find the approximate values z x and z y of function f(x, y 0 ) and f(x 0 , y) at points x' and y', respectively. It can be seen that this is really two approximate evaluation problems of univariate functions. Thus, we further imagine that if there is respectively an adjacent point (x*, y 0 ) and (x 0 , y*) in the x-direction and y-direction of the point (x 0 , y 0 ), as shown in Figure 3-1, whose corresponding function values f(x*, y 0 ) and f(x 0 , y*) are known, then the approximate value z x of f(x', y 0 ) can be got by utilizing f(x*, y 0 ), and the approximate value z y of f(x 0 , y') can be got by utilizing f(x 0 , y*), just like that of the previous unary function. Thus, we firstly get separately the approximation-degree 0 ( ′ ) and 0 ( ′ ) , then set 0 z = 0 ( ′ ) , and 0 z = 0 ( ′ ); and then, substitute them separately into inverse function 0 z 1 of 0 z and get two pairs of candidate approximations, then taking separately f(x*, y 0 ) and f(x 0 , y*) as reference, choose z x and z y from respective candidate values (as shown in Figure 3-1).
Having got the approximate values z x and z y , how can we further get the approximate value z we need?
Let z 1 =(z x +z y )/2 be the average of z x and z y . It can be seen from Figure 3-2 that z 1 can actually be viewed as an approximate value of f(x, y) at midpoint (denoted by (x 1 , y 1 )) between (x', y 0 ) and (x 0 , y'). We can see from the figure that z 1 <z 0 , i.e., the varying trend of function values from z 0 to z 1 is decreasing. Set z 0 z 1 =c 1 (c 1 is the length of segment BC in Figure 3-3(a)), then z 1 =z 0 c 1 . Then, according to the varying trend of function values from z 0 to z 1 (i.e., the slope of segment AB in Figure 3-3(a)), also, taking into account that point (x 1 , y 1 ) is just the midpoint of segment joining points (x', y') and (x 0 , y 0 ), that is,
′, ′ − ( 0 , 0 ) =2 1 , 1 − ( 0 , 0 )
, so we infer that the approximate value of function at point (x', y') can be z 0 2c 1 (as shown in Figure 3-3(a)). Thus, it follows that
z=z 0 2c 1 =z 0 2(z 0 z 1 )=z 0 2[z 0 (z x +z y )/2] = z x +z y z 0
Of course, z 1 may also be greater than z 0 or equal to z 0 . If z 1 >z 0 , then the varying trend of function values from z 0 to z 1 is increasing (as shown in Figure 3-3(b)). Set z 0 z 1 =c 2 (c 2 is the length of segment BC in Figure 3-3(b)), then z 1 =z 0 c 2 . Then, according to the varying trend of function values from z 0 to z 1 (i.e., the slope of segment AB in Figure 3-3(b)), we infer that the value of function at point (x', y') can be z 0 2c 2 . Thus, z=z 0 +2c 2 =z 0 +2(z 1 z 0 )=z 0 +2[ (z x +z y )/2z 0 ] = z x +z y z 0
(x 0 , y*) (x*, y 0 ) (x 0 , y 0 ) (x 0 , y') (x', y 0 ) (x', y') z x z 0 z 1 z y z y x (x 0 , y*) (x', y 0 ) (x 0 , y') (x', y') (x*, y 0 ) (x 0 , y 0 )
The third case: z 1 =z 0 . This indicates that the values of function remain unchanged from z 0 to z 1 . Thus, we can take z=z 0 . And by z 0 =z 1 = (z x +z y )/2, it follows that 2z 0 =z x +z y . Thus,
z=2z 0 z 0 = z x +z y z 0
In summary, we see that, no matter what relationship may be between the average of z x and z y and the z 0 , or no matter how the value of the function varies from z 0 to z 1 , the approximate value of the function at point (x', y') can always be taken as
z= z x +z y z 0 (3-4)
This equation is also the calculation model of the approximate value of function of two variables, z= f(x, y). It can be seen that this is actually decomposing the approximate evaluation of a function of two variables into the approximate evaluation of two functions of one variable, firstly, then, synthesizing two obtained approximate values into a value as an approximate value of the original function of two variables. Extending this technique of "first splitting then synthesizing" to the approximate evaluation of a function of 3 variables, u= f(x, y, z), we obtain the formula synthesizing approximate value of the function is u = u x +u y +u z 2u 0 (3-5) And then, for a function of n variables, y= f(x 1 , x 2 , … , x n ), the corresponding formula synthesizing approximate value of the function is
y= 1 + 2 + ⋯ + − ( − 1) 0 or y= n i x y n y i 1 0 ) 1 ( (3-6)
For convenience of narration, we may as well refer to the Equations (3-4), (3)(4)(5) and (3)(4)(5)(6) as the sum-times-difference formula.
Interpolation Based on Approximation-Degree
Let y=f(x) be a function (relation) from [a, b] to [c, d]. A set of pairs of corresponding values of function y=f(x), {(x 1 , y 1 ), (x 2 , y 2 ), … , (x n , y n )}, is known, where x 1 <x 2 < , … , <x n . Now the
(a) (b) B z y x z 0 z 1 z A (x', y') (x 1 , y 1 ) (x 0 , y 0 ) C y x z z z 1 z 0 C A B (x 1 , y 1 ) (x', y') (x 0 , y 0 )
question is: in the case that the expression of function f(x) is unknown or not used, construct an interpolating function g(x) such that g(
x i )= f(x i ) (i=1, 2, … , n), and for other x[a, b], g(x) f(x)
. This is the usual interpolation problem. We now use the approach that finding approximate value of a function above to solve the interpolation problem.
Let ax 1 , x n b, then x 1 , x 2 , … , x n is a group of interpolation base points (or nodes). We definite the approximation region of x 1 as [x 1 , x 2 ], the approximation region of x i as [x i1 , x i1 ] (i=2, 3, … , n1), and the approximation region of x n as [x n1 , x n ], and then definite separately the approximation-degree functions of base point x 1 , x i , and x n as
1 ( )= − 2 1 − 2 , x[ 1 , 2 ] (4-1) − −1 − −1 , x[ 1 , ] ( )= (4-2) − +1 − +1 , x[ , 1 ] ( )= − −1 − −1 , x[ 1 , ] (4-3)
Note that it is not hard to see from the above expressions of approximation-degree function that when x[ 1 ,
x 1 +x 2 2 ], [ x i−1 +x i 2 , x i +x i+1 2 ], or [ x n −1 +x n 2
, ], the corresponding approximation-degrees 1 ( ), , and ( ) are always 0.5. This means that x is closer to the corresponding base point x 1 , x i or x n . We then define the approximation-degree functions of y i (i=1, 2, … , n) in the same principle and way.
( )= − −1 − −1 = 1 − −1 − −1 − −1 , y[ 1 , ] (4-4) ( )= −1 − −1 − = 1 − −1 − −1 − −1 , y[ , 1 ] (4-5) ( )= +1 − +1 − = 1 − +1 − +1 − +1 , y[ , +1 ] (4-6) ( )= − +1 − +1 = 1 − +1 − +1 − +1 , y[ +1 , ](4-7)
Obviously, in the four expressions above, (4-4) = (4-5) and (4-6) = (4-7). Thus, the 4 functional expressions can be reduced as two expressions:
( ) = 1 − −1 − y −1 − −1 ( )= 1 − +1 − y +1 − +1
And then, we get the inverse expressions of these two functional expressions:
( ) 1 =d y (y i -y i1 )+y i1 (4-8) ( ) 1 =d y (y i -y i+1 )+y i+1(( ) 1 = − −1 − −1 (y i y i1 )+y i1 = − −1 − −1 + −1 − −1 − −1 namely y= − −1 − −1 + −1 − −1 − −1 , x[ −1 + 2 , ] (4-10)
Similarly, the Expression (4-9) of the inverse function
( ) 1 becomes y= − +1 − +1 + x i y i+1 −x i+1 y i x i −x i+1
, x[ , ].
+
Actually，Equations (4-10) and (4-11) are two interpolation formulas. In this way, we actually derive an interpolation approach by using the approximate evaluation of function based on approximation-degree. We call this approach to be the approximation-degree-based interpolation, or ADB interpolation for short.
Specifically, the practice of ADB interpolation is: take base points a= 1 , 2 , … , = as points of view, according to base points and their approximation regions to partition interval
[a, ]=[ 1 ,
] into 2n2 subintervals as shown in Figure 4- And the effect is shown in Figure 4-2.
From the interpolation method of the univariate function and the method of finding the approximate value of the function of n variables above, we obtain a general n-dimensional ADB interpolation method, that is: decompose an n-dimensional interpolation into n one-dimensional interpolation, then use one-dimensional ADB interpolation to find the corresponding approximate values, respectively, and finally synthesize the n approximate values by using sum-times-difference formula into one value as the approximation value of is the approximate value of function that obtained by one-dimensional ADB interpolation for (i=1,2,…,n). Actually, it is not difficult to see that the equation (4)(5)(6)(7)(8)(9)(10)(11)(12) can also be said to be the formula of multidimensional ADB interpolation, which is also a calculation model of high-dimensional interpolation. (Note: due to the space limit, ADB interpolation with scattered data points will be introduced in another article.)
Instance-Based Learning Using ADB interpolation
In the above, we develop a new interpolation approach, ADB interpolation, from finding approximate value of a function. Since ADB interpolation is a local interpolation, it can be used for instance-based machine learning. In the following, we present two learning algorithms.
( , y j ), z ij ) , as example data:
x i : 20, 18, 16, … , 2, 0, 2, … , 16, 18, 20.
y j : 20, 18, 16, … , 2, 0, 2, … , 16, 18, 20.
z ij : x i 2 y j 2 and take the following data points, (x, y), as query data:
x: 20, 20, 20, 19. The effect drawing is shown in Figure 5-1.
Example 5-2 Figure 5-2 below shows an effect drawing of learning using ADB interpolation. The example data are taken from the peaks function in MATLAB, and the query data is also designed according to this function. Limited by space, these data are omitted. Where the grid curve is the graph formed by example data from function z=x 2 y 2 , and the red circles indicate the points obtained by learning using ADB interpolation.
As can be seen from the above, this instance-based learning using ADB interpolation has the following characteristics:
The learning method takes data points (x l ) of training examples ((x l , f(x l )) as centers to set approximation regions and compute approximation-degrees.
ADB interpolation is a local interpolation, the training examples involved in interpolation are related to the position of the currently queried data point x' relative to its nearest data point x k , the number of which is related to the dimension of the vector x, n-dimensional x only involves 1+n training examples ((x k , f(x k )) and (x 1 , f(x 1 )), (x 2 , f(x 2 )), … , (x n , f(x n )) ). But since the point x' is only approximate to the point x k , the corresponding (j=1,…, n) are most affected by the example (x k , f(x k )), and the final synthesized value ( ') is also most affected by (x k , f(x k )).
If distributed storage and parallel processing (including parallel lookup and parallel computation) are used, the time complexity of corresponding algorithm is independent of the dimension of vector x, and its efficiency is almost equal to that of one-dimensional interpolation at all time.
The interpolation formulas (including sum-times-difference formula) are derived entirely by the mathematical method, so they have definite mathematical basis.
The sum-times-difference formula is actually a linear combination of coordinate components of an interpolation point, and the denominators of the coefficients before each coordinate component are separately the difference between the coordinate component and the corresponding coordinate component of corresponding base point, that is, the distance between the two, so, these coefficients happen to also have a function of the weight values. Thus, viewed from the form, the sum-times-difference formula is a linear weighted regression formula. That is to say, the sum-times-difference formula here coincides with the traditional local weighted linear regression model. However, in local weighted linear regression, these coefficients are determined by searching, i.e., learning, while in our ADB interpolation, these coefficients are determined by looking up. The former is guided and constrained by error ), and the latter by approximation-degree function (e.g. ( )). In the sense of approximation-degree, the approximate value ( ') of the function obtained from the sum-times-difference formula is always the most accurate. The accuracy of the returned approximate value ( ') of a function is positively related to the approximation-degree of x' to x k .
Comparing the learning using ADB interpolation with the deep learning, the deep learning is to approximate objective function with the strategy of deepening vertically [9] , yet the learning using ADB interpolation can approximate objective function with the strategy of increasing density horizontally. So, the learning using ADB interpolation and the deep learning can complement each other, and can even have an effect of "different approaches but equal results" in the case of sufficient samples. Figure 5-3, and the small boxes surrounding them are their respective strict approximation regions; and the white circles represent data points to be classified. Using the learning algorithm that using ADB interpolation to classify these data points, the classifying results are shown in the figure. As you can see, there are two queried data points are respectively classified to two classes to which the corresponding training example data points belong, because they are located respectively in the strict approximation regions of the corresponding data points, while the two queried data points outside the small boxes are not classified. It can be seen that for classification problems, the learning using ADB interpolation has the following advantages:
Although similar to the traditional 1-NN algorithm, the approximation regions in the algorithm are aimed at the data points of training examples, and which are strict approximation regions of square.
Since the approximation and approximation regions involved in the algorithm are strict approximation and strict approximation regions, for classification problems, learning using ADB interpolation avoids, from the source, the problem in traditional instance-based learning (e.g. k-NN algorithm) that some datum objects with similar partial attributes are classified into a class.
The classifying result of learning using ADB interpolation is unique, and there is no the phenomenon like k-NN algorithm, that classifying result may change with the change of k's value.
Summary
In this paper, started from finding approximate value of a function, we introduced the measure of approximation-degree between two numerical values, proposed the concepts of "strict approximation" and "strict approximation region", then, derived the corresponding one-dimensional interpolation methods and formulas, and then presented a calculation model called "sum-times-difference formula" for high-dimensional interpolation, thus we developed a new interpolation approach approximation-degree-based interpolation, i.e., ADB interpolation. ADB interpolation was applied to the interpolation of actual functions with satisfactory results. Viewed from principles and examples, the approach is of novel idea, and it has the advantages of simple calculations (they are all arithmetic), stable accuracy (benefitting from local interpolation and that the approximation-degrees are always not less than 0.5); especially, the approach facilitates parallel processing, very suiting for high-dimensional interpolation, and easy to be extended to the interpolation of vector valued functions.
Applying ADB interpolation to instance-based learning, we obtained a new instance-based learning method learning using ADB interpolation, and we also gave several examples of the learning. Viewed from principles and examples, the learning method is of unique technique (e.g., taking data points of training examples as centers to set approximation regions that are also strict approximation regions and compute approximation-degrees). Besides the advantages of ADB interpolation, the learning method has also the advantages of definite mathematical basis, implicit distance weights, avoiding misclassification (guaranteed by strict approximation), high efficiency (benefitting from distributed storage and parallel processing), and wide range of applications (which can be applied to the regression or classification problems, and can be used for large sample learning or small sample even single sample learning), as well as being interpretable, etc. In principle, this method is a kind of learning by analogy, which and the deep learning that belongs to inductive learning can complement each other, and for some problems, the two can even have x 2 an effect of "different approaches but equal results" in big data and cloud computing environment. Thus, the learning using ADB interpolation can also be regarded as a kind of "wide learning" that is dual to deep learning.
0 Figure 2 - 1
021Definition 2-3 Let R be real number field, x 0 [a, b]R, and [ 0 , 0 ][a, b] be the approximation region of x 0 . Set x An illustration of the relation between strict approximation region and ordinary approximation regionWhere the square region is the strict approximation region of point x 0 , and the circular region is its ordinary approximation region.
Figure 3 -
32 Illustration-1 of synthesizing z x and z y into z
Figure 3 - 1
31Utilizing the values of the function at points (x*, y 0 ) and (x 0 , y*) to determine the approximate values of f(x', y 0 ) and f(x 0 , y'), respectively
Figure 3 - 3
33Illustration-2 of synthesizing z x and z y into z
Figure 4 - 2
42The effect drawing of ADB interpolation for function y=sin x
Figure 4
4of n variables. The n pairs of one-dimensional interpolation formulas are required for n-dimensional ADB interpolation, they are as follows:
Figure 4 - 3
43The effect drawing of ADB interpolation for function z = x 2 y 2 Where (a) is the functional graph before interpolating, and (b) is the functional graph after interpolating Example 4-3 Using ADB interpolation to do interpolation for function of three variables, u= − 3 − 3 − 3 , the effect (slice chart) is shown in Figure 4-4.
Figure 4 - 4
441) A leaning algorithm for regression problems (in which training examples are regularly distributed) ------------------------------------ Put samples in sample set X={( , ( ))} =1 (x=(x 1 , x 2 … , x n )) into a corresponding data structure S in centralized or distributed manner (as training examples); For the currently being queried x'=(x 1 ', x 2 ', … , x n '): According to its coordinate components x 1 ', x 2 ', … , x n ' look up in S sequentially or in parallel to determine a x k (k{1,2,…,m}) to which the approximation-degree of x' is highest; Take x k as the center, and according to the position of x' relative to x k , choose n corresponding nearest neighbors x 1 , x 2 , … , x n (The data points here are renumbering.) from S, then construct the corresponding one-dimensional interpolation formulas = g(x j k j x , f(x k ), l j x , f(x l )) (j=1,…, n; l{1,2,…,n}) and then compute 1 , 2 ,…, sequentially or in parallel; The effect drawing (slice chart) of ADB interpolation for function of three variables, u= − 3 − 3 − 3Where (a) is the functional graph before interpolating, and (b) is the functional the following pairs of corresponding values of function z=x 2 y 2 , ((x i
Figure 5 - 1
51An effect drawing of the learning using ADB interpolation
Figure 5 - 2
52An illustration of the effect of learning using ADB interpolationWhere the left is the functional graph formed by example data, and the right is the functional graph obtained by learning using ADB interpolation.
( 2 )
2A learning algorithm for classification problems ------------------------------------ Put samples in sample set X={( , ( ))} =1 (x=(x 1 , x 2 … , x n ), f(x) is a class label ) into a corresponding data structure S in centralized or distributed manner; (as training examples) For the currently being queried x'=(x 1 ', x 2 ', … , x n '): According to its coordinate components x 1 ', x 2 ', … , x n ' look up in S sequentially or in parallel to determine a x k (k{1,2,…,m}) to which the approximation-degree of x' is highest; If such a x k is found, return ( ') f(x k ) Else exit. ------------------------------------Example 5-3 Suppose that the data points of training examples of a classification problem are shown as the black circles in
Figure 5
5Figure 5-3 An illustration of applying the learning using ADB interpolation to classification
] is the approximation-degree of y to y i .4-9)
Here d y = ( )[0, 1Now, set
d y =d x =
Also, considering that on the sub interval [ −1 +
2
, ], the interpolated function y=f(x) may
being increasing, decreasing, or a constant; while when y=f(x) is increasing, certainly y i-1 <y i ,
so the corresponding y[ −1 +
2
, ]; when y=f(x) is decreasing, certainly y i <y i-1 , so the
corresponding y[ , −1 +
2
]; and when y=f(x) is a constant, y i =y i-1 , so the corresponding
y=y i [ −1 +
2
, ] as well as y[ , −1 +
2
]. Thus, when x[ −1 +
2
, ], the corresponding
y i and y i-1 are adjacent, and here
=
x−x i−1
x i −x i−1
, thus the Expression (4-8), the expression of
the corresponding inverse function
( ) 1 , becomes
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| []
|
[
"A Non-Intrusive Low-Rank Approximation Method for Assessing the Probabilistic Available Transfer Capability",
"A Non-Intrusive Low-Rank Approximation Method for Assessing the Probabilistic Available Transfer Capability"
]
| [
"Ieee Transactions ",
"Smart Grid ",
"Vol ",
"No "
]
| []
| []
| In this paper, a mathematical formulation of the probabilistic available transfer capability (PATC) problem is proposed to incorporate uncertainties from the large-scale renewable energy generation (e.g., wind farms and solar PV power plants). Moreover, a novel non-intrusive low-rank approximation (LRA) is developed to assess PATC, which can accurately and efficiently estimate the probabilistic characteristics (e.g., mean, variance, probability density function (PDF)) of the PATC. Numerical studies on the IEEE 24-bus reliability test system (RTS) and IEEE 118-bus system show that the proposed method can achieve accurate estimations for the probabilistic characteristics of the PATC with much less computational effort compared to the Latin hypercube sampling (LHS)-based Monte Carlo simulations (MCS). The proposed LRA-PATC method offers an efficient and effective way to determine the available transfer capability so as to fully utilize the transmission assets while maintaining the security of the grid. | null | [
"https://arxiv.org/pdf/1810.08156v1.pdf"
]
| 69,270,886 | 1810.08156 | ea2bd5685d2d45b11dcb31e875cde0773ef588fe |
A Non-Intrusive Low-Rank Approximation Method for Assessing the Probabilistic Available Transfer Capability
2018
Ieee Transactions
Smart Grid
Vol
No
A Non-Intrusive Low-Rank Approximation Method for Assessing the Probabilistic Available Transfer Capability
12018Index Terms-Available transfer capability (ATC)Copulalow-rank approximation (LRA)Nataf transformationpolyno- mial chaos expansion (PCE)transmission reliability margin (TRM)total transfer capability (TTC)
In this paper, a mathematical formulation of the probabilistic available transfer capability (PATC) problem is proposed to incorporate uncertainties from the large-scale renewable energy generation (e.g., wind farms and solar PV power plants). Moreover, a novel non-intrusive low-rank approximation (LRA) is developed to assess PATC, which can accurately and efficiently estimate the probabilistic characteristics (e.g., mean, variance, probability density function (PDF)) of the PATC. Numerical studies on the IEEE 24-bus reliability test system (RTS) and IEEE 118-bus system show that the proposed method can achieve accurate estimations for the probabilistic characteristics of the PATC with much less computational effort compared to the Latin hypercube sampling (LHS)-based Monte Carlo simulations (MCS). The proposed LRA-PATC method offers an efficient and effective way to determine the available transfer capability so as to fully utilize the transmission assets while maintaining the security of the grid.
I. INTRODUCTION
A VAILABLE transfer capability (ATC) is a crucial index designed to describe how much more electric power (MW) is available to buy or sell in a specified period in a competitive electric power market. It is calculated based on a set of assumed operating conditions well before the system approaches that operational state. The U.S. Federal Energy Regulatory Commission (FERC) requires the ATC to be periodically calculated and posted on the Open Access Same-time Information System (OASIS) for public access [1].
In 1996, the North American Electric Reliability Council (NERC) proposed a uniform definition for ATC and the associated terminologies [2]. By definition, ATC is a measure of the transfer capability remaining in the physical transmission network for further commercial activity over and above already committed uses. Mathematically, ATC is defined as: ATC = TTC -TRM -ETC -CBM (1) where TTC denotes the total transfer capability, TRM denotes the transmission reliability margin, ETC denotes the existing transmission commitments (base case) which include retail customer service, and CBM denotes the capacity benefit margin (CBM). More detailed explanations for the terminologies are presented below.
• TTC is the amount of electric power that can be transferred over the interconnected transmission network in a reliable manner while meeting all of a specific set of defined pre-and post-contingency system conditions (i.e., thermal, voltage, and stability limits). The TTC is the same as the first contingency total transfer capability (FCTTC) defined in [3].
• TRM is the amount of transmission transfer capability necessary to ensure that the interconnected transmission network is secure under a reasonable range of uncertainties in system conditions. The TRM can be considered as the difference of TTC between cases with and without consideration of uncertainties brought about by, for instance, forecasted load demand and the integration of renewable energy resources (RES). TRM can be determined by assuming a fixed percentage reduction in TTC (typically too conservative). Reference [4] summarized four methods to determine the TRM, of which the probabilistic approaches is preferable considering the timevariant system conditions.
• CBM is the amount of transmission transfer capability reserved by load-serving entities to ensure access to generation from interconnected systems to meet generation reliability requirements. Typical CBM could be a multiple of the largest generation unit within the transmission system [5].
• ETC is the sum of existing transfer capability between the source and the sink in the base case. Since we are interested in the probabilistic characteristics of the additional transfer capability on top of the base case (i.e., TTC -ETC). Without loss of generality, we assume CBM = 0 and use the term probabilistic ATC (PATC) to represent ATC plus TRM (i.e., TTC -ETC) in the rest of the paper.
From the above definitions, it can be seen that the nucleus of assessing ATC is to calculate the PATC and its cumulative distribution function (CDF), based on which TRM can be estimated. As a result, the ATC can be computed directly from
ATC = E[PATC] − TRM(2)
Indeed, the main contribution of the paper is to propose a mathematical formulation for PATC and to develop a computationally efficient yet accurate algorithm for the PATC calculation.
In the previous work, traditional deterministic ATC calculation methods can be categorized into four classes: i) Linear approximation methods (e.g., [6]); ii) Repeated power flow (RPF) methods (e.g., [7]); iii) Continuation power flow (CPF) methods (e.g., [8], [9]); and iv) Optimal power flow (OPF) methods (e.g., [10]). However, deterministic ATC calculation does not consider the uncertainties of the renewable energy generation and their intrinsic dependencies.
To overcome the limitation of deterministic ATC methods, various probabilistic ATC assessment methods have been discussed in the previous literature, which can be categorized into 1) Monte-Carlo simulation; 2) Bootstrap method; 3) point estimation method; 4) stochastic programming. Monte-Carlo simulation method is the most widely used method due to its simplicity. It has been used in combination with all deterministic ATC solvers mentioned above. However, it suffers from a high computational effort that may hamper its practical applications even with efficient sampling methods like the Latin Hypercube sampling method [11]. Clustering methods [12] were applied to speed up Monte-Carlo simulation yet at the expense of accuracy. The Bootstrap method is powerful in constructing confidence region of ATC but is time-consuming due to the re-sampling procedure [13]. Attempting to release the computational burden, some analytical methods have been developed utilizing mathematical approximations. In [14], the point estimation method was proposed to estimate the variance of the TTC to determine the TRM and thus the ATC.
Apart from the methods mentioned above, another class of methods for uncertainty quantification is the meta-modeling, which aims to build a statistically-equivalent functional representation for the desired response (e.g., the PATC in this study) using a small number of model evaluations. A representative method is the polynomial chaos expansion (PCE) which has been applied in the context of power systems to study the probabilistic power flow [15], the load margin problem [16], and the available delivery capability problem [17]. Another emerging method, an alternative to PCE, is the canonical low-rank approximations (LRA), which employs the canonical decomposition to express the desired response as a sum of rank-one functions [18]. The original idea of canonical decomposition dates back to 1927 [19], and the method has recently become attractive for uncertainty quantification in structural vibration problems [20].
In this paper, we propose a mathematical formulation of the PATC problem considering the uncertainties from the wind power, the solar PV and the loads. More importantly, we apply a novel LRA approach to solve the important PATC problem. The main contribution of this paper are summarized as follows:
• A mathematical formulation of PATC problem is developed for transmission system integrating renewable energy resources and loads that may follow various marginal distributions. This formulation considers both the pre-contingency and post-contingency cases.
• A novel efficient yet accurate algorithm-LRA is proposed to evaluate the PATC. Particularly, random variables with diverse marginal distributions and correlation can be accommodated using proper polynomial basis and Nataf transformation.
• Accurate probabilistic characteristics (probabilistic density function (PDF), cumulative distribution function (CDF), mean, standard deviation, etc.) of the PATC can be achieved with much less computational efforts compared to the LHS-based MCS. Such probabilistic information can be further used to compute a more realistic TRM, leading to a more fully utilized transmission assets. The rest of the paper is organized as follows. Section II introduces the probabilistic models of the randomness in the PATC problem. Section III proposes the mathematical formulation of the PATC problem. Section IV elaborates the low-rank approximation method and its implementation in PATC calculation. The detailed algorithm to assess the PATC is summarized in Section V. The simulation results are presented in Section VI. Conclusions and perspectives are given in Section VII.
II. MODELING UNCERTAINTIES IN THE PROBABILISTIC ATC PROBLEM
The ATC calculation is typically applied to estimate the transfer capability for a pre-specified future period before the system approaches that operational state. To obtain a reasonable and dependable ATC, the base case condition, the target transactions, a credible contingency list, the physical and operational limits, and the network response should be well defined and estimated. The variability of diverse renewable energy sources and loads result in uncertainties in the base case, which therefore requires careful modeling.
A. Projection of the Wind Generation in Base Case
In this study, the wind farms are modeled as an aggregated wind turbine with equivalent parameters. The Weibull distribution provides a great fit for the wind speed in many locations around the world [21], the probability density function of which is
fV (v) = k c v c k−1 exp − v c k(3)
where v is the wind speed, k and c are the equivalent shape and scale parameters, respectively. As a result, the active power output P w can be calculated by the piece-wise wind speedpower output relation [22] Pw
(v) = 0 v ≤ vin or v > vout v − vin v rated − vin Pr vin < v ≤ v rated Pr v rated < v ≤ vout(4)
where v in , v out and v rated are the cut-in, cut-out, and rated wind speed (m/s), P r is the rated wind power (kW ). The reactive power can be determined according to the speed control type of the wind turbine [23] since the wind turbine can be modelled as either a constant P-Q bus or a constant P-V bus with given Q-limits.
B. Projection of the Solar Generation in Base Case
Similar to the wind farms, large solar PV power plants can be modeled as an aggregated solar PV with equivalent parameters. Typically, the solar radiation is represented by the Beta distribution [24]
fR(r) = Γ(α + β) Γ(α)Γ(β) r rmax α−1 1 − r rmax β−1 (5)
where α and β are the shape parameters of the distribution, Γ denotes the Gamma function. The parameters are typically obtained from fitting historical solar radiation data. r and r max (W/m 2 ) are the respective actual and maximum solar radiations. The active power P pv corresponding to the solar radiation r is determined by the piece-wise function [22] Ppv(r) =
r 2 rcr std
Pr 0 ≤ r < rc r r std Pr rc ≤ r ≤ r std Pr r > r std (6) where r c is a certain radiation point typically set as 150 W/m 2 , r std is the solar radiation in the standard environment, P r is the rated power of the solar PV. Solar generation is required to be injected into the power grid at unity power factor [25], and hence Q pv is assumed to be zero in this study.
C. Projection of the Forecasted Load Demand in Base Case
By nature, load demand is uncertain in power systems. It is a common practice to model the load uncertainty by Gaussian distribution due to the forecasting error [26]
f (PL) = 1 √ 2πσP exp − (PL − µP ) 2 2σ 2 P(7)
where the mean value µ P of P L is provided by load forecaster, and σ P denotes the forecasting error. Generally, only the active power is predicted, whereas the reactive power is calculated under the assumption of constant power factor [27].
III. MATHEMATICAL FORMULATION OF PROBABILISTIC ATC PROBLEM In this section, we present a CPF-based mathematical formulation of the probabilistic ATC problem. The near-future base case conditions are formulated as a probabilistic power flow problem in which uncertainties of the RES and loads are incorporated. Besides, target inter-area transactions are modeled as the load-generation variation vector. The deterministic power flow equations of a N -bus transmission system can be represented as
f (x) = PGi − PLi − Pi(x) QGi − QLi − Qi(x) = 0 (8) with Pi(x) = Vi N j=1
Vj (Gij cos θij + Bij sin θij )
Qi(x) = Vi N j=1
Vj(Gij sin θij − Bij cos θij)
(9) where x = [θ, V ]
T , e.g., voltage angles and magnitudes for all buses; P Gi and Q Gi are the total active and reactive generation power at bus i; P Li and Q Li are the total active and reactive load power at bus i; G ij and B ij are the real and imaginary part of the entry Y ij in the bus admittance matrix.
Let v, r and P L be the random vectors representing wind speeds, solar radiations and load variations in the pre-specified period, respectively. The resulting base case condition of a Nbus system can be described as a set of probabilistic power flow (PPF) equations. Specifically, for P-Q type buses, the PPF equations are:
PGi + Pwi(vi) + Ppvi(ri) − PLi(PLi) − Pi(x) = 0 QGi + Qwi(vi) − QLi(PLi) − Qi(x) = 0(10)
For P-V type buses, the corresponding PPF equations are:
PGi + Pwi(vi) + Ppvi(ri) − PLi(PLi) − Pi(x) = 0 Vi = Vi0 QGi = −Qwi(vi) + QLi(PLi) + Qi(x) Qmin,i ≤ QGi ≤ Qmax,i(11)
where P wi (v i ), P pvi (r i ), P Li and P Gi are the real power injection from the wind farm, the solar PV power plant, the load, and the conventional generator at bus i; Q wi (v i ), Q Li and Q Gi are the corresponding reactive power injections. If Q Gi exceeds its limits, then the terminal bus switches from P-V to P-Q with Q Gi fixed at the violated limit. Besides, the target transactions under the study can be described as a load-generation variation vector b of the system in the form of
b = ∆PG − ∆PL −∆QL(12)
In order to study how much power can be transferred along the direction specified by vector b, the set of equations (10)- (12) can be formulated as a set of probabilistic CPF equations in the following compact form:
f (x, µ, λ, U ) = f (x, µ, U ) − λb = 0(13)
where x is the vector of state variables, µ is the vector of control parameters such as the tap ratio of transformers, U = [v, r, P L ] is the random vector describing the wind speed, the solar radiation, and the load active power. It is obvious that the set of the parameterized power flow equations become the base-case power flow equation if λ = 0.
For reliable operation of a power system, ATC calculation is required to account for both normal operating state and the state when a contingency occurs. Typically, only the N − 1 contingencies are of interest in ATC calculation. However, more complex multiple contingencies may be required by the reliability criteria in a certain individual system. Enumeration of all contingencies is unnecessary and usually leads to a too conservative ATC value, hence it is a common practice to obtain a credible contingency list from the Security Analysis (SA) module in the Energy Management System (EMS). Therefore, the probabilistic ATC formulation considering a credible contingency list can be formulated as below:
λAT C = min{λ (0) , λ (1) , ..., λ (N C ) }(14)
in which the transfer capability λ (v) for the v-th case is determined by
maximize λ (v) subject to f (v) (x, µ, U ) − λ (v) b = 0, (a) V (v) min,i ≤ Vi(x, µ, λ (v) , U ) ≤ V (v) max,i , (b) S (v) ij (x, µ, λ (v) , U ) ≤ S (v) ij,max , (c) Pmin,i ≤ PGi(x, µ, λ (v) , U ) ≤ Pmax,i, (d) Qmin,i ≤ QGi(x, µ, λ (v) , U ) ≤ Qmax,i, (e)(15)
where v = 0 represents the base case, and v = 1, ..., N C represent the contingency cases. Specifically, f (0) corresponds to the pre-contingency (base case) network configuration, while
f (v) , v = 1, ..., N C corresponds to the v-th post-contingency network configuration. Similarly, [V (0) min,i , V(0[V (v) min,i , V (v) max,i ] and S (v)
ij,max (v = 1, ..., N C ) are the emergency voltage and thermal limits applied to the contingency cases. λ is the normalized load margin under the given load-generation variation vector. The maximum value of λ that could be achieved without the violation of (15) gives the ATC. Note that λ is a random variable due to the random input U . Equation (a) specifies that the solution must satisfy the parameterized power flow equations (13); Equations (b)-(e) imply that the solution has to satisfy typical operational and electrical constraints.
For each realization of U , there are N C + 1 (N C is the size of the contingency list) deterministic cases of (15) to be solved. As a result, it is extremely time-consuming to run Monte-Carlo simulations to estimate PATC using a large number of samples. It is imperative and essential to develop an efficient and accurate method to estimate the PATC.
IV. CANONICAL LOW-RANK APPROXIMATION USING POLYNOMIAL BASIS
This section presents a general framework of the low-rank approximation of a multivariate stochastic response function. For simplicity, we first consider a scalar response function of independent inputs. The case of dependent inputs will be addressed in Section IV-E.
A. Low-rank Approximation with Polynomial Basis
Consider a random vector ξ = (ξ 1 , ξ 2 , ..., ξ n ,) with joint probability density function (PDF) f ξ and marginal distribution functions f ξi , i = 1, ..., n (ξ i is related with the random variables U i in (15), see Section III), then the canonical rankr approximation [18] of the target stochastic response (e.g., PATC in this study) Y = g(ξ) can be represented by:
Y ≈Ŷ =ĝ(ξ) = r l=1 b l ω l (ξ)(16)
in which b l , l = 1, ..., r are normalizing weighting factors, and ω l is a rank-one function of ξ in the form of
ω l (ξ) = n i=1 v (i) l (ξi)(17)
where v (i) l denotes the i-th dimensional univariate function in the l-th rank-one function. For most applications, the number r of rank-one terms is usually small (under 5), hence (16) and (17) represent a canonical low-rank approximation.
In order to obtain the rank-r approximation, a natural choice is expanding v (i) l on a polynomial basis {φ (i) k , k ∈ N } that is orthogonal to f Xi , the resulting rank-r approximation takes the form:Ŷ
=ĝ(ξ) = r l=1 b l n i=1 p i k=0 z (i) k,l φ (i) k (ξi)(18)
where φ (i) k denotes the k-th degree univariate polynomial in the i-th random input, p i is the maximum degree of φ (i) and z
e −x 2 /2 Hermite (-∞,∞) Uniform 1 2 Legendre [-1,1] Beta (1−x) α (1+x) β 2 α+β+1 B(α+1,β+1) Jacobi [-1,1] Exponential e −x Laguerre (0,∞) Gamma x α e −x Γ(
B. Selection of the Univariate Polynomial Basis
It is crucial to choose a proper polynomial φ i for the ith random input ξ i , i = 1, 2, ..., n to avoid low convergence rate and/or higher degree of expansion [28], which will hamper the capability of LRA in dealing with high-dimensional problems. Table I shows a set of typical continuous distributions and the respective optimal Wiener-Askey polynomial basis, which can ensure the exponential convergence rate. In case that f ξi is out of the list in Table I, the discretized Stieltjes procedure is adopted to numerically construct a set of univariate orthogonal polynomial basis [29].
C. Calculation of the Coefficients and Weighing Factors
Based on an experiment design of size M C , i.e., a set of samples of ξ C = {ξ (1) , ξ (2) , ..., ξ (MC ) } and the corresponding response y C = {y (1) , y (2) , ..., y (MC ) } evaluated by deterministic tools, different algorithms have been proposed in the literature for solving the LRA coefficients and the weighting factors in a non-intrusive manner [30], [31]. The sequential correction-updating scheme presented in [20] is employed in this study due to its efficiency and capability of constructing low-rank approximation using less sample evaluations. In the r-th correction step, a new rank-one function ω r is built, while in the r-th updating step, the set of weighing factors {b 1 , ..., b r } is determined. This process continues until the applied error index (the relative empirical error in this study) stop decreasing [20].
Correction step: the r-th correction step aims to find a new rank-one tensor ω r , which can be obtained by solving the following minimization problem: (19) where W represents the space of rank-one tensors, e r−1 = (g −ĝ r−1 ) is the approximation error of the response Y at the (r − 1)-th step, . 2 represents the norm 2 of the residual after the new rank-one tensor w is applied, and the subscript ξ C indicates that the minimization is carried over the whole set of samples in the experiment design (ξ C , y C ).
ωr(ξ) = arg min ω∈W er−1 − ω 2 ξ C = arg min ω∈W M C m=1 y (m) −ĝr−1(ξ (m) ) − ω(ξ (m) ) 2
By exploiting the retained tensor-product form of the univariate polynomial basis, as shown in (18), typical scheme for solving equation (19) is the alternated least-square (ALS) minimization, which involves sequential minimization along each dimension i = 1, ..., n to solve the corresponding polynomial coefficients z
(i) r = (z (i) 0,r , ..., z (i)
pi,r ). The total number of coefficients to be solved in each correction step is n i=1 (p i + 1), which grows linearly as the number of random inputs n increases. Since ω r is the product of v (i) l (ξ) as shown in (17), v (i) l (ξ i ) can be initialized by setting to 1.0.
In the minimization along the i-th dimension, the polynomial coefficients corresponding to all other dimensions are "frozen" at their current values and the coefficients z
(i) r = (z (i) 0,r , ..., z (i)
pi,r ) can be determined by: 20) where C i is a scalar
z (i) r = arg min ζ∈R (p i +1) er−1 − Ci p i k=0 ζ k φ (i) k 2 ξ C(Ci = j =i v (j) r (ξj) = j =i p j k=0 z (j) k,r φ (j) k (ξj)(21)
Updating step: After the r-th correction step is completed, the algorithm proceed to the r-th updating step to determine the weighing factor b r of the newly solved rank-one function ω r (ξ), meanwhile, the set of existing weighing factors b = (b 1 , ..., b r−1 ) are updated too. The updating step can be achieved by solving the following minimization problem
b = arg min β∈R r g − r l=1 β l ω l 2 ξ C(22)
Stop criteria: The correction-updating scheme successively adds new rank-one function to improve the accuracy of the approximation (18). Hence, error reduction in two successive iterations becomes a natural stop criteria for this process. In this study, the relative empirical error is employed which is given by:ê
r = er−1 − ωr 2 ξ C V(yC )(23)
where V(y C ) denotes the empirical variance of the desired response over the experimental design. Remark: (i) It is worth pointing out that the minimization problems in (20) and (22) can be efficiently solved with the ordinary least-squares (OLS) technique because the dimension of unknowns are small. (ii) When the LRA (18) for the desired response is built up, the response of any new samples can be evaluated efficiently by directly substituting to (18) instead of solving the original complex problem (e.g., the PATC problem (15)).
D. Selection of optimal rank and polynomial degree
Currently, there is no systematic way to identify the optimal rank r and the polynomial degree p i for individual random input beforehand. In this study, we specify a candidate set {1, 2, 3, 4, 5} for the rank r, and another one {2, 3, 4, 5} for all the univariate polynomials. The rank selection is performed by progressively increasing the rank and applying the corrected error (23) to select the best one. The selection of optimal degree can be implemented similarly.
E. Integration of Dependent Random Inputs
So far, the random inputs are assumed to be mutually independent as requested by the LRA method. To accommodate dependent random inputs with correlation matrix ρ, the Nataf transformation [32], [33] and the isoprobabilistic transformation can be employed to build up a mapping between U and the independent standard random variables ξ: u = T (ξ), where T is invertible. Therefore, the set of samples of ξ can be transformed back into samples of U to evaluate the corresponding responses Y , after which the desired response y = g(T −1 (u)) can be expanded onto the polynomial basis with ξ using the aforementioned method [18].
F. Moments of a Low-Rank Approximation
Due to the orthogonality of the univariate polynomials that form the LRA basis (see (18)), the mean and the variance of the meta-model can be obtained analytically in terms of the polynomial coefficients and the weighting factors. In particular, the mean and variance of the LRA response are given respectively by [18]:
µy = E [ĝ(ξ)] = r l=1 b l n i=1 z (i) 0,l σ 2 y = r l=1 r m=1 b l bm n i=1 p i k=0 z (i) k,l z (i) k,m − z (i) 0,l z (i) 0,m(24)
Hence, if only mean and variance are of interests, (24) can be applied directly without evaluating a large size of samples, which in contrast is required by most of the simulation-based methods.
V. COMPUTATION OF PROBABILISTIC AVAILABLE TRANSFER CAPABILITY
In this section, a step-by-step description of the LRA method for PATC calculation is summarized below:
Step 1: Input the network data, the transactions to be studied, the credible contingency list, the probability distribution and the parameters of the random inputs U , i.e., the wind speed, the solar radiation, the active load power, and their correlation matrix ρ.
Step 2: Build the load-generation pattern (i.e., vector b in (15)-(a)) according to the transactions under study.
Step 3: Choose the independent standard variable ξ i and the corresponding univariate polynomial φ i for each random input U i .
Step 4: Generate an experimental design of size M C : (1) , ξ (2) , ..., ξ (MC ) ) in the standard space by the LHS method.
• i) Generate M C samples ξ C = (ξ
• ii) Transform ξ C into the physical space by the inverse Nataf transformation u C = T −1 (ξ C ).
• iii) Apply the deterministic ATC solver to evaluate the accurate response y C (i.e., PATC) of u C . For each realization, the overall ATC is computed by applying the parallel scheme [8] which enables an early stop in CPF calculation when the first violation is encountered in either normal or contingency cases. Pass the sampleresponse pairs (ξ C , y C ) to Step 5.
Step 5: Apply the algorithm in Section IV-C to build the lowrank approximation (18) for the PATC. If the LRA for PATC has reached the prescribed accuracy, go to Step 7; otherwise, go to Step 6.
Step 6: Generate additional ∆M C new samples and evaluate them, then go back to Step 5 using the enriched experiment design (ξ C + ∆ξ C , y C + ∆y C ).
Step 7: Calculate the mean and standard deviation of the response through (24).
Step 8: Sample ξ extensively, e.g., M S samples, and apply the solved functional approximation (18) to evaluate the corresponding response y S for all samples. Then compute the statistics of interest (PDF/CDF of PATC in this study).
Step 9: Compute the TRM value and the resulting ATC (AT C = µ P AT C − T RM ) for the given confidence level p cl %.
Step
VI. NUMERICAL STUDIES
In this section, we apply the proposed LRA method to investigate the probabilistic ATC of the modified IEEE 24bus reliability test system (RTS) and IEEE 118-bus systems [34]. The LHS-based Monte Carlos simulation serves as a benchmark for validating the accuracy and the performance of the proposed method. In addition, a comparison between the LRA and the sparse PCE [17] is also presented.
In this study, we assume that the probability distributions and the associated parameters of all random inputs are available from up-front modeling. Particularly, the wind speed follows Weibull distribution; the solar radiation follows Beta distribution; the load power follows Normal distribution. For each individual load, the mean value is set to be its base case value and the variance is equal to 5% of its mean value. The univariate polynomial basis used in (18) for Weibull, Beta and Normal distributions are chosen appropriately according to Table I. For simplicity, the linear correlation coefficient ρ ij between component i and j of wind speed v, solar radiations r and load power P L are 0.8040, 0.5053 and 0.4000, respectively. We first apply the CPF-based ATC solver to the deterministic system (i.e., without uncertainty). As shown in Table IV, the binding limit is the thermal limit at branch 7-8 in the base case, leading to an overall ATC 83.0207 MW. The contingencies do not deteriorate the transfer capability potentially because the given limits of branches are larger in emergency compared to the normal operating condition.
A. The Modified IEEE 24-Bus RTS
Next, we exploit the proposed LRA method to assess the probabilistic characteristics (e.g., mean, variance, PDF and CDF) of the PATC and compare the results with those of the LHS-based MCS and with those of the PCE method. Applying the proposed algorithm, 125 simulations are required in Step 4-5 to build up the LRAs (18) of the PATC, which consist of 1 rank-one function with the highest polynomial degrees p i = 2. The total number of coefficients plus weighting factors is 76 (i.e., Table V. Furthermore, 2000 samples are generated in Step 8 to assess Once we have the statistics of the PATC, a reasonable amount of TRM can be obtained from the CDF of PATC. Table VI shows the TRM with different confidence levels and the resulting ATC values. For example, if 95% confidence level is requested, i.e., P (AT C actual >= (µ AT C − T RM )) = 0.95, the corresponding ATC is 61.8815 MW.
B. The Modified 118-bus System
The IEEE-118 bus system is a simplified representation of the Midwest U.S. transmission system in 1962, which contains 19 generators, 35 Likewise, we first apply the CPF-based ATC solver to compute the maximum transfer without considering the uncertainties of RES and loads. The deterministic ATC is 24.1835 MW. Next, we exploit the proposed LRA-PATC method to assess the probabilistic characteristics of the PATC and compare the results with those of the LHS-based MCS and the sparse PCE method.
Applying the proposed algorithm, 556 simulations are needed in Step 4-5 to build up the LRAs (18) of the PATC, which consist of 1 rank-one function with the highest degree p i = 2. The total number of coefficients plus weighting factors is 334. Eventually, the mean and standard deviation of the responses are computed in Step 7 and are compared with those from the LHS-based MCS and from the sparse PCE method as shown in Table VII. All these results and comparisons clearly demonstrate that the LRA method can provide accurate estimation for the probabilistic characteristics of the PATC solutions. However, to get comparable accuracy, the LHSbased MCS needs to run 10000 simulations (i.e., solving (15)), taking 9110s, while the LRA takes only 480s, i.e., about 1 19 of the computational time required by MCS. Clearly, the LRA is much more efficient than the MCS.
The TRM and the resulting overall ATC for different confidence levels are shown in Table VIII, which empowers the system operator to determine a reasonable amount of TRM to make full use of the transmission assets.
VII. CONCLUSION AND PERSPECTIVES
In this paper, we have proposed a mathematical formulation for the probabilistic ATC (PATC) problem in which the uncertainties from RES and loads are incorporated. Moreover, we have proposed a novel LRA method which can assess the PATC accurately and efficiently by building up a statistically-equivalent low-rank approximation. Numerical studies show that the proposed method can accurately estimate the probabilistic characteristics of the PATC with much less computational effort compared to the LHS-based MCS.
The PATC provides important insights into how the uncertainties may affect the transfer capability of the power system. More importantly, the proposed method can help determine the TRM and thus the ATC in a more efficient manner to fully utilize the transmission assets while maintaining the security and reliability of the grid.
In the future, we plan to develop control measures to reduce the variance of PATC to decrease the TRM for a full utilization of transmission assets.
This work is supported by Natural Sciences and Engineering Research Council (NSERC) under Discovery Grant NSERC RGPIN-2016-04570. The authors are with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC H3A 0G4, Canada. (e-mail: [email protected], [email protected])
l is the coefficient of φ (i) k in the l-th rank-one function. Building the low-rank approximation for desired response in (18) requires: (i) choose an appropriate univariate polynomial for each random input; (ii) solve the polynomial coefficients z (i) k,l as well as the weighing factors b l . This process relies on a set of samples and their corresponding accurate response which are usually termed as experimental design (ED).
10 .
10Generate the result report. Remark: the number of samples M C in Step 4 is usually much smaller than M S in Step 8. Unlike MCS, LRA does not solve power flow equations for all M S samples in Step 8, and hence it is more efficient. The main computational effort of LRA lies in Step 4.
,4,9} in area 1 (sink). The contingency list under study contains four N −1 outages {G1#1, L2−4, L3−24, L9−11}.
2+1)+1). Once the coefficients and the weighting factors are computed, the mean and the standard deviation of the responses are computed in Step 7 and are compared with those of the LHS-based MCS and with those of the sparse PCE method as shown in
TABLE I STANDARD
IFORMS OF CLASSICAL CONTINUOUS DISTRIBUTIONS AND THEIR CORRESPONDING ORTHOGONAL POLYNOMIALS[28] Distribution
Density Function
Polynomial
Support
Normal
1
√
2π
The IEEE 24-bus RTS is composed of 4 areas, containing 33 generators, 32 branches and 17 loads. In this paper, we modify this test system by adding 4 wind farms of 80 MW at bus {15, 18, 21, 23} and 4 solar PV power plants of 60 MW at bus {1, 2, 7, 16}, respectively. Their corresponding parameters {W 1, W 2, W 3, W 4} and {S1, S2, S3, S4} are shown in Table II-III. Together with 17 stochastic loads, there are totally 25 random inputs. The existing transmission commitments (base case) is the first-day peak load as defined in [35], and the new transaction under study is to transfer 75 MW from generator bus {7} in area 2 (source) to load
TABLE II WIND
IISPEED AND WIND TURBINE PARAMETERS[21] N o.
c
k
Vr
V in
Vout
W 1
8.0063
2.1182
13.50
3.50
25.00
W 2
11.5762
2.7022
13.80
3.50
25.00
W 3
11.2441
3.6322
13.00
5.00
25.00
W 4
12.4813
3.2465
12.90
5.00
24.00
W 5
11.1533
3.2895
12.00
5.50
24.00
W 6
8.8261
2.6511
10.00
3.50
20.00
TABLE III
SOLAR RADIATION AND SOLAR PV PARAMETERS [36]
N o.
α
β
r min
rmax
Rc
R std
TABLE IV THE
IVDETERMINATION OF ATC CONSIDERING NORMAL ANDCONTINGENCY CASES
Case
Outage
ATC w.r.t limits (MW)
Overall
No.
Facility
Voltage
limit
Thermal
limit
Voltage
collapse
ATC
0
Base case
280.3934
83.0207
511.9046
83.0207
1
G1-1
469.0375
128.0214
508.8632
2
L2-4
201.7742
127.8677
451.6651
3
L3-24
183.7445
127.5457
355.2315
4
L9-11
395.0070
127.8440
460.6714
TABLE V COMPARISON
VOF THE ESTIMATED STATISTICS OF THE OVERALL ATC BY THE MCS, PCE AND LRA METHODS TABLE VI THE ESTIMATED TRM AND RESULTING ATC FOR DIFFERENTthe PDF and the CDF of the response.Fig. 1shows the PDF and CDF of the PATC computed by the LHS-based MCS, the sparse PCE, and the solved LRA, respectively. These results clearly demonstrate that the LRA-PATC method can provide accurate estimation for the probabilistic characteristics of the PATC.Fig. 1. The distribution of the PATC computed by the MCS, the PCE and the LRA. They are almost overlapped. The TRM for 95% confidence level is 21.1497 MW and the resulting ATC is 61.8815 MW.Indices
MCS
PCE
LRA
∆P CE
M CS %
∆LRA
M CS %
µ
83.0312
83.2369
83.2226
0.2478
0.2305
σ
15.5418
14.4898
15.4278
-6.7692
-0.7340
CONFIDENCE LEVELS
Confid. Level
E(PATC)
TRM (MW)
ATC (MW)
99.0%
83.0312
26.2326
56.7986
98.0%
83.0312
24.4386
58.5926
95.0%
83.0312
21.1497
61.8815
90.0%
83.0312
17.9604
65.0707
80.0%
83.0312
12.9772
70.0540
40
60
80
100
120
140
160
PATC (MW)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
MCS
PCE
LRA
40
60
80
100
120
140
160
PATC (MW)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cumulative Distribution Function
MCS
PCE
LRA
No uncertainty
synchronous condensers, 177 transmission lines, 9 transformers and 91 loads. Six wind farms, each with 50 MW, are connected to bus {10, 25, 26, 49, 65, 66} using the parameters {W 1, W 2, W 3, W 4, W 5, W 6} in Table II respectively. Six solar PV parks, each with an installed capacity of 30 MW, are connected to bus {12, 59, 61, 80, 89, 100} using parameters {S1, S2, S3, S4, S5, S6} in Table III. Besides,
TABLE VII COMPARISON
VIIOF THE ESTIMATED STATISTICS OF THE OVERALL ATC BY THE MCS, PCE AND LRA METHODS TABLE VIII THEESTIMATED TRM AND RESULTING ATC FOR DIFFERENT there are 99 stochastic loads, leading to 111 random inputs in total. The total load in the base case is 4242 MW and 1438 Mvar. The new transaction under study is to transfer 60 MW from bus 89 to bus 91. The contingency list contains six N −1 outages {L88−89, L89−90, L90−91, L89−92, L91− 92, L92 − 94} surrounding the source and sink buses.Indices
MCS
PCE
LRA
∆P CE
M CS %
∆LRA
M CS %
µ
23.9278
23.7476
23.7550
-0.7530
-0.7220
σ
4.8779
4.5220
4.8617
-7.2973
-0.3327
CONFIDENCE LEVELS
Confid. Level
E(PATC)
TRM (MW)
ATC (MW)
99.0%
23.9278
10.1839
13.7439
98.0%
23.9278
8.9942
14.9336
95.0%
23.9278
7.4975
16.4303
90.0%
23.9278
6.1159
17.8119
80.0%
23.9278
4.3121
19.6157
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| []
|
[
"Zee-Babu type model with U (1) L µ −L τ gauge symmetry",
"Zee-Babu type model with U (1) L µ −L τ gauge symmetry"
]
| [
"Takaaki Nomura \nSchool of Physics\nKIAS\n02455SeoulKorea\n",
"Hiroshi Okada \nPhysics Division\nNational Center for Theoretical Sciences\n300HsinchuTaiwan\n"
]
| [
"School of Physics\nKIAS\n02455SeoulKorea",
"Physics Division\nNational Center for Theoretical Sciences\n300HsinchuTaiwan"
]
| []
| We extend the Zee-Babu model introducing local U (1) Lµ−Lτ symmetry with several singlycharged bosons. We find a predictive neutrino mass texture in a simple hypothesis that mixings among singly-charged bosons are negligible. Also lepton flavor violations are less constrained compared with the original model. Then we explore testability of the model focussing on a doublycharged boson physics at the LHC and the ILC. | 10.1103/physrevd.97.095023 | [
"https://arxiv.org/pdf/1803.04795v2.pdf"
]
| 119,075,820 | 1803.04795 | 014e32c5310ec51c8e5a42ac56f833bb938bc2bd |
Zee-Babu type model with U (1) L µ −L τ gauge symmetry
7 May 2018 (Dated: May 8, 2018)
Takaaki Nomura
School of Physics
KIAS
02455SeoulKorea
Hiroshi Okada
Physics Division
National Center for Theoretical Sciences
300HsinchuTaiwan
Zee-Babu type model with U (1) L µ −L τ gauge symmetry
7 May 2018 (Dated: May 8, 2018)
We extend the Zee-Babu model introducing local U (1) Lµ−Lτ symmetry with several singlycharged bosons. We find a predictive neutrino mass texture in a simple hypothesis that mixings among singly-charged bosons are negligible. Also lepton flavor violations are less constrained compared with the original model. Then we explore testability of the model focussing on a doublycharged boson physics at the LHC and the ILC.
I. INTRODUCTION
Radiative seesaw models are one of the promising candidates not only to establish neutrino mass matrix radiatively but also to have high testability for new physics at current and future experiments. Zee-Babu model is the minimal scenario that does not require any additional fermions but only two charged bosons; singly-charged one (h ± ) and doubly-charged one (k ±± ) [1], where the neutrino mass matrix is arisen at the two loop level. It is found a phenomenological prediction on neutrinos; a massless active neutrino is predicted due to antisymmetry of the neutrino mass structure. The serious analyses are found in refs. [2,3], considering lepton flavor violations (LFVs). 1 Also one could find new signals of new bosons, especially, the mass of k ±± and its related interacting couplings can be constrained by the LEP experiment [15] and can be well-measured by current LHC experiments [16,17] and by future international linear collider (ILC) experiment [13,14] applying the chiral polarizations of beam, which might be distinguished from those in the other models [18].
In this letter, we extend the original Zee-Babu model by imposing a gauged U(1) Lµ−Lτ symmetry with several singlet bosons having L µ − L τ charge where the Yukawa couplings associated with neutrino mass generation are constrained by the symmetry realizing predictable mass strructure 2 . In addition, this gauge symmetry has phenomenologically interesting properties; gauge anomaly is canceled [21,22], excess of muon anomalous magnetic dipole moment (muon g − 2) can be explained [23][24][25], lepton flavor non-universality in semileptonic B-meson decays can be addressed with some extensions [26][27][28][29][30], and other interesting studies can be found in [31][32][33][34][35][36][37][38][39][40][41][42][43]. In our analysis we find that restricted Yukawa couplings coming from the additional symmetry lead to a predictive neutrino texture in a simple hypothesis that mixings among singly-charged bosons are negligibly tiny. Furthermore LFV constrains are much relaxed compared to the original model in such a small mixing scenario. Then we will focus on analyzing doubly-charged boson at the collider experiments such as the LHC and the future ILC, and discuss testability of our model taking into account current constraints at the LHC and the LEP. This paper is organized as follows. In Sec. II, we show our model, and formulate neutrino mass matrix. In Sec. III, we analyze the doubly-charged boson at colliders and show results.
Fields H h + −1 h + 0 h + +1 k ++ ϕ L e L µ L τ e R µ R τ R SU (2) L 2 1 1 1 1 1 2 2 2 1 1 1 U (1) Y 1 2 1 1 1 2 0 − 1 2 − 1 2 − 1 2 −1 −1 −1 U (1) Lµ−Lτ 0 −1 0 1 0 1 0 1 −1 0 1 −1under SU (2) L × U (1) Y × U (1) Lµ−Lτ .
Finally We conclude and discuss in Sec. IV.
II. A MODEL
In this section we introduce our model in which neutrino masses are generated at two-loop level and U(1) Lµ−Lτ gauge symmetry is imposed. Fermion sector is the same as the SM one where leptons have U(1) Lµ−Lτ charge as shown In Table I. In scalar sector, we introduce three singly charged scalar and one doubly charged scalar fields which are SU(2) singlet; singly charged scalar fields have U(1) Lµ−Lτ charge +1, 0 and −1 while doubly charged scalar field does not have L µ − L τ charge. Here we write singly charged scalars as h + Q Lµ−Lτ with electric (L µ − L τ ) charge +1(Q Lµ−Lτ ) and complex conjugate is defined as (h + Q Lµ−Lτ ) * = h − −Q Lµ−Lτ . In addition we introduce SM singlet scalar ϕ with L µ − L τ charge to break U(1) Lµ−Lτ gauge symmetry and to give mass to Z ′ boson from the new U(1). The SM Higgs H and ϕ develop VEVs and we write these fields as
H = w + v H +h+iz √ 2 , ϕ = 1 √ 2 (v ϕ + ϕ R + iz ′ ) (II.1)
where w + , z and z ′ are Nambu-Goldstone(NG) bosons absorbed by W + , Z and Z ′ .
The Yukawa couplings associated with charged scalar fields are given by
L Y =f eµL c Le (iσ 2 )L Lµ h + −1 + f µτL c Lµ (iσ 2 )L Lτ h + 0 + f eτL c Le (iσ 2 )L Lτ h + +1 + g eeē c R e R k ++ + g µτμ c R τ R k ++ + h.c. , (II.2)
where σ 2 is the second Pauli matrix. Note that the coupling f ab is anti-symmetric due to nature of anti-symmetry under SU(2) L indices in the corresponding operators [1] 3 . The scalar potential of our model is given by where we have omitted to write quartic terms containing only charged scalar fields and the coupling constants are assumed to be real for simplicity.
V =µ 2 H H † H + λ H (H † H) 2 + µ 2 ϕ |ϕ| 2 + λ ϕ |ϕ| 4 + M 2 k ++ k ++ k −− + M 2 h + −1 h + −1 h − +1 + M 2 h + 0 h + 0 h − 0 + M 2 h + +1 h + +1 h − −1 + (µ kh k ++ h − 0 h − 0 +μ kh k ++ h − −1 h − +1 + µ ϕh ϕh + −1 h − 0 +μ ϕh ϕ * h + +1 h − 0 + c.c.) + (λ ϕhk ϕk ++ h − −1 h − 0 +λ ϕhk ϕ * k ++ h − +1 h − 0 + c.c.) + λ Hk ++ (H † H)(k ++ k −− ) + λ Hh + −1 (H † H)(h + −1 h − +1 ) + λ Hh + +1 (H † H)(h + +1 h − −1 ) + λ Hh + 0 (H † H)(h + 0 h − 0 ) + λ ϕk ++ |ϕ| 2 (k ++ k −− ) + λ ϕh + −1 |ϕ| 2 (h + −1 h − +1 ) + λ ϕh + +1 |ϕ| 2 (h + +1 h − −1 ) + λ ϕh + 0 |ϕ| 2 (h + 0 h − 0 ) +
CP-even scalar sector: After gauge symmetry breaking, we have two neutral physical scalar fieldsh and ϕ R . The mass matrix for them is given by
L ⊃ 1 4 h ϕ R T λ H v 2 H λ Hϕ v H v ϕ λ Hϕ v H v ϕ λ ϕ v 2 ϕ T h ϕ R . (II.4)
Diagonalizing the mass matrix, we obtain mass eigenvalues and corresponding mass eigenstate {h, φ} such that
m 2 h,φ = λ H v 2 H + λ ϕ v 2 ϕ 4 ± 1 4 λ H v 2 H − λ ϕ v 2 ϕ 2 + 4λ 2 Hϕ v 2 H v 2 ϕ , (II.5) h φ = cos θ sin θ − sin θ cos θ h ϕ R , tan 2θ = 2λ Hϕ v H v ϕ λ H v 2 H − λ ϕ v 2 ϕ . (II.6)
In this paper we do not give further analysis for neutral scalar bosons where we assume mixing between φ and the SM Higgs is small to satisfy experimental constraints. Phenomenology of the neutral scalar sector without charged Higgs bosons is given in ref. [43].
Charged scalar bosons: Here we first consider mass of singly charged scalars where mass 3 The flavor-diagonal term ofL c Le (iσ 2 )L Le h + 0 vanishes due to the nature of anti-symmetry, although this term is allowed by
SU (2) L × U (1) Y × U (1) Lµ−Lτ .
terms are given by
L M h + = M 2 h + −1 + 1 2 λ Hh + −1 v 2 H + 1 2 λ ϕh + −1 v 2 ϕ h + −1 h − +1 + M 2 h + 0 + 1 2 λ Hh + 0 v 2 H + 1 2 λ ϕh + 0 v 2 ϕ h + 0 h − 0 + M 2 h + +1 + 1 2 λ Hh + +1 v 2 H + 1 2 λ ϕh + +1 v 2 ϕ h + +1 h − −1 + (m 2 A h + −1 h − 0 + m 2 B h + +1 h − 0 + c.c.), (II.7) where we have defined m 2 A ≡ µ ϕh v ϕ / √ 2 and m 2 B ≡μ ϕh v ϕ / √ 2.
In our scenario, we take
m A,B ≪ M h + −1 ,h + 0 ,h + +1
so that mixing among singly charged scalars is negligibly small. Moreover we obtain two-zero texture of neutrino mass matrix for small mixing case as we will see below. Therefore mass eigenstates of singly charged scalars are {h + −1 , h + 0 , h + +1 } whose masses are given by
m 2 h + −1,0,+1 ≃ M 2 h + −1,0,+1 + 1 2 λ Hh + −1,0,+1 v 2 + 1 2 λ ϕh + −1,0,+1 v 2 ϕ . (II.8)
On the other hand mass of doubly charged scalar k ±± is given by
m 2 k ±± = M 2 k ++ + 1 2 λ Hk ++ v 2 + 1 2 λ ϕk ++ v 2 ϕ . (II.9)
Z' boson: After U(1) Lµ−Lτ symmetry breaking, we have massive Z ′ boson. The mass of Z ′ is given by
m Z ′ = g ′ v ϕ , (II.10)
where g ′ is the U(1) Lµ−Lτ gauge coupling constant and we have ignored U(1) kinetic mixing assuming it is negligibly small. Gauge interactions of Z ′ and the SM fermions are written as
g ′ Z ′ ν (L µ γ ν L µ −L τ γ ν L τ +μ R γ ν µ R −τ R γ ν τ R ). (II.11)
Muon g − 2: In our model, Z ′ and h + −1,0 bosons can provide contributions to muon g − 2, ∆a µ , at one-loop level. These contributions are given by [45]
∆a µ = ∆a Z ′ µ + ∆a h ± µ ∆a Z ′ µ = g ′2 8π 2 1 0 dx 2m 2 µ x 2 (1 − x) x 2 m 2 µ + (1 − x)m 2 Z ′ , ∆a h ± µ ≃ − 4m 2 µ 96π 2 |f eµ | 2 m 2 h + −1 + |f µτ | 2 m 2 h + 0 . (II.12)
Note that charged scalar contribution provides negative sign contribution while Z ′ gives positive one. Neutrino mass matrix: In our model, neutrino masses are generated at two loop level as the original Zee-Babu model [1]. From the two loop diagrams in Fig. 1, non-zero components of neutrino mass matrix are obtained such that
M 11 = 8μ kh f eµ m µ g * µτ m τ f τ e I m h + −1 , m h + +1 , m k ++ , m µ , m τ , (II.13) M 12 = 4 √ 2λ ϕhk v ϕ f eµ m µ g * µτ m τ f τ µ I m h + −1 , m h + 0 , m k ++ , m µ , m τ , (II.14) M 13 = 4 √ 2λ ϕhk v ϕ f eτ m τ g * µτ m µ f µτ I m h + +1 , m h + 0 , m k ++ , m τ , m µ , (II.15) M 23 = 8μ kh f µe m e g * ee m e f eτ I m h + −1 , m h + +1 , m k ++ , m e , m e ,(II.16)
where the I(m 1 , m 2 , m 3 , m ℓ i , m ℓ j ) is the loop integral factor given by [46]
I(m 1 , m 2 , m 3 , m ℓ i , m ℓ j ) = d 4 p (2π) 4 d 4 q (2π) 4 1 p 2 + m 2 1 1 p 2 + m 2 ℓ i 1 q 2 + m 2 2 1 q 2 + m 2 ℓ j 1 (q + p) 2 + m 2 3 .
(II.17)
We thus obtain two-zero texture of the neutrino mass matrix in which M 33 ≃ M 22 ≃ 0 for small mixing among singly charged scalar bosons [47].
Notice that the mass matrix can not be written as a product of f and g in contrast to the original Zee-Babu model, and we can have three non-zero neutrino masses. The loop integral factor is typically given by
I(m 1 , m 2 , m 3 , m ℓ i , m ℓ j ) ≃ C I (4π) 4 1 M 2 , (II.18)
where C I is O(1) numerical factor [45] and M is the largest scalar mass inside the loop diagram.
III. PHENOMENOLOGY OF THE MODEL
In this section, we discuss phenomenology of our model. Firstly we consider experimental constraints on the Yukawa couplings associated with charged scalar bosons. The coupling g ee is constrained by the LEP data regarding e + e − → e + e − scattering and the upper limit is written by [18] |g ee | √ 4πm k ±± 8.6 TeV .
(III.1)
In addition, LFV process τ ± → µ ∓ e ± e ± gives the constraint [18,48] |g µτ g * ee | 0.007
m k ±± TeV 2 , (III.2)
where the other LFV processes can be suppressed taking small mixing among singly charged scalar bosons. Thus Yukawa couplings associated with singly charged scalars are less constrained in the small mixing case.
We then discuss requirements from neutrino mass matrix. Here we do not give detailed fitting to the neutrino oscillation data and use the result in ref. [49] where the mass matrix elements are required to be ∼ O(10 −11 ) GeV. The most stringent requirement comes from M 23 element since it is proportional to m 2 e in our case. Using Eq. (II.18) for rough estimation, we obtain
|M 23 | ∼ 3 × 10 −12 GeV × |f µe | 10 |g * ee | 0.4 |f eτ | 10 C Iμ kh TeV TeV M 2 . (III.3)
Thus we should require |f µe,eτ | ∼ 10,μ kh ∼ 10M and M to not be much larger than TeV scale to obtain |M 23 | ∼ O(10 −11 ) GeV, taking into account the LEP constraint for g ee in Eq. (III.1). Moreover we find g ee ≫ g µτ comparing M 11 and M 23 elements where the ratio of these couplings is roughly m µ m τ /m 2 e . Then LFV constraint Eq. (III.2) is easily satisfied as g µτ should be much smaller than g ee . In general, we can easily get required values of the other matrix elements as we have sufficient free parameters. Thus we focus on collider phenomenology for the doubly charged scalar boson associated with the coupling g ee in our following analysis.
Before we move to collider physics, we comment on the muon g − 2 in our model. Since we require large magnitude of Yukawa coupling |f µe |, ∆a h + µ get large negative contribution:
∆a h + µ ∼ −4.7 × 10 −9 |f µe | 10 2 TeV m h + 0 2 , (III.4)
where we have omitted contribution from f µτ coupling assuming it is subdominant. This negative contribution can still be compensated by the Z ′ contribution. For example, we obtain ∆a Z ′ µ ≃ 6.5 × 10 −9 for {m Z ′ , g ′ } = {9 MeV, 0.0008} which is allowed by the neutrino trident experiment [44]. Thus it is possible to explain muon g − 2, ∆a µ = (26.1 ± 8.0) × 10 −10 [50], by the Z ′ contribution even if we have negative contribution from singly charged scalar loop. 4
A. Doubly charged scalar production at hadron collider
The doubly charged scalar can be produced at the LHC via the process pp → Z/γ * → k ++ k −− and dominantly decays into e ± e ± due to the requirement for the Yukawa couplings discussed above. Here we estimate the production cross section using CalcHEP [51] implementing relevant interactions with CTEQ6L PDF [52]. In Fig. 2, we show the cross section for LHC 13 TeV which is compared with the current LHC limit obtained from data of ∼ 36 fb −1 integrated luminosity [17]. Then the doubly charged scalar should be larger than ∼ 650
GeV. We can test heavier mass region with more integrated luminosity at the LHC or High luminosity LHC. If we take m k ±± = 1 TeV the pair production cross section is around 0.014(0.019) fb for √ s = 13 (14) TeV. Thus the LHC data with 300(3000) fb −1 integrated luminosity would provide few (few ×10) number of events of e + e − e + e − signature from decay of doubly charged scalar bosons. Detailed simulation analysis is omitted here as it is beyond the scope of this paper. TeV.
B. Testing doubly charged scalar Yukawa coupling at lepton collider
Here we discuss test of doubly charged Yukawa coupling at the ILC with √ s = 250 GeV.
Although doubly charged scalar can not be produced directly at the ILC, we can test the coupling observing deviation from the SM prediction for the e + e − → e + e − scattering. The relevant effective interaction is written by
L ef f = g 2 ee 2m 2 k ++ (ēγ µ P R e)(ēγ µ P R e), (III.5)
where Fierz transformation is applied to get the operator. We then apply the analysis in ref. [18] based on polarized initial state at the ILC, considering the process
e − (k 1 , σ 1 )e + (k 2 , σ 2 ) → ℓ − (k 3 , σ 3 )ℓ + (k 4 , σ 4 ), (III.6)
where k i indicates 4-momentum of each particle and we explicitly show the helicities of initialand final-state leptons σ i = ±. Combining the SM and k ±± contributions, total helicity amplitudes for the process of e − (σ 1 )e + (σ 2 ) → e − (σ 3 )e + (σ 4 ) denoted by M σ i = M(σ 1 σ 2 σ 3 σ 4 ) are given by
M(+ − +−) = −e 2 (1 + cos θ) 1 + s t + c 2 R s s Z + s t Z + 2s α(Λ e RR ) 2 , (III.7) M(− + −+) = −e 2 (1 + cos θ) 1 + s t + c 2 L s s Z + s t Z , (III.8) M(+ − −+) = M(− + +−) = e 2 (1 − cos θ) 1 + c R c L s s Z , (III.9) M(+ + ++) = M(− − −−) = 2e 2 s t 1 + c R c L t t Z , (III.10) where Λ e RR ≡ 4πm 2 k ±± /g 2 ee , t = (k 1 − k 3 ) 2 = (k 2 − k 4 ) 2 = −s(1 − cos θ)/2, s = (k 1 + k 2 ) 2 = (k 3 + k 4 ) 2 , s Z = s − m 2 Z + im Z Γ Z , t Z = t − m 2 Z + im Z Γ Z , e 2 = 4πα
with α being the QED coupling constant, c R = tan θ W , c L = − cot 2θ W , and cos θ is the scattering polar angle.
The differential cross-section for purely-polarized initial-state σ 1,2 = ±1, is obtained using the amplitudes such as
dσ σ 1 σ 2 d cos θ = 1 32πs σ 3 ,σ 4 M {σ i } 2 . (III.11)
Then the partially-polarized differential cross section is defined as dσ(P e − , P e + ) d cos θ = σ e − ,σ e + =± 1 + σ e − P e − 2 1 + σ e + P e − 2 dσ σ e − σ e + d cos θ , (III. 12) where P e − (e + ) is the degree of polarization for the electron(positron) beam and we sum up the helicity of final states. We then apply the following two cases as realistic values at the ILC for polarized cross sections σ L,R :
dσ R d cos θ = dσ(0.8, −0.3) d cos θ , dσ L d cos θ = dσ(−0.8, 0.3) d cos θ . (III.13)
Applying the polarized cross sections, we study the sensitivity to k ±± boson in e + e − → e + e − scattering via the measurement of a forward-backward asymmetry at the ILC, which is given by Then we compare this quantity with a statistical error of the asymmetry which is given by assuming only SM contribution:
A F B (σ L,R ) = N F (σ L,R ) − N B (σ L,R ) N F (σ L,R ) + N B (σ L,δ SM A F B (σ L,R ) = 1 − (A SM F B (σ L,R )) 2 N SM F (σ L,R ) + N SM B (σ L,R )
.
(III. 16) In Fig. 3, we show ∆A F B (σ R ) and ∆A F B (σ L ) by solid and dashed curves respectively as a function of g ee where we apply integrated luminosity of 1000 fb −1 as a reference value.
The curves are compared with the values 5δ SM A F B and 2δ SM A F B which are respectively given by ∼ 7.2×10 −3 and ∼ 1.8×10 −3 . Thus we find that g ee 0.12 can be tested with the integrated luminosity of 1000 fb −1 with 2 σ level and 5 σ significance can be obtained for g ee 0.18 for polarized cross section σ R . On the other hand, ∆A F B (σ L ) is much smaller than that for σ R . Therefore comparing two polarized cross section cases we can test right-handed property of the Yukawa coupling. Moreover applying larger integrated luminosity as 2000 fb −1 we can test the whole region of g ee > 0.1 which is preferred by the neutrino mass matrix.
IV. CONCLUSION
We have proposed a model providing the neutrino mass and mixing by extending the Zee-Babu model imposing gauged U(1) Lµ−Lτ symmetry and introducing several charged scalar fields. Due to restricted Yukawa couplings resulting from the additional symmetry, we have found a predictive neutrino texture in a simple hypothesis in which mixings among singlycharged scalar bosons are negligibly tiny. In addition, LFV constrains are much relaxed compared to the original model in the small mixing scenario. The structure of neutrino mass matrix also constrains the relative values of the Yukawa couplings associated with doubly-charged scalar and the masses of charged scalars are preferred to be around TeV scale. We have also shown that anomalous muon magnetic dipole moment can be explained by Z ′ boson contribution even if singly charged scalars give opposite sign contributions.
Then we have focussed on analyzing phenomenology of doubly-charged scalar boson in the LHC and the future ILC experiments. The doubly-charged scalars can be produced in pair at the LHC if its mass is around TeV and e + e − e + e − signature will be dominant since electron Yukawa coupling is required to be the largest from our neutrino mass matrix. With sufficient integrated luminosity, doubly-charged Higgs can be discovered at the LHC and the high luminosity LHC experiments. On the other hand effect of doubly-charged scalar can be explored by measuring e + e − → e + e − scattering at the ILC although direct production is not kinematically allowed. Considering forward backward asymmetry in the process with polarized beam, we have estimated the sensitivity to the doubly-charged scalar interaction at the ILC with √ s = 250 GeV. We then find that a Yukawa coupling g ee up to ∼0.1 can be tested for m k ±± ≈ 1 TeV where those parameter region are preferred to get viable neutrino mass matrix. Therefore doubly-charged scalar interactions in our model can be tested in future collider experiments in which the LHC and the ILC will give complementary results.
FIG. 1 :
1Two loop diagrams generating neutrino mass matrix.
FIG. 2 :
2The blue curve show cross section for pp → k ++ k −− as a function of m k ±± at the LHC 13
FIG. 3 :
3R ) , N F (B) (σ L,R ) = L 0.5(0) 0(−0.5) d cos θ dσ L,R d cos θ , (III.14) where L is an integrated luminosity, and a bound of integral ±0.5 is chosen to maximize the sensitivity. Then the forward-backward asymmetry is estimated for cases with only the The blue solid(dashed) curve shows ∆A F B defined as Eq. (III.15) as a function of g ee for σ R(L) . The statistical error in the SM, δ SM A FB given by Eq. (III.16), is estimated to be ∼ 0.36 × 10 −3 both σ R and σ L . SM gauge boson contributions, and with both SM and k ±± boson contributions, in order to explore the sensitivity to k ±± interaction. We thus obtain N SM F (B) (σ L,R ) and A SM F B (σ L,R ) for the former case, and N SM +k ±± F (B) (σ L,R ) and A SM +k ±± F B (σ L,R ) for the latter case. Finally the sensitivity to k ±± interaction is estimated by ∆A F B (σ L,R ) = |A SM +k ±± F B (σ L,R ) − A SM F B (σ L,R )|. (III.15)
TABLE I :
IField contents of bosons and fermions in the lepton sector and their charge assignments
There are several variation models applying Zee-Babu model[4][5][6][7][8][9][10][11][12].2 This gauge symmetry with different types of radiative seesaw models are found in refs.[19,20].
Note here that Z ′ does not affect the effective number of neutrino species and cosmology even if it is light since Z ′ decouples from thermal bath earlier than neutrinos due to small gauge coupling and decays into neutrinos where the life time of Z ′ with {m Z ′ , g ′ } = {9 MeV, 0.0008} is of the order 10 −15 second that is much shorter than the time of Big Bang nucleosynthesis.
AcknowledgmentsH. O. is sincerely grateful for all the KIAS members.
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| []
|
[
"Flavorful Two Higgs Doublet Models with a Twin",
"Flavorful Two Higgs Doublet Models with a Twin"
]
| [
"Wolfgang Altmannshofer \nDepartment of Physics\nUniversity of California Santa Cruz\n1156 High St95064Santa CruzCAUSA\n\nSanta Cruz Institute for Particle Physics\n1156 High St95064Santa CruzCAUSA\n",
"Brian Maddock \nDepartment of Physics\nUniversity of California Santa Cruz\n1156 High St95064Santa CruzCAUSA\n\nSanta Cruz Institute for Particle Physics\n1156 High St95064Santa CruzCAUSA\n"
]
| [
"Department of Physics\nUniversity of California Santa Cruz\n1156 High St95064Santa CruzCAUSA",
"Santa Cruz Institute for Particle Physics\n1156 High St95064Santa CruzCAUSA",
"Department of Physics\nUniversity of California Santa Cruz\n1156 High St95064Santa CruzCAUSA",
"Santa Cruz Institute for Particle Physics\n1156 High St95064Santa CruzCAUSA"
]
| []
| A two Higgs doublet model with flavorful Yukawa structure, in which the two doublets give mass to the third and the first two generations respectively, is combined with the twin Higgs mechanism to stabilize the Higgs mass against radiative corrections. We consider both a mirror twin and fraternal twin setup. We identify Higgs signal strength measurements and the decay B s → µµ as the most important indirect constraints on the parameter space of the model. We explore the collider phenomenology of the model and find that the heavy Higgs in the visible sector can give a sizable number of displaced decays into b-jets in regions of parameter space where the SM-like Higgs and the twin Higgs do not provide any striking signatures.I. INTRODUCTIONThe absence of clear evidence for new degrees of freedom at the electroweak scale from the Large Hadron Collider (LHC) challenges "traditional" solutions to the hierarchy problem that predict new colored degrees of freedom at the TeV scale. One elegant way to address the hierarchy problem that largely avoids constraints from direct searches at the LHC, is the twin Higgs mechanism [1] and its variations[2,3]. In the twin Higgs model the SM Higgs exists as part of an enlarged, approximately SU (4) symmetric, scalar sector. The symmetry is broken resulting in Higgs doublets both in the visible sector and in an additional "twin" sector. The original twin Higgs model prescribed a mirror symmetry, resulting in an exact copy of the SM in the twin sector (see e.g. also[4,5]). The twin fermions are not charged under the SM gauge symmetries and therefore very hard to search for experimentally. The | 10.1103/physrevd.98.075005 | [
"https://arxiv.org/pdf/2003.01320v1.pdf"
]
| 118,812,477 | 2003.01320 | fbcb63be80e78c0f405df9434cd9e5bef0a410f0 |
Flavorful Two Higgs Doublet Models with a Twin
Wolfgang Altmannshofer
Department of Physics
University of California Santa Cruz
1156 High St95064Santa CruzCAUSA
Santa Cruz Institute for Particle Physics
1156 High St95064Santa CruzCAUSA
Brian Maddock
Department of Physics
University of California Santa Cruz
1156 High St95064Santa CruzCAUSA
Santa Cruz Institute for Particle Physics
1156 High St95064Santa CruzCAUSA
Flavorful Two Higgs Doublet Models with a Twin
A two Higgs doublet model with flavorful Yukawa structure, in which the two doublets give mass to the third and the first two generations respectively, is combined with the twin Higgs mechanism to stabilize the Higgs mass against radiative corrections. We consider both a mirror twin and fraternal twin setup. We identify Higgs signal strength measurements and the decay B s → µµ as the most important indirect constraints on the parameter space of the model. We explore the collider phenomenology of the model and find that the heavy Higgs in the visible sector can give a sizable number of displaced decays into b-jets in regions of parameter space where the SM-like Higgs and the twin Higgs do not provide any striking signatures.I. INTRODUCTIONThe absence of clear evidence for new degrees of freedom at the electroweak scale from the Large Hadron Collider (LHC) challenges "traditional" solutions to the hierarchy problem that predict new colored degrees of freedom at the TeV scale. One elegant way to address the hierarchy problem that largely avoids constraints from direct searches at the LHC, is the twin Higgs mechanism [1] and its variations[2,3]. In the twin Higgs model the SM Higgs exists as part of an enlarged, approximately SU (4) symmetric, scalar sector. The symmetry is broken resulting in Higgs doublets both in the visible sector and in an additional "twin" sector. The original twin Higgs model prescribed a mirror symmetry, resulting in an exact copy of the SM in the twin sector (see e.g. also[4,5]). The twin fermions are not charged under the SM gauge symmetries and therefore very hard to search for experimentally. The
original twin Higgs model includes light twin fermions and a massless twin photon. These light degrees of freedom lead to the mirror twin Higgs model having tension with early universe cosmology [1,6].
Twin Higgs models can be reconciled with cosmological bounds for example in nonstandard cosmologies [6][7][8][9], or by relaxing the mirror symmetry so that there are no light degrees of freedom in the twin sector. One realization of the second approach is the fraternal twin Higgs (FTH) model [10]. In this model the twin sector is constructed with the minimal amount of new physics needed in order to solve the little hierarchy problem in a consistent way.
The minimal twin sector required to stabilize the Higgs up to a scale of O(10) TeV contains a twin Higgs doublet, the twin third generation of fermions, and a twin SU (3) c × SU (2) L gauge symmetry.
Twin Higgs models have been explored extensively in recent years. For example, the collider phenomenology of such models have been studied in [10][11][12][13][14]. Distinct collider signatures arise due to the fact that the SM sector and the mirror sector are only connected through the Higgs portal (see however [15,16]). Twin Higgs models also lead to interesting dark matter phenomenology [17][18][19][20][21][22][23][24][25][26], they can be used to model baryogenesis [27], and can give rise to exotic astrophysical signatures [28].
In the fraternal twin Higgs model the third generation and the first and second generations are inherently treated differently. We wish to motivate the distinction between these generations. We propose that the visible sector is actually realized as a 2 Higgs doublet model (2HDM) with a flavorful Yukawa structure [29,30]. One Higgs doublet is responsible for the mass of the third generation fermions and the other doublet is responsible for the mass of the first and second generations. In such a flavorful 2HDM (F2HDM), the mass of the first and second generation of fermions is set by the vacuum expectation value (vev) of the second Higgs that can be considerably smaller than the vev of the first Higgs. Combining the flavorful 2HDM with the twin Higgs mechanisms thus offers the possibility to partially address the hierarchical structure of the quark and charged lepton masses and, at the same time, to stabilize the electroweak scale up to O(10) TeV.
We consider two setups of this "twinned" flavorful two Higgs doublet model. In the fist setup, the twin sector is realized in a similar fashion to the mirror twin Higgs model, with a fully mirrored 2HDM structure. In the second setup, we consider a minimal twin sector similar to that of a fraternal twin Higgs model. We show under which conditions these two setups can be mapped onto each other.
The paper is organized as follows: we briefly summarize twin Higgs models in sec. II; in sec. III we describe the details of the setup of our twin F2HDM and discuss the resulting physical Higgs mass eigenstates and their couplings to both the SM and twin sector particles; in sec. IV we discuss the bounds on the model from Higgs signal strength measurements and the most important flavor constraint, the B s → µµ decay; finally, in sec. V we look at the phenomenology of this model, particularly focusing on displaced decays occurring in regions of parameter space that are unique to this setup; we conclude in sec. VI.
II. TWIN HIGGS MODELS
The twin Higgs mechanism stabilizes the Higgs mass up to some moderate scale, Λ, usually considered to be around 10 TeV. Above this scale some additional new physics is invoked to protect the Higgs mass up to the Planck scale. The largest contributions to the Higgs mass are the 1-loop top quark correction, the 1-loop SU (2) L correction, and the 2-loop QCD correction. In the twin Higgs model a twin sector exists with new degrees of freedom which cancel these contributions. Here we briefly review two versions of the twin sector: the mirror model and the fraternal model. More detailed discussions of these models and the underlying protection mechanism can be found in [1] and [10], respectively.
The twin Higgs mechanism is based on an approximate SU (4) symmetry that is respected by the scalar sector. An SU (4) fundamental scalar Φ contains two doublets φ andφ, parameterized as
Φ = φ φ = φ + (v + S + iη)/ √ 2 φ + (v +Ŝ + iη)/ √ 2 ,(1)
with the potential
V (φ,φ) = −µ 2 |Φ| 2 + λ|Φ| 4 + κ|φ| 4 +κ|φ| 4 − σµ 2 |φ| 2 .(2)
Besides the SU(4) symmetric mass term µ 2 and the quartic coupling λ, the potential includes a soft SU (4) breaking term σ, which allows a misalignment of the SM and twin vevs, v and v, and the parameters κ andκ are hard breaking terms, which help to reduce fine tuning [12]. We identify φ as the SU (2) L Higgs doublet in the SM sector andφ is the corresponding doublet in the twin sector.
After symmetry breaking and rotating to the physical mass eigenstates results in two physical scalar bosons that we identify as a SM-like Higgs (h) and a twin Higgs (ĥ) which are mixed states of S andŜ. The mixing angle is of order O(v/v).
The particle content of the twin sector is where the mirror and fraternal realizations of the twin Higgs mechanism differ. We first consider the mirror twin Higgs model where the twin sector is an exact copy of the SM sector containing the same forces, particles, and couplings that the SM does.
The Higgs mass receives loop contributions from both fermions and twin fermions, as shown in fig. 1. The twin top contribution comes with a relative minus sign as compared to the top contribution causing these two diagrams to cancel. In a similar fashion to the top quarks the twin contributions from the weak gauge bosons and two loop gluon contributions to the Higgs mass are exactly the same as the SM contribution with a relative minus sign. This is the fundamental mechanism that stabilizes the Higgs mass in twin Higgs models. The total correction to the Higgs mass from these loops is [10]
δm 2 h = f N f Λ 2 4π 2 (y 2 f −ŷ 2 f ) + 9Λ 2 32π 2 (g 2 (Λ) 2 −ĝ 2 (Λ) 2 ) + 3y 2 t Λ 2 4π 4 (g 3 (Λ) 2 −ĝ 3 (Λ) 2 ),(3)
whereĝ 2 is the strength of the twin SU (2) L ,ĝ 3 is the strength of the twin SU (3), andŷf are the twin Yukawa couplings. The color factor N c = 3 for quarks and N c = 1 for leptons. In the mirror twin Higgs model the couplings in the twin sector and the visible sector are set to be equal, thus leading to δm 2 h above being zero. However, the many light degrees of freedom in the mirror twin sector (in particular the light twin fermions and the massless twin photon) lead to tensions with cosmology. This inspired a minimal version of the twin Higgs model known as the fraternal twin Higgs model.
The fraternal twin Higgs model adds the minimum new physics in the twin sector necessary to stabilize the Higgs. The particle content in the twin sector consists of a twin top, a twin SU (2) L and a twin SU (3). In this setup the couplings of these particles are free parameters.
From eq. (3) we see that to ensure that the Higgs mass is not significantly tuned up to Λ ∼ 10 TeV one requires
ŷ t (Λ) − y t (Λ) y t (Λ) 0.01, ĝ 2 (Λ) − g 2 (Λ) g 2 (Λ) 0.1, g 3 (Λ) −ĝ 3 (Λ) g 3 (Λ) 0.1.(4)
In order for the twin SU To ensure the twin fermions other than the top do not reintroduce large corrections to the Higgs mass one has to demand that
y 2 f 4π 2 3Λ 2 δm 2 h + y 2 f ∼ 0.002 × 10 TeV Λ 2 δm h 125 GeV 2 ,(5)
where in the last step we neglected the small SM Yukawas y f . The above criterion translates intoŷf being no larger than ∼ 0.05, with the precise value depending on the maximum acceptable choice for δm h .
III. TWIN TWO HIGGS DOUBLET MODELS
Both the mirror and fraternal twin Higgs models successfully stabilize the Higgs mass up to order Λ. However, the mirror twin Higgs needs additional physics which can reconcile the model with cosmology, while the fraternal twin Higgs model leaves us with no explanation for the lack of the first two generations in the twin sector. Here we describe how the addition of a new source of mass generation in the form of a second Higgs doublet might provide a resolution to these issues. We also show how the mirror and fraternal version of a 2HDM setup can be mapped onto one another.
A. Mirror Setup
A well studied setup that provides additional sources of mass generation and distinguishes between the first two generations and the third generation is the flavorful 2HDM [29,30].
This model contains a SM-like doublet which primarily provides mass to the third generation fermions, and an additional doublet that primarily provides mass to the first and second generations. We propose a mirror twin Higgs inspired model where both the visible sector and the twin sector are realized as flavorful 2HDMs.
In this realization we have four doublets φ 1 ,φ 1 , φ 2 and,φ 2 , where φ 1 and φ 2 live in the visible sector andφ 1 andφ 2 live in the twin sector. The fields are arranged into SU (4)
multiplets Φ 1 = φ 1 φ 1 , Φ 2 = φ 2 φ 2 .(6)
We will consider a scenario in which φ 1 andφ 1 couple to the third generation particles in the visible and twin sector, respectively, and φ 2 andφ 2 couple to the first two generations in the visible and twin sector, respectively. The most generic potential for Φ 1 and Φ 2 looks like
V (Φ 1 , Φ 2 ) = −µ 2 1 |Φ 1 | 2 − µ 2 2 |Φ 2 | 2 + λ 1 (Φ † 1 Φ 1 ) 2 + λ 2 (Φ † 2 Φ 2 ) 2 + λ 3 |Φ 1 | 2 |Φ 2 | 2 + m 2 12 φ † 1 φ 2 + h.c. +m 2 12 φ † 1φ 2 + h.c. + λ 4 |Φ † 1 Φ 2 | 2 + λ 5 2 (Φ † 1 Φ 2 ) 2 + h.c. + κ 1 (φ † 1 φ 1 ) 2 +κ 1 (φ † 1φ 1 ) 2 + κ 2 (φ † 2 φ 2 ) 2 +κ 2 (φ † 2φ 2 ) 2 + σ 1 µ 2 1 (φ † 1 φ 1 ) + σ 2 µ 2 2 (φ † 2 φ 2 ) ,(7)
where the soft breaking terms σ andσ and the hard breaking terms κ 1 , κ 2 ,κ 1 , andκ 2 are introduced in analogy to the usual twin Higgs setup. The terms containing m 2 12 andm 2 12 are mass parameters that mix the doublets φ 1 andφ 1 with φ 2 andφ 2 , respectively. m 2 12 =m 2 12 is another source of soft symmetry breaking.
As shown in [31] this setup is a self consistent extension of the twin Higgs model and provides the same cancellations as in the traditional twin Higgs setup. However, we now have extra sources of mass generation in the SM sector from φ 2 and the twin sector fromφ 2 .
Constraints from cosmology (in particular N ef f ) can be avoided by making the twin degrees of freedom sufficiently heavy, i.e. heavier than O(1 GeV).
The first two generations in the visible and twin sector have masses
m f 1,2 = y f 1,2 v 2 √ 2 , mf 1,2 = y f 1,2v 2 √ 2 .(8)
Due to the mirror symmetry the only way to make the first two generations of twin fermions heavy is to make the vacuum expectation valuev 2 much larger than v 2 (see [32] for a different mechanism to raise the masses of the fermions in the twin sector.) Characteristic values for v 1 /v 2 ≈ 10 (motivated to explain the hierarchy between the third and the second generation of SM fermions) and requiring that the lightest mirror particles to be at least O(1 GeV), leads tov 2 ≈ 10 TeV. We thus envision the following set of vevs
v 2 ∼ O(10 GeV) , v 1 ∼ O(100 GeV) ,v 1 ∼ O(1 TeV) ,v 2 ∼ O(10 TeV) .(9)
We can approximate the amount of fine tuning needed to put the vevs in this hierarchical structure as [10,33] v
1 − Tuning ∼ 2v 2 1 v 2 1 , v 2 − Tuning ∼ 2v 2 2 v 2 2 .(10)
This means the tuning of v 1 vs.v 1 is order percent level, but the tuning of v 2 vs.v 2 is substantial, of order 10 −6 .
In addition to the fermions, also the twin photon needs to be sufficiently heavy to avoid cosmological bounds. Two options to do this are: breaking electromagnetism in the mirror sector, or simply removing the U (1) hypercharge in the twin sector. In both cases the mirror symmetry of the model is weakened. In the following we will follow the scenario where there is no twin U (1) hypercharge.
The setup we have described so far leads to a large number of O(1 GeV) particles in the twin sector resulting in a complicated, yet rich set of dynamics. We leave a detailed discussion of this scenario to future work. Instead, we focus on a simplified setup which takes the twin Yukawas as free parameters in order to make the first and second generation twin fermions sufficiently heavy to be irrelevant for the Higgs phenomenology that we will discuss below.
By takingŷ f to be free parameters (up to the bound imposed by eq. (5)) we can push the masses of the twin first and second generation particles to O(1 TeV), forv 2 ∼ 10 TeV. In such a setup, the low energy phenomenology will be determined by the twin third generation, the twin SU (3) andφ 1 , while all the other twin states are effectively decoupled.
B. Fraternal Setup
Another approach to a twin flavorful 2HDM is to construct a model inspired by the fraternal twin Higgs. Starting from a flavorful 2HDM with doublets φ 1 and φ 2 , we add a third doubletφ 1 , with φ 1 andφ 1 being part of an approximate SU (4) The twin sector consists of a Higgs doublet, a twin SU (2), a twin SU (3), and the third generation of twin fermions.
C. Twin F2HDM
The two approaches mentioned above both result in the same particle content and forces at low scales. Regardless of the high scale setup we will refer to the low energy simplified model as the twin F2HDM. 1 The potential for the twin F2HDM can be derived from eq. (7) withφ 2 integrated out.
This leaves an effective three Higgs doublet potential for the fields
Φ 1 = φ 1 φ 1 = φ + 1 (v 1 + S 1 + iη 1 )/ √ 2 φ + 1 (v 1 +Ŝ 1 + iη 1 )/ √ 2 , φ 2 = φ + 2 (v 2 + S 2 + iη 2 )/ √ 2 ,(11)V (Φ 1 , φ 2 ) = −µ 2 1 |Φ 1 | 2 − µ 2 2 |φ 2 | 2 + λ 1 (Φ † 1 Φ 1 ) 2 + λ 2 (φ † 2 φ 2 ) 2 + λ 3 |Φ 1 | 2 |φ 2 | 2 + m 2 12 φ † 1 φ 2 + h.c. + λ 4 |φ † 1 φ 2 | 2 + λ 5 2 (φ † 1 φ 2 ) 2 + h.c.
1 One difference between the two discussed setups (mirror and fraternal twin sector) is that the mass of the twin weak gauge bosons will be set by a combination ofv 1 andv 2 in the mirror setup, but only set byv 1 in the fraternal setup. Generically, becausev 2 v 1 then twin weak gauge bosons in the mirror setup will be much heavier than in the fraternal setup. However, in both cases the twin weak gauge bosons will be O(1 TeV) or heavier, leaving no noticeable difference in the low energy phenomenology we will discuss in the remainder of this paper.
+ σµ 2 1 φ † 1 φ 1 + κ 1 (φ † 1 φ 1 ) 2 + κ 2 (φ † 2 φ 2 ) 2 .(12)
After electroweak symmetry breaking we are left with 6 massive modes: three scalar Higgs bosons, two charged Higgs bosons, and one pseudoscalar Higgs boson. The three scalars S 1 , S 1 , S 2 are related to the mass basis counterparts h 1 ,ĥ 1 , and h 2 (identified as the SM-like, twin, and heavy Higgs) by
S 1 = c α 1 c α 2 h 1 + c α 1 s α 2 c α 3 + s α 1 s α 3 h 2 + s α 1 c α 3 − c α 1 s α 2 s α 3 ĥ 1 ,(13a)S 1 = −s α 1 c α 2 h 1 + c α 1 s α 3 − s α 1 s α 2 c α 3 h 2 + c α 1 c α 3 + s α 1 s α 2 s α 3 ĥ 1 ,(13b)S 2 = −s α 2 h 1 + c α 2 c α 3 h 2 − c α 2 s α 3ĥ 1 . (13c)
where the three mixing angles (s α i = sin(α i ), c α i = cos(α i )) are approximately given by
sin(α 1 ) v 1 λ 1 v 1 (κ 2 + λ 1 ) , sin(α 2 ) − v 2 v 1 , sin(α 3 ) − v 2 λ 3 2v 1 (κ 2 + λ 1 )
.
The three Higgs boson masses are approximately
m 2 h 1 2v 2 1 κ 1 + λ 1 κ 2 κ 2 + λ 1 , m 2 h 2 m 2 12 v 1 v 2 , m 2 h 1 2v 2 1 (λ 1 + κ 2 ) .(15)
The SM-like Higgs mass can be set by fixing κ 1 , κ 2 , and λ 1 . The heavy Higgs mass is primarily set by the parameter m 2 12 , and the twin Higgs mass is primarily set byv 1 , both of which can be taken as free parameters.
The most generic Yukawa Lagrangian can be written as
−L twin-F2HDM ⊃ i,j λ u 1,ij (q i u j )φ 1 + λ d 1,ij (q i d j )φ 1 + λ e 1,ij (¯ i e j )φ 1 + i,j λ u 2,ij (q i u j )φ 2 + λ d 2,ij (q i d j )φ 2 + λ e 2,ij (¯ i e j )φ 2 + ŷt(qt)φ 1 +ŷb(qb)φ 1 +ŷν(¯ ν)φ 1 +ŷτ (¯ τ )φ 1 + h.c. .(16)
The Yukawa matrices in the SM sector λ f i are determined by the flavor structure imposed on φ 1 and φ 2 in the flavorful setup [29,30]. The couplings λ u 1 , λ d 1 , and λ e 1 are rank one matrices, providing mass only to the third generation, while λ u 2 , λ d 2 , and λ e 2 have full rank and provide mass for the remaining fermions as well as CKM mixing in the quark sector. We find that the couplings of the Higgs bosons to the up-type quarks in the fermion mass eigenstate basis are given by
Y h 1 u i u j = δ ij m u i v s β (c α 1 c α 2 ) − m u i u j v s β c β (c β c α 1 c α 2 + s β s α 2 ) ,(17a)Y h 2 u i u j = δ ij m u i v s β (s α 3 s α 1 + c α 1 s α 2 c α 3 ) − m u i u j v s β c β (−c α 2 c α 3 s β + c α 1 c α 3 s α 2 c β + s α 1 s α 3 c β ) , (17b) Yĥ 1 u i u j = δ ij m u i v s β (c α 3 s α 1 − c α 1 s α 2 s α 3 ) + m u i u j v s β c β (c α 1 s α 2 s α 3 c β − c α 3 s α 1 c β − c α 2 s α 3 s β ) , (17c)
where v = v 2 1 + v 2 2 = 246 GeV and s β = sin β, c β = cos β with tan β = v 1 /v 2 . The mass parameters m u i u j are given by the Yukawa couplings λ u 2 in the fermion mass eigenstate basis. For the flavor indices i or j equal to 1, the mass parameters m u i u j are of the order of the up quark mass and of the order of the charm quark mass otherwise (see [30] for their explicit expressions).
The above expressions for the couplings hold analogously for the down-type quarks and leptons. The couplings of the SM fermions to the charged Higgs bosons are the same as in the standard versions of the F2HDMs [29]. In our setup discussed here, the scalar Higgs bosons (and charged Higgs bosons) couple in addition also to the twin sector fermions as
Y h 1 f = −ŷf √ 2 c α 2 s α 1 ,(18a)Y h 2 f = −ŷf √ 2 (c α 3 s α 1 s α 2 − c α 1 s α 3 ) ,(18b)Yĥ 1 f =ŷf √ 2 (c α 1 c α 3 + s α 1 s α 2 s α 3 ) .(18c)
Finally, the couplings of the Higgs bosons to the vector bosons (hW W and hZZ) are given by the following expressions
Y h 1 V Y SM V = c α 1 c α 2 s β − s α 2 c β ,(19a)Y h 2 V Y SM V = (c α 1 c α 3 s α 2 + s α 1 s α 3 )s β − (c α 3 s α 1 s α 2 − c α 1 s α 3 )c β ,(19b)Yĥ 1 V Y SM V = (c α 3 s α 1 − c α 1 s α 2 s α 3 )s β + (c α 1 c α 3 + s α 1 s α 2 s α 3 )c β .(19c)
where Y SM V are the corresponding couplings of the Higgs boson in the Standard Model.
IV. CONSTRAINTS
The introduction of two additional Higgs doublets alters the couplings of the SM-like Higgs boson h 1 as shown in eqs. (17)- (19). The ATLAS and CMS experiments at the LHC have taken measurements of the production and decays of the Higgs boson and we must ensure that our model is consistent with the existing experimental results. Additionally, we will also consider the impact of projected sensitivities from the high luminosity (HL) LHC.
To determine these constraints we construct a χ 2 function
χ 2 = i,j (σ × BR) exp i (σ × BR) SM i − (σ × BR) BSM i (σ × BR) SM i (σ × BR) exp j (σ × BR) SM j − (σ × BR) BSM j (σ × BR) SM j cov −1 ij ,(20) where (σ × BR) exp i , (σ × BR) SM i , and (σ × BR) BSM i
are the experimental measurements, the Standard Model predictions, and our BSM predictions for the production cross sections times branching ratio of the various measured channels.
We use the SM predictions from [34]. As in [30], [44,45], and top associated production [46,47]. The projected sensitivities are taken from [48], and correspond to 3000 fb −1 of data collected at 14 TeV. the twin Higgs, as seen in eq. (14). In addition, there is also weak dependence of the Higgs couplings on the mass parameters m f i f j (see eq. (17) and text below). We let those mass parameters vary up to a factor of 3 around their expected values, as was also done in [30].
Generally, the sensitivities that are expected at the HL-LHC can potentially constrain the twin vevv 1 much stronger than the current bound. Previous studies found the constraint v 1 3v, while future experiments favorv 1 to be closer to an order of magnitude larger than v, at least for moderate values of tan β.
As shown in [29], the flavorful structure we impose on the φ 1 and φ 2 couplings leads to flavor violating Higgs couplings for the SM-like Higgs and the heavy Higgs. The SU (2) 5 flavor symmetry, that is preserved by the rank 1 Yukawa couplings of the doublet φ 1 , protects flavor changing transitions between the first and second generation of quarks and leptons that typically plague 2HDMs without flavor conservation [49]. However, we still find strong and robust constraints from the rare decay B s → µµ. 2 In the limitv 1 v, the expression for the B s → µµ branching ratio in our model can be easily generalized from the expression in [30] with α → α 2 .
The SM prediction and the current experimental measurements are [50] BR(B s → µµ) SM = (3.67 ± 0.15) × 10 −9 , For the future experimental sensitivities to B s → µµ, we assume that the central value for the branching ratio stays consistent with the current experimental value, while we take an uncertainty of ±0.16 × 10 −9 [51]. It is important to note that there is some tension (at the 2σ level) between the SM prediction and current experimental value, and assuming that the experimental central value holds there will be very significant discrepancy from future experiments.
BR(B s → µµ) exp = (2.67 +0.45 −0.35 ) × 10 −9 .(21)
The current constraint (left) and future sensitivity (right) from B s → µµ is shown in the plane of the heavy Higgs mass m h 2 vs. tan β in fig. 3, with the Higgs fit constraints overlayed in red. Based on current constraints masses as low as 300 GeV are consistent with both B s → µµ and Higgs signal strengths measurements for a moderate tan β 5. However, future projections push this lower bound on the mass up to around 450 GeV. For somewhat larger tan β 10, the expected bound on the heavy Higgs mass is around 700 GeV. This is significant as the production of the heavy Higgs becomes quickly suppressed as its mass m h 2 increases.
Our model can moderate the tension between the theoretical prediction and experimental value of the B s → µµ branching ratio by the additional contributions of the heavy and pseudoscalar Higgs. In particular the pseudoscalar Higgs contribution interferes destructively with the SM amplitude and thus can lower the B s → µµ rate, reconciling the theoretical prediction with the experimental central value. This is evident in the shape of the allowed region of the plots in fig. 3, where the band represents the region of parameter space that removes unwanted tension. In the scenario that tan β becomes too large the rate of B s → µµ also becomes too large, violating the 2σ bound. While if m h 2 becomes too large, our theoretical prediction matches back onto the SM prediction and is disfavored.
In addition to Higgs fit and the B s → µµ constraints, there exists constraints from heavy Higgs searches performed by ATLAS and CMS. The most relevant constraints come currently from H → µµ searches [52,53], but are weak compared to the B s → µµ and Higgs signal strength measurements as shown in previous work [30]. For a detailed description of the twin bottomonium and glueball spectrum see [10]. We will assume that the the twin taus and neutrinos are sufficiently heavy such that the bottomonia and glueballs do not decay into them. Some of the bottomonia and glueballs (in particular the lightest glueball) can mix with the Higgs bosons in the visible and therefore decay into SM particles. The lifetime of the glueballs can be sizeable and one often finds displaced decays in the discussed scenario.
Displaced events occur as a result of the production of twin sector states (bottomonia and/or glueballs) through one of the three scalar Higgs bosons or the pseudoscalar Higgs boson. We assume that the twin spectrum is such that there are glueball states with mass below half the mass of the twin bottomonia mĜ < m [bb] /2 and assume that all decays in the twin sector result in at least one lightest glueballĜ 0 . The lightest glueball has the same quantum numbers as the SM-like Higgs allowing it to mix back into the visible sector and decay, in particular to bb. In the viable region of our parameter space, the corresponding lifetime of the glueball can be approximated as [10] cτ ≈ 18m × 10GeV mĜ
0 7 v 1 750GeV 4 ,(22)
which depends very sensitively on the glueball mass.
We can break down the phenomenology of this scenario into three distinct regions: SM-like Higgs dominated, twin Higgs dominated, and heavy Higgs dominated. The SM-like Higgs dominates the phenomenology when the twin vevv 1 and twin bottom Yukawaŷb take values such that the twin bottomonia and the twin glueballs are lighter than half the Higgs mass, m [bb] < m h 1 /2. As the SM-like Higgs is produced at the LHC at a much higher rate than the heavy Higgs or twin Higgs, displaced decays from the SM-like Higgs dominate the phenomenology 3 . In this case the phenomenology of our model is similar to that of the original FTH model. For this reason we forgo an analysis of this scenario here and instead point the reader to [10,11].
The twin Higgs dominates the displaced phenomenology whenv 1 andŷb take on values such that m [bb] > m h 1 /2, while the twin Higgsĥ 1 is still moderately light. In this case the twin Higgs is produced at a high enough rate that its production of twin sector hadrons is much larger than that of the heavy Higgs, so again the phenomenology follows a similar path of the original FTH model, and the addition of the heavy Higgs has little impact on the phenomenology. For this reason we forgo an analysis of this scenario here as well, and instead point the reader to [10,12,13,54].
The final region of parameter space is characterized by a phenomenology that is dominated by the heavy Higgs. This happens whenv 1 andŷb take on values such that m [bb] > m h 1 /2 and at the same time the twin Higgsĥ 1 is very heavy (motivated byv 1 being large). In this regime both the heavy Higgs and the twin Higgs participate in producing twin sector particles, but the production rate of the twin Higgs becomes very small. Asv 1 rises, the decay of the heavy Higgs to the twin sector is also suppressed. However, the production rate of the twin Higgs drops more quickly than the branching ratio of heavy Higgs decays into GeV. The production cross sections and decay rates are rather robust against order one changes to the parameters of this benchmark point other than λ 3 . We can see from eq. (14) that λ 3 controls the mixing of the heavy Higgs with the twin sector and therefore has a substantial impact on the branching ratio into the twin sector BR(H →bb). We choose a large value of λ 3 = 5 as a representative example. For the Yukawa coupling of the twin bottom we chooseŷb = 3y SM b , within the range allowed by naturalness arguments, see eq. (5). The branching ratios in this model are similar to that of the type 1B F2HDM [30], with the addition of a small, but important, branching ratio to the twin bottom. In the left plot we show the production cross sections of the heavy Higgs. Also the cross sections are similarly to the type 1B F2HDM and we see that over most of parameter space the main production modes are charm-charm fusion, gluon-gluon fusion, and vector boson fusion. Generally, gluon-gluon fusion is dominant at small tan β and charm-charm fusion is dominant at high tan β. For moderate values of tan β, the production cross section for a 500 GeV heavy Higgs can be around 100 fb. absent. In that case, whenv 1 is larger than about 2000 GeV, we see that the lifetime of the glueballs is of the order of at least millimeters which, given a typical boost factor of a few, falls into the decay lengths of interest for displaced signatures at the LHC.
We see that for the lower mass choices of the heavy Higgs 500 GeV (at the LHC) and 700
GeV (at the HL-LHC) O(10)s of events could occur with O(few mm) displaced decays. As we push the scale ofv 1 to larger values, we see that this number drops down to a handful of decays. The HL-LHC will generically produce more displaced decays at a given heavy Higgs mass, but the stronger expected constraints on the parameter space roughly balance out the increase. So, for masses that are not indirectly probed by flavor constraints or Higgs coupling strength measurements, we see a similar amount of expected displaced decays. Similarly such an observation also holds for the higher mass scenarios that we considered. We see that for heavier Higgs mass m h 2 the estimated number of events is reduced to several and below as the production of the heavy Higgs is suppressed at these higher masses.
Searches for long lived particles have been explored to some degree at the LHC [55,56].
The existing searches do currently not put strong constraints on the displaced decays we have considered in this model. For sizable displacements of the order O(1 cm − 1 m) the expected sensitivities from the LHC could cover sizable regions of parameter space (see [54] for a detailed study of the fraternal twin Higgs model). However, in the scenarios discussed in this paper, the displacement is typically of the order O(few mm) 4 , making it much more challenging to search for the displaced signatures, for example due to triggering difficulties (see e.g. [10,56]). Future improvements in searching for displaced decays with O(few mm) displacement would be necessary to further explore the models described in this work.
VI. CONCLUSION
The little hierarchy problem and the SM flavor puzzle are two longstanding problems in particle physics. We have discussed a setup which attempts to address both of them (at least partially). We considered a 2HDM with a flavorful Yukawa structure, where one Higgs doublet is responsible for the mass of the third generation fermions and the other doublet is responsible for the mass of the first and second generations. A hierarchy in vevs can explain the mass hierarchy between the third and first two generations. We combined this setup with the twin Higgs mechanism which stabilizes the Higgs mass up to O(10 TeV), considering both a mirror twin and fraternal twin setup.
In the visible sector, the flavorful Yukawa structure of this model leads to modifications of the B s → µµ branching ratio. Large values of tan β are already strongly constrained.
We showed that the current mild tension that exists between the SM prediction and the experimental results can be solved in our setup for moderate values of tan β. This is of particular interest in view of the expected future sensitivities to B s → µµ from LHCb, which could turn the current tension into a very significant discrepancy. The prediction of this scenario are slightly displaced decays at length scales of few millimeters which are challenging to detect experimentally. We find that for a twin vevv 1 of at least 2000 GeV that the heavy Higgs can naturally dominate the displaced phenomenology with as many as O(30) displaced decays predicted to have taken place at the LHC already.
Anticipating improved indirect constraints on the model parameter space from future experimental results on Higgs signal strengths measurements and the B s → µµ decay, we find that there is still viable region of parameter space which can produce O(30) displaced decays at the HL-LHC.
Figure 1 .
1Diagrams showing the loop contributions to the Higgs mass from the top quark (left) and from the twin top (right).
( 3 )
3to be anomaly free there must exist a right handed twin bottom. For the twin SU (2) to be anomaly free one need an SU (3) neutral SU (2) doublet, which contains the left handed twin tau and twin neutrino. In order to make the twin tau and neutrino massive one also introduces a right handed twin tau and twin neutrino. Thus, the minimal particle content of the fraternal twin Higgs model contains a twin Higgs doublet, a full third generation of twin fermions, and twin gauge interactions based on the gauge groups SU (2), and SU (3).
symmetry. The doublets φ 1 andφ 1 are responsible for the mass generations of the third generation particles in their respective sectors. The visible sector doublet φ 2 provides mass to the first and second generation fermions in the visible sector, which have no counterparts in the mirror sector.
The couplings to the SM-like Higgs boson are primarily determined by tan β andv 1 . From the expressions in eq. (17) and eq.(19) we can see that generically, large values of tan β and v 1 correspond to couplings of the Higgs to fermions and vector bosons that are SM-like. This can be clearly seen infig. 2where we show the Higgs signal strength constraints. We show the 2σ excluded regions based on the current LHC results and HL-LHC projections. The parameters λ 1 , λ 3 , κ 1 , and κ 2 that enter the Higgs couplings through the mixing angles in eq. (14) are set to λ 1 = 1, λ 3 = 5, κ 2 = 1 with κ 1 being set to reproduce the SM-like Higgs mass as in eq.(15). The results are fairly robust to the choice of these parameters, only being modified slightly by the choice of λ 1 which scales the mixing of the SM-like Higgs with
Figure 2 .
2The 2σ constraints on the twin vevv 1 vs. tan β based on Higgs signal strength measurements at the LHC are shown in the shaded gray region and projections from the HL-LHC are denoted by the black, dotted line. The relevant Higgs potential parameters are set to λ 1 = 1, λ 3 = 5, κ 1 = −3/8, and κ 2 = 1. The mass parameters in the flavorful Yukawa couplings are allowed to vary up to a factor of 3 around their expected values.
Figure 3 .
3Constraints on the heavy Higgs mass m h 2 and tan β based on current experiments (left)and expected sensitivities (right). We set λ 1 = 1, λ 3 = 5, κ 1 = −3/8, κ 2 = 1, andv 1 = 2500 GeV.
sector in the discussed scenario contains a number of states, the lightest ones being the twin bottom, tau and tau neutrino, as well as twin gluons. The twin gluons and twin bottoms hadronize, with the lightest bound states being bottomonia [bb] and glueballsĜ.
twin sector particles. The result of this is a region of parameter space where the heavy Higgs plays the most important role in the phenomenology.The mass of the twin bottom as a function of the coupling twin bottom Yukawa coupling yb and the twin vevv 1 is shown infig. 4. The gray region denotes where the twin bottom is light enough, such that bottomonium can be produced by the SM-like Higgs. The phenomenology of this region is analogous to that of traditional twin Higgs models. In the following we focus on the region with heavy twin bottoms mb ∼ O(100 GeV).
Figure 4 .
4The mass of the twin bottom as a function of the twin bottom Yukawa coupling yb and twin vevv 1 . In the gray region twin bottoms are sufficiently light such that the SM-like Higgs can decay into twin bottomonia. To better understand the production of the twin bottom via the heavy Higgs boson we look at the production and decay of the heavy Higgs boson in fig. 5. The right plot shows the branching ratios of the heavy Higgs with mass m h 2 = 500 GeV as a function of tan β, for a benchmark parameter point defined by κ 1 = −3/8, κ 2 = 1, λ 1 = 1, λ 3 = 5, andv 1 = 2500
Figure 5 .
5The production cross section at 13 TeV proton-proton collisions (left) and branching ratios (right) of the heavy Higgs with mass m h 2 = 500 GeV as function of tan β. We setκ 1 = −3/8, κ 2 = 1, λ 1 = 1, λ 3 = 5,v 1 = 2500 GeV, andŷb = 3y SM b .The number of expected events with displaced decays as a function of the twin vevv 1 and tan β is shown infig. 6. We assume that each decay of the heavy Higgs into twin bottoms results in at least one long lived glueball that decays back into the SM through mixing with a Higgs. The left two plots show the number of events produced after run 3 of the LHC(300/fb) for a heavy Higgs mass of 500 GeV (top) and 800 GeV (bottom), while the right two plots shows the number produced displaced decay events after the conclusion of the HL-LHC (3000/fb) for a heavy Higgs mass of 700 GeV (top) and 1000 GeV (bottom). The chosen heavy Higgs masses correspond to choices which still exhibit a fair amount of freedom in other parameters, such as tan β as shown in fig. 3. The gray shaded region corresponds to the parameter space ruled out by either the B s → µµ constraint (dashed boundary), or Higgs signal strength measurements (solid boundary). The blue region shows where the phenomenology is dominated by the twin Higgs. The vertical black lines show the proper lifetime cτ of the twin glueballs. The lifetime is primarily determined byv 1 and the mass of the glueballs, see eq. (22). In fig 6 we set the glueball mass to 70 GeV, such that decays of the SM-like Higgs to glueballs are completely
Figure 6 .
6The allowed parameter space for the process pp → H →bb in the plane of twin Higgs vevv 1 vs. tan β. The left two plots show the current constraints as well as the prediction for the number of events (green contours) after run 3 of the LHC for a heavy Higgs mass of 500 GeV (top) and 800 GeV (bottom). The right two plots show the expected constraints and predictions of number of events at the HL-LHC with 3000 fb −1 for a heavy Higgs mass of 700 GeV (top) and 1000 GeV (bottom). The gray shaded region with a solid black boundary shows the exclusion due to Higgs signal strength fits and the gray shaded region with the dashed black boundary shows the constraints from the B s → µµ decay. In the blue shaded region the number of displaced decays coming from the twin Higgs exceeds that of the heavy Higgs. The vertical black contours show the proper lifetime of twin glueballs with mass 70 GeV.
The second (heavy) Higgs doublet in the visible sector also provides interesting phenomenology in the form of displaced signatures at the LHC. This heavy Higgs can decay into the twin sector, in particular twin bottomonia and twin glueballs (which we assume to be the lightest states in the twin sector in this work) that can subsequently decay back to the visible sector through mixing with the Higgs bosons. This often leads to displaced signatures, in particular displaced b-jets. This can happen in regions of parameter space in which displaced signatures of the SM-like Higgs and the twin Higgs are suppressed or even completely absent. The corresponding parameter space is characterized by a heavy twin sector, where the production cross section of the twin Higgs is small and the SM-like Higgs is kinematically excluded from decaying to the twin sector. We have shown that in such a scenario the heavy Higgs boson can be light enough to be produced with a sizeable cross section and heavy enough to decay into the twin sector, thus offering the possibility to probe broader regions of parameter space with searches for displaced signatures.
Other flavor constraints, in particular from B meson oscillations are much less robust, as they depend strongly on the details of the Yukawa couplings of the second Higgs doublet to down type quarks λ d 2 , see[30].
Note, however, that twin hadrons that are produced from the heavy Higgs or the twin Higgs are much more energetic than those produced from the SM-like Higgs. Therefore, even a small number of displaced decays of the heavy Higgs or twin Higgs might be as prominent as a larger number of displaced SM-like Higgs decays. We do not study such a scenario in detail.
Note however that the displacement depends strongly on the assumed glueball mass, see eq.(22). Reducing the glueball mass slightly, say to 60 GeV, increases the lifetime of the glueballs by a factor of (7/6) 7 3.
ACKNOWLEDGEMENTSWe would like to thank Stefania Gori for useful discussions. The research of WA and BM is supported by the National Science Foundation under Grant No. PHY-1912719.
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| []
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[
"The Gor'kov and Melik-Barkhudarov correction to the critical field transition to Fulde-Ferrell-Larkin-Ovchinnikov states",
"The Gor'kov and Melik-Barkhudarov correction to the critical field transition to Fulde-Ferrell-Larkin-Ovchinnikov states"
]
| [
"Heron Caldas \nDepartamento de Ciências Naturais\nUniversidade Federal de São João Del Rei\nPraça Dom Helvécio 74, São João Del Rei36301-160MGBrazil\n",
"Qijin Chen \nShanghai Branch\nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n\nZhejiang Institute of Modern Physics\nDepartment of Physics\nZhejiang University\n310027HangzhouZhejiangChina\n"
]
| [
"Departamento de Ciências Naturais\nUniversidade Federal de São João Del Rei\nPraça Dom Helvécio 74, São João Del Rei36301-160MGBrazil",
"Shanghai Branch\nNational Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina",
"Zhejiang Institute of Modern Physics\nDepartment of Physics\nZhejiang University\n310027HangzhouZhejiangChina"
]
| []
| The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, characterized by Cooper pairs condensed at finitemomentum are, at the same time, exotic and elusive. It is partially due to the fact that the FFLO states allow superconductivity to survive even in strong magnetic fields. We calculate the effects of induced interactions at zero temperature, and find that the critical field at which the quantum phase transition to an FFLO state occurs is strongly suppressed in imbalanced Fermi gases. Possible impurity effects are also considered. * | 10.1002/andp.202000222 | [
"https://arxiv.org/pdf/1912.10215v1.pdf"
]
| 209,444,329 | 1912.10215 | 9774584c996d770eb61282152122cd3e7759b3e1 |
The Gor'kov and Melik-Barkhudarov correction to the critical field transition to Fulde-Ferrell-Larkin-Ovchinnikov states
Heron Caldas
Departamento de Ciências Naturais
Universidade Federal de São João Del Rei
Praça Dom Helvécio 74, São João Del Rei36301-160MGBrazil
Qijin Chen
Shanghai Branch
National Laboratory for Physical Sciences at Microscale and Department of Modern Physics
University of Science and Technology of China
201315ShanghaiChina
Zhejiang Institute of Modern Physics
Department of Physics
Zhejiang University
310027HangzhouZhejiangChina
The Gor'kov and Melik-Barkhudarov correction to the critical field transition to Fulde-Ferrell-Larkin-Ovchinnikov states
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, characterized by Cooper pairs condensed at finitemomentum are, at the same time, exotic and elusive. It is partially due to the fact that the FFLO states allow superconductivity to survive even in strong magnetic fields. We calculate the effects of induced interactions at zero temperature, and find that the critical field at which the quantum phase transition to an FFLO state occurs is strongly suppressed in imbalanced Fermi gases. Possible impurity effects are also considered. *
I. INTRODUCTION
Fermionic particles with different spins occupying states of momenta with equal size but in opposite directions close to their common Fermi surface form Cooper pairs, when subject to a pairing interaction. This is successfully explained by the BCS theory of superconductivity [1]. The presence of an imbalance between the two spin configurations prevents this mechanism, since there are now two Fermi surfaces that do not coincide so that pairing with zero total momentum for the BCS state is energetically unfavorable since the formation of Cooper pairs imply equal densities of the two spin species [2][3][4].
At the mean-field (MF) level, for small asymmetries between the two spin species, and at zero temperature T , the system persists as a superfluid. However, when the imbalance between the two Fermi surfaces is too large, superfluid pairing is broken apart and the system undergoes a quantum phase transition to the normal state. Therefore, for a given imbalance, there exists a lower threshold of pairing strength for the BCS pairing solution to exist. On the other hand, for a given interaction strength, there exists an upper bound for the imbalance before pairing is broken. The existence of such a transition at a critical value of the polarization was first realized by Clogston [5] and Chandrasekhar [6], who independently predicted the occurrence of a first-order phase transition from the superfluid to the normal state. This is known as the CC limit of superfluidity, and was originally proposed in the context of conventional superconductivity.
Stability analysis based on energetic considerations reveal that the mean-field BCS solutions at T = 0 is not stable in the presence of imbalance until the system enters the BEC regime where the gap and hence the condensation energy become large [7,8]. Indeed, the momentum of the minority fermions would have to be lifted up to match that of their majority partners, but the energy cost is larger than the condensation energy gain when the pairing gap is small. As a consequence, thermal smearing of the Fermi surfaces leads to possible intermediate temperature superfluidity, at both the mean-field level and with fluctuations included.
Experimental investigations with ultracold imbalanced Fermi gases, where the numbers of atoms in the two spin states are different, have shown that the first order transition between the superfluid at equal spin population and the imbalanced normal mixture brings about a phase separation between coexisting normal and superfluid phases [9,10]. Recent experiments using tomographic techniques, have found a sharp separation between a superfluid core and a partially polarized normal phase [11].
Thus, the exploration of a two-component Fermi gas with imbalanced populations is a current and active area of research in the field of ultracold atoms in both theory [2,[12][13][14] and experiment [15][16][17][18][19], which gives to this field the unprecedented opportunity for mimicking and simulating condensed matter systems [20,21].
To address pairing with a Fermi surface mismatch, Fulde and Ferrell (FF) [22], and Larkin and Ovchinnikov (LO) [23], in independent proposals, considered the possibility for the Cooper pairs in an s-wave superconductor in the presence of a Zeeman field to have a non-zero total momentum q, with a spatially modulated superfluid order parameter. In this intriguing pairing mechanism, superfluidity "perseveres" in the form of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with finite pair momenta. In the last 55 years many groups have tried to find the FFLO phase experimentally, and some have found only indirect signatures as, for example, in the heavy fermion superconductor CeCoIn 5 [24][25][26].
It has been shown recently that the FFLO states suffer unavoidable pairing fluctuations, and because of that FFLO superfluids cannot exist due to their intrinsic instability in three and two dimensions [27]. Besides this intrinsic instability, here we investigate other drawbacks of FFLO.
To obtain the correct superfluid transition temperature T c of the BEC-BCS crossover, there is another effect of particlehole fluctuations that affects the superfluid state. Namely, there is a change in the coupling of the interaction due to screening of the interspecies (or induced) interaction, known as the Gor'kov and Melik-Barkhudarov (GMB) cor-rection [28]. On the BEC side of the unitarity, fluctuations in the pairing channel are dominant, while the GMB fluctuations become weaker towards the BEC side and usually is taken as vanishing in this region due to the disappearance of the Fermi surface.
The effect of induced interactions was first considered by Gor'kov and Melik-Barkhudarov, who found that in a dilute three-dimensional (3D) spin-1/2 Fermi gas the (overestimated) MF transition temperature is suppressed by a factor (4e) 1/3 ≈ 2.2 [28]. The effects of the induced interactions on the transition temperature of a quasi-2D balanced Fermi gas of atoms in the BCS region is reduced by a factor ≈ 2.72 [29,30].
Quite generally, the calculation of the GMB correction has been restricted to the balanced case, with the Zeeman field h = (µ ↑ − µ ↓ )/2 = 0. Previous investigations with h = 0 were focused on the effects of the induced interactions on the tricritical point (P t , T t /T F ) of imbalanced Fermi gases in 3D [31] and 2D [30]. Here P t and T t are the polarization and the temperature of the tricritical point, and T F is the Fermi temperature. More sophisticated calculations involving self-consistent feedback effect from both particle-particle and particle-hole channels can be found in Ref. [32].
In this work we investigate at the mean-field level the effects of the GMB correction on the FFLO transition that may occur in Fermi gases with imbalanced spin populations. We study the continuous phase transition that is triggered by an increase in the chemical potential imbalance h, in homogeneous 3D systems. We find the GMB correction to the critical chemical potential imbalance h s responsible for the phase transitions from the partially polarized (PP) FFLO phase to a fully polarized (FP) normal state.
We also consider the effect of a finite lifetime of the quasiparticles states in the FP-PP instability. We show that short lifetimes necessarily decrease h s and, consequently, the predicted FFLO region of existence.
The paper is organized as follows. In Sec. II we calculate the generalized pair susceptibility, associated with the onset of the instability of the PP normal phase. In Sec. III we obtain the induced interactions and find its effects on the critical chemical potential imbalance h s which sets the transition to the FFLO phase. In Sec. IV we show how finite lifetime effects affect h s . Finally, we conclude in Sec. V.
II. THE INTERMEDIATE NORMAL-MIXED PHASE
As mentioned in Ref. [33], a very important, and still open issue, is the precise nature of the ground state in the regime h c < h < h s , where h c = ∆ 0 / √ 2 sets the CC transition. Let us now investigate the possible FFLO phase that may arise in the intermediate region. Suppose we are in the normal FP phase at some h > h s , and the "field" h is decreased until it enters the PP phase. In order to have a qualitative and quantitative description of this picture, usually one expands the action in fluctuations |∆ q | a la Landau, since the transition from the FP to the normal-mixed phase is continuous [34].
We then expand the action up to second order in the order parameter |∆ q |, and obtain
S eff = q,ω0 α(| q|, ω 0 )|∆ q | 2 + O |∆| 4 ,(1)
where α(| q|, ω 0 ) = 1 g − χ(| q|, ω 0 ) with g being the pairing interaction strength, and χ(| q|, ω 0 ) is the generalized pair susceptibility without feedback effect,
χ(| q|, ω 0 ) = k 1 − f (ξ k− q/2,↑ ) − f (ξ k+ q/2,↓ ) ξ k− q/2,↑ + ξ k+ q/2,↓ − ω 0 ,(2)
where ξ k,↑↓ = k 2 /2m−µ ↑↓ are the single particle dispersions and f (ξ k ) = 1/(e βξ k + 1) is the Fermi function with β ≡ 1/k B T . Evaluation of the equation above is straightforward. At zero temperature we find that χ(| q|, ω 0 ) is given by:
χ(| q|, 0) = N (0) 1 + ln 2ω c 2h − 1 2 ln 1 −q 2 + 1 q ln 1 +q 1 −q ,(3)
where ω c is an energy cutoff, N (0) = mk F 2π 2 is the density of states at the Fermi level,q ≡ v F q 2h is the dimensionless "measure" of the pair momentum, with q ≡ | q| and v F is the Fermi velocity.
The Thouless criterion for pairing instability, which corresponds to the divergence of the T-matrix,
T −1 = − 1 g + χ(| q|, 0) = 0,(4)
yields,
h ∆ 0 = e 2|1 +q| 1 +q 1 −q q−1 2q = 1 2 +q 2 12 + O(q 3 ),(5)
where ∆ 0 = 2ω c exp(−1/N (0)g) is the zero temperature BCS gap. Note that the exact expression for ∆ 0 is not crucial here, although this specific expression is appropriate only for the weak coupling BCS regime. At the same time, it has been known that possible FFLO phases only exist on the BCS side of unitarity [35,36]. We determine the critical reduced momentumq c by imposing an extremal condition on the pair susceptibility. Thus, extremizing χ(| q|, 0) with respect toq yields
2q c = ln 1 +q c 1 −q c .(6)
A numerical solution of the equation above givesq c = 0, and q c 1.2. However, the locus of continuous transitions may be determined from the value ofq c at which α(q c ) is both minimized and passes through zero, and this happens only for q c 1.2 [34] or, equivalently, at a wave-vector
q c 2.4h/v F . Theq =q c limit of χ(q c ) gives N (0) 1 − 0.59 + ln 2ωc 2h , so that α(q =q c ) = 0 is −1/g + χ(q =q c ) = −1/g + N (0) 0.41 + ln 2ωc 2h = 0,
which leads to h s 0.75∆ 0 , for the location of the FFLO transition, agreeing with the findings of Shimahara [37], Burkhardt and Rainer [38], and Combescot and Mora [39].
The critical h s in turn yields the magnitude of the wavevector q c 1.8∆ 0 /v F . These results also agree with the ones obtained in Ref. [37] by a variational approach for a threedimensional FF superconductor with a spherical symmetric Fermi surface. The FFLO window is then h c < h < h s , where the phase transition at h c = h CC ≈ 0.71∆ 0 , is of first order, and that at h s 0.75∆ 0 is of second order. The same results, and conclusions, are obtained if the calculations are performed with the interaction g replaced by the dimensionless parameter 1/k F a which is appropriate for a short range interaction [40], as it should be.
Note that in 2D the critical field and wave-vector are h s = ∆ 0 and q c 2∆ 0 /v F , respectively [37,41]. This means that the FFLO phase is enhanced in 2D due to the two-dimensional structure of the Fermi surface.
It should be mentioned that the original Clogston derivation equates the free energy of the superfluid state at zero-field (i.e., y = 0) with that of a polarized normal state at the threshold y c , both at T = 0. This approach is expected to be valid for the small ∆ 0 case in the perturbative sense. However, we argue that the balanced case and the imbalanced case are really distinct and cannot connect to each other continuously. This can be told from the fact that in the BCS regime, an arbitrarily small but nonzero population imbalance is sufficient to destroy superfluidity at precisely T = 0 in the 3D homogeneous case (when stability is taken into account) [7]. For a finite ∆ 0 , the "magnetic field" h would have to jump from 0 of the balanced case to a value comparable to ∆ 0 , implying that h should not be treated perturbatively. Furthermore, there is no guarantee that the normal state in Clogston's approach is a solution of the BCS gap equation in the zero gap limit. To check the CC limit, we calculate for a 3D homogeneous Fermi gas the gap ∆ 0 in the balanced case at zero T and the field h in the imbalanced case when the mean-field T c approaches 0, both as a function of pairing strength. The result is shown in Fig. 1, which indicates that the exact mean-field solution yields h c /∆ GMB 0 = 0.5 in the BCS regime, substantially different from 1/ √ 2 given by CC, and this ratio increases to about 0.733 at unitarity. This result suggests that exact calculation is needed in order to obtain quantitatively accurate value for h c . This corresponds to the q = 0 limit of Eq. (5) and is not stable. The difference between this result and that of CC can be attributed likely to the possibility that the CC normal state does not satisfy the Thouless criterion while the present case does.
In the inset of Fig. 1, we show the stable LOFF phase at the mean-field level, as the yellow shaded region. The upper boundary is given by the zero gap solution with a finite q vec- tor, which separates the LOFF phase from the normal Fermi gases. The lower phase boundary is given by the instability condition of the LOFF phase against phase separation. Both boundary lines were taken from Ref. [35]. Next to but on the right side of this boundary are phase separated states. It is clear that the Sarma mean-field T c curve line lies completely within the stable LOFF phase, in agreement with the fact that the states along this curve are unstable against LOFF. Interestingly, it turns out that, in the BCS limit, the ratio h/∆ 0 along the lower boundary is close to 1/ √ 2, in agreement with Ref. [47]. Meanwhile, the ratio along the upper boundary is close to 0.75. This leaves us with roughly the same LOFF window of 0.71 < h/∆ 0 < 0.75 in the absence of the induced interactions.
-5 -4 -3 -2 -1 0 1/k F a 0 0.2 0.4 0.6 0.8 h(T * LOFF =0)/E F h LOFFstability /E F -3 0 1/k F a 0 1 p LOFF instability T * LOFF = 0 T c MF = 0 h(T c MF =0)/∆ 0 ∆ 0 / E F h (
III. EFFECTS OF THE INDUCED INTERACTION ON THE FFLO WINDOW
The induced interaction was obtained originally by GMB in the BCS limit by the second-order perturbation [28]. For a scattering process with p 1 + p 2 → p 3 + p 4 , the induced interaction for the diagram in Fig. 2 is expressed as
U ind (p 1 , p 4 ) = −g 2 χ ph (p 1 − p 4 ),(7)
where p i = ( k i , ω li ) is a four vector in the space of wavevector k and fermion Matsubara frequency ω l = (2l + 1)π/β. Including the induced interaction, the effective pairing inter-action between atoms with different spins is given by
U eff (p 1 , p 4 ) ≡ U eff = g + U ind (p 1 , p 4 ) (8) = g − g 2 χ ph (p 1 − p 4 ).
The polarization function χ ph (p ) is given by
χ ph (p ) = p G 0b (p)G 0a (p + p ) = d 3 k (2π) 3 f b k − f a k+ q iΩ l + ξ k,a − ξ k+ q,b ,(9)
where f σ k ≡ f (ξ k,σ ), p = ( q, Ω l ), Ω l = 2lπ/β is the Matsubara frequency of a boson, and G 0σ (p) = 1/(iω l − ξ kσ ) is the Matsubara Green's function of a non-interacting Fermi gas. This means that U eff is a function of momentum and frequency. The static polarization function is then,
χ ph (q, h) = − 2m (2π) 2 dk k 2 2qk f ↓ k ln q 2 − 4mh + 2kq q 2 − 4mh − 2kq + f ↑ k ln q 2 + 4mh + 2kq q 2 + 4mh − 2kq ,(10)
where q ≡ | q|. The above expression is usually computed in the zero temperature limit, with f ↓,↑
k → Θ(k ↓,↑ F − k), where Θ(x)
is the step function, such that the induced correction to the coupling g is a (temperature independent) constant.
χ(q, h) ≡ χ(q, h) ↓ + χ(q, h) ↑ (11) = − m (2π) 2 q k ↓ F 0 dk k ln q 2 − 4mh + 2kq q 2 − 4mh − 2kq − m (2π) 2 q k ↑ F 0 dk k ln q 2 + 4mh + 2kq q 2 + 4mh − 2kq .
Equation (11) shows that the static polarization function in the case of a spin imbalanced Fermi gas separates into contributions from the spin-down and the spin-up like susceptibilities. The integration in k gives
χ(q, h) ↓ = − m 8π 2 q ln q 2 − 4mh + 2qk ↓ F q 2 − 4mh − 2qk ↓ F k ↓ F 2 − q 2 − 4mh 2q 2 + k ↓ F q 2 − 4mh q ,(12)χ(q, h) ↑ = − m 8π 2 q ln q 2 + 4mh + 2qk ↑ F q 2 + 4mh − 2qk ↑ F k ↑ F 2 − q 2 + 4mh 2q 2 + k ↑ F q 2 + 4mh q .(13)
The equations above can be put in a more convenient form,
χ(x, y) ↓ = − N (0) 2 1 2 1 − y 1 − y x 2 − ln √ 1 − y + y 2x − x √ 1 − y − y 2x + x 1 4x 1 − y − x 2 1 − y 2x 2 2 (14) = −N (0)L(x, y) ↓ ,
and
χ(x, y) ↑ = − N (0) 2 1 2 1 + y 1 + y x 2 − ln √ 1 + y − y 2x − x √ 1 + y + y 2x + x 1 4x 1 + y − x 2 1 + y 2x 2 2 (15) = −N (0)L(x, y) ↑ ,
where N (0) = mk F 2π 2 , x ≡ q 2k F , and y ≡ h µ . This allows us to write the polarization function of an imbalanced Fermi gas as
χ ph (x, y) = −N (0)L(x, y),(16)
where L(x, y) ≡ (L(x, y) ↓ + L(x, y) ↑ )/2 is the generalized Lindhard function.
Notice that in the y → 0 limit, k ↓ F = k ↑ F = k F , such that χ(x, 0) ↓ = χ(x, 0) ↑ = χ(x) and we obtain the well-known (balanced) result
χ(x) = − m 4π 2 q k F q − ln q 2 − 2qk F q 2 + 2qk F k F 2 − q 2 4 = −N (0)L(x),(17)
where L(x) is the standard Lindhard function,
L(x) = 1 2 − 1 4x (1 − x 2 ) ln 1 − x 1 + x .(18)
In the scattering process the conservation of total momentum implies that k 1 + k 2 = k 3 + k 4 , with k 1 = − k 2 and k 3 = − k 4 . q is equal to the magnitude of k 1 + k 3 ,
so q = ( k 1 + k 3 ).( k 1 + k 3 ) = k 2 1 + k 2 3 + 2 k 1 . k 3 = k 2 1 + k 2 3 + 2| k 1 || k 3 | cos φ,
where φ is the angle between k 1 and k 3 . Since both particles are at the Fermi surface, | k 1 | = | k 3 | = k F = √ 2M µ, thus, q = k F 2(1 + cos φ), and consequently x = 2(1 + cos φ)/2, which sets 0 ≤ x ≤ 1.
The s-wave part of the effective interaction is approximated by averaging the polarization function χ ph (q) over the Fermi sphere, which means an average of the angle φ [31,32,[42][43][44][45][46],
χ ph (x, y) = 1 2 1 −1 d cos φ χ ph (x, y) = −N (0) 1 2 1 −1 d cos φ L(x, y) ≡ −N (0)L(y),(19)
where we have made use of Eq. (16). Taking into account the GMB correction, the divergence of the T-matrix in Eq. (4) is now given by
T −1 = − 1 g − χ ph (x, y) + χ(| q|, 0) = 0,(20)
which can be obtained by replacing g in Eq. (4) by U ef f , as given in Eq. (8). This expression has been shown to be correct when the more complicate T -matrix in the particle-hole channel is included self-consistently [32]. This yields a GMB
h s ∆ 0 = h s ∆ 0 MF e −L(ys) ,(21)
where y s = h s /µ, and hs ∆0 MF 0.75 is the MF result without the GMB corrections. In Fig. 3 we show how the averaged functionL(y) depends on 0 ≤ y ≤ 1. As y increases from 0 to 1,L(y) increases by over a factor of 2, from about 0.65 to 1.6.
It is well known that the zero temperature BCS pairing gap (at y = 0) is modified due to the particle-hole channel effect (or GMB correction) as [32,42]
∆ GMB 0 = ∆ 0 (4e) 1/3 = 8 e 2 1 (4e) 1/3 µe −π/2k F |a| = 2 e 7/3 µe −π/2k F |a| .(22)
In order to find the appropriate y for consistently evaluating Eq. (21), we take the expression for the GMB gap at unitarity, 1/k F a = 0, and obtain
∆ GMB 0 = 2 e 7/3 µ 0.49µ(23)
for y = 0. Note that this is a rough estimate, since the BCSlike solution Eq. (22) is a good approximation only in the BCS regime. Nonetheless, it is very close to the more complete solution of ∆ GMB 0 = 0.42E F = 0.50µ with µ = 0.837E F , calculated with the full particle-hole T -matrix included at the G 0 G 0 level [32]. According to Eq.
which agrees with Eq. (24). On the other hand, with the shifted interaction strength, the CC limit is modified to
h c ∆ 0 = 1 (4e) 1/3 1 √ 2 = 0.3191.(26)
Combining Eq. (24) and Eq. (26), we conclude that the screening of the medium (i.e., the induced interactions) has shrunk the FFLO window to 0.32 ≤ h/∆ 0 ≤ 0.34.
IV. FINITE LIFETIME EFFECTS
Besides the GMB corrections that, as we have seen in Section III, narrow the FFLO window, there is another effect that may also act against this state, which are the lifetime effects. In quasi-one-dimensional organic superconductors, for instance, the issue of lifetime effects arise from non-magnetic impurities or defects [48].
In this section we show that the LOFF window may be affected by impurity effects. Impurities lead to a spectral broadening γ of the fermions. In Fig. 4, we compare with ratio h/∆ 0 as a function ofq for both the clean limit (maroon curve) and the dirty case with γ/h = 1 (blue dashed line). The clean case has a maximum value forq =q c 1.2, which gives h/∆ 0 = h s /∆ 0 0.75, as obtained by the analytical calculations in Section II. In the dirty case, the maximum has shifted toward lowerq, and the maximum ratio h s /∆ 0 has decreased significantly. This inevitably narrows the LOFF window. When this ratio drops below 1/ √ 2, the LOFF window will be gone and thus the LOFF phase will disappear. Detailed analysis is given below.
Considering the finite lifetime of the quasi-particle states in the momentum representation [49], the real part of the particle-particle dynamic susceptibility can be written as,
Reχ(| q|, γ) = N (0) 1 + ln 2ω c 2h + 1 4q (1 −q) ln (1 −q) 2 + γ 2 4h 2 − 1 4q (1 +q) ln (1 +q) 2 + γ 2 4h 2 ,(27)
where γ = τ −1 is the inverse of the lifetime of a quasiparticle q-state in the normal phase. This approach is formally close to that used to investigate the effect of non-magnetic impurities in one dimensional imbalanced Fermi gases [50], and in two [51] and three [52]
h ∆ 0 = e 2 (1 −q) 2 + γ 2 4h 2 1−q 4q (1 +q) 2 + γ 2 4h 2 −( 1+q 4q ) .(28)
Notice that in the limit γ → 0 in Eqs. (27) and (28), the standard results in Eqs. (3) and (5) are recovered.
In Fig. 4, it is depicted the numerical solutions of h/∆ 0 as a function ofq from Eqs. (5) and (28), which shows that for γ = 0, solid curve, h/∆ 0 has its maximum value forq =q c 1.2, which gives h/∆ 0 = h s /∆ 0 0.75, as obtained by the analytical calculations. However, for finite lifetime, dashed curve, the maximum value of h/∆ 0 is greatly reduced.
The critical reduced momentumq c instead of being a solution of Eq. (6) is now given by the solution of in agreement with Fig. 4. However, this value is beyond the critical valueγ c = 0.3, which gives h s /∆ 0 0.71 = h c /∆ 0 , closing the FFLO window. This means that for this critical value of γ the system undergoes a first-order quantum phase transition from the BCS to the polarized normal phase. Conversely, for infinite life time (γ = 0) the FFLO window remains opened with the "unperturbed" limits h c < h < h s .
V. CONCLUSIONS AND OUTLOOK
In summary, we have investigated in homogeneous 3D systems the GMB correction to the chemical potential difference h/∆ 0 , responsible for the transition to the FFLO phase. We found at the mean-field level that the window for the FFLO phase to exist has been reduced by a factor of (4e) −1/3 . This shall make the evasive FFLO states even more difficult to find experimentally.
We have also considered the presence of (non-magnetic) impurities or defects, in which case a finite lifetime τ = 1/γ of the quasi-particle excitations may be considered. As a consequence, for strong disorder there is a reduction of the critical field h s of the continuous phase transition between the FFLO and the normal phase.
FIG. 1 .
1Gap ∆0 (red, as labeled) and h along different lines, all in units of EF , as well as their ratio h(T MF c = 0)/∆0 (blue dashed line), as a function of 1/kF a for a 3D homogeneous Fermi gas. The calculation was done at the mean-field level without the particle-hole channel, which does not affect the ratio. Shown in the inset is the phase diagram of the stable LOFF phase. The upper boundary (green curve) is given by T * LOFF = T LOFF c (∆ = 0), and the lower boundary (magenta curve) is given by the instability condition against phase separation. The Sarma T MF c = 0 line (black dashed) lies within the LOFF phase. The h value along these lines are shown in the main figure with the same color coding and line shapes.
FIG. 2 .
2The lowest-order diagram representing the induced interaction U ind (p1, p4). Arrowed and dashed lines describe fermionic propagators and the coupling g between the fermionic atoms.
∆
GMB (y) = ∆(y)e −L(y) , with ∆ 0 ≡ ∆(y = 0, T = 0) and ∆ GMB 0 = ∆ GMB (y = 0, T = 0). This amounts to
2
(22), when taking into account the GMB correction, h c /∆ 0 = 0.3455 at unitarity.In Eq. (21), we approximate y s ≡ h s /µ by y c inL(y s ),
FIG. 3 .
3Behavior of the averaged L as a function of y = h/µ.
c ) 2 +γ 2 (1 −q c ) 2 +γ 2 , whereγ ≡ γ/2h. Forγ = 1, for instance, the numerical solution of the equation above givesq c = 0, andq c 1.01. This leads to h s 0.61∆ 0 for the location of the FFLO transition, FIG. 4. The ratio h/∆0 as a function of q. h/∆0. The solid curve is for γ = 0, and attains its maximum value 0.75, atq =qc 1.2. The dashed curve is for γ/h = 1 and shows a significant reduction of the maximum value of h/∆0.
dimensional FFLO superconductors. With Reχ(| q|, γ) from Eq. (27) the divergence of the Tmatrix now gives
ACKNOWLEDGMENTSHC wish to thank CNPq and FAPEMIG for partial financial support, and QC is supported by NSF of China under grant No. 11774309.
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| []
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[
"On saturation of Berge hypergraphs",
"On saturation of Berge hypergraphs"
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| [
"Dániel Gerbner \nAlfréd Rényi Institute of Mathematics\n\n",
"Balázs Patkós \nAlfréd Rényi Institute of Mathematics\n\n\nMoscow Institute of Physics and Technology\n\n",
"Zsolt Tuza \nAlfréd Rényi Institute of Mathematics\n\n\nDepartment of Computer Science and Systems Technology\nUniversity of Pannonia\n\n",
"Máté Vizer \nAlfréd Rényi Institute of Mathematics\n\n"
]
| [
"Alfréd Rényi Institute of Mathematics\n",
"Alfréd Rényi Institute of Mathematics\n",
"Moscow Institute of Physics and Technology\n",
"Alfréd Rényi Institute of Mathematics\n",
"Department of Computer Science and Systems Technology\nUniversity of Pannonia\n",
"Alfréd Rényi Institute of Mathematics\n"
]
| []
| A hypergraph H = ( | 10.1016/j.ejc.2021.103477 | [
"https://arxiv.org/pdf/2103.08437v1.pdf"
]
| 232,233,566 | 2103.08437 | 08256fdfefd251d7142b479689593b27d6d60ae2 |
On saturation of Berge hypergraphs
15 Mar 2021
Dániel Gerbner
Alfréd Rényi Institute of Mathematics
Balázs Patkós
Alfréd Rényi Institute of Mathematics
Moscow Institute of Physics and Technology
Zsolt Tuza
Alfréd Rényi Institute of Mathematics
Department of Computer Science and Systems Technology
University of Pannonia
Máté Vizer
Alfréd Rényi Institute of Mathematics
On saturation of Berge hypergraphs
15 Mar 2021arXiv:2103.08437v1 [math.CO]
A hypergraph H = (
Introduction
Given a family F of (hyper)graphs, we say that a (hyper)graph G is F -free if G does not contain any member of F as a subhypergraph. The obvious question is how large an F -free (hyper)graph can be, i.e. what is the maximum number ex(n, F ) of (hyper)edges in an F -free n-vertex (hyper)graph is called the extremal/Turán problem. A natural counterpart to this well-studied problem is the so-called saturation problem. We say that G is F -saturated if G is F -free, but adding any (hyper)edge to G creates a member of F . The question is how small an F -saturated (hyper)graph can be, i.e. what is the minimum number sat(n, F ) of (hyper)edges in an F -saturated n-vertex (hyper)graph.
In the graph case, the study of saturation number was initiated by Erdős, Hajnal, and Moon [5]. Their theorem on complete graphs was generalized to complete uniform hypergraphs by Bollobás [2]. Kászonyi and Tuza [9] showed that for any family F of graphs, we have sat(n, F ) = O(n). For hypergraphs, Pikhurko [10] proved the analogous result that for any family F of r-uniform hypergraphs, he proved that we have sat(n, F ) = O(n r−1 ).
For some further types of saturation ("strongly F -saturated" and "weakly F -saturated" hypergraphs) the exact exponent of n was determined in [11] for every forbidden hypergraph F .
In this paper, we consider some special families of hypergraphs. We say that a hypergraph H is a Berge copy of a graph F (in short: H is a Berge-F ) if V (F ) ⊂ V (H) and there is a bijection f : E(F ) → E(H) such that for any e ∈ E(F ) we have e ⊂ f (e). We say that F is a core graph of H. Note that there might be multiple core graphs of a Berge-F hypergraph and F might be the core graph of multiple Berge-F hypergraphs.
Berge hypergraphs were introduced by Gerbner and Palmer [8], extending the notion of hypergraph cycles in Berge's definition [1]. They studied the largest number of hyperedges in Berge-F -free hypergraphs (and also the largest total size, i.e. the sum of the sizes of the hyperedges). English, Graber, Kirkpatrick, Methuku and Sullivan [3] considered the saturation problem for Berge hypergraphs. They conjectured that sat r (n, Berge-F ) = O(n) holds for any r and F , and proved it for several classes of graphs. Here and throughout the paper the parameter r in the index denotes that we consider r-uniform hypergraphs, and we will denote sat r (n, Berge-F ) by sat r (n, F ) for brevity. The conjecture was proved for 3 ≤ r ≤ 5 and any F in [4]. In this paper we gather some further results that support the conjecture.
English, Gerbner, Methuku and Tait [4] extended this conjecture to hypergraph-based Berge hypergraphs. Analogously to the graph-based case, we say that a hypergraph H is a Berge copy of a hypergraph F (in short: H is a Berge-F ) if V (F ) ⊂ V (H) and there is a bijection f : E(F ) → E(H) such that for any e ∈ E(F ) we have e ⊂ f (e). We say that F is a core hypergraph of H. The conjecture in this case states that if F is a u-uniform hypergraph, then sat r (n, F ) = O(n u−1 ).
For a hypergraph H = (V (H), E(H)) and a family of hypergraphs F we say that H is F -oversaturated if for any hyperedge h ⊂ V (H) that is not in H, there is a copy of a hypergraph F ∈ F that consists of h and |E(F )| − 1 hyperedges in E(H). Let osat r (n, F ) denote the smallest number of hyperedges in an F -oversaturated r-uniform hypergraph on n vertices. Proposition 1.1. For any u-uniform hypergraph F and any r > u, we have osat r (n, F ) = O(n u−1 ). Moreover, there is an r-uniform hypergraph H with O(n u−1 ) hyperedges such that adding any hyperedge to H creates a Berge-F such that its core hypergraph F 0 (which is a copy of F ) is not a core hypergraph of any Berge-F in H.
We remark that in the case u = 2, the linearity of osat r (n, F ) follows from either of the next two theorems, as they imply sat r (n, K k ) = O(n). Indeed, if v is the number of vertices of any graph in F , then any Berge-K v -saturated hypergraph is obviously Berge-Foversaturated.
Theorem 1.2. For any r, s ≥ 2 and any sequence of integers 1 ≤ k 1 ≤ k 2 ≤ · · · ≤ k s+1 we have sat r (n, K k 1 ,k 2 ,...,k s+1 ) = O(n).
Let F be a fixed graph on v vertices with degree sequence d 1 ≤ d 2 ≤ · · · ≤ d v . Set δ := d 2 − 1. We say that F is of type I if there exist vertices u 1 , u 2 with d F (u 1 ) = d 1 , d F (u 2 ) = d 2 that are joined with an edge. Otherwise F is called of type II. Observe that any regular graph is of type I.
Proofs
Proof of Proposition 1.1. We define V (H) as the disjoint union of a set R of size r − u and a set L of size n − r + u. We take an F -saturated u-uniform hypergraph G with vertex set L that contains O(n u−1 ) hyperedges. Such a hypergraph exists by the celebrated result of Pikhurko [10]. As another alternative, one may take an oversaturated hypergraph with O(n u−1 ) hyperedges, whose existence is guaranteed by [11,Theorem 1]. Then we let the hyperedges h of H be the r-sets with the property that h ∩ L is a hyperedge of G or has at most u − 1 vertices from L.
Obviously H has O(n u−1 ) hyeredges. Clearly we have that every r-set h that is not a hyperedge of H contains a u-element subset e of L that is not a hyperedge of G. Then e creates a copy of F . We let f (e) = h, and for each other edge e ′ of that copy of F , we let f (e ′ ) = e ′ ∪ R. This shows that this copy of F is the core of a Berge-F .
Proof of Theorem 1.2. We consider two cases according to how large the uniformity r is compared to the sum of class sizes k 1 , k 2 , . . . , k s+1 . We set N := s i=1 k i − 1. For brevity, we write K for K k 1 ,k 2 ,...,k s+1 .
Case I. r ≤ s+1 i=1 k i − 3 Let (C, B 1 , B 2 , . . . , B m , R) be a partition of [n] with |C| = N, |B i | = k s+1 for all i ≤ m where m = ⌊ n−N k s+1 ⌋, and |R| ≡ n − N (mod k s+1 ), |R| < k s+1 . Consider the family G = {A ∈ [n] r : A ⊂ C ∪ B i for some i}. Observe that G is Berge-K-free.
Indeed, a copy of a Berge-K must contain a vertex v in the smallest s classes of the core from outside C. But then, if v ∈ B i , either the whole copy is in C ∪ B i or C must contain all classes of the core of the copy. As none of these are possible, G is indeed Berge-K-free.
Next observe that adding any r-set G to G that contains two vertices u and v from different This shows that the additional hyperedges of any K-saturated family that contains G are subsets of C ∪ B i ∪ R for some i, and hence there is only a linear number of them. As G also contains a linear number of hyperedges, the total size of such K-saturated families is O(n).
B i s, say u ∈ B i , v ∈ B j (i = j), would create a copy of a Berge-K. Indeed, the assumption r ≤ s+1 i=1 k i − 3 ensures that there exist bijections f i : C∪B i 2 → C∪B i r with e ⊂ fCase II. r ≥ s+1 i=1 k i − 2 Once again, we define a partition (C, B 1 , B 2 , . . . , B m , R) of [n] with |C| = N, but now with |B i | = r − 2 and |R| ≡ n − N (mod r − 2), |R| < r − 2. Let x 1 , x 2 , . .
. , x N be the elements of C, let e 1 , e 2 , . . . , e ( N 2 ) be the edges of the complete graph on C, and finally let π 1 , π 2 , . . . , π ( N 2 ) be permutations of C such that the endvertices of e i are the values π i (j), π i (j + 1) for some j.
Then let us define the family G as
G = m i=1 {{x π h (j) , x π h (j+1) } ∪ B i : 1 ≤ j ≤ N, h ≡ i (mod N 2 )}.
First, we claim that G is Berge-K-free. Indeed, there are only N vertices with degree at least N + 1.
Next, observe that if we add a family F to G that contains a Berge-
S k s+1 (r−2) (a star with k s+1 (r − 2) leaves) with core completely in ∪ m i=1 B i , then G ∪ F contains a Berge-K. Indeed,
if v is the center of the star, then C ∪ {v} plays the role of the smallest classes of K, and k s+1 leaves that belong to distinct B i s can play the role of the largest class of K. Here, we use the facts that every edge in C 2 is contained in an unbounded number of hyperedges of G as n tends to infinity and that for any vertex u ∈ ∪ m i=1 B i , G contains a Berge-star with center u and core C ∪ {u}; and if u and u ′ belong to different B i 's, then the hyperedges of these Berge-stars are distinct.
Let F be such that F ∪ G is Berge-K-free. Then by the above,
F ′ = {F ∩ (∪ m i=1 B i ) : F ∈ F } is Berge-S k s+1 (r−2) -free. Note that |F | ≤ 2 |C∪R| |F ′ |, thus showing that |F ′ | = O(n)
finishes the proof. It is well-known that forbidding a Berge-star (or any Berge-tree) results in O(n) hyperedges, but for sake of completeness we include a proof for stars.
Observe that F ′ being Berge-S k s+1 (r−2) -free is equivalent to the condition that for every x ∈ ∪ m i=1 B i the family {F \ {x} : x ∈ F ∈ F ′ } is not disjointly k s+1 (r − 2)-representable, i.e. there do not exist y 1 , y 2 , . . . , y k s+1 (r−2) and sets F 1 , F 2 , . . . ,
F k s+1 (r−2) ∈ F ′ x with y α ∈ F α \ ∪ k s+1 (r−2)
j=1,j =α F j . By a well-known result of Frankl and Pach [7] if all sets of a family H with this property have size at most r, then |H| is bounded by a constant depending only on r and k s+1 (r − 2). That is, |F ′ x | is bounded by the same constant independently of x, and therefore the size of F ′ , and thus the size of G is linear. We obtained that any k-saturated family G ′ with G ⊂ G ′ has O(n) hyperedges.
Proposition 2.1. Let F be a graph with no isolated vertex and with an isolated edge (u 1 , u 2 ). Then for any r ≥ 3 we have sat r (n, F ) = O(n).
Proof. Let U be a set of size n. Let v denote the number of vertices of F , let F ′ be the graph obtained from F by removing the edge (u 1 , u 2 ) and let C be a (v − 2)-subset of U. Suppose first r ≤ v − 1, and let G 0 be a Berge copy of F ′ with core C and G 0 ⊆ G C,1 ⊆ U r , where G C,1 is the set of r-sets that contain at most one vertex from U \ C. Note that G C,1 contains a linear number of r-subsets. Then let G be an r-graph with G 0 ⊆ G ⊆ G C,1 such that any H ∈ G C,1 \ G creates a Berge copy of F with G. Then G has linearly many hyperedges and is clearly F -saturated since if G contains at least two vertices from U \ C, then G can play the role of (u 1 , u 2 ) and together with the Berge copy of F ′ they form a Berge-F .
If r ≥ v, then any G with e(F ) − 1 r-subsets sharing v − 2 common elements (denote their set by C) is F -saturated. Indeed, any additional r-set G contains at least 2 vertices not in C, so those two vertices can play the role of u 1 and u 2 , G can play the role of the edge (u 1 , u 2 ), and the r-sets of G form a Berge copy of F ′ with core C.
Observe that if F is of type I, then it cannot contain isolated vertices, and since graphs with an isolated edge are covered by Proposition 2.1, we may and will assume that d 2 − 1 = δ ≥ 1 holds.
Proof of Theorem 1.3. Let F be a graph of type I on v vertices and let u 1 , u 2 be a pair of vertices of F showing the type I property. Set d := |N(u 1 ) ∩ N(u 2 )| and let F ′ denote the subgraph of F on N(u 1 ) ∩ N(u 2 ) spanned by the edges incident to u 1 or u 2 with the edge (u 1 , u 2 ) removed. Our strategy to prove the theorem is to construct a Berge-F -free r-graph G with O(n) hyperedges such that any F -saturated r-graph G ′ ⊃ G contains at most a linear number of extra hyperedges.
Let us say that G is F -good if its vertex set V can be partitioned into V = C ∪ B 1 ∪ B 2 ∪ · · · ∪ B m ∪ R such that |C| = v − 2, all B i 's have equal size b at most r, |R| < b and the following properties hold:
1. every hyperedge of G not contained in C is of the form A ∪ B i for some i = 1, 2, . . . , m with A ⊂ C,
2. every vertex u ∈ ∪ m i=1 B i has degree δ in G,
3. for every 1 ≤ i < j ≤ m and y ∈ B i , y ′ ∈ B j , the sub-r-graph {G ∈ G : y ∈ G} ∪ {G ∈ G : y ′ ∈ G} contains a Berge-F ′ with y, y ′ being the only vertices of the core not in C and y, y ′ playing the role of u 1 , u 2 ,
4. for any 1 ≤ i < j ≤ m there exist v−2 2 hyperedges G 1 , G 2 , . . . , G ( v−2 2 ) ∈ G that are disjoint from B i ∪ B j and if e 1 , e 2 , . . . , e ( v−2
2 ) is an enumeration of the edges of the complete graph on C, then e h ⊂ G h for all h = 1, 2, . . . , v−2 2 , i.e., these hyperedges form a Berge-K v−2 with core C. Claim 2.2. If G is F -good, then G is Berge-F -free and any F -saturated supergraph G ′ of G contains at most a linear number of extra edges compared to G.
Proof of Claim. Observe first that G is Berge-F -free as the core of a copy of a Berge-F should contain at least two vertices not in C, both of degree δ < d 2 .
Next, we claim that for any hyperedge H meeting two distinct B's, say B i and B j , the r-graph G ∪ {H} contains a Berge-F . Indeed, let y ∈ B i ∩ H, y ′ ∈ B j ∩ H. Then by item 3 of the F -good property, y can play the role of u 1 and y ′ can play the role of u 2 , H can play the role of the edge (u 1 u 2 ), and item 4 of the F -good property ensures that the other vertices of C can play the role of the rest of the core of F .
Finally, let G ′ be any F -saturated r-graph containing G. Then by the above, any hyperedge in G ′ \ G meets at most one B i , and thus is of the form P ∪ Q with P ⊂ C ∪ R, Q ⊂ B i for some i. The number of such sets is at most 2 b 2 v−2+b m = O(n). Claim 2.3. For any type I graph F on v ≥ 7 vertices with δ > 0 and any integer r ≥ 6 there exists an F -good r-graph G with O(n) hyperedges.
Proof of Claim. We fix a set D ⊂ C of size d.
Case I. r ≤ v − 4
Then putting all r-subsets of C into G ensures item 4 of the F -good property. We set b = r−2, so all further sets will meet C in 2 vertices. Observe that
d 1 −1−d+δ−d ≤ v−2−d is equivalent to 2(d 1 − 1 − d) + δ − d 1 + 1 ≤ v − 2 − d. Let D = {x 1 , x 2 , . .
. , x d }, and y 1,1 , y 1,2 , y 2,1 , y 2,2 , . . . ,
y d 1 −1−d,1 , y d 1 −1−d,2 , z 1 , z 2 , . . . , z δ−d 1 +1 be distinct vertices in C. Note that d 1 − 1 ≤ δ implies that δ − d 1 + 1 is non-negative. Then let G 0 := {{x ℓ , x ℓ+1 } : 1 ≤ ℓ ≤ d}∪{{y ℓ,1 , y ℓ,2 } : 1 ≤ ℓ ≤ d 1 −1−d}∪{{z ℓ , z ℓ+1 } : 1 ≤ ℓ ≤ δ−d 1 +1},
where addition is always modulo the underlying set, i.e., G 0 consists of two cycles and a matching. Let us put all sets of the form A ∪ B h with A ∈ G 0 and 1 ≤ h ≤ m into G. Then items 1 and 2 of the F -good property are satisfied, thus we need to check item 3.
For any 1 ≤ i < j ≤ m, D plays the role of N(u 1 ) ∩ N(u 2 ) and the hyperedges B i ∪ {x ℓ , x ℓ+1 }, B j ∪ {x ℓ , x ℓ+1 } play the role of the edges connecting u 1 , u 2 to vertices of D, respectively. Vertices y ℓ,1 play the role of vertices in N(u 1 ) \ (N(u 2 ) ∪ {u 2 }), while vertices y ℓ,2 with ℓ = 1, 2, . . . , d 1 − 1 − d and z ℓ ′ with ℓ ′ = 1, 2, . . . , δ − d 1 + 1 play the role of N(u 2 ) \ (N(u 1 ) ∪ {u 1 }). The use of hyperedges as edges is straightforward.
Case II. r > v − 4 Then we set b = r − (v − 4) and thus every hyperedge meets C in c := v − 4 = |C| − 2 vertices. Consequently, |R| is the residue of n − v + 2 modulo b. By v ≥ 7, we obtain c ≥ 3. Let e 1 , e 2 , . . . , e ( v−2 2 ) be an enumeration of the edges of the complete graph on C. Then for any 1 ≤ h ≤ m, we will put a hyperedge of the form
A 1,h ∪ B h with e α ⊂ A 1,h ⊂ C where α ≡ h (mod v−2
2 ). As n tends to infinity, so does m, and this will ensure item 4 of the F -good property.
Suppose first d > 0 and observe that (
d 1 − 1) + δ ≤ v − 2 − d, as d 1 − 1 is the size of N(u 1 ) \ (N(u 2 ) ∪ {u 2 }) and δ is the size of N(u 2 ) \ (N(u 1 ) ∪ {u 1 }). For any 1 ≤ h ≤ m,
we define A 1,h , A 2,h , . . . , A δ,h and put A ℓ,h ∪ B h into G for all 1 ≤ ℓ ≤ δ as follows. Let x 1 , x 2 , . . . , x d be the elements of D, and A 1,h be a (v − 4)-element set containing x 1 and e α (with α defined in the previous paragraph), and for 2 ≤ ℓ ≤ d let A ℓ,h be an arbitrary (v − 4)-element subset of C containing x 1 , x ℓ . (We need v − 4 ≥ 3 to be able to make the choice of A 1,h .) Finally, let A d+1,h , A d+2,h , . . . , A δ,h be distinct (v − 4)-element subsets of C \ {x 1 }. There are v − 3 such subsets, each missing one element of C \ {x 1 }. We take them one by one, starting with those that miss an element from D \ {x 1 }. The choice of A 1,h verifies item 4 of the F -good property and items 1 and 2 hold by definition.
To see item 3, let 1 ≤ i < j ≤ m. We need to create a copy of a Berge-F ′ . Vertices of D will play the role of N(u 1 ) ∩ N(u 2 ), A 1,i ∪ B i , A 2,i ∪ B i , . . . , A d,i ∪ B i will play the role of the edges connecting u 1 to all the vertices of D and similarly A 1,j ∪ B j , A 2,j ∪ B j , . . . , A d,j ∪ B j will play the role of the edges connecting u 2 to all the vertices of D. To finish the Berge copy of F ′ we will connect both u 1 and u 2 to all the vertices in C \ D (thus in fact we present a Berge copy of K 2,v−2 , which clearly contains F ′ ). We will use the hyperedges A i,d+1 , A i,d+2 , . . . , A i,d 1 −1 , A i,d+1 to connect u 1 to the vertices in C \ D. As they each contain
u 1 , it is enough to show an injection f from A i,d+1 , A i,d+2 , . . . , A i,d 1 −1 , A i,d+1 , A j,d+1 , A j,d+2 , . . . , A j,d 1 −1 , A j,d+1 to C \ D such that f (H)
∈ H for all sets. All we need to check is whether Hall's condition holds: as for any two distinct sets, their union contains C \ D, the only problem can occur if A i,ℓ ∩ (C \ D) = A j,ℓ ′ ∩ (C \ D) = C \ D and |C \ D| = 1 or 2. But then by v − 2 ≥ 5, we have d ≥ 3 and thus all choices of A i,ℓ , A j,ℓ contain C \ D by the assumption that we picked those such subsets first that miss another element of D apart from x 1 . Suppose next d = 0. Then for any 1 ≤ h ≤ m let us fix π h , a permutation z 1 , z 2 , . . . , z v−2 of vertices of C with z 1 , z 2 being the endvertices of the edge e α . Now let A 1,h , A 2,h , . . . , A δ,h be cyclic intervals of length v − 4 of π h with e α ⊂ A 1,h . Then putting the sets of the form A ℓ,h ∪ B h to G will satisfy items 1 and 2 by definition, item 4 by the choice of A 1,h , and item 3 by a similar Hall-condition reasoning as in the case of d > 0. Now we are ready to prove the theorem. If δ = 0, then F contains an isolated edge, and we are done by Proposition 2.1. Otherwise by Claim 2.3 there exists an F -good hypergraph G with O(n) hyperedges, and by Claim 2.2 any F -saturated extension of G has a linear number of hyperedges.
Concluding remarks
For any graph F , integer r ≥ 2 and enumeration π : G 1 , G 2 , . . . , G ( n r ) of [n] r we can define a greedy algorithm that outputs a Berge-F -saturated r-uniform hypergraph G as follows: we let G 0 = ∅, and then for any i = 1, 2, . . . , n r we let G i = G i−1 ∪ {G i } if G i−1 ∪ {G i } is Berge-F -free, and G i = G i−1 otherwise. Clearly, G π,r = G ( n r ) is Berge-F -saturated.
Theorem 1 . 3 .
13For any graph F of type I and any r ≥ 3 we have sat r (n, F ) = O(n).
(e). Then vertices of C and u can play the role of the s smallest classes of K, and {v} ∪ B i \ {u} can play the role of the largest class of K.
of Vizer was supported by the János Bolyai Research Fellowship. and by the New National Excellence Program under the grant numberÚNKP-20-5-BME-45. Research of Patkós was supported by the Ministry of Educational and Science of the Russian Federation in the framework of MegaGrant no. 075-15-2019-1926.
We claim that the graph G will consist of δ(F )−1 universal vertices, ⌊ n−δ(F )+1 v(F )−δ(F ) ⌋ cliques of size v(F ) − δ(F ), and one possible extra clique of size s, where s is the residue of n − δ(F ) + 1 modulo v(F ) − δ(F ). F without having degree at least δ(F ). The universal vertices with any clique of size v(F ) − δ(F ) form a clique of size v(F ) − 1, so any new vertex is joined to the universal vertices. and the next v(F ) − δ(F ) vertices can form a clique because of the condition δ(F ) = κ(F )Indeed, a new vertex cannot help creating a copy of F without having degree at least δ(F ). We claim that the graph G will consist of δ(F )−1 universal vertices, ⌊ n−δ(F )+1 v(F )−δ(F ) ⌋ cliques of size v(F ) − δ(F ), and one possible extra clique of size s, where s is the residue of n − δ(F ) + 1 modulo v(F ) − δ(F ). The universal vertices with any clique of size v(F ) − δ(F ) form a clique of size v(F ) − 1, so any new vertex is joined to the universal vertices, and the next v(F ) − δ(F ) vertices can form a clique because of the condition δ(F ) = κ(F ).
Observe that the Erdős-Rényi random graph G(k, 1/2) satisfies the condition of Proposition 3.1 with probability tending to 1 as k tends to infinity. So, for almost all graphs F , the greedy algorithm outputs an F -saturated graph G. with a linear number of edgesObserve that the Erdős-Rényi random graph G(k, 1/2) satisfies the condition of Propo- sition 3.1 with probability tending to 1 as k tends to infinity. So, for almost all graphs F , the greedy algorithm outputs an F -saturated graph G with a linear number of edges.
so we can assume that F is not a star. Let F u denote the component of u in F \ {(u, w)} and F w denote the component of w in F \ {(u, w)}. Let us consider a partition B 0. B 1 , B 2B m of an n-element set U with |B 1 | = |B 2 | = · · · = |B m | = r + 1 and |B 0 | ≤ r. For i = 1, 2, . . . , m, let G 0,i consist ofProof. It is known that the saturation number of stars is linear, so we can assume that F is not a star. Let F u denote the component of u in F \ {(u, w)} and F w denote the component of w in F \ {(u, w)}. Let us consider a partition B 0 , B 1 , B 2 , . . . , B m of an n-element set U with |B 1 | = |B 2 | = · · · = |B m | = r + 1 and |B 0 | ≤ r. For i = 1, 2, . . . , m, let G 0,i consist of
= ∪ m i=1 G 0,i . Clearly, G 0 is Berge-F -free as its components and only. Note that the degree of any edge e is at least e(F u ) − 2 and if e 1 and e 2 are disjoint edges of F u , then in B their neighborhood is G 0,i . Therefore the only problem that can occur is if F u is a star with e(F ) − 1 leaves. If the center of F u is u, then F is also a star, contradicting our assumption. If the center c of F u is not u, then a vertex u ′ ∈ B i that is contained in all= ∪ m i=1 G 0,i . Clearly, G 0 is Berge-F -free as its components and only if e ⊂ G. Note that the degree of any edge e is at least e(F u ) − 2 and if e 1 and e 2 are disjoint edges of F u , then in B their neighborhood is G 0,i . Therefore the only problem that can occur is if F u is a star with e(F ) − 1 leaves. If the center of F u is u, then F is also a star, contradicting our assumption. If the center c of F u is not u, then a vertex u ′ ∈ B i that is contained in all
C Berge, Hypergraphes: combinatoire des ensembles finis. Gauthier-VillarsC. Berge, Hypergraphes: combinatoire des ensembles finis, Gauthier-Villars, 1987.
On generalized graphs. B Bollobás, Acta Mathematica Academiae Scientiarum Hungaricae. 16B. Bollobás, On generalized graphs, Acta Mathematica Academiae Scientiarum Hun- garicae, 16 (1965) 447-452.
S English, N Graber, P Kirkpatrick, A Methuku, E C Sullivan, Saturation of Berge hypergraphs. 342S. English, N. Graber, P. Kirkpatrick, A. Methuku, E. C. Sullivan, Saturation of Berge hypergraphs, Discrete Mathematics, 342 (2019) 1738-1761.
Linearity of saturation for Berge hypergraphs. S English, D Gerbner, A Methuku, M Tait, European Journal of Combinatorics. 78S. English, D. Gerbner, A. Methuku, M. Tait, Linearity of saturation for Berge hyper- graphs, European Journal of Combinatorics, 78 (2019) 205-213.
A problem in graph theory. P Erdős, A Hajnal, J Moon, American Mathematical Monthly. 71P. Erdős, A. Hajnal, J. Moon, A problem in graph theory, American Mathematical Monthly, 71 (1964) 1107-1110.
A survey of minimum saturated graphs. J R Faudree, R J Faudree, J R Schmitt, The Electronic Journal of Combinatorics. 1819J. R. Faudree, R. J. Faudree, J. R. Schmitt, A survey of minimum saturated graphs, The Electronic Journal of Combinatorics, 18 (2011) DS19.
On disjointly representable sets. P Frankl, J Pach, Combinatorica. 4P. Frankl, J. Pach, On disjointly representable sets, Combinatorica, 4 (1984) 39-45.
Extremal Results for Berge hypergraphs. D Gerbner, C Palmer, SIAM Journal on Discrete Mathematics. D. Gerbner, C. Palmer, Extremal Results for Berge hypergraphs, SIAM Journal on Discrete Mathematics, 31 (2017) 2314-2327.
Saturated graphs with minimal number of edges. L Kászonyi, Zs, Tuza, Journal of Graph Theory. 10L. Kászonyi, Zs. Tuza, Saturated graphs with minimal number of edges, Journal of Graph Theory, 10 (1986) 203-210.
Results and open problems on minimum saturated hypergraphs. O Pikhurko, Ars Combinatoria. 72O. Pikhurko, Results and open problems on minimum saturated hypergraphs, Ars Com- binatoria, 72 (2004) 111-127.
Asymptotic growth of sparse saturated structures is locally determined. Zs, Tuza, Discrete Mathematics. 108Zs. Tuza, Asymptotic growth of sparse saturated structures is locally determined, Dis- crete Mathematics, 108 (1992) 397-402.
| []
|
[
"Minimal self-contained quantum refrigeration machine based on four quantum dots",
"Minimal self-contained quantum refrigeration machine based on four quantum dots"
]
| [
"Davide Venturelli \nNEST\nScuola Normale Superiore and Istituto Nanoscienze-CNR\nPiazza dei Cavalieri 7I-56127Pisa, Italy\n\nQuantum Artificial Intelligence Laboratory\nNASA Ames Research Center\n94035-1000Moffett FieldCA\n",
"Rosario Fazio \nNEST\nScuola Normale Superiore and Istituto Nanoscienze-CNR\nPiazza dei Cavalieri 7I-56127Pisa, Italy\n",
"Vittorio Giovannetti \nNEST\nScuola Normale Superiore and Istituto Nanoscienze-CNR\nPiazza dei Cavalieri 7I-56127Pisa, Italy\n"
]
| [
"NEST\nScuola Normale Superiore and Istituto Nanoscienze-CNR\nPiazza dei Cavalieri 7I-56127Pisa, Italy",
"Quantum Artificial Intelligence Laboratory\nNASA Ames Research Center\n94035-1000Moffett FieldCA",
"NEST\nScuola Normale Superiore and Istituto Nanoscienze-CNR\nPiazza dei Cavalieri 7I-56127Pisa, Italy",
"NEST\nScuola Normale Superiore and Istituto Nanoscienze-CNR\nPiazza dei Cavalieri 7I-56127Pisa, Italy"
]
| [
"PACS numbers: 03.65.Yz, 03.67.-a, 73.63.Kv, 73.23"
]
| We present a theoretical study of an electronic quantum refrigerator based on four quantum dots arranged in a square configuration, in contact with as many thermal reservoirs. We show that the system implements the minimal mechanism for acting as a self-contained quantum refrigerator, by demonstrating heat extraction from the coldest reservoir and the cooling of the nearby quantum-dot. | 10.1103/physrevlett.110.256801 | [
"https://arxiv.org/pdf/1210.3649v2.pdf"
]
| 9,746,313 | 1210.3649 | ac1da54cac49030bb995e77fcfbc2ef73363bd19 |
Minimal self-contained quantum refrigeration machine based on four quantum dots
22 Jun 2013
Davide Venturelli
NEST
Scuola Normale Superiore and Istituto Nanoscienze-CNR
Piazza dei Cavalieri 7I-56127Pisa, Italy
Quantum Artificial Intelligence Laboratory
NASA Ames Research Center
94035-1000Moffett FieldCA
Rosario Fazio
NEST
Scuola Normale Superiore and Istituto Nanoscienze-CNR
Piazza dei Cavalieri 7I-56127Pisa, Italy
Vittorio Giovannetti
NEST
Scuola Normale Superiore and Istituto Nanoscienze-CNR
Piazza dei Cavalieri 7I-56127Pisa, Italy
Minimal self-contained quantum refrigeration machine based on four quantum dots
PACS numbers: 03.65.Yz, 03.67.-a, 73.63.Kv, 73.23
22 Jun 2013(Dated: May 3, 2014)
We present a theoretical study of an electronic quantum refrigerator based on four quantum dots arranged in a square configuration, in contact with as many thermal reservoirs. We show that the system implements the minimal mechanism for acting as a self-contained quantum refrigerator, by demonstrating heat extraction from the coldest reservoir and the cooling of the nearby quantum-dot.
The increasing interest in quantum thermal machines has its roots in the need to understand the relations between thermodynamics and quantum mechanics [1,2]. The progress in this field may as well have important applications in the control of heat transport in nano-devices [3]. In a series of recent works [4][5][6] the fundamental limits to the dimensions of a quantum refrigerator have been found. It has been further demonstrated that these machines could still attain Carnot-efficiency [5] thus launching the call for the implementation of the smallest possible quantum refrigerator. Refs. [4][5][6] considered selfcontained thermal machines defined as those that perform a cycle without the supply of external work, their action being grounded on the steady-state heat transfer from thermal reservoirs at different temperatures. The major difficulty in the realization [7,8] of self-contained refrigerators (SCRs) is the engineering of the crucial three-body interaction enabling the coherent transition between a doubly excited state in contact with a hot (H) and cold (C) reservoir, and a singly-excited state coupled to an intermediate (or "room" -R) temperature bath. We get around this problem by proposing an experimentally feasible implementation of a minimal SCR with semiconducting quantum dots (QDs) operating in the Coulomb blockade regime. We are thus able to establish a connection between the general theory of quantum machines and the heat transport in nanoelectronics [3].
QDs contacted by leads were proposed as ideal systems for achieving high thermopower [9][10][11] or anomalous thermal effects [12]. Here we study a four-QD planar array (hereafter named a "quadridot" for simplicity) coupled to independent electron reservoirs as shown in Fig. 1; with proper (but realistic) tuning of the parameters, we will show that the quadridot acts as a SCR which pumps energy from the high temperature reservoir H and the low temperature reservoir C to the intermediate temperature reservoirs R 1 , R 2 . Furthermore we will analyze the conditions under which the quadridot is able to cool the dot QD 2 which is directly connected to the bath C, at an effective temperature that is lower than the one it would have had in the absence of the other reservoirs. This will lead us to introduce an operative definition of the local effective temperature depending on the measurement The four quantum dots QD1, QD2, QD3, and QD4 are weakly coupled to the reservoirs R1, C, R2, and H, respectively, which are all grounded and maintained at temperatures TH > (TR 1 = TR 2 = TR) > TC. Tunneling is allowed only between QD1 and QD4, and between QD2 and QD3 (t being the gauging parameter).
setup, and to predict the existence of working regimes where, for instance, the refrigeration is not accompanied by the cooling of QD 2 . We start analyzing the system Hamiltonian, identifying the conditions that allows us to mimics the behavior of the SCR of Ref. [4].
In the absence of the coupling to the leads, the quadridot shown in Fig. 1
is described by the Hamitonian
H QD = i=1,··· ,4 ǫ i n i + i =j U ij 2 n i n j − t(c † 1 c 4 + c † 2 c 3 + h.c.),
where for i = 1, · · · , 4, c † i , c i , and n i = c † i c i represent respectively the creation, annihilation and number operators associated with the i-th QD. In this expression the quantities ǫ i gauge the single particle energy levels, t defines the tunneling coupling between the dots, and U ij describes the finite-range contribution of the Coulomb repulsion. To reduce the maximum occupancy in each QD to one electron, we will assume the on-site repulsion terms U ii to be the largest energy scale in the problem. Furthermore, in order to mimic the dynamics of [4] we will take U 12 = U 34 = U ⊥ and U 23 = U 14 = U , both much larger than the "diagonal" terms U 24 = U 13 = U d , and tune the single-electron energy level of the upperright dot (which will be coupled to the cool reservoir C) so that ǫ 2 = ǫ 1 + ǫ 3 − ǫ 4 . These choices ensure that in the absence of tunneling (t = 0), the "diagonal" two-particle states |d = |1, 0, 1, 0 and |d = |0, 1, 0, 1 shown in Fig. 2 are degenerate (the charge states are labeled according to the occupation of the four dots |n 1 , n 2 , n 3 , n 4 ). These are the only states of the two-electron sector which play an active role in the system evolution, mimicking the role of the vectors |01 , |10 of [4]. Due to the presence of U ⊥ or U the other configurations are indeed much higher in energy to get permanently excited in the process. Still the states |u = |1, 1, 0, 0 and |l = |0, 0, 1, 1 play a fundamental role in the SCR as their presence generates (via a Schrieffer-Wolff transformation [13,14] and the non-zero hoppings t) an effective coupling term between |d and |d of the form H ef f = g dd (|d d | + |d d|) with
g dd ≃ 2t 2 (U d − U ⊥ ) (U d − U ⊥ ) 2 − (ǫ 4 − ǫ 1 ) 2 ≪ t.
(1)
In our model g dd is the analogous of the perturbative parameter g of [4]. Its role is to open a devoted channel which favors energy exchanges between the couple H-C and the couple R 1 -R 2 by allowing two electrons to pass from the first to the second through the mediation of the quadridot states |d [which is in contact with H and C] and |d [which is connected to R 1 and R 2 ]. For proper temperature imbalances this is sufficient to establish a positive heat flux from C to QD 2 even if T C is the lowest of all bath temperatures. The mechanisms can be heuristically explained as follows: if T H is sufficiently higher than the other bath temperatures, then the dot which has more chances of getting populated by its local reservoir is QD 4 . When this happens, the large values of U ⊥ , U will prevent QD 1 and QD 3 from acquiring electrons too. On the contrary, while QD 4 is populated, QD 2 is allowed to accept an electron from its reservoir C [U d being much smaller than U ⊥ , U ] creating |d . The coupling provided by H ef f will then rotate the latter to |d giving the two electrons [the one from H and the one from C] a chance of being absorbed by R 1 -R 2 . The opposite process (creation of |d by absorption of a couple of electrons from R 1 -R 2 , rotation to |d , and final emission toward H-C) is statistically suppressed due to the (relatively) low probability that QD 1 or QD 3 will get an electron form their reservoir before QD 4 gets its own from H: the net result is a positive energy flux from H-C to R 1 -R 2 .
To verify this picture we explicitly solve the open dynamics of the quadridot and study its asymptotic behavior. Specifically, we model our four local baths H, C, R 1 and R 2 as independent electron reservoirs (leads) characterized by their own chemical potential µ i and their own temperature [both quantities entering in the Fermi-Dirac occupation functions f i (ǫ) associated with the reservoir]. For the sake of the simplest correspondence with the model of Ref. [4] in this study all µ i will be set to be identical and fixed to a value that will be used as reference for the single particle energies of the system [e.g. setting ǫ i = 0 in the Hamiltonian corre- sponds to have the i-th quadridot level at resonance with the Fermi energy of the reservoirs]. Furthermore as detailed in the caption of Fig. 1 the temperatures will be chosen to satisfy the relation T C < T R1 = T R2 < T H [15]. Within these assumptions, the only required external action is exerted in maintaining the local equilibrium temperatures and chemical potential, in accordance to the standard definition of self-containance. The quadridotbath couplings (parametrized by the amplitudes Γ (k)
i ) are hence expressed as tunneling terms of the form
H T = i=1,··· ,4 k Γ (k) i c † i a i,k + h.c.(2)
with a i,k being an annihilation operator which destroy an electron of momentum k in the lead i. In the Born-Markov-Secular limit [17,18] these extra terms, give rise to a Lindblad equation for the reduced density matrix ρ of the quadridot. The presence of t plus the rotation into the low-energy sector eliminates degeneracies among all possible energy transitions between the eigenstates of the quadridot. The evolution of ρ is then determined bẏ
ρ = i D i [ρ] ,(3)
where for each reservoir D i represents associated Lindblad dissipator. This equation can be solved in the steady-state regime (ρ = 0) yielding the asymptotic configuration ρ ∞ , from which the heat currents J Q,i flowing through the i-th reservoir are then computed as [17],
J Q,i = Tr H QD D i [ρ ∞ ] .(4)
If our implementation of the SCR is correct, we should see a direct heat flow from the hot H and cold C reservoirs to the reservoirs R 1 and R 2 , while the dot QD 2 should reach an occupation probability corresponding to an effective temperature which is lower than the one dictated by its local reservoir C (See Fig. 3-b). We have verified this by setting the system parameters to be consistent with those presented in [4] -making sure however that for such choice no additional degeneracies are introduced into the system due to the larger dimension of our physical model. While the performances of the device do not change qualitatively when varying the parameters according to above prescriptions, in the following we focus on a specific scenario where we fixed ǫ 1 = 2.1, ǫ 3 = 2.9, ǫ 4 = 4.0, and U ⊥ = 12.0 [U instead is taken to be infinitely large for simplicity as its effect could be absorbed in the energy level renormalization after the Schrieffer-Wolff transformation]. The value of g dd is finally taken to be -0.001 determining t ( < ∼ 0.1) through Eq. (1), while the couplings terms Γ (k) i which link the quadridot to the reservoirs via Eq. (2) are chosen to provide effective dissipation rates of order ∼ 0.0001 [19]. Solving numerically the steady state equation (3) we observe that for each T C < T R , there exists a minimal threshold value for T H above which the SCR indeed extracts heat from the cold reservoir C. This is shown in Fig. 3-a for T R =2 and different values of U d , the quadridot works as a SCR in the blue region. Consistently with the second principle of thermodynamics the threshold value of T H (black curve in the plot) is always greater than T R = 2 (for T H below T R the machine cannot produce work from H to pump heat from C), and that the region above this threshold gets larger as U d gets smaller. The existence of a threshold for T H implies also that, for given T H > T R , there is a minimal temperature T * C for the cold reservoir under which the SCR cannot work. Interestingly for T H /T R → ∞ the value of T * C appears to asymptotically converge toward a finite non-zero temperature which depends upon the engine microscopic parameters and which can be interpreted as the emergent absolute zero of the model. An approximate analytical expression for T * C can be derived exploiting the recent general theory of genuine, maximally-efficient self-contained quantum thermal machines [6]. This is done by interpreting the quadridot as a composite system, consisting of an "effective" virtual qubit formed by the states |0, 0, 0, 1 and |d which (through g dd ) mediates the interaction between QD 2 and the reservoirs H, R 1 and R 2 . The average occupations of the virtual qubit levels (determined by the coupling with the reservoirs H, R 1 and R 2 ) defines the effective (average) temperature of H, R 1 and R 2 which is perceived by QD 2 : such temperature competes with T C in cooling down the dot and can be identified with the value of T * C of our model. Observing that the energy levels of |0, 0, 0, 1 and |d are ǫ 4 , ǫ 1 + ǫ 3 + U d respectively, from [6] we get
T * C ≃ T R T H ǫ 1 + ǫ 3 + U d − ǫ 4 T H (ǫ 1 + ǫ 3 + U d ) − T R ǫ 4 ,(5)
which fits pretty well our numerical results (See Fig. 3-a) and which for T H → ∞ yields
T R (1 − ǫ 4 /[ǫ 1 + ǫ 3 + U d ])
as emergent absolute zero of the model. Blue/Red background color intensity is proportional to the actual heat pumped to/extracted from C. c) Comparison between heat extraction and single particle occupation for U d = 2. In regions I/II the SCR is working (heat is extracted from C) while in regions III/IV the C bath receives heat. In regions I/III we have an effective decrease of the occupation number of QD2 (i.e. n2 < n 0 2 ). d) Efficiency of the SCR compared to the Carnot efficiency (dashed black line) for TC=1.0. The curves represents U d = 0 (blue), U d = 1 (magenta) and U d = 2 (brown). The dashed horizontal line indicates maximum limit of efficiency for the quadridot computed (for U d = 0) as in [4].
Following [5], we evaluate the ratio η = J Q,2 / J Q,4 between the heat current through the cold and hot reservoirs comparing it with the upper bound (1 − TR TH )/( TR TC − 1) posed by the Carnot limit, and with the theoretical value η th =(ǫ 1 + ǫ 3 − ǫ 4 )/ǫ 4 of [4] applied to the quadridot for U d = 0 [20]. The dependence of η upon T H is plotted in Fig. 3-d for different values of U d . We noticed that in the case U d =0 the efficiency of the quadridot converges indeed towards the theoretical value η th of [4] at least for large enough T H .
Measurements and effective local temperatures:-An important question is whether this refrigeration effect is accompanied with a cooling of QD 2 , namely whether its effective local temperature T (ef f ) C decreases as T H increases, for sufficiently high T H , in analogy with the qubit-cooling described in Ref. [4]. While for such idealized qubit model the definition of the local temperature is relatively straightforward, in nanoscale systems out of equilibrium, local temperatures must be operationally defined [21]. The most common way to proceed is to introduce a probe reservoir P (a "thermometer") which is weakly coupled to that part of the system we are interested in (the dot QD 2 in our case) and identify the effective temperature of the latter with the value of the temperature T P of the probe which nullifies the heat flow through P. This procedure yields a natural way of measuring the effect we are describing and can be implemented easily in our model by adding an extra term in (2) that connects the new reservoir P to QD 2 with a tunnel amplitude Γ P which is much smaller than those associated with the other reservoirs of the system (in the calculation we set the ratio between Γ P and Γ i of the other reservoirs to be of the order 10 −3 : this make sure that the presence of P does not perturb the system). The obtained values of T (ef f ) C are presented in Fig. 3-a where it is shown that, according to this definition of the local temperature, the conditions for cooling of QD 2 (i.e. T (ef f ) C < T C ) are the same for the SCR to work (implying incidentally that in this case T (ef f ) C is always greater than the emergent zero-temperature of the systemT C ).
The quantity T (ef f ) C introduced above has a clear operational meaning and according to the literature it is a good candidate to define the effective temperature of QD 2 . Still it is important to acknowledge that in experiments the cooling of QD 2 can also be detected by using the non-invasive techniques of e.g. Ref. [22] to look at the decrease of the mean asymptotic occupation number of QD 2 , n 2 = 0, 1, 0, 0|ρ ∞ |0, 1, 0, 0 + d|ρ ∞ |d ), with respect to the same quantity computed when the SCR is "turned off" (e.g. n 0 2 = 0, 1, 0, 0|ρ ∞ 0 |0, 1, 0, 0 + d|ρ ∞ 0 |d where now ρ ∞ 0 is the asymptotic stationary state of the system reached when all the reservoirs but C are disconnected, i.e. Γ i =C = 0). We notice however that the cooling condition hereby defined does not coincide with the same pictured in Fig. 3-a. We indeed exemplify in Fig. 3-c for U d =3 that according to this new definition different operating regimes are possible for the SCR. The QD 2 might be either colder (n 2 < n 0 2 in zone I) or hotter (n 2 > n 0 2 , in region II) when the device extract heat from the C reservoir. Conversely, we might achieve a colder QD 2 also when the quadridot pumps heat into the colder bath (III). In region IV none of the refrigeration effects are active. Similar regimes emerge with other activation prescriptions, such as defining n 0 2 as the occupation for T H = T R = T C while maintaining all tunnel couplings as constant.
Conclusions:-We conclude with experimental considerations. Quadridots in GaAs/AlGaAs heterostructures have been implemented for Cellular-Automata computation [23] and for single-electron manipulation [24]. Strongly capacitively-coupled QDs with interdot capacitance energy (U ⊥ and U ) up to 1/3 of the intra-dot charging energy (taken to be infinite in our model) can be fabricated with current lithographic techniques [25]. The diagonal inter-dot term U d is expected to be at most U / √ 2 ≃ U ⊥ / √ 2 from geometrical considerations, but practically it is expected to be much smaller [24]. The local charging energy can be as big as 1 meV, and usually represents about 20% of the confinement energy [26], which is the typical tunable values of the single-particles levels ǫ i . Charging effects are expected to be further enhanced by the presence of a significant magnetic field, due to the emergence of the incompressible antidot regime in the dots [27], possibly allowing the working conditions to be achieved even more easily. In this high-field regime, the spin/orbital-Kondo effect [28,29] is suppressed [30], as the transport becomes spin-polarized, so our effective description is expected to be valid. A final ingredient for the quadridot to act as a SCR is quantum coherence. In QDs it is known that the main source of decoherence comes from 1/f noise arising from background charge fluctuations [31] (however coherent manipulation of QDs have been reported in several experiments, e.g. see Ref. [32]). Accordingly Eq. (3) acquires an extra contribution whose effect (see Supplemental Material [33]) is to modify the steady state populations. In our setup as long as the new rates are of the same order of the ones due to the coupling to the leads the quadridot will still work as a SCR (note, indeed, that the bounds to the blue region in Fig. 3 a) do not depend on these rates). Possibly the only serious challenge is posed by the need that the induced broadening should not be too large with respect to t. For the sake of simplicity we adopted small values of this parameter, however it is very much possible that higher values will help the efficiency of the SCR by speeding up the |d , |d rotations. We finally observe that the maximum thermal energies involved should not exceed the large charging energies (i.e. < ∼ 10K). We acknowledge useful discussions with C.W.J. Beenakker, M. Carroll, F. Giazotto, F. Mazza, and J. Pekola and support from MIUR through the FIRB-IDEAS project RBID08B3FM, by EU through the Project IP-SOLID, and by Sandia National Laboratories.
FIG. 1 :
1[Color online] The quadridot.
FIG. 2 :
2[Color online] Pictorial view of the low-energy electronic charged states (a black circle indicates occupation by an electron). Due to the hoppings terms t and g dd the eigenstates of the low-energy Hamiltonian are bonding-antibonding states |d ± |d and the four bonding-antibonding delocalized single particles states (the completely empty state is not shown). For t = 0 the two electron states |d and |d are resonant, while |u and |l are the high-energy virtual states responsible for the effective interaction g dd coupling |d , |d .
FIG. 3 :
3[Color online] a) Panels refer to U d = 0 (left), U d = 1 (center), U d = 2 (right). When TH approaches the black curve, i.e. at values very close to the analytical value of Eq. (5) (dashed line), a change of sign in the heat flow occurs. For TH above the threshold (blue region) the machine works as pictured in b), extracting heat from the C,H-reservoirs and pumping it into R. In the opposite regime (red region) heat cannot be extracted from C. Black dashed line above the grey region indicates TH = TR = 2.
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| []
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[
"Landau damping of surface plasmons in metal nanostructures",
"Landau damping of surface plasmons in metal nanostructures"
]
| [
"Tigran V Shahbazyan \nDepartment of Physics\nJackson State University\n39217JacksonMSUSA\n"
]
| [
"Department of Physics\nJackson State University\n39217JacksonMSUSA"
]
| []
| We develop a quantum-mechanical theory for Landau damping of surface plasmons in metal nanostructures of arbitrary shape. We show that the electron surface scattering, which facilitates plasmon decay in small nanostructures, can be incorporated into the metal dielectric function on par with phonon and impurity scattering. The derived surface scattering rate is determined by the local field polarization relative to the metal-dielectric interface and is highly sensitive to the system geometry. We illustrate our model by providing analytical results for surface scattering rate in some common shape nanostructures. Our results can be used for calculations of hot carrier generation rates in photovoltaics and photochemistry applications. | 10.1103/physrevb.94.235431 | [
"https://arxiv.org/pdf/1611.08670v3.pdf"
]
| 119,078,749 | 1611.08670 | a6f08672b935c8d89fe1dc5315105c3b00efba00 |
Landau damping of surface plasmons in metal nanostructures
24 Dec 2016
Tigran V Shahbazyan
Department of Physics
Jackson State University
39217JacksonMSUSA
Landau damping of surface plasmons in metal nanostructures
24 Dec 2016
We develop a quantum-mechanical theory for Landau damping of surface plasmons in metal nanostructures of arbitrary shape. We show that the electron surface scattering, which facilitates plasmon decay in small nanostructures, can be incorporated into the metal dielectric function on par with phonon and impurity scattering. The derived surface scattering rate is determined by the local field polarization relative to the metal-dielectric interface and is highly sensitive to the system geometry. We illustrate our model by providing analytical results for surface scattering rate in some common shape nanostructures. Our results can be used for calculations of hot carrier generation rates in photovoltaics and photochemistry applications.
I. INTRODUCTION
Surface plasmons are collective electron excitations that provide unprecedented means for energy concentration, conversion, and transfer at the nanoscale [1][2][3]. Plasmons can be resonantly excited in metal-dielectric nanostructures giving rise to strong oscillating local fields that underpin numerous plasmon-enhanced spectroscopy phenomena, including surface-enhanced Raman scattering [4], plasmon-enhanced fluorescence and energy transfer [5], or plasmonic laser (spaser) [6]. Among key characteristics that impact many plasmonics applications [7][8][9][10] is the plasmon lifetime, which, depending on the plasmonic system size, is governed by several decay mechanisms [11][12][13][14][15]. While in large systems, the plasmon lifetime is mostly limited by radiation [16], in systems with characteristic size L < c/ω, where c and ω are, respectively, the light speed and frequency, the dominant decay mechanism is excitation of electron-hole (e-h) pairs by the plasmon local field accompanied by phonon and impurity scattering or, for small systems, surface scattering [17]. Recently, plasmon decay into e-h pairs has attracted intense interest as a highly efficient way of hot carrier generation and transfer across the interfaces with applications in photovoltaics [18][19][20][21][22][23][24][25][26][27] and photochemistry [28][29][30][31][32]. Plasmon-assisted hot carrier generation is especially efficient in smaller plasmonic systems, where light scattering is relatively weak and extinction is dominated by resonant plasmon absorption. In such systems, carrier excitation is enhanced due to strong surface scattering that provides a new momentum relaxation channel [17].
Surface-assisted plasmon decay (Landau damping) has been extensively studied experimentally [33][34][35][36][37][38][39][40][41][42][43] and theoretically [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] since the pioneering paper by Kawabata and Kubo [44], who have shown that, for a spherical particle of radius a, the surface scattering rate is γ sp = 3v F /4a, where v F is the electron Fermi velocity. In subsequent quantum-mechanical studies carried within random phase approximation (RPA) [45][46][47][48][49][50][51] and timedependent local density approximation (TDLDA) [52][53][54][55][56][57][58][59] approaches, a more complicated picture has emerged involving the role of confining potential and nonlocal effects. These are dominant at the spatial scale ξ nl = v F /ω that defines the characteristic length for nonlocal effects [60,61] (e.g., for noble metals, v F /ω < 1 nm in the plasmon frequency range), whereas for larger systems with L ≫ v F /ω (i.e., several nm and larger), they mainly affect the overall magnitude of γ sp , while preserving intact its size dependence [56,58]. The latter implies that in a wide size range v F /ω ≪ L ≪ c/ω, which includes most plasmonic systems used in applications, the detailed structure of electronic states is unimportant, and the confinement effects can be reasonably described in terms of electron surface scattering, which can be incorporated, along with phonon and impurity scattering, in the metal dielectric function ε(ω) = ε ′ (ω) + iε ′′ (ω). Here, we adopt the Drude dielectric function ε(ω) = ε i (ω)−ω 2 p /ω(ω+iγ), where ε i (ω) describes interband transitions, ω p is the plasma frequency, and γ is the scattering rate. Thus, it is expected that, for systems in the above size range, the scattering rate should be modified as γ = γ 0 + γ s , where γ 0 is the bulk scattering rate and γ s is the surface scattering rate. In particular, the standard expression, in terms of metal dielectric function, for the plasmon decay rate [3],
Γ = 2ε ′′ (ω) ∂ε ′ (ω) ∂ω −1 ,(1)
should describe plasmon damping due to both bulk and surface-assisted processes if surface-modified ε(ω) is used instead. For example, for ω well below the onset of interband transitions, the rate (1) coincides with (modified) Drude scattering rate: Γ ≈ γ = γ 0 + γ s . The major roadblock in the way of carrying this program forward has so far been the lack of any quantummechanical model for evaluation of γ s in a nanostructure of arbitrary shape. Due to the complexity of electronic states in general-shape confined systems, calculations of γ s were performed, within RPA [44][45][46][47][48][49][50][51] and TDLDA [52][53][54][55][56][57][58][59] approaches, only for some simple (mostly spherical) geometries. For general shape systems, the surface scattering rate was suggested, within the classical scattering (CS) model [62][63][64][65][66], in the form γ cs = Av F /L, where L is interpreted as the ballistic scattering length in a classical cavity, while the phenomenological constant A accounts for the effects of surface potential, electron spillover, and dielectric environment. However, the unreasonably wide range of measured A (0.3÷1.5 for spher-ical particles [17]) raised questions about the CS model validity [67], while recent measurements of plasmon spectra in nanoshells [38], nanoprisms [41], nanorods [42], and nanodisks [43] revealed significant discrepancies with its predictions. Furthermore, the CS approach is questionable on physical grounds as well since it involves carrier scattering across the entire system even for L ≫ v F /ω, i.e., when the nonlocal effects are expected to be weak.
On the other hand, surface scattering should depend sensitively on the local fields accelerating the carriers towards the metal-dielectric interface. This dependence was, in fact, masked in all previous quantum-mechanical studies of simple-shape systems [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], where a specific functional form of the local field, appropriate for the given geometry, was adopted, while it is completely missing in the CS approach. Moreover, for the most widely studied spherical geometry, the local field is uniform inside the particle (apart from surface effects), which further obscured its importance. Note, however, that our recent RPA calculations of the surface plasmon lifetime in spherical metal nanoshells with dielectric core [68] revealed the crucial role of local fields; for thin shells, the field is pushed out of the metal region, resulting in a reduction of the plasmon decay rate. This result contrasts sharply with the CS model predictions but, in fact, is consistent with the measured light-scattering spectra of single nanoshells [38]. Furthermore, recent measurements of plasmon spectra in nanorods and nanodisks revealed strong sensitivity of plasmon modes' linewidth to the local field polarization relative to the system symmetry axis [43]. For general-shape systems, the local field orientation relative to the interface can strongly affect the surface scattering rate, and, therefore, must be properly accounted for in any consistent theory of surface-assisted plasmon decay.
In this paper, we present a quantum-mechanical theory for surface-assisted e-h pair excitation by alternating local electric field Ee −iωt in metal nanostructures of general shape. We note that intraband absorption of energy ω takes place in a region of size v F /ω (see Fig. 1) and, therefore, can be viewed as a local process in systems with characteristic size L ≫ v F /ω. We show that, within RPA, surface scattering can be included into the Drude dielectric function by modifying the scattering rate as γ = γ 0 + γ s . We derive the surface scattering rate γ s as
γ s = A v F dS|E n | 2 dV |E| 2 ,(2)
where E n is the local field component normal to the interface and the integrals are carried over the metal surface (numerator) and volume (denominator). The constant A has the value A = 3/4 for hard-wall confining potential, but can be adjusted to account for surface and nonlocal effects. The full plasmon decay rate, including bulk and surface contributions, has still the form (1), but with modified ε(ω) that now includes the surface scattering rate (2). Surface scattering is highly sensitive to the local field polarization relative to the metal-dielectric interface, leading to distinct rates for different plasmon modes, as we illustrate here for some common geometries. The paper is organized as follows. In Sec. II, we outline our approach to plasmon Landau damping in metal nanostructures. In Sec. III, we derive an explicit expression for surface-assisted absorbed power and the corresponding scattering rate in systems of arbitrary shape. In Sec. IV, we present analytical and numerical results for surface scattering rates in some common nanostructures. In Sec. V, we discuss the effect of confining potential profile on the surface scattering rate, and the Appendices detail some technical aspects of our calculations.
II. DECAY RATE OF SURFACE PLASMONS IN METAL NANOSTRUCTURES
In this section, we outline our approach to calculation of the plasmon decay rate for a metal nanostructure embedded in dielectric medium. For simplicity, we restrict ourselves by metal structures occupying some volume V with a single surface S, so that the local dielectric function ε(ω, r) = ε ′ (ω, r) + iε ′′ (ω, r) equals ε(ω) and ε d in the metal and dielectric regions, respectively. For systems with characteristic size L ≪ c/ω, the retardation effects are unimportant, and plasmon modes are determined by the Gauss law ∇ · [ε ′ (ω l , r)E l (r)] = 0, where E l (r) is the slow component of plasmon local field and ω l is the plasmon mode frequency. For brevity, we omit the mode index l hereafter. The general expression for plasmon decay rate Γ has the form [69]
Γ = Q U ,(3)
where U is the mode energy [70],
U = ω 16π ∂ε ′ (ω) ∂ω dV |E| 2 ,(4)
and Q is the absorbed power (loss function)
Q = ω 2 Im dV E * · P.(5)
Here, P(r) is the electric polarization vector and the star stands for complex conjugation. In the classical (local) picture, the polarization vector is proportional to the local field, P loc (r) = E(r)[ε(ω, r) − 1]/4π, yielding the absorbed power due to the bulk processes [70]
Q = ωε ′′ (ω) 8π dV |E| 2 ,(6)
which, along with the mode energy (4), leads to the standard form (1) of the plasmon damping rate. Surface contribution to the absorbed power, Q s , comes from the momentum relaxation channel provided by carrier scattering from the metal-dielectric interface. Since surface scattering introduces nonlocality, Q s must be evaluated microscopically. The general expression for Q s can be obtained by relating P(r) to the electron polarization operator P (ω; r, r ′ ) via the induced charge density:
ρ(r) = e dr ′ P (r, r ′ )Φ(r ′ ) = −∇ · P(r),(7)
where local potential Φ(r) is defined as eE(r) = −∇Φ(r) (e is the electron charge). With the help of Eq. (7), integration of Eq. (5) by parts yields
Q s = ω 2 Im dV dV ′ Φ * (r)P (ω; r, r ′ )Φ(r ′ ),(8)
where P (ω; r, r ′ ) includes only the electronic contribution, i.e., without phonon and impurity scattering effects. Within RPA, P (ω; r, r ′ ) is replaced by the polarization operator for noninteracting electrons [71], yielding
Q s = πω αβ |M αβ | 2 [f (ǫ α ) − f (ǫ β )] δ(ǫ α − ǫ β + ω),(9)
where M αβ = dV ψ * α Φψ β is the transition matrix element of local potential Φ(r) calculated from the wave functions ψ α (r) and ψ β (r) of electron states with energies ǫ α and ǫ β separated by ω, f (ǫ) is the Fermi distribution function, and spin degeneracy is accounted for.
In terms of Q s , the surface-assisted contribution to the plasmon decay rate, i.e., the Landau damping (LD) rate, has the form
Γ s = Q s U ,(10)
where U is given by Eq. (4). Note that often in the literature, Γ s is identified with the standard first-order transition probability rate, given by the expression similar to Eq. (9) but divided by the factor ω/2. We stress that in a system with dispersive dielectric function, where the mode energy is U rather than ω [70], the standard transition rate must by rescaled by the factor ω/2U [69]. Calculation of Q s (and, hence, of Γ s ) hinges on the transition matrix element M αβ , which has so far been evaluated, either analytically or numerically, only for several simple geometries permitting separation of variables [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]68]. For general-shape systems, evaluation of M αβ presents an insurmountable challenge of finding, with a good accuracy, the three-dimensional electron wave functions oscillating rapidly, with the Fermi wavelength period λ F , on the system size scale L ≫ λ F . However, as we demonstrate in the following section, this difficulty can be bypassed and even turned into an advantage as Q s is derived in a closed form for any nanostructure larger than the nonlocality scale, i.e., for L ≫ v F /ω.
III. ABSORBED POWER AND SURFACE SCATTERING RATE
In this section, we derive the surface contribution Q s to the absorbed power due to e-h pair excitation by alternating local field Ee −iωt created in the metal either by plasmons or as a response to an external field.
We start with the transition matrix element M αβ = dV ψ * α Φψ β , where ψ α (r) is the eigenfunction of the Hamiltonian H = −( 2 /2m)∆ for an electron with energy ǫ α in a hard-wall cavity (this approximation is discussed later). We consider the case when excitation energy ω is much larger than the electron level spacing, so that, in the absence of phonon and impurity scattering, the electron transition to the state ψ β (r) with energy ǫ β = ǫ α + ω requires momentum transfer to the interface. A direct evaluation of M αβ , so far carried out only for some simple geometries [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]68], requires the knowledge of ψ α in the entire system volume. We note, however, that for a typical plasmon frequency ω ≪ E F , where E F is the Fermi energy in the metal, the momentum transfer q ∼ ω/v F takes place in a region of size ξ nl ∼ /q ∼ v F /ω, so that, for characteristic system size L ≫ v F /ω, the e-h pair excitation takes place in a close proximity to the interface (see Fig. 1). It is our observation that, for an electron in a hard-wall cavity, the boundary contribution to M αβ can be extracted as a surface integral of the form,
M s αβ = −e 4 2m 2 ǫ 2 αβ dS[∇ n ψ α (s)] * E n (s)∇ n ψ β (s),(11)
where ∇ n ψ α (s) is the wave-function normal derivative at a surface point s, E n (s) is the corresponding normal field component, ǫ αβ = ǫ α − ǫ β is the e-h pair excitation energy, and m is the electron mass. The derivation of Eq. (11) is given in Appendix A. Using the above matrix element, Eq. (9) can be recast as
Q s = e 2 4 4πm 4 ω 3 dSdS ′ E n (s)E * n ′ (s ′ )F ω (s, s ′ ),(12)
where F ω (s, s ′ ) stands for e-h surface correlation function defined as
F ω (s, s ′ ) = dǫf ω (ǫ)ρ nn ′ (ǫ; s, s ′ )ρ n ′ n (ǫ + ω; s ′ , s).(13)
Here, the function f ω (ǫ) = f (ǫ) − f (ǫ + ω) restricts the electron initial energy to the interval ω below E F , and
ρ nn ′ (ǫ; s, s ′ ) = ∇ n ∇ ′ n ′ ImG(ǫ; s, s ′ )(14)
is the normal derivative of the electron cross density of states ρ(ǫ; r, r ′ ) = ImG(ǫ; r, r ′ ) at surface points, where
G(ǫ; r, r ′ ) = α ψ α (r)ψ * α (r ′ ) ǫ − ǫ α + i0(15)
is the Green function of a confined electron. Note that neither the Green function G(ǫ; r, r ′ ) nor the correlation function F ω (s, s ′ ) can be evaluated with any reasonable accuracy for a general-shape cavity. However, an explicit expression for Q s in terms of local fields can still be derived by exploiting the difference in the length scales characterizing electron and plasmon excitations. Namely, while the electron wave-functions oscillate with the Fermi wave length period λ F , the local fields significantly change on the much larger system scale L ≫ λ F . Below we outline the main steps of our derivation of Q s and refer to Appendix B for details. First, we note that since excitation of an e-h pair with energy ω near the Fermi level takes place in a region of size v F /ω, the correlation function F ω (s, s ′ ) peaks in the region |s − s ′ | v F /ω ≪ L and rapidly oscillates outside of it (see below). On the other hand, in such a region, the local field E n is nearly constant, i.e., E n (s) ≈ E n (s ′ ), and so Q s takes the form
Q s = e 2 4 4πm 4 ω 3 dS|E n (s)| 2F ω (s),(16)
whereF
ω (s) = dS ′ F ω (s, s ′ ) is, for L ≫ v F /ω, indepen- dent of the surface point s.
Evaluation ofF ω is based upon multiple-reflection expansion for the electron Green function G(ǫ; s, s ′ ) in a hard-wall cavity [72]. For L ≫ λ F , the direct and single-reflection paths provide the dominant contribution, while higher-order reflections are suppressed as powers of λ F /L ≪ 1 (see Appendix B), and we obtain
ρ nn ′ (ǫ; s, s ′ ) = 2∇ n ∇ ′ n ImG 0 (ǫ, s − s ′ )(17)
where
G 0 (ǫ, r) = m 2π 2 e ikǫr r , k ǫ = √ 2mǫ ,(18)
is the free electron Green function, and factor 2 reflects equal contributions of direct and reflected paths at a surface point. It is now easy to see that, for ǫ ∼ E F and ω/E F ≪ 1, the integrand of Eq. (13) peaks in the region
|s − s ′ | 1 k ǫ+ ω − k ǫ ≈ v F ω(19)
and rapidly oscillates outside of it. This sets up the length scale v F /ω for correlation function F ω (s, s ′ ) in Eq. (12) and leads to Eq. (16). The final step is to compute the normal derivatives in Eq. (17) which, for L ≫ v F /ω, is accomplished by approximating the surface by the tangent plane at the surface point (see Appendix B), yieldingF
ω = ω 2m 4 E 2 F π 8 .(20)
Substituting thisF ω into Eq. (16), we finally arrive at the surface contribution to the absorbed power
Q s = e 2 2π 2 E 2 F ( ω) 2 dS|E n | 2 = 3v F 32π ω 2 p ω 2 dS|E n | 2 . (21)
The above expression for the surface absorbed power Q s , which is our central result, is valid for any metal nanostructure with characteristic size L ≪ c/ω in an alternating electric field with frequency ω ≫ v F /L.
The surface contribution (21) should be considered in conjunction with the bulk contribution to the absorbed power. In fact, both contributions can be combined in the general expression (6) for absorbed power by modifying the scattering rate in the Drude dielectric function ε
(ω) = ε i (ω) − ω 2 p /ω(ω + iγ) as γ = γ 0 + γ s , where γ s = 3v F 4 dS|E n | 2 dV |E| 2 ,(22)
is the surface scattering rate. Then, the surface contribution Q s , Eq. (21), is obtained as the first-order term in the expansion of full absorbed power Q, Eq. (6), over γ s , implying that the surface scattering rate enters on par with its bulk counterpart into the metal dielectric function. While γ s is independent of the local field strength, it does depend strongly on its polarization relative to the interface and, in fact, represents the averaged over the surface local scattering rate. Finally, let us show that the full plasmon decay rate Γ due to both bulk and surface scattering is still given by the general expression (1), but with modified dielectric function ε(ω) that now includes the surface scattering rate (22). Indeed, using Eqs. (21) and (4), the surface contribution to Γ, i.e., the LD rate, takes the form
Γ s = Q s U = 2ω 2 p γ s ω 3 ∂ε ′ (ω) ∂ω −1 .(23)
The same expression is obtained by expanding Eq. (1) [with modified ε(ω)] to the first order in γ s . For ω well below the interband transitions onset, the LD rate and surface scattering rate coincide, Γ s ≈ γ s .
IV. EVALUATION OF SURFACE SCATTERING RATES FOR SPECIFIC GEOMETRIES
The strong polarization dependence of the surface absorbed power Q s and surface scattering rate γ s makes it possible to manipulate, in a wide range, the hot carrier excitation efficiency by realigning the electric field orientation [10]. This effect can be described by surface enhancement factor M defined as the ratio of the full absorbed power, Q = Q b + Q s , to the bulk one, Q b . Within RPA, the enhancement factor takes a simple form M = 1 + γ s /γ 0 , i.e., it is completely determined by the surface scattering rate.
Using our model, surface scattering rates for nanostructure of arbitrary shape can be evaluated directly from the local fields, without further resorting to quantummechanical calculations. In this section, we employ our main result Eq. (22) to evaluate γ s for some common structures: spherical particles, cylindrical wires, and spheroidal particles (nanorods and nanodisks).
We start with recasting the surface scattering rate (22) as the ratio of two surface integrals,
γ s = 3v F 4 dS|∇ n Φ| 2 dSΦ * ∇ n Φ ,(24)
where the real part of the denominator is implied. This representation is especially useful for systems, whose shape permits separation of variables, and, as we show below, it yields analytical results for some common structures, such as nanorods and nanodisks, which so far eluded any attempts of quantum-mechanical evaluation of γ s .
A. Spherical particles and cylindrical wires
Let us first apply Eq. (24) to the simplest case of a sphere of radius a. In the quasistatic limit, the potentials inside the sphere are given by regular solutions of the Laplace equation, Φ ∝ r l Y lm (r), where r is the radial coordinate and Y lm (r) are the spherical harmonics (l and m are, respectively, the polar and azimuthal numbers). Then, a straightforward evaluation of Eq. (24) recovers the surface scattering rate for the lth mode [45]:
γ l sp = 3lv F 4a .(25)
The same rate is obtained for the lth transverse mode in an infinite cylindrical nanowire of radius a.
B. Nanorods and nanodisks
Nanorods and nanodiscs are often modeled by prolate and oblate spheroids, respectively. Here we distinguish between longitudinal and transverse modes oscillating along the symmetry axis and within the symmetry plane, characterized, respectively, by semiaxes a and b (see Fig. 2). Using Eq. (24), the surface scattering rate for all modes can be found in an analytical form (see Appendix C), but here only the results for the dipole modes are presented. For a nanorod (prolate spheroid) with aspect ratio b/a < 1, we obtain the following rates for longitudinal and transverse polarizations, respectively: where α = arccos(b/a) is the angular eccentricity. For a nanodisk (oblate spheroid) with b/a > 1, the rates (26) apply with α = i arccosh(b/a). Note that the CS rate for a spheroidal particle is [65]
γ L s = 3v F 4a 3 2 tan 2 α 2α sin 2α − 1 , γ T s = 3v F 4a 3 4 sin 2 α 1 − 2α tan 2α ,(26)γ cs = v F S 4V = 3v F 8a 1 + 2α sin 2α .(27)
C. Numerical Results
Here we present calculated surface scattering rates for spheroidal particles as the system shape evolves, with changing aspect ratio b/a, from a needle to a pancake. In Fig. 2, we plot the rates (26) normalized by the dipole mode rate γ sp = 3v F /4a for spherical particle of radius a. At the sphere point a = b, the normalized rates for prolate and oblate spheroids continuously transition into each other (e.g., PL to OL and PT to OT), and depending on the mode polarization, exhibit dramatic differences in behavior with changing aspect ratio. The normalized rate for the PL mode decreases with reducing b/a, in sharp contrast to the CS rate, which shows the opposite trend. In the needle limit b/a ≪ 1, the PL mode rate depends linearly on b,
γ PL s ≈ 9πv F b 16a 2 ,(28)
while both the PT mode rate and CS rate are inversely proportional to b,
γ PT s ≈ 9πv F 32b , γ cs ≈ 3πv F 16b .(29)
The similar behavior γ PT s and γ cs for b/a ≪ 1 originates the fact that random ballistic scattering is dominated by the shortest system length. Note, however, that the former exceeds the latter (γ PT s /γ cs → 3/2) since directional scattering is more efficient than random one.
For nanodisks (b/a > 1), the above trends are reversed: With increasing b/a, as the nanodisk flattens, the normalized rates are increasing for the longitudinal (OL) mode and decreasing for the transverse (OT) mode (see Fig. 2). In the pancake limit b/a ≫ 1, the OL mode rate and CS rate are dominated by the pancake height a, which is now the shortest length,
γ OL s ≈ 9v F 8a , γ cs ≈ 3v F 8a ,(30)
with their ratio γ OL s /γ cs → 3, while the OT mode rate exhibits a more complicated behavior:
γ OT s ≈ 9v F a 16b 2 ln 2b a 2 − 1 .(31)
To highlight the role of the local fields in surface scattering, we show in Fig. 3 the evolution, with changing b/a, of γ s for all modes, normalized by the CS rate γ cs .
Here, we have γ s < γ cs for the field polarization mostly tangential to the system boundary (PL and OT modes), and γ s > γ cs for mostly normal polarization (PT and OL modes). Note that recent measurements [43] in cylindershaped nanorods and nanodisks revealed strong polarization dependence of the plasmon spectrum linewidth.
V. CONCLUSIONS
In conclusion, let us discuss the assumptions and approximations we made in deriving the surface scattering rate (2). First, we assumed that the metal nanostructure is characterized by a single metal-dielectric interface. Our model can be straightforwardly extended to systems with two or more interfaces, such as, e.g., core-shell particles of various shapes or onion-like structures, by including each interface contribution in the matrix element M s αβ [see Eq. (11)]. Importantly, the surface contribution Q s to the absorbed power, containing |M s αβ | 2 [see Eq. (9)], will now include interference terms due to carrier scattering between the interfaces. For two interfaces that are sufficiently close to each other, such interference terms would lead to coherent oscillations (quantum beats), with the period v F /ω, of the surface scattering rate with changing interface separation. Such oscillations were recently identified and studied in detail for spherical metal nanoshells with dielectric core [68].
We considered systems with characteristic size L significantly larger than the nonlocality scale v F /ω [60,61] (i.e., with L at least several nm large), and with electron level spacing much smaller than the optical energy ω. Correspondingly, we disregarded quantum confinement effects that dominate optical response of small metal clusters. Specifically, the large electron level spacing in nanometer-sized clusters leads to oscillatory behavior of the resonance width [52,53] (not to be confused with the above coherent oscillations [68]) that should be visible, e.g., in small aspect ratio behavior of γ s in Figs. 2 and 3. Such effects are best described within the TDLDA approach [52][53][54][55][56][57][58][59] and are out of scope of this paper.
Finally, let us discuss the effect of realistic confining potential profile on the surface scattering rate. While the hard-wall approximation is often used for systems larger than several nm [44][45][46][47][48][49][50][51], recent TDLDA calculations for spherical particles indicate that, even for relatively large systems, deviations of the surface barrier from rectangular shape do affect the overall magnitude of the plasmon decay rate [55,56,58]. Importantly, the potential profile has distinct effects on the rapidly-oscillating electron wave-functions and slowly-varying plasmon local fields, which both determine the transition matrix element (11). While, within TDLDA, the Kohn-Sham wave functions are directly determined by the (self-consistent) confining potential, the local fields are, instead, defined solely by the induced charge density via the (screened) Coulomb potential and, therefore, depend on the confining potential indirectly. Hence, the deviation of E n from its classical behavior across the interface is determined by the electron density spillover over the classical (hard wall) boundary [73], and, therefore, is largely independent of the system overall shape. Furthermore, recent TDLDA studies of relatively large (up to 10 nm) nanoparticles revealed [55,56] that the main impact on plasmon linewidth comes precisely from the electron density tail and dielectric environment, implying that it is the plasmon local field near the interface, rather than electron wave functions, that chiefly determines the plasmon decay rate magnitude in real structures. We now note that we employed the hard-wall approximation only for evaluation of the e-h correlation function (13), while retaining explicit local field dependence in the surface scattering rate. Therefore, for general shape systems, the latter can still be obtained, in a good approximation, from Eq. (2) using the classical local fields, but with the constant A calculated self-consistently for some specific (e.g., spherical) system geometry, i.e., A ≈ 0.32 [55,56].
In summary, we developed a quantum-mechanical theory for Landau damping of surface plasmons in metal nanostructures of arbitrary shape. We derived an explicit expression for the surface scattering rate that can be included, on par with the bulk scattering rate, in the metal dielectric function. The rate is strongly dependent on the local field polarization, and is highly sensitive to the system geometry. Our results can be used for calculations of hot carrier generation rates in photovoltaics and photochemistry applications.
Evaluation ofFω
To evaluate ρ nn ′ (ǫ; s, s ′ ) = 2ImG 0 nn ′ (s − s ′ ), we use the fact that the size of characteristic region dominating surface integrals in the correlation function F is |s−s ′ | ∼ v F /ω ≪ L, and compute normal derivatives of G 0 (r −r ′ ) with respect to the tangent plane z = 0,
G zz ′ (ǫ, s − s ′ ) = 2 ∂ ∂z ∂ ∂z ′ G 0 (ǫ, r − r ′ ) z,z ′ =0 = −2 ∂ 2 ∂z 2 G 0 (ǫ, r − r ′ ) z,z ′ =0 . (B5)
Introducing notations r = √ s 2 + z 2 , we write
∂ 2 ∂z 2 G 0 (ǫ, r) = ∂r ∂z 2 ∂ 2 ∂r 2 + ∂ 2 r ∂z 2 ∂ ∂r G 0 (ǫ, r), (B6)
and, in the limit z = 0, we obtain
∂ 2 ∂z 2 G 0 (ǫ, r) z=0 = 1 s ∂ ∂s G 0 (ǫ, s),(B7)
yielding
ρ zz (ǫ, s) = m π 2 s ∂ ∂s sin k ǫ s s . (B8)
To evaluateF ω , we note that for L ≫ v F /ω, the surface integral can be replaced by integral over the tangent plane,
F ω = m 2 π 2 4 dǫf ω (ǫ) d 2 s s 2 ∂ ∂s sin k ǫ s s ∂ ∂s sin k ǫ+ ω s s = m 2 4π 4 dǫf ω (ǫ) k ǫ k ǫ+ ω k 2 ǫ + k 2 ǫ+ ω (B9) − k 2 ǫ+ ω − k 2 ǫ 2 arctanh k ǫ k ǫ+ ω .
The function f ω (ǫ) = f (ǫ)−f (ǫ+ ω) restricts the energy integral to the interval of width ω, and, after rescaling the integration variable, we obtain
F ω = ω 2m 4 E 2 F π 8 g( ω/E F ),(B10)
where the function g(ξ) = is normalized to unity, g(0) = 1. Then, we obtain
Q s = e 2 2π 2 E 2 F ( ω) 2 g( ω/E F ) dS|E n | 2 .(B12)
Finally, for optical frequency well below the Fermi energy, ω/E F ≪ 1, and using the relation ω 2 p = 4πe 2 n/m = 4e 2 k 3 F /3πm, where n is the electron concentration, we arrive at surface contribution to the absorbed power:
Q s = 3v F 32π ω 2 p ω 2 dS|E n | 2 . (B13)
Appendix C: Scattering rate for separable shapes
For system geometries that allow separation of variables, we present the potential as Φ(r) = R(ξ)Σ(η, ζ), where ξ is the radial (normal) coordinate and the pair (η, ζ) parametrizes the surface. With surface area element dS = h η h ζ dηdζ and normal derivative ∇ n = h −1 ξ (∂/∂ξ), where h i are the scale factors (i = ξ, η, ζ), the surface scattering rate takes the form
γ s = 3v F 4 R ′ (ξ) R(ξ) dηdζ(h η h ζ /h 2 ξ )|Σ| 2 dηdζ(h η h ζ /h ξ )|Σ| 2 .(C1)
Below we evaluate γ s for a spheroidal particle. Spheroidal metal nanoparticles exhibit longitudinal and transverse plasmon modes with electric field oscillating, respectively, along the axis of symmetry (semiaxis a) and within the symmetry plane (semiaxis b). Inside the prolate spheroid (b/a < 1), the potential has the form Φ n (r) ∝ P |m| l (ξ)Y lm (η, φ), where P m l (x) is the Legendre function of the first kind. Spheroid surface corresponds to ξ = a/f where f = √ a 2 − b 2 is half the distance between the foci, and the scale factors are given by
h ξ = f ξ 2 − η 2 ξ 2 − 1 , h η = f ξ 2 − η 2 1 − η 2 , h φ = f (ξ 2 − 1)(1 − η 2 ).(C2)
The surface area and volume of the prolate spheroid are
S = 2π b 2 + abα sin α , V = 4π 3 b 2 a,(C3)
are the normalized (to spherical shape) rates. Within the CS model, the decay rate has the form γ cs = v F S/4V = (3v F /4a)f cs , where f cs = aS 3V = 1 2 1 + 2α sin 2α
.
(C6)
The rates for the oblate spheroid (b/a > 1) are described by the above expressions with α = i arccosh(b/a).
FIG. 1 .
1Schematics for surface-assisted excitation of an e-h pair with energy ω. (a) An external optical field incident on a metal nanostructure of characteristic size L, (b) excites a surface plasmon that decays into an e-h pair, (c) accompanied by momentum relaxation via carrier surface scattering in a small region of size vF /ω ≪ L.
FIG. 2 .
2Normalized surface scattering rates for prolate and oblate spheroids are shown with changing aspect ratio b/a along with the CS rate. Insets: Schematics of plasmon modes' polarizations.
FIG. 3 .
3Surface scattering rates for prolate and oblate spheroids normalized by the CS rate are shown with changing aspect ratio b/a.
where α = arccos(b/a) is the angular eccentricity. A straightforward evaluation of Eq. (C1) yields: and transverse dipole modes, i.e., (lm) = (10) and (lm) = (11), respectively, we obtain γ L,Ts = (3v F /4a)f L,T , where f L = 3 2 tan 2 α 2α sin 2α − 1 , f T = 3 4 sin 2 α 1 − 2α tan 2α ,
Appendix A: Transition matrix elementTo extract surface contribution to the matrix elementwe first apply the Hamiltonian H = −( 2 /2m)∆ to the wave function product aswhere ǫ αβ = ǫ α −ǫ β is the excitation energy, and summation over repeating indices µ = (x, y, z) is implied. After integrating by parts, the matrix element takes the formwhere eE µ = −∇ µ Φ are the electric field components, and we used that ∇ µ E µ = 0 inside the metal and ψ α vanish at the boundary S. Applying again the Hamiltonian H to Eq. (A3), we writeIntegrating the first term by parts yields the surface contributionwhile the rest represents the bulk contribution, which can be manipulated to the formSince the local fields change smoothly on the Fermi wavelength scale, the bulk contribution is negligibly small. Noting that only normal derivatives of ψ α do not vanish at the (hard-wall) boundary S, the surface contribution to matrix element takes the form The electron Green function in a hard-wall potential well can be presented as an infinite series in reflections from the boundary as (suppressing energy dependence)[72]G(r,dSG n (r, s)G 0 (s−r ′ ), (B1)where G n (r, s) ≡ ∇ n G(r, s) is the normal derivative of the Green function at surface point s on the boundary inner side, satisfying(B2) Here, G 0 (ǫ, r) = (m/2π 2 )e ikǫr /r, with k ǫ = √ 2mǫ/ , is the free electron Green function, andis its symmetric normal derivative at the inner (s − ) and outer (s + ) boundary sides. Iterations of this system lead to the multiple-reflection expansion[72]. For characteristic cavity size L ≫ λ F , the leading contribution comes from the direct and single-reflection paths [first term in Eq. (B2)], while the higher-order terms account for multiple reflections due to the surface curvature R ∼ L, and are suppressed by powers of λ F /R[72]. Since the Fermi wavelength in metals is small, λ F < 1 nm, the higher-order terms can be disregarded. The equation for G nn ′ (s, s ′ ) is obtained by taking the normal derivative of Eq. (B2). Keeping only the first term, we obtain G nn ′ (s, s ′ ) = 2G 0 nn ′ (s − s ′ ).
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| []
|
[
"Origins of De Novo Genes in Human and Chimpanzee",
"Origins of De Novo Genes in Human and Chimpanzee"
]
| [
"J Ruiz-Orera ",
"J Hernandez-Rodriguez ",
"C Chiva ",
"E Sabidó ",
"I Kondova ",
"R Bontrop "
]
| []
| [
"PLoS Genet"
]
| The birth of new genes is an important motor of evolutionary innovation. Whereas many new genes arise by gene duplication, others originate at genomic regions that did not contain any genes or gene copies. Some of these newly expressed genes may acquire coding or noncoding functions and be preserved by natural selection. However, it is yet unclear which is the prevalence and underlying mechanisms of de novo gene emergence. In order to obtain a comprehensive view of this process, we have performed in-depth sequencing of the transcriptomes of four mammalian species-human, chimpanzee, macaque, and mouse-and subsequently compared the assembled transcripts and the corresponding syntenic genomic regions. This has resulted in the identification of over five thousand new multiexonic transcriptional events in human and/or chimpanzee that are not observed in the rest of species. Using comparative genomics, we show that the expression of these transcripts is associated with the gain of regulatory motifs upstream of the transcription start site (TSS) and of U1 snRNP sites downstream of the TSS. In general, these transcripts show little evidence of purifying selection, suggesting that many of them are not functional. However, we find signatures of selection in a subset of de novo genes which have evidence of protein translation. Taken together, the data support a model in which frequently-occurring new transcriptional events in the genome provide the raw material for the evolution of new proteins.process. Here we have examined RNA-Seq data from 8 mammalian species in order to define a set of putative newly-born genes in human and chimpanzee and investigate what drives their expression. This is the largest-scale project to date that tries to address this scientific question. We have found thousands of transcripts that are human and/or chimpanzee-specific and which are likely to have originated de novo from previously nontranscribed regions of the genome. We have observed an enrichment in transcription factor binding sites in the promoter regions of these genes when compared to other species; this is consistent with the idea that the gain of new regulatory motifs results in de novo gene expression. We also show that some of the genes encode new functional proteins expressed in brain or testis, which may have contributed to phenotypic novelties in human evolution. | 10.1371/journal.pgen.1005721 | null | 434,606 | 1507.07744 | 1f385dde548fbe27fc31355354e142ec80101b92 |
Origins of De Novo Genes in Human and Chimpanzee
2015
J Ruiz-Orera
J Hernandez-Rodriguez
C Chiva
E Sabidó
I Kondova
R Bontrop
Origins of De Novo Genes in Human and Chimpanzee
PLoS Genet
11121005721201510.6084/m9Received: July 28, 2015 Accepted: November 11, 2015RESEARCH ARTICLE 1 / 24 Editor: James Noonan, Yale University, UNITED STATES Data Availability Statement: Sequencing data is deposited in the Gene Expression Omnibus under accession number GSE69241. De novo gene annotation files are available at Figshare entries Funding: The main grant was BFU2012-36820 from the Spanish Government, which was co-funded by the European Regional Development Fund (FEDER). Another grant was from Instituto de Salud Carlos III, Gobierno de España, grant number PT13/0001. We also received funds from Agència de Gestió d'Ajuts
The birth of new genes is an important motor of evolutionary innovation. Whereas many new genes arise by gene duplication, others originate at genomic regions that did not contain any genes or gene copies. Some of these newly expressed genes may acquire coding or noncoding functions and be preserved by natural selection. However, it is yet unclear which is the prevalence and underlying mechanisms of de novo gene emergence. In order to obtain a comprehensive view of this process, we have performed in-depth sequencing of the transcriptomes of four mammalian species-human, chimpanzee, macaque, and mouse-and subsequently compared the assembled transcripts and the corresponding syntenic genomic regions. This has resulted in the identification of over five thousand new multiexonic transcriptional events in human and/or chimpanzee that are not observed in the rest of species. Using comparative genomics, we show that the expression of these transcripts is associated with the gain of regulatory motifs upstream of the transcription start site (TSS) and of U1 snRNP sites downstream of the TSS. In general, these transcripts show little evidence of purifying selection, suggesting that many of them are not functional. However, we find signatures of selection in a subset of de novo genes which have evidence of protein translation. Taken together, the data support a model in which frequently-occurring new transcriptional events in the genome provide the raw material for the evolution of new proteins.process. Here we have examined RNA-Seq data from 8 mammalian species in order to define a set of putative newly-born genes in human and chimpanzee and investigate what drives their expression. This is the largest-scale project to date that tries to address this scientific question. We have found thousands of transcripts that are human and/or chimpanzee-specific and which are likely to have originated de novo from previously nontranscribed regions of the genome. We have observed an enrichment in transcription factor binding sites in the promoter regions of these genes when compared to other species; this is consistent with the idea that the gain of new regulatory motifs results in de novo gene expression. We also show that some of the genes encode new functional proteins expressed in brain or testis, which may have contributed to phenotypic novelties in human evolution.
Author Summary
For the past 20 years scientists have puzzled over a strange-yet-ubiquitous genomic phenomenon; in every genome there are sets of genes which are unique to that particular species i.e. lacking homologues in any other species. How have these genes originated? The advent of massively parallel RNA sequencing (RNA-Seq) has provided new clues to this question, with the discovery of an unexpectedly high number of transcripts that do not correspond to typical protein-coding genes, and which could serve as a substrate for this Introduction New genes continuously arise in genomes. Recent evolutionary 'inventions' include small proteins that have functions related to the adaptation to the environment, such as antimicrobial peptides or antifreeze proteins, which have independently evolved in different groups of organisms [1,2]. A well-studied process for the formation of new genes is gene duplication and subsequent sequence divergence [3,4]. However, in recent years another important mechanism for the birth of new functional genes has been discoveredde novo gene emergence [5][6][7]. As deduced by comparisons to the genomic syntenic regions in other species, these genes derive from previously non-genic regions of the genome [8][9][10][11][12][13][14]. Genes that have recently evolved de novo are characterized by their lack of homologous genes in other species and, contrary to duplicated genes, they can evolve without the limitations which constrain sequences that have high similarity to a pre-existing gene [15]. Despite their recent origin, it has been shown that de novo Drosophila genes can quickly become functionally important [13,16].
Species or lineage-specific genes, which are often called orphan genes, have been described in a wide range of organisms, including yeast [9,17,18], primates [12,[19][20][21], rodents [10,11,22], insects [8,[23][24][25], and plants [26,27]. These studies based on annotated proteincoding genes have revealed that orphan genes tend to have a simple gene structure, a short protein size, and are preferentially expressed in one tissue [28,29]. As orphans lack homologues in other species, many of these genes are likely to have arisen de novo. Some of these proteins have been functionally characterized. One example is the hominoid-specific antisense gene, NCYM, which is over-expressed in neuroblastoma; this gene inhibits the activity of glycogen synthase kinase 3β (GSK3β), which targets NMYC for degradation [30].
Massively parallel RNA sequencing (RNA-Seq) has revealed that a large fraction of the genome extending far beyond the set of annotated genes is transcribed [31,32] and possibly translated [33][34][35][36][37]. Many genes that are annotated as long non-coding RNAs (lncRNAs) are lineage-specific and display high transcriptional turnover [38,39]. The high transcriptional activity of the genome provides abundant raw material for the birth of new genes. Indeed, the use of transcriptomics data has led to the discovery of an unexpectedly high number of recently emerged genes in yeast [33] and Drosophila [40,41]. As most of these genes show little evidence of selection, they have been called 'protogenes' [33]. The products resulting from the expression of protogenes become exposed to natural selection. If useful, they will be retained and continue to evolve under selective constraints [29,42,43].
Here we use transcriptomics data from four mammalian species to quantify the amount of transcription that is human and/or chimpanzee-specific and investigate the molecular mechanisms driving the expression of these transcripts. The data is used to assemble transcripts and identify both annotated and novel genes. The majority of de novo genes originate from regions with conserved genomic synteny in macaque. Analysis of these regions reveals that the expression of the genes is associated with the gain of novel regulatory motifs in the promoter region and U1snRNP splice sites downstream of the transcription start site. We also show that at least a subset of the newly evolved genes is likely to encode functional proteins.
Results
Assembly of annotated and novel transcripts from strand-specific RNA-Seq data We used strand-specific sequencing of polyadenylated RNA (polyA+ RNA-Seq) from several tissues from human, chimpanzee, macaque, and mouse, to perform transcript assembly with Cufflinks [44]. The total number of RNA-Seq datasets was 43, of which 26 were generated in this study and the rest were public datasets from previous studies [20,38,45]. The set of tissues sampled included testis and brain; these tissues have been found to be enriched in de novo genes [20,46]. In this study, we will use the term 'gene' to refer to the set of transcripts merged into a single locus by Cufflinks. Any genome unmapped reads were assembled de novo with Trinity for the sake of completeness [47].
Subsequently, we selected transcripts longer than 300 nucleotides (nt). This excluded any sequencing artifacts resulting from one single amplified paired end read (2x100 nt). We also filtered out all genes with a per-base read coverage lower than 5 to ensure transcript completeness (see Materials and Methods). A negative control lacking reverse transcriptase in the library construction step (RT-) indicated that the probability of a transcript to have resulted from DNA contamination was very low, virtually 0 in the case of multiexonic transcripts. To ensure a highly robust set of transcripts we filtered out intronless genes. This also removed possible promoter-or enhancer associated transcripts (PROMPTS and eRNAs). As a result of this process, we recovered 99,670 human, 102,262 chimpanzee, 93,860 macaque and 85,688 mouse transcripts merged in 34,188 human, 35,915 chimpanzee, 34,427 macaque, and 31,043 mouse gene loci. This included a large fraction of the long multiexonic genes annotated in Ensembl plus a significant number of non-annotated genes (Fig 1a). The number of annotated genes was much higher in human and mouse than in chimpanzee and macaque, mostly due to differences in the number of annotated lncRNAs. About 48% of the human genes not annotated by Ensembl matched genes assembled in recent large-scale RNA-Seq studies [38,48] (S1 Fig). Unsurprisingly, novel genes were shorter and expressed at lower levels than annotated genes (Fig 1b and 1c, respectively). In humans, unannotated genes represented 0.5-2% of the transcriptional cost depending on the tissue, as measured in terms of sequencing reads.
Identification of de novo genes in human and chimpanzee
Next, we used BLAST-based sequence similarity searches [49] to identify the subset of de novo genes that could have originated in human, chimpanzee, or the common ancestor of these two species since the divergence from macaque (hominoid-specific genes). These genes lacked homologues in other species after exhaustive searches against the transcript assemblies described above, the transcript assemblies obtained using previously published non-stranded single read RNA-Seq data for nine vertebrate species [50], Ensembl gene annotations for the same set of species, and the complete expressed sequence tag (EST) and non-redundant (nr) protein databases from NCBI. We also employed genomic alignments to discard any transcripts expressed in syntenic regions in other species that could have been missed by BLAST ( S2 Fig). This pipeline identified 634 human-specific genes (1,029 transcripts), 780 chimpanzee-specific genes (1,307 transcripts), and 1,300 hominoid-specific genes (3,062 transcripts). Taken together, the total number of candidate de novo genes was 2,714 (5,398 transcripts) (Fig 2a). The rest of genes will be referred to as conserved genes.
As we used strand-specific RNA sequencing, we could unambiguously identify a large number of antisense transcripts. Many of them were located within intronic regions (38.31%) and others partially overlapped exonic regions of other genes (10.62%). The rest of de novo transcripts were located in intergenic regions (51.07%). These percentages were similar for human, chimpanzee, and hominoid-specific genes (Fig 2b). Eight de novo genes from human and/or chimpanzee matched annotated protein-coding genes (S1 Table). One example was GTSCR1 (Gilles de la Tourette syndrome chromosome region, candidate 1), encoding a 137 amino acid long protein with proteomics evidence. Curiously, the human protein-coding genes in this set, including GTSCR1, were annotated as long non-coding RNAs (lncRNAs) in a subsequent Ensembl version (77). About 20% of de novo genes matched annotated lncRNAs or sequence entries in the 'EST' or 'nr' databases (Fig 2c). De novo transcripts had a similar distribution along the chromosomes than the rest of assembled transcripts (S3 Fig). Transcripts from de novo genes were shorter and expressed at lower levels than those from conserved genes (S4 Fig). These biases have also been noted in young annotated primate proteincoding genes [12,20]. In general, de novo genes were located in regions with conserved synteny in macaque (> 75% S5 Fig), the proportion being similar to that observed for phylogenetically Fig 2. Identification and characterization of de novo genes in human and chimpanzee. a) Simplified phylogenetic tree indicating the nine species considered in this study. In all species we had RNA-Seq data from several tissues. Chimpanzee, human, macaque and mouse were the species for which we performed strand-specific deep polyA+ RNA sequencing. We indicate the branches in which de novo genes were defined, together with the number of genes. b) Categories of transcripts in de novo genes based on genomic location. Intergenic, transcripts that do not overlap any other gene; Overlapping antisense, transcripts that overlap exons from other genes in the opposite strand; Overlapping intronic, transcripts that overlap introns from other genes in the opposite strand, with no exonic overlap. c) Classification of de novo genes based on existing evidence in databases. Annotated; genes classified as annotated in Ensembl v.75; EST/nr; nonannotated genes with BLAST hits (10 −4 ) to expressed sequence tags (EST) and/or non-redundant protein (nr) sequences in the same species. Novel; rest of genes. d) Patterns of gene expression in four tissues. Brain refers to frontal cortex. Transcripts with FPKM > 0 in a tissue are considered as expressed in that tissue. In red boxes, fraction of transcripts whose expression is restricted to that tissue (τ > 0.85, see Methods). Chimp conserved, transcripts assembled in chimpanzee not classified as de novo. Human conserved, transcripts assembled in human not classified as de novo. e) Number of testis GTEx samples with expression of de novo and conserved genes. We considered all annotated genes with FPKM > 0 in at least one testis sample. Conserved, genes sampled from the total pool of annotated genes analyzed in GTEx with the same distribution of FPKM values than in annotated de novo genes (n = 200). conserved genes. De novo transcripts were enriched in transposable elements; about 20% of their total transcript length was covered by transposable elements, whereas only 8% was covered in conserved genes (S6 Fig). An enrichment in transposable elements was previously observed in primate-specific protein-coding genes [12] as well as in lncRNAs in general [51].
De novo genes are enriched in testis
We determined which genes were expressed in different human and chimpanzee tissues using the RNA-Seq data. The vast majority of de novo transcripts were expressed in testis (93.8-94.5%), as were transcripts from phylogenetically conserved genes (Fig 2d). In contrast, in brain, liver and heart, transcripts from de novo genes were underrepresented when compared to transcripts from conserved genes. This enrichment in testis has also been observed for mammalian lncRNAs [38,45,52]. It does not appear to be the result of increased capacity to detect weakly expressed genes in this tissue, as deduced from the overall distribution of gene expression values in testis compared to other tissues (S7 Fig). It was previously reported that young human protein-coding genes were enriched in the brain [46], but we did not detect a similar bias in our data.
As a result of the aforementioned differential expression patterns, de novo genes were twice as likely to show testis-restricted expression than the rest of genes (94.1%-96.4% as opposed tõ 64% of all assembled transcripts, see Material and Methods). The use of gene expression data from GTEx, although limited to human annotated transcripts, produced consistent results (S8 Fig). The majority of de novo genes were detected in all or nearly all the 60 individuals with testis sequencing data in GTEx [53], indicating that they are expressed in a stable manner in the population (Fig 2e).
Signatures of transcription initiation and elongation in de novo genes
Divergent transcription from bidirectional promoters is widespread in eukaryotic genomes [54,55] and leads to the expression of numerous transcripts in antisense orientation, most of them poorly conserved in other species and generally lacking coding potential [56]. It has been proposed that the reuse of existing promoters can be a driving force of new gene origination [57]. We searched for bidirectional promoters by scanning the genome for transcription start sites of antisense transcripts at a distance < 1 Kb. Our hits had an average distance between the two TSSs of about 100 bp, consistent with the presence of a bidirectional promoter (S9 Fig). However, de novo genes were not enriched in bidirectional promoters with respect to the rest of genes (20% versus 29.81%), indicating that this is not the predominant mechanism for de novo gene formation.
Comparison of GC content in the region surrounding the TSSs clearly revealed that de novo genes are more A/T-rich than conserved annotated genes (S10 Fig). We searched for overrepresented transcription factor binding sites in the promoters of de novo genes using the programs PEAKS [58] and HOMER [59] (Fig 3a and 3b). With PEAKS we identified a clear enrichment of sites for CREBP, RFX, and JUN in the first 100 bp upstream of the TSS (p-value < 10 −5 , motif frequency > 20% higher than in other sequence bins). While CREBP (cAMP-responsive element binding protein) and JUN (transcription factor AP1) are general transcriptional activators, RFX (regulatory factor X) has been associated with expression in testis [60,61]. With HOMER we identified the same three motifs as well as two additional motifs (M1, M2) enriched in the first 100 bp upstream of the TSS. M1 and M2 matched the transcription factor TFIIB (RNA polymerase II complex) downstream element (BREd), which has the consensus sequence G/A-T-T/G/A-T/G-G/T-T/G-T/G [62].
We argued that, if the expression of de novo human and chimpanzee genes was at least partly due to the co-option of genomic sequences as active promoters, we should observe a lower frequency of the relevant TFBS in the corresponding syntenic regions in macaque. This is exactly what we found for the five motifs mentioned earlier, whereas no differences in motif frequencies existed for conserved genes (Fig 3c, S11 Fig). This was consistent with the gain of new transcription factor binding sites in the hominoid branches after the split from macaque in the de novo genes. We also noted that the occurrence of transposable elements (Fig 3d) tended to decrease near the TSS of all gene classes except for endogenous retrovirus-derived Overrepresented transcription factor binding sites (TFBS) in the region -100 to 0 with respect to the transcription start site (TSS) in de novo genes. The region from -300 to +300 with respect to the TSS was analysed (n = 3,875). Color code relates to normalized values (highest value is yellow). b) Fine-grained motif density 200bp upstream of the TSS is shown. c) Comparison of motif density in genomic syntenic regions in macaque for de novo transcripts (n = 3,116) and conserved transcripts (n = 4,323, randomly taken human and chimpanzee annotated transcripts not classified as de novo). Significant differences between human/chimpanzee and macaque are indicated; Fisher-test; *, pvalue < 0.05; **, p-value < 0.01. d) Density of the main human transposable elements (TE) families around the TSS of de novo and conserved transcripts. Regions -3 kB to +3 kB with respect to the TSS were analyzed. LTR frequency is higher in the region -100 to +100 in de novo genes when compared to conserved genes (Fisher-test p-value < 10 −18 ). e) Comparison of motif density in promoters with and without long terminal repeat (LTR) in the region -500 to 0 with respect to the TSS. Significant differences in motif density in the -100 bp window are indicated. f) Signatures of transcription elongation in de novo and conserved genes. Density of U1 and PAS motifs in the 500bp region upstream and downstream of the TSS. Comparison of U1 and PAS motif density in genomic syntenic regions in macaque for de novo transcripts (n = 3,116) and conserved transcripts (n = 4,323). There is an increase of U1 motifs in de novo transcripts when compared to macaque (indicated by a black arrow, Fisher-test, p-value = 0.016 for the region +100 to +200). New Genes in Humans long terminal repeats (LTRs), which on average overlapped 13% of the proximal promoters of de novo genes compared to 5% in conserved genes. Further analyses indicated that LTRs tend to contribute CREB motifs (Fig 3e).
Transcription elongation is highly dependent on the presence of U1 small nuclear ribonucleoprotein recognition sites downstream of the TSS, whereas poly(A) sites (PAS) cause transcription termination [63]. The sequences bound by U1 correspond to 5' splice sites (5'ss). As in standard multiexonic mRNAs, de novo genes showed enrichment of U1 sites and depletion of PAS downstream of the TSS. As U1 sites suppress the effect of PAS sites, we predicted that if transcription elongation is restricted to hominoids, we should see an underrepresentation of U1 sites in the corresponding macaque syntenic regions, but not necessarily of PAS sites. We indeed observed this pattern in de novo genes, whereas no differences were detected for conserved genes (Fig 3f). This is consistent with the idea that the gain of U1 sites contributes to the stabilization of de novo genes.
De novo originated proteins
Most de novo genes were not annotated in the databases and their coding status was unclear. We analyzed two coding properties in de novo genes as well as in other sequences: ORF length and ORF coding score. The latter score was based on hexanucleotide frequencies in bona fide sets of coding and non-coding sequences (see Methods). The median length of the longest ORF of each de novo gene was 52 amino acids. De novo predicted proteins were shorter than proteins encoded by annotated coding RNAs (codRNA) with the same transcript length distribution as the set of de novo genes, and comparable to ORFs from similarly sampled intronic sequences (Fig 4a and 4b). In contrast, the coding score of the longest ORF was higher in de novo genes than in intronic ORFs (Wilcoxon test, p-value < 10 −10 ) and comparable to the score for proteins shorter than 100 amino acids in the set of annotated protein-coding genes.
Next we searched for experimental evidence of proteins produced by de novo genes. We employed mass-spectrometry data from a recent study [64], limiting the searches to the same tissues we used for transcript assembly to increase specificity (testis, brain, heart, and liver), and also searched in Proteomics DB [65]. We identified uniquely mapping peptides in 6 de novo genes; 1 human and 5 hominoid-specific genes ( Table 1). All 6 were expressed in testis; one was preferentially expressed in heart. In addition, we detected signatures of translation in 5 human and 10 hominoid-specific de novo genes using available ribosome profiling sequencing data from human brain [66]. Overall, 21 de novo genes had evidence of translation.
Closer inspection of the genes with experimental protein evidence showed that their size (median 76 amino acids) and coding potential (median 0.0414) were in line with the values observed in the rest of de novo genes (Fig 4c and 4d). Specific examples of proteins encoded by de novo genes are shown in Fig 4e and 4f. Two thirds of the ORFs in these genes were truncated in the syntenic region in macaque and none of them were detected in the syntenic region in mouse, consistent with absence of the proteins in these species (S12 Fig). These genes showed significant signatures of purifying selection (Table 2); this was assessed by calculating the fraction of nucleotide substitutions in different gene regions (introns, exons, ORF) with respect to the corresponding macaque syntenic genomic sequences. We tested whether the sequences had a lower number of substitutions than sequences evolving in a neutral or nearly neutral manner (introns), which would indicate purifying selection. We have to consider that this is a conservative test, as selection is not expected to have acted in the macaque branch in de novo genes, and positive selection may increase the number of substitutions counteracting the effect of negative selection. Despite this, signatures of purifying selection could be clearly distinguished in ORFs from the de novo genes with evidence of translation when compared to intronic regions (Fisher-test, p-value < 0.005), as it occurs in coding sequences encoding functional proteins ( Table 2). In contrast, in de novo genes in general there was not a significant decrease in the number of substitutions in the longest ORF when compared to neutrally evolving sequences, suggesting that the majority of these transcripts do not encode functional proteins.
Discussion
We performed a large-scale transcriptomics-based investigation on the emergence of new genes in hominoids. Our strategy was annotation-independent, which allowed us to recover many novel (non-annotated) genes and compare species for which the level of annotation varies greatly. The approach was entirely different from that employed in previous studies in which the initial datasets were composed of annotated protein coding genes in humans that lacked homologous proteins in other species [12,[19][20][21]. We instead focused on new transcriptional events and subsequently analyzed the properties of the transcripts including coding potential and purifying selection signatures. We assembled the transcriptomes from different species to account for differences in the level of annotation, being able to recover a large number of genes likely to have originated very recently.
We employed a polyadenylated RNA sequencing strategy that was based on a combination of high sequencing depth and strand-specific sequencing, with an average of 115 Million mapped reads per sample. After performing exhaustive sequence similarity searches, we identified 2,714 genes which were specific of human, chimpanzee, or their hominoid ancestor. This is more than one order of magnitude greater than the number of human or primate-specific . CodRNA (short), annotated coding sequences sampled as to have the same transcript length distribution as de novo transcripts (n = 1,952). Intron, longest ORF in intronic sequences from annotated genes sampled as to have the same transcript length distribution as novo transcripts (n = 5,000); Proteogenomics-ORFs in de novo transcripts with peptide evidence by mass-spectrometry; Ribosome profiling-ORFs in de novo transcripts with ribosome association evidence in brain. e) Example of hominoidspecific de novo gene with evidence of protein expression from proteogenomics, with RNA-Seq read profiles in two human samples. genes reported in previous studies [12,[19][20][21]. The de novo origin of these genes is supported by the lack of genes expressed in the corresponding syntenic genomic regions of closely related species. We employed a carefully chosen per-base read coverage threshold, which allowed for the full recovery of complete sequences while permitting the detection of transcripts which were expressed at low levels. Our analysis was based on multiexonic genes but we have to consider that many recently evolved genes may not have yet acquired the capacity to be spliced, as shown by several examples in Drosophila [41]. Therefore, there are probably many more de novo genes than those studied here. The de novo genes constituted about 4% of all expressed hsa_00400469 Human Brain 42 novel a Proteogenomics, detection is based on the identification of mass spectrometry peptides with a unique match to an ORF and corrected p-value (q-value) < 0.01 (brain, heart, liver and testis data from [64]). b Ribosome profiling, detection is based on the presence of ribosome profiling reads overlapping the ORF (brain data from [66]).
c Age refers to whether the gene is human-specific or hominoid-specific. d The tissue with preferential expression is indicated, using the RNA-Seq data generated here for human brain, heart, liver and testis. multiexonic genes in human and chimpanzee. This fraction is consistent with similar transcriptomics-based studies in insects [40,24]. As these genes are short and expressed at low levels, their associated transcriptional cost is relatively small. De novo genes showed characteristic promoter and splicing signals and were expressed in a consistent manner across different individuals. However, they had very weak purifying selection signatures in general. This is interesting because it means that even if these genes are expressed in a stable manner, many of them are likely to lack functionality and thus can be considered protogenes [33]. The proportion of de novo genes with conserved genomic synteny in macaque was comparable to that of conserved genes. Given the low number of nucleotide differences in neutrally evolving regions between these two species (~6%), we could reliably use syntenic alignments to examine transcription-related sequence features. Relative to the corresponding genomic regions in macaque, we found an enrichment of transcription factor binding sites and U1snRNP motifs in de novo genes in human and chimpanzee; this is consistent with the idea that the gain of regulatory motifs underlies de novo gene origination. This scenario had been proposed for the formation of a new gene in mouse [7,10] but until now it had not been considered at a genome-wide scale. Interestingly, in addition to general activators and polymerase II binding sites we found an enrichment in RFX motifs in de novo gene promoters. Although there are several members of the RFX transcription factor family that bind to similar sequences, many of the sites in our sequences may be recognized by RFX2, which is highly expressed in testis and has been involved in spermiogenesis [61].
Several studies have found an excess of genes of very recent origin when compared to older gene classes [40,24]. This suggests that many young genes are subsequently lost, which is consistent with the relatively constant number of genes observed in a taxon. Our finding that signatures of purifying selection are generally very weak for de novo genes is indeed consistent with a scenario in which many of these genes are dispensable. However, a subset of genes with evidence of translation do display significant signatures of purifying selection, indicating that they correspond to functional genes. Studies in Drosophila indicate that directional selection determines the fate of some de novo genes from the very early stages [41]. While we focused primarily on possible coding functions, some of the genes may have also acquired non-coding functions. This is especially relevant in the case of antisense transcripts which can potentially influence the expression of the transcript in the opposite orientation [67]. It is important to consider that the annotations alone may not suffice to differentiate between coding and noncoding transcripts, as many annotated lncRNAs may translate short peptides according to ribosome profiling data [34,36,37]. LncRNAs tend to have small open reading frames and display Table 2. Divergence with macaque syntenic regions. Estimated number of substitutions per Kb (PAML). Dataset 3 corresponds to the genes in Table 1. ORF in datasets 1 and 2 is the longest ORF in the transcript. Introns refers to sampled intronic regions of size 500 bp from the same set of transcripts. We tested for differences between complete exons and introns, and ORF and introns with the Fisher test. limited phylogenetic conservation [37,68] and it has been previously proposed they may act as precursors of new protein-coding genes [13,21,37]. An interesting observation was that the coding score of de novo genes was clearly non-random. One possible explanation is that natural selection rapidly eliminates transcripts that produce toxic peptides [35], as one could expect such peptides to often have unusual amino acid compositions.
Here we detected 20 putative new human proteins using ribosome profiling from brain tissue [66]. Considering that the expression of most de novo genes was restricted to testis for which no ribosome profiling data has yet been published, we expect this number to increase substantially in the future. Mass-spectrometry has important limitations for the detection of short peptides [69], but we could nevertheless detect 8 putative proteins, mostly from testis. Our results indicate that the expression of new loci in the genome takes place at a very high rate and is probably mediated by random mutations that generate new active promoters. These newly expressed transcripts would form the substrate for the evolution of new genes with novel functions.
Materials and Methods
Ethics statement
Chimpanzee and macaque samples were obtained from the Primate Bio-Bank of the Biomedical Primate Research Center (BPRC). BPRC offers state-of-the-art animal facilities (AAALAC accredited) and is fully compliant with regulations on the use of non-human primates for medical research. BPRC's Primate Tissue Bank is one of the biggest non-human primate banks in Europe and it is involved in the framework of the EuprimNet Bio-Bank (www.euprim-net.eu). The EUPRIM-Net Bio-Bank is conducted and supervised by the scientific government board along all lines of EU regulations and in harmonization with Directive 2010/63/EU on the Protection of Animals Used for Scientific Purposes. The animals used for tissue collection in all cases are diagnosed with cause of death other than their participation in this study and without any relation to the tissues used.
Library preparation and strand-specific polyA+ RNA-Seq protocol
Human and mouse total RNA was purchased from Amsbio. Chimpanzee and macaque total RNA was extracted using a miRNeasy Mini kit from tissue samples obtained at the Biomedical Primate Research Centre (BPRC, Netherlands). Mouse samples were from a pool of 3 males and 3 females (Balb/C strain).
Libraries were prepared using the TruSeq Stranded mRNA Sample Prep Kit v2 according to the manufacturer's protocol. PolyA+ RNA was purified from 250-500 mg of total RNA using streptavidin-coated magnetic beads (AMPure XP) and subsequently fragmented to~300 bp. cDNA was synthesized using reverse transcriptase (SuperScript II, Invitrogen) and random primers. We did not add reverse transcriptase to one of the human testis replicate samples to use it as a control for DNA contamination (RT-). The strand-specific RNA-Seq library preparation was based on the incorporation of dUTP in place of dTTP in the second strand of the cDNA. Double-stranded DNA was further used for library preparation. Such dsDNA was subjected to A-tailing and ligation of the barcoded Truseq adapters. Library amplification was performed by PCR on the size selected fragments using the primer cocktail supplied in the kit. Sequencing was done with an Illumina HiSeq 2000 sequencer in a paired-end configuration (2x100 nt) according to the manufacturer's instructions. Library preparation and sequencing were done at the Genomics Unit of the Center for Regulatory Genomics (CRG, Barcelona, Spain).
RNA-Seq datasets
The polyA+ RNA-Seq included 96 sequencing datasets for 9 different species: 43 strand-specific paired end data (~3 billion reads) and 53 single read data (~3.2 billion reads). The strandspecific data was employed for the assembly of reference transcripts for human, chimpanzee, macaque, and mouse (Fig 1 for a summary of results). For comparative purposes, we used the same tissues and number of biological samples for human and chimpanzee (liver, heart, brain, and testis; two biological replicates per tissue). For macaque and mouse, we added available strand-specific RNA-Seq data from other tissues: adipose, skeletal muscle for macaque [20], and ovary and placenta for mouse [38,45]. The single read data corresponded to 5 primate species (human, chimpanzee, gorilla, orangutan, and macaque) and 4 additional vertebrates (mouse, chicken, platypus, and opossum) in 6 different tissues (brain, cerebellum, heart, kidney, liver, and testis) [50]. While these experiments were based on single reads and had lower coverage than the strand-specific RNA-Seq data, they were used to increase the number of species with expression data for sequence similarity searches. More information about the samples can be found in S1 Dataset. Sequencing data generated for this study is deposited in the Gene Expression Omnibus under accession number GSE69241.
Read mapping and transcriptome assembly
RNA-Seq sequencing reads underwent quality filtering using Condetri (v.2.2) [70] with the following settings (-hq = 30 -lq = 10). Adapters were trimmed from filtered reads if at least 5 nucleotides of the adaptor sequence matched the end of each read. In all experiments, reads below 50 nucleotides or with only one member of the pair were not considered. We retrieved genome sequences and gene annotations from Ensembl v. 75 [71]. We aligned the reads to the correspondent reference species genome with Tophat (v. 2.0.8) [72] with parameters -N 3, -a 5 and -m 1, and including the correspondent parameters for paired-end and strand-specific reads whenever necessary. Multiple mapping to several locations in the genome was allowed unless otherwise stated.
We performed gene and transcript assembly with Cufflinks (v 2.2.0) [44] for each individual sample. Per-base read coverage and FPKM (fragments per kilobase of transcript per million mapped fragments) values were calculated for each transcript and gene as described by [44]. We only considered assembled transcripts that met the following criteria: a) the transcript was covered by at least 4 reads, b) Abundance was higher than 1% of the most abundant isoform of the gene and, c) <20% of reads were mapped to multiple locations in the genome.
Subsequently, we used Cuffmerge [44] to build a single set of assembled transcripts for each species, always keeping the strand-specific and the single read based RNA-Seq experiments separate. We compared our set of assembled transcripts with gene annotation files from Ensembl (gtf format, v.75) with Cuffcompare [44] to identify transcripts corresponding to annotated genes. This included the categories ' = ' (complete match), 'c' (contained), 'j' (novel isoform), "e", and "o" (other exonic overlaps in the same strand). Genes for which none of the assembled transcripts matched an annotated gene were labeled 'novel'. In human, 82% of the total annotated protein-coding and 44.5% of the non-coding genes (lincRNA, antisense and processed transcripts) were recovered.
Additionally, we ran Trinity [47], which reconstructs transcripts in the absence of a reference genome, with all unmapped reads in each species (read length > = 75 nucleotides). Before running Trinity, unmapped reads were normalized by median using Khmer (parameters -C 20, -k 20, -N 4). This allowed the recovery of any transcripts falling into non-assembled parts of the genome. We selected transcripts with a minimum size of 300 nucleotides.
We obtained a set of reference transcripts from the strand-specific RNA-Seq data using a per-nucleotide read coverage > = 5. This choice was based on the relationship between read coverage and the percentage of fully reconstructed annotated coding regions (CDS, longest one per gene) for the subset of genes mapping to annotated protein coding genes (Ensembl v.75) using only the categories ' = ' and 'c' in Cuffcompare (18,694 protein-coding genes). For values higher than 5 there was no substantial increase in the percentage of fully reconstructed CDS (coverage > = 5: 87.8%; coverage > = 10: 88.5%; coverage > = 20: 89.4%). The selection was based on coding regions and not complete transcripts because of the prevalence of alternative transcription start sites in many annotated transcripts, causing uncertainty in the latter parameter [73]. Very similar results were obtained for CDS shorter than 500 nucleotides or genes with only one annotated CDS, indicating that protein length or gene complexity has little effect on the suitability of this threshold.
Transcript assembly with the RT-control (see above) resulted in 22,803 different sequences that presumably corresponded to genomic DNA contamination, resulting from regions resistant to DNAse treatment. Except for the reverse transcriptase, all other reagents were added in the same concentration as in the other samples. Therefore, the number of contaminant fragments must be considered an upper boundary, as in a normal RNA-Seq experiment these fragments are probably sequenced much less efficiency as they have to compete with the genuine RT products. The sequences obtained in the RT-control did not contain any introns and the majority of them were shorter than 300 nucleotides (98.58%).
Genomic comparisons
Reference transcripts were classified into three categories depending on their location with respect to transcripts from other genes: a) Intergenic: Transcripts that did not overlap any other assembled locus. b) Overlapping intronic: Transcripts located within introns of other assembled genes on the opposite strand. c) Overlapping antisense: Transcripts partially or completely overlapping exons from other assembled genes on the opposite strand.
We downloaded long interspersed element (LINE), short interspersed element (SINE), and long terminal repeat (LTR) annotations in the human and chimpanzee genomes from Repeat-Masker (same genome versions than in Ensembl v.75) [74]. We used BEDTools [75] to identify any overlap between transcripts and/or genomic elements.
We downloaded human-chimpanzee, human-macaque, human-mouse, chimpanzeemacaque and chimpanzee-mouse pairwise syntenic genomic alignments, obtained by blastz [76], from UCSC. We developed an in-house Python script to recover syntenic regions corresponding to a given human or chimpanzee transcript, or to regions upstream and downstream of a human or chimpanzee transcription start site (TSS), using these alignments.
We scanned the human and chimpanzee genomes to identify transcripts with bidirectional promoters. We recovered any antisense pairs in which the distance between the two TSSs was < 1 kb). We estimated that 29.81% of the conserved genes and 20% for de novo genes were expressed from bidirectional promoters. This was significantly higher than the number expected by chance (5,31%, Binomial Test, p-value << 10 −5 ). The location of different types of genes in the human chromosomes was visualized with Circos [77].
Identification of de novo genes
We developed a pipeline to identify de novo genes in human and chimpanzee based on the lack of homologues in other species. We first selected multiexonic transcripts from the reference transcriptome assemblies. Then, we performed exhaustive sequence similarity searches against sequences from other species with the BLAST suite of programs. Subsequently, we searched for overlapping transcripts in genomic syntenic regions.
Sequence similarity searches, using reference human or chimpanzee transcripts as query, were performed against the complete transcriptome assemblies from the nine different vertebrate species, gene annotations from Ensembl v.75 for the same species, and the EST and nonredundant protein "nr" [78] NCBI databases. We employed both BLASTN and TBLASTX programs [49], with an E-value threshold of 10 −4 . All BLAST searches were performed with the filter of low-complexity regions activated; we discarded all transcripts for which self-hits were not reported. Species-specific genes were those for which no transcripts (or transcripts of any paralogs) had sequence similarity hits to transcripts in any other species. To identify syntenybased homologues we took advantage of the existing pairwise syntenic genomic alignments from UCSC. We used data from human, chimpanzee, macaque, and mouse. If two transcripts overlapped (> = 1bp) in a syntenic region we considered it as evidence of homology. We reclassified the de novo genes accordingly.
We identified 634 human-specific genes (1,029 transcripts) and 780 chimpanzee-specific genes (1,307 transcripts). In the case of hominoid-specific genes we allowed for hits to gorilla and orangutan in addition to human and chimpanzee; this yielded 1,300 hominoid-specific genes (3,062 transcripts). About one third of them (221 genes and 1,016 transcripts) were reference transcripts in both species (multiexonic, coverage > = 5) and the rest were identified via the complete transcriptome assemblies, EST, and/or nr databases. Due to the fact that not all of these genes were detected as reference transcripts in both species the number of hominoid-specific genes is different for human and chimpanzee (604 and 916, respectively). Annotation files of de novo genes in GTF format are available at Figshare
Tissue gene expression
We analyzed the patterns of tissue expression in assembled transcripts, considering a transcript as expressed in one tissue if FPKM > 0. We measured the number of tissue-restricted transcripts using a previously proposed metric [79]:
t ¼ X i¼n i¼1 ð1 À x i Þ n À 1
Where n is the number of tissues and x i is the FPKM expression value of the transcript in the sample normalized by the maximum expression value over all tissues. We classified cases with a τ > 0.85 as preferentially expressed in one tissue or as tissue-restricted.
For de novo genes annotated in Ensembl v.75 we obtained expression data from the GTEx project, which comprises a large number of human tissue samples. We used this data to calculate the number of genes showing tissue-restricted expression as well as the number of testis samples with detectable expression of a given gene.
Motif analysis
We searched for significantly overrepresented motifs in de novo and conserved genes using computational approaches. We employed sequences spanning from 300 bp upstream to 300 bp downstream of the transcription start site (TSS). Redundant TSS positions were only considered once. With PEAKS [58] we identified three TRANSFAC motifs [80] enriched in de novo genes, corresponding to CREB, JUN, RFX. HOMER [59], a tool for motif discovery, also detected these motifs plus two additional motifs (M1, M2). The five motifs were enriched in the first 100 bp upstream of the TSS (p-value < 10 −5 , minimum 30 motif occurrences and enrichment > 20% when compared to other regions). M1 and M2 matched the transcription factor TFIIB (RNA polymerase II complex) downstream element (BREd), which has the consensus sequence G/A-T-T/G/A-T/G-G/T-T/G-T/G [62].
For graphical representation of the results, we computed the relative motif density in 100 bp windows upstream and downstream of the TSS in human and chimpanzee, and the corresponding genomic syntenic regions in macaque and mouse. We used MEME [81] to scan the sequences for the occurrence of motifs (matches to weight matrices with a p-value < 10 −5 ). The average number of motif occurrences (motif density) was normalized to values between 0 and 1, where 1 corresponded to the highest density of a given motif in a sequence window.
It has been previously proposed that new genes tend to gain new U1 sites and lose PAS sites as they become more mature [63]. We used MEME with the same parameters as described above to search for U1 (U1 snRNP 5' splice site consensus motif) and PAS (poly-adenylation signals) sites 500 bp upstream and downstream of the TSS (see supplementary material for weight matrices). PAS motifs found < 500bp downstream of a U1 site were not considered since the PAS effect is abolished by snRNPs bound to these U1 motifs at such distances.
Coding score
We defined an open reading frame (ORF) in a transcript as any sequence starting with an ATG codon and finishing at a stop codon (TAA, TAG or TGA). In addition we require it to be at least 75 nucleotides long (24 amino acids), which is the size of the smallest complete human polypeptide found in genetic screen studies [82].
In each ORF we computed a coding score based on hexamer frequencies in bona fide coding and non-coding sequences [37]. Specifically, we first computed one coding score (CS) per nucleotide hexamer:
CS hexamerðiÞ ¼ log freq coding ðhexamerðiÞÞ freq nonÀcoding ðhexamerðiÞÞ
The coding hexamer frequencies were obtained from all human transcripts encoding experimentally validated proteins. The non-coding hexamer frequencies were calculated using the longest ORF in intronic regions which were selected randomly from expressed protein-coding genes. The hexamer frequencies were computed separately for ORFs with different lengths to account for any possible length-related biases (24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50)(51)(52)(53)(54)(55)(56)(57)(58)(59), >60 amino acids). Next, we used the following statistic to measure the coding score of an ORF:
CS ORF ¼ X i¼n i¼1 CS hexamerðiÞ n
Where i is each hexamer sequence in the ORF, and n is the number of hexamers considered.
The hexamers were calculated in steps of 3 nucleotides in frame (dicodons). We did not consider the initial hexamers containing a Methionine or the last hexamers containing a STOP codon. Given that all ORFs were at least 75 nucleotides long, the minimum value for n was 22.
In coding RNAs (CodRNA all) the annotated ORF was selected for further analysis. To account for any possible bias due to transcript length, we randomly selected a subset of protein-coding transcripts (CodRNA short) with the same transcript length distribution as the de novo transcripts. In sequences with no annotated coding sequence (introns and transcripts from de novo genes), we chose the longest ORF considering all three possible frames. The only exception was when the longest ORF in another frame had a higher coding score than expected for non-coding sequences (0.0448 if ORF < 40 aa; 0.0314 if 60 aa > length ORF > = 40 aa; 0.0346 if length ORF > = 60 aa; p-value < 0.05) or if it was longer than expected for non-coding sequences (> = 134 aa, p-value < 0.05). In this very small number of cases (3.4%) we selected this other ORF.
Ribosome profiling data
We downloaded data from ribosome profiling experiments in human brain tissue [66]. Ribosome profiling reads were filtered as described previously [37]. We then used Bowtie2 [83] to map the reads to the human assembled transcripts with no mismatches. We considered each strand independently since the RNA-Seq data was strand-specific. RNA-Seq reads from the same experiment were also mapped to de novo transcripts to determine how many of them were expressed (FPKM > 0). Because of the low detectability of ribosome association at low FPKM expression values [37], two ribosome profiling reads mapping to a predicted ORF were deemed sufficient for the signal to be reported.
Mass spectrometry data
We used available mass-spectrometry data from human frontal cortex, liver, heart, and testis [64,65] to identify any putative peptides produced by de novo genes. Mass-spectrometry data was analyzed using the Proteome Discoverer software v.1.4.1.14 (Thermo Fisher Scientific, United States) using MASCOT v2.5 [84] as a search engine. The database we used contained the human entries in SwissProt [85], the most common contaminants, and putative peptides derived from the translation of transcripts from de novo genes. Carbamidomethylation for cysteines was set as fixed modification whereas acetylation in protein N-terminal and oxidation of methionine were set as variable modifications. Peptide tolerance was 7 ppm in MS and 20mmu in MS/MS mode, maximum number of missed cleavages was set at 3. The Percolator [86] algorithm implemented in the Proteome Discoverer software was used to estimate the qvalue and only peptides with qvalue < 0.01 and rank = 1 were considered as positive identifications. Lastly, we considered unique peptides matching young transcripts by using BLAST with short query parameters to search the candidate peptides against all predicted ORFs in assembled transcripts. Additionally, we searched for any matching peptides in Proteomics DB [65]. We found 6 de novo genes with proteomics evidence; two of them were annotated in Ensembl as lncRNAs and expressed in !55 testis samples from GTEx. Details of the results can be found in the supplementary material.
Calculation of substitution rates
We estimated the number of substitutions per Kb in human-macaque genomic alignments with the maximum likelihood method 'baseml' from the PAML package [87] with model 4 (HKY85). We only analyzed transcripts with complete synteny in both species. We compared the number of substitutions with respect to sequence length in different sequence sets using the Fisher exact test.
Statistical data analyses and plots
The analysis of the data, including generation of plots and statistical test, was done using R [88]. genes. The existence of full or partial synteny was assessed using pairwise genomic alignments from UCSC. Hominoid (inner circle) refers to human when chimpanzee is the reference species and to chimpanzee when human is the reference species. The proportion of de novo and conserved transcripts with full or partial synteny decreases with phylogenetic distance. The proportion of transcripts from de novo genes with complete genomic synteny in macaque was comparable to that of transcripts from conserved genes. 3a in main manuscript file). b) Same data for conserved annotated genes. c) Relative motif frequencies in de novo genes including motifs overrepresented in conserved annotated genes in general but not in de novo genes (NRF, MAZ, EGR-1, E2F). d) Data for the same motifs for conserved annotated genes. (PNG) S12 Fig. Conservation of ORFs in syntenic genomic regions corresponding to de novo genes with experimental evidence of translation. The existence of full or partial synteny was assessed using pairwise genomic alignments from UCSC. Hominoid (inner circle) refers to human when chimpanzee is the reference species and to chimpanzee when human is the reference species. Only ORFs in de novo genes with evidences of preoteogenomics or ribosome profiling are displayed. Non-truncated ORFs are the ones in which the frame, the start codon and the stop codon are conserved in the other syntenic genomic region; otherwise the ORF is truncated. (PNG) S1 Table. De novo genes annotated as protein-coding in Ensembl v. 75. Identification of annotated genes in the set of de novo genes was based on the comparison of the genomic coordinates of the assembled transcripts and the genomic coordinates of annotated genes using Cuffcompare. All these genes were hominoid-specific (expressed both in human and chimpanzee). ( Ã ) refers to the same orthologous gene in human and chimpanzee. Note that all human coding genes had been annotated as different classes of long non-coding RNAs (lncRNAs) in Ensembl v. 77. (DOC) S1 Dataset. Samples and sequence data. It contains five different datasheets (T1-T5). T1. Detailed information on the RNA-Seq samples from this study. T2. Stranded assemblies, information on the transcript assemblies obtained using strand-specific RNA-Seq data. T3. Single assemblies, information on the transcript assemblies obtained using single read RNA-Seq data. T4. Weight matrices, relative nucleotide frequencies of the motif weight matrices used in this study. T5. Mass spectrometry, information on the peptides identified by proteomics. (XLS)
Supporting Information
Fig 1 .
1Global properties of assembled transcriptomes. a) Percentage of annotated and novel genes and transcripts using strand-specific deep polyA+ RNA sequencing. Classification is based on the comparison to reference gene annotations in Ensembl v.75. 70.65 and 87.77% of annotated genes in human and mouse are classified as protein-coding, respectively. Number of genes identified: human 34,188; chimpanzee, 35,915; macaque 34,427; mouse 31,043. Number of transcripts identified: human 99,670; chimpanzee 102,262; macaque 93,860; mouse 85,688. b) Cumulative density of nucleotide length in annotated and novel assembled transcripts. c) Cumulative density of expression values in logarithmic scale in annotated and novel assembled transcripts. Expression is measured in fragments per kilobase per million mapped reads (FPKM) values, selecting the maximum value across all samples. doi:10.1371/journal.pgen.1005721.g001
doi:10.1371/journal.pgen.1005721.g002
Fig 3 .
3Recent signatures of transcription in de novo genes. a)
doi:10.1371/journal.pgen.1005721.g003
Fig 4 .
4Coding potential of de novo genes. a-d) ORF length and coding score for ORFs in different sequence types. De novo gene, longest ORF in de novo transcripts (n = 1,933). CodRNA (all), annotated coding sequences from Ensembl v.75 (n = 8,462)
(f) Example of hominoid-specific de novo gene with RNA-Seq and ribosome profiling read profiles. Predicted coding sequences are highlighted with red boxes and the putative encoded protein sequences displayed. doi:10.1371/journal.pgen.1005721.g004
e
Annotation refers to the classification of the transcripts as novel or annotated in Ensembl v.75. doi:10.1371/journal.pgen.1005721.t001
S1
Fig. Comparisonof human genes assembled in this study and in other published datasets. 'Ruiz-Orera' is this study.'Necsulea' represents genes that match lncRNAs annotated in"Necsulea A, Soumillon M, Warnefors M, Liechti A, Daish T, et al. (2014) The evolution of lncRNA repertoires and expression patterns in tetrapods. Nature 505: 635-640" [38]. 'Iyer' refers to genes that match lncRNAs annotated in "Iyer MK, Niknafs YS, Malik R, Singhal U, Sahu A, et al. (2015) The landscape of long noncoding RNAs in the human transcriptome. Nat Genet 47: 199-208" [48]. (PNG) S2 Fig. Summary of the filters applied to obtain the final list of de novo genes specific of human or chimpanzee. Transcript homology: genes discarded because of homology to transcriptomes (assemblies or annotations) from other species using sequence similarity searches. Synteny: genes discarded because they overlapped other transcripts in genomic syntenic regions. EST/nr: genes discarded because they matched sequences from the EST or nr databases. (PNG) S3 Fig. Circos plot showing the distribution of different types of sequences in the human chromosomes. De novo genes include both human-and hominoid-specific genes. Pseudogenized retrocopies correspond to genes annotated as "processed pseudogenes" in Ensembl. (PNG) S4 Fig. Properties of de novo transcripts when compared to all annotated and novel transcripts. a) Cumulative density of length in species-specific, hominoid-specific, annotated and novel assembled transcripts. b) Log2 cumulative density of expression values in species-specific, hominoid-specific, annotated and novel assembled transcripts. Expression is measured in fragments per kilobase per million mapped reads (FPKM) values, selecting the maximum value across all samples. Collectively, de novo genes had a median size of 595 nucleotides and median expression of 0.31 FPKM. Species-specific transcripts are significantly shorter (Wilcoxon test, p-value <10 −16 ) than hominoid-specific transcripts, but no differences in expression levels are observed. (PNG) S5 Fig. Conservation of syntenic genomic regions corresponding to de novo or conserved
. De novo genes are enriched in transposable elements. Transcrips covered by transposable elements (TEs) considering all annotated transcripts, hominoid-specific genes or species-specific genes (human-or chimpanzee-specific genes). CDS is the annotated coding sequence in annotated protein-coding transcripts and the longest ORF in de novo transcripts.Classes of TEs: LINEs; long interspersed elements; LTRs, long terminal repeats; SINEs, short interspersed elements. a) Average fraction of transcript length covered by TEs. b) Number of transcripts covered by TEs (> = 1bp overlap). (PNG) S7 Fig. Distribution of expression values in assembled genes across tissues. Log10 cumulative density of expression values in assembled genes. Expression is measured in fragments per kilobase per million mapped reads (FPKM) values, selecting the maximum value across all samples. Testis does not show a lack of highly expressed transcripts (actually the opposite is observed for human) that could explain why we detect so many transcripts being expressed in this tissue. (PNG) S8 Fig. Human annotated transcripts from de novo genes are enriched in testis according to GTEx data. Data is for annotated transcripts in the GTEx catalog which are preferentially expressed in one tissue (tissue-restricted), as measured by a tissue preferential expression index higher than 0.85 (see Methods online for more details on this index). (PNG) S9 Fig. Distance between the transcription start site (TSS) of transcripts from de novo genes and the nearest TSS from another transcript, for genes with divergent transcription. These were defined as antisense genes with the TSSs separated by less than 1 kb, potentially sharing a bidirectional promoter. Negative values imply overlap between the transcripts. There is a strong peak at around 100 nucleotides. (PNG) S10 Fig. De novo genes have a low GC content when compared to conserved annotated genes. Nucleotide frequencies 300 bp upstream and 300 downstream of the transcription start site (TSS) were calculated for different sets of transcripts. Conserved: 4,323 randomly taken human and chimpanzee annotated transcripts not classified as de novo. (PNG) S11 Fig. Regulatory motif frequencies around the TSS. a) Number of matches of overrepresented motifs in 100 bp windows in de novo genes and in the corresponding macaque syntenic regions (corresponds to Fig
Table 1 .
1Human de novo genes with evidence of protein translation.Detection
Technique
Assembly gene
ID
Assembly transcript ID
Age c
Tissue d
Protein
length
Annotation e
Proteogenomics a
XLOC_175402
hsa_00362506
Hominoid
Heart
36
LncRNA
(ENSG00000223485)
XLOC_068697
hsa_00142705
Hominoid
Testis
37
Novel
XLOC_085716
hsa_00181285
Hominoid
Testis
64
Novel
XLOC_088783
hsa_00187116, hsa_00187117,
hsa_00187118
Hominoid
Testis
148, 136, 61
LncRNA
(ENSG00000263417)
XLOC_105288
hsa_00223807
Hominoid
Testis
199
Novel
XLOC_196865
hsa_00404039
Human
Testis
49
Novel
Ribosome profiling b
XLOC_002919
hsa_00006742, hsa_00006743,
hsa_00006744
Hominoid
Brain,
Heart
68, 64, 58
Novel
XLOC_031861
hsa_00068400
Human
Brain
58
LncRNA
(ENSG00000273409)
XLOC_042102
hsa_00090118
Hominoid
Brain
90
LncRNA
(ENSG00000257061)
XLOC_050821
hsa_00107269
Human
Brain
56
Novel
XLOC_057303
hsa_00119633
Hominoid
Testis,
Brain
52
Novel
XLOC_073846
hsa_00154236
Hominoid
Brain
54
Novel
XLOC_082421
hsa_00173626, hsa_00173627
Hominoid
All 4
tissues
95, 95
LncRNA
(ENSG00000265666)
XLOC_085590
hsa_00181107, hsa_00181108
Hominoid
Brain,
Testis
89, 83
Novel
XLOC_104066
hsa_00221170
Hominoid
Brain
68
Novel
XLOC_106910
hsa_00227119
Human
Brain
36
LncRNA
(ENSG00000228999)
XLOC_152506
hsa_00317537
Hominoid
Brain
53
LncRNA
(ENSG00000251423)
XLOC_160844
hsa_00333276, hsa_00333277
Hominoid
Brain
65, 65
Novel
XLOC_168602
hsa_00348960
Hominoid
Brain
29
LncRNA
(ENSG00000228408)
XLOC_184660
hsa_00380291
Human
Brain
101
LncRNA
(ENSG00000236197)
XLOC_195038
, http://dx.doi.org/10.6084/m9. figshare.1604892 (human) and http://dx.doi.org/10.6084/m9.figshare.1604893 (chimpanzee).
PLOS Genetics | DOI:10.1371/journal.pgen.1005721 December 31, 2015
PLOS Genetics | DOI:10.1371/journal.pgen.1005721 December 31, 2015 9 / 24
PLOS Genetics | DOI:10.1371/journal.pgen.1005721 December 31, 2015 24 / 24
AcknowledgmentsWe thank the three reviewers of the manuscript for their useful comments. We are also grateful for discussions with many colleagues during the progress of the work. We acknowledge Will Blevins for advice with the text.Author Contributions
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. R Team, R Development Core Team. R A Lang Environ Stat ComputTeam R (2013) R Development Core Team. R A Lang Environ Stat Comput.
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"A consistent discretization of the single-field two-phase momentum convection term for the unstructured finite volume Level Set / Front Tracking method",
"A consistent discretization of the single-field two-phase momentum convection term for the unstructured finite volume Level Set / Front Tracking method"
]
| [
"Jun Liu es:[email protected] \nMathematical Modeling and Analysis\nTechnische Universität Darmstadt\n\n",
"Tobias Tolle [email protected] \nMathematical Modeling and Analysis\nTechnische Universität Darmstadt\n\n",
"Dieter Bothe [email protected] \nMathematical Modeling and Analysis\nTechnische Universität Darmstadt\n\n",
"Tomislav Marić [email protected] \nMathematical Modeling and Analysis\nTechnische Universität Darmstadt\n\n"
]
| [
"Mathematical Modeling and Analysis\nTechnische Universität Darmstadt\n",
"Mathematical Modeling and Analysis\nTechnische Universität Darmstadt\n",
"Mathematical Modeling and Analysis\nTechnische Universität Darmstadt\n",
"Mathematical Modeling and Analysis\nTechnische Universität Darmstadt\n"
]
| []
| We propose the ρLENT method, an extension of the unstructured Level Set / Front Tracking (LENT) method[1,2], based on the collocated Finite Volume equation discretization, that recovers exact numerical stability for the two-phase momentum convection with a range of density ratios, namely ρ − /ρ + ∈ [1, 10000]. We provide the theoretical basis for the numerical inconsistency in the collocated finite volume equation discretization of the single-field two-phase momentum convection. The cause of the numerical inconsistency lies in the way the cell-centered density is computed in the new time step (ρ n+1 c ). Specifically, if ρ n+1 c is computed from the approximation of the fluid interface at t n+1 , and it is not computed by solving a mass conservation equation (or its equivalent), the two-phase momentum convection term will automatically be inconsistently discretized. We provide the theoretical justification behind the auxiliary mass conservation equation introduced by Ghods and Herrmann[3], and used by[4,5,6,7,8]. The evaluation of the face-centered (mass flux) density we base on the fundamental principle of mass conservation, used to model the single-field density, contrary to the use of different weighted averages of cell-centered single-field densities and alternative reconstructions of the mass flux density by other contemporary methods. This results in a theoretical relationship between the mass flux density and the phase indicator. Implicit discretization of the two-phase momentum convection term is achieved, removing the CFL stability criterion. Numerical stability is demonstrated in terms of the relative L ∞ velocity error norm with realistic viscosity and strong surface tension forces. The proposed ρLENT method is additionally applicable to other two-phase flow simulation methods that utilize the collocated unstructured Finite Volume Method for discretizing single-field two-phase Navier-Stokes Equations. | null | [
"https://arxiv.org/pdf/2109.01595v3.pdf"
]
| 246,904,904 | 2109.01595 | fb44a9f626a1a37ed86bb7b205bcfd20a6545f55 |
A consistent discretization of the single-field two-phase momentum convection term for the unstructured finite volume Level Set / Front Tracking method
17 Feb 2022
Jun Liu es:[email protected]
Mathematical Modeling and Analysis
Technische Universität Darmstadt
Tobias Tolle [email protected]
Mathematical Modeling and Analysis
Technische Universität Darmstadt
Dieter Bothe [email protected]
Mathematical Modeling and Analysis
Technische Universität Darmstadt
Tomislav Marić [email protected]
Mathematical Modeling and Analysis
Technische Universität Darmstadt
A consistent discretization of the single-field two-phase momentum convection term for the unstructured finite volume Level Set / Front Tracking method
17 Feb 2022(Tomislav Marić)level setfront trackingfinite volumeunstructured meshhigh density ratios * Corresponding author
We propose the ρLENT method, an extension of the unstructured Level Set / Front Tracking (LENT) method[1,2], based on the collocated Finite Volume equation discretization, that recovers exact numerical stability for the two-phase momentum convection with a range of density ratios, namely ρ − /ρ + ∈ [1, 10000]. We provide the theoretical basis for the numerical inconsistency in the collocated finite volume equation discretization of the single-field two-phase momentum convection. The cause of the numerical inconsistency lies in the way the cell-centered density is computed in the new time step (ρ n+1 c ). Specifically, if ρ n+1 c is computed from the approximation of the fluid interface at t n+1 , and it is not computed by solving a mass conservation equation (or its equivalent), the two-phase momentum convection term will automatically be inconsistently discretized. We provide the theoretical justification behind the auxiliary mass conservation equation introduced by Ghods and Herrmann[3], and used by[4,5,6,7,8]. The evaluation of the face-centered (mass flux) density we base on the fundamental principle of mass conservation, used to model the single-field density, contrary to the use of different weighted averages of cell-centered single-field densities and alternative reconstructions of the mass flux density by other contemporary methods. This results in a theoretical relationship between the mass flux density and the phase indicator. Implicit discretization of the two-phase momentum convection term is achieved, removing the CFL stability criterion. Numerical stability is demonstrated in terms of the relative L ∞ velocity error norm with realistic viscosity and strong surface tension forces. The proposed ρLENT method is additionally applicable to other two-phase flow simulation methods that utilize the collocated unstructured Finite Volume Method for discretizing single-field two-phase Navier-Stokes Equations.
Introduction
A variety of natural and industrial two-phase flow processes involve gas/liquid flows, characterized by density ratios ρ − /ρ + ≥ 1000 1 , such as the atomization of fuel jets [9], sloshing tank [10], mold filing [11], water flooding [12]. Large density ratios at the fluid interface cause severe challenges for numerical simulations [13]: the pressure Poisson equation becomes ill-conditioned due to the discontinuity, and spurious numerical errors in the solution of the momentum equation accumulate because of inconsistencies between mass and momentum advection. Ghods and Herrmann [3] point out that for level set methods mass and momentum are typically transported in different, inconsistent ways. While mass is transported by a solution of the level set equation, momentum is obtained from solving a non-conservative form of a momentum balance equation. Hence, a large non-physical change in the momentum can be generated by a small error in the interface position when the density ratio is high. Nangia et al. [4] state that the abrupt change in density often introduces notable shear at the interface and adds difficulties in the discretization of governing momentum equations at the interface, which further leads to higher stiffness of the linear equation system.
Many researchers have addressed these problems, and some indicated further that because of the sizeable numerical error resulting from high-density ratios, some flow algorithms or solvers can only be used to solve low density-ratio cases with ρ − /ρ + ∈ [1, 10] [5]. However, in engineering applications, density ratios usually range from 100 to 1000, and even 10000 for molten metals or water-water vapor systems. Hence, a solution algorithm with the ability to handle a broader range of density ratio problems is required to simulate real-world engineering problems.
A pioneering attempt to alleviate numerical instability of the VOF method caused by highdensity ratios was made by Rudman [14]. Rudman [14] has used a sub-mesh with a doubled mesh resolution for advecting volume fractions, compared to the mesh used for the momentum and pressure equations. The goal of this two mesh approach was the reduction of small errors in the discrete momentum that cause large errors in the velocity. However, an additional higher mesh resolution for the volume fractions requires a discrete divergence free velocity on the finer mesh. Furthermore, using an additional mesh for the volume fractions increases the computational costs significantly, and it is not applicable to general unstructured meshes. Rudman [14] demonstrates qualitatively a reduction of parasitic currents for the stationary droplet case with ρ − /ρ + = 100, and improved results for more complex cases. Another important finding of Rudman [14] is the role of the densities used in the mass flux and the momentum flux in ensuring numerical consistency of the two-phase momentum advection.
Bussmann et al. [15] extended the work of Rudman [14] for the unstructured collocated finite volume method. Bussmann et al. [15] employ the conservative form for the momentum convection. At first, the momentum advection is solved separately, using an explicit Euler time integration scheme. Bussmann et al. [15] use the unstructured unsplit Volume-of-Fluid method of Rider and Kothe [16], which enables the simplification of the numerical consistency requirement for the density and momentum equations. Specifically, the solution of the volume fraction equation results in phasespecific volumes at face centers. Those phase-specific volumes are then used to compute the volume fractions at face centers. These volume fractions are used by Bussmann et al. [15], together with a simple average of cell densities, and velocities calculated by the least squares reconstruction technique, to compute the momentum fluxes at face centers. Since the velocity is continuous at the interface, the least squares approximation is acceptable. However, calculating face-centered densities by an average does not yield numerical stability in all cases. Contrary to Rudman [14], Bussmann et al. [15] do not require an additional finer mesh. They do, however, limit the solution to first-order accuracy in time and introduce the CFL condition by solving the momentum advection equation explicitly. Bussmann et al. [15] introduce the important case of a translating droplet in a quiescent ambient fluid. This test case can be used to demonstrate numerical consistency in the momentum transport. Their solutions show accurate results for high density ratios, especially considering the fact that even the unsplit VOF method distorts the interface during the translation [17]. However, for ρ − /ρ + ∈ [1,100], the constant translation velocity is modified by the solution of the pressure and momentum equations, which implies a remaining numerical inconsistency in this approach.
Sussman et al. [18] employ the CLSVOF method [19] for obtaining a robust and stable solution for the density ratio of 1000 by extrapolating the liquid velocities into the gas domain. The interface is advected using the extrapolated liquid velocity field only.
Raessi and Pitsch [5] propose a 2D staggered discretization of conservative single-field form of two-phase Navier-Stokes equations for handling high density ratios. Like Bussmann et al. [15] did, Raessi and Pitsch [5] first solve the momentum advection equation, using second-order (or higher) explicit integration schemes, and upwinding for the velocity near the interface. The density used in the momentum convective term is computed as a weighted combination of signed distances from the old and the new time step. For the partially submerged line segments bounding 2D rectangular cells, intersection between the mesh and the zero level set (iso-surface) is performed using the marching cubes algorithm. Raessi and Pitsch [5] point out that there is still an inconsistency between the face-centered density and the momentum transport, as the Level Set equation remains decoupled / inconsistent with the momentum transport. The verification of numerical stability was done using the translating droplet case from Bussmann et al. [15], and results demonstrate qualitative improvement for the density ratio ρ − /ρ + = 10 6 . Other density ratios have not been verified. A viscous oscillating droplet case demonstrates quantitative improvement in terms of the improved amplitude decay rate, compared to non-conservative form of the momentum equation.
Le Chenadec and Pitsch [20] extend their forward/backward Lagrangian tracking and Eulerian remapping VOF method [20] for handling high density ratios. Equivalent to volume fractions in [20], the density and the momentum are advected in the Lagrangian forward/backward tracking step by observing the control volume as a material volume and moving the mesh forward / backward with the flow velocity. While the content of material volumes does not change on the continuum level, this condition cannot be discretely ensured and is a source of conservation errors. In the Eulerian re-mapping step, physical properties are transferred from the Lagrangian to the Eulerian mesh, and the geometrical intersections between the PLIC interface on the forward/backward image of the mesh, and the background mesh, are another source of volume conservation errors. Ensuring numerical consistency further requires the transfer of velocities located at the center of mass. Since the velocities associated with the cell centroids are used, an inconsistency is introduced. Qualitative results show significant improvements for the stationary droplet with ρ − /ρ + = 10 9 , and quantitative improvement is shown for the standing wave by Prosperetti [21] with ρ − /ρ + = 850.
Ghods and Herrmann [3] have developed a Consistent Rescaled momentum transport (CRMT) method. The CRMT method discretizes the conservative form of the single-field Navier-Stokes equations using a collocated unstructured Finite Volume method. To increase the numerical stability for high density ratio, CRMT solves a mass conservation equation using a mass flux either by upwinding the face-centered density in the interface cells and their face-neighbors (defined by a volume fraction tolerance), or by averaging the densities elsewhere. The same discretization scheme used for the face-centered density is also applied to the mass flux in the convective term of the momentum equation. A difference is therefore introduced in the mass flux of the continuity equation and the mass flux in the convective term of the momentum equation when upwinding is used, because the upwinded face-centered density in the continuity equation uses the face-centered velocity, while the upwinded mass flux in the momentum equation includes both the upwind velocity and density. We show that any difference in the discretization of the mass flux to be a source of numerical inconsistency for the two-phase momentum advection. Like Bussmann et al. [15], the explicit discretization of the momentum convective term introduces the CFL condition, limiting the time step for convection-dominated multiphase flows, where high density ratios play a major role. Using upwind schemes makes the discretization first-order accurate. The droplet translation case [15], with ρ − /ρ + = 10 6 , is compared in terms of the droplet shape, that remains stable. Other density ratios are not reported for this verification case. It is our opinion, that the droplet shape errors may result from the interface advection scheme 2 , and should be generally substituted by the L ∞ norm of the velocity error to demonstrate numerical consistency.
Vaudor et al. [22] base their approach on a CLSVOF code from Aniszewski et al. [23], which can switch between LS-based and VOF-based mode to calculate momentum fluxes. They chose the VOF-based momentum fluxes calculation mode and implemented the framework of Rudman's method [14] but with more accurate interpolation schemes for velocities and velocity gradients on faces of staggered meshes to ensure consistency. This method is developed in two-dimensions and exploits two sets of meshes. To provide a more widely applicable method, Vaudor et al. [24] advanced the method in their more recent study. In contrast to the previous work [22], the LS method tracks the interface, while the VOF method is utilized to update density. They exploited the identical scheme to discretize conservative convective term in mass and momentum equation. In addition, the mass flux is also identical in both discretized equations. A new strategy that leverages half cell-faces' and half cells' quantities of volume fraction and density to couple staggered mass cells and momentum cells is introduced to avoid the need for a refined mesh in the original method by Rudman [14]. A prominent feature of this new method is that it can be used to simulate threedimensional applications. Besides, comparing with the method from Rudman [14], the new method shows relatively low computational cost when simulating the same 2D application.
Owkes and Desjardins [25] presented a three-dimensional, unsplit, second-order semi-Lagrangian VOF scheme that conserves mass and momentum and ensures consistency between the mass (volume fraction) and momentum fluxes. The volume fractions are geometrically transported near the fluid interface using the method from [26]. As in [14], Owkes and Desjardins [25] introduce an additional refined mesh for the calculation of semi-Lagrangian fluxes. The motivation for the refined mesh is to enforce the consistency between semi-Lagrangian mass and momentum fluxes, similar to Rudman [14]. Results confirm mass and momentum conservation, and stability of the momentum convection. The method proposed by Owkes and Desjardins [25] relies on the staggered variable arrangement and this, together with the use of the additional finer mesh, makes this approach inapplicable to unstructured finite volume meshes.
Orazzo et al. [6], similarly to Rudman [14], resolve the volume fraction function on twice finer sub-cells and update density from the volume fraction. After that, they update face-centered density on mass cells by averaging density on sub-cells, and then evaluate the mass flux on the faces of standard staggered momentum cells. These density and mass flux values are used to initialize and calculate interim momentum and velocity during the prediction step. Zuzio et al. [7] made no changes and applied Orazzo's method [6]. Besides, they further verified and validated this method with new, different cases and analyzed the results of simulations using more novel ways. Yang et al. [27] notice that the high-density ratio has a profound effect on robustly simulating two-phase flows at high Reynolds numbers. To mitigate the problem, they adopt the consistent framework from Nangia et al. [4] and replace the interface-capturing method in [4], which is standard LS, with CLSVOF method [19] to ensure mass conservation.
Patel and Natarajan [28] employ the method of Ghods and Herrmann [3], a high-resolution scheme called Cubic Upwind Interpolation (CUI) for the convective terms of momentum and volume fraction transport equations, and the solution of a momentum equation in the face-normal direction. The face-normal momentum equation leads to a combined collocated/staggered variable arrangement, that requires the use of nonlinear solvers, as this equation is a non-linear algebraic equation. Patel and Natarajan [28] demonstrate the balanced nature of their discretization for the stationary droplet using exact curvature and density ratios ρ − /ρ + ∈ [10,1000]. Numerical stability is demonstrated with reduced parasitic currents when the curvature is approximated numerically for ρ − /ρ + = 10, We = 1. For the verification test case of the two-phase momentum advection problem, ρ − /ρ + = 10 6 is used without surface tension and viscous forces and qualitative results show slight deformations of the interface shape, the L ∞ norm of the velocity error is not reported. With enabled surface tension and viscous forces and exact curvature prescribed, and density ratios ρ − /ρ + = 1, 1000, the velocity error in the L ∞ norm lies within [10 −3 , 10 −2 ].
Manik et al. [29], similarly to [28], attempt to enforce numerical consistency by applying the similar discretization scheme on the conservative form of the volume fraction advection equation and the momentum conservation equation. Manik et al. [29] are using a collocated unstructured Finite Volume method for the equation discretization and the CUBISTA scheme (Alves et al. [30]) to discretize convective terms. The verification of the numerical consistency for the two-phase momentum advection is done using the droplet translation case of Bussmann et al. [15] and density ratios ρ − /ρ + = 10 3 , 10 6 , that demonstrates qualitative improvement compared to a naive discretization of the momentum convective term with the upwind method. The qualitative evaluation is based on the shape of the droplet, given by the 0.5 iso-surface of the volume fraction. Although the proposed method demonstrates improvement w.r.t. an obviously inconsistent approach, some shape deformation is still visible, so one can conclude that L ∞ (v) = 0 and some non-zero velocities are still generated.
A recent second-order accurate LS method is proposed by Nangia et al. [4], extending the work from Ghods and Herrmann [3] that is first-order accurate. Similar to the method proposed by Ghods and Herrmann [3], an additional mass conservation equation is solved, and the identical mass flux is used for both mass and momentum transport. Two techniques are employed: one is the third-order accurate Koren's limited CUI, which is modified to consistently discretize the convective term of both mass and momentum equation. This scheme satisfies the convection-boundedness criterion (CBC) and is total variation diminishing (TVD). The second technique is the solution of an update equation for the face-centred densities. In this step, a third-order accurate strong stability preserving Runge-Kutta (SSP-RK3) scheme is used for time integration. The update is performed in every fixpoint iteration, and the updated face-centered density is then employed to solve the discretized momentum equation.
Zuzio et al. [7] also follow Ghods and Herrmann [3] by solving an auxiliary continuity equation for increasing the numerical consistency in discretizing the two-phase momentum convection term. Their Consistent Mass-Momentum (CMOM) transport method utilizes a staggered Carte-sian variable arrangement and utilizes the two-phase incompressible Navier-Stokes equations in the conservative form, solved using Chorin's projection method together with the CLSVOF method for tracking the fluid interface. The solution of the auxiliary density equation requires the evaluation of staggered (face-centered) densities, by constructing staggered control volumes, and evaluating the densities using sub-grid quadtree (octree in 3D) refinement and intersection with the PLIC interfaces. This aspect of CMOM shows the importance of evaluating the densities at face-centeres that are required for the solution of the auxiliary continuity equation. Momentum flux reconstruction scales the fluxed phase-specific volume from the VOF method. Finally, the two-phase momentum is advected in the staggered cells, and scaled with the corresponding density to obtain velocity components in all spatial directions. Zuzio et al. [7] demonstrate significant improvements in numerical stability in a very detailed way, reporting shape, position and kinetic energy errors for canonical verification and validation cases. The kinetic energy for the dense translating droplet [15] with a density ratio of 10 6 is reported, and CMOM recovers a numerically stable solution.
Arrufat et al. [31] consider the conservative form of the advection equation of a discontinuous property to enforce numerical consistency of the advected two-phase momentum, using face averages that are derived by integrating the advection equation in space and time. Since the discontinuity of the property introduced by the interface complicates the evaluation of the face averages, two additional equations are introduced, one for each phase. The method is derived for the MAC staggered variable arrangement. Results demonstrate a numerically stable droplet shape when it is advected with a constant velocity, however, the authors consider this case to only test the consistency of the implementation and not the numerical consistency of the method -we consider it important for both -so the results are not quantified in terms of kinetic energy or L ∞ velocity errors. Still, the method shows significant improvements for realistic multiphase flows with high density ratios.
The high-density ratio is also challenging for other numerical methods for two-phase flows, like the phase-field and lattice Boltzmann. The corresponding surveys are beyond the scope of this work, more details can be found in [32,33,34,35,36,37]. Contrary to the numerical two-phase methods mentioned so far, the difficulties with high density ratios are far less pronounced for Front Tracking methods [38] because the marker field (phase-indicator) is not as sharp as in the unstructured Volume-of-Fluid method [39] and the unstructured Level Set / Front Tracking method [1,2].
The methods of Bussmann et al. [15], Ghods and Herrmann [3], Patel and Natarajan [28], Manik et al. [29] utilize the unstructured Finite Volume equation discretization, other above-mentioned methods utilize a staggered variable arrangement that is not applicable to unstructured meshes. Compared to contemporary collocated Finite Volume methods, our proposed ρLENT method achieves the numerical consistency in the two-phase momentum advection exactly. We derive the requirement for the auxiliary mass conservation equation introduced by Ghods and Herrmann [3], and derive the requirement for the face-centered (flux) density from the mass conservation principle. Compared to a similar observation by [31], we avoid the integration in time that complicates the evaluation of face-centered quantities, as demonstrated in detail below. Although hybrid Level Set / Front Tracking LENT method [1] is used for interface capturing, the ρLENT solution algorithm can be used with other interface capturing methods, where there is a discrepancy in the evaluation of the collocated density. Compared to all contemporary collocated methods, ρLENT allows an implicit discretization of the two-phase momentum convective term which removes the CFL stability condition.
Ω − (t) ∂Ω n Σ Ω + (t) Σ(t)
Mathematical model
As shown in fig. 1, the overall flow domain Ω ⊂ R 3 is filled with two phases which occupy subdomains Ω − (t) and Ω + (t), such that Ω = Ω + (t) ∪ Ω − (t) ∪ Σ(t). The unit normal vector n Σ of the interface Σ(t) is oriented outwards for the subdomain Ω − (t). These two phases have different material properties that change sharply across the interface Σ(t), separating the two subdomains. To identify the phase at a particular location x and time t, the phase indicator is utilized and defined as
χ(x, t) := 1, x ∈ Ω − (t), 0, x ∈ Ω + (t).(1)
In this work, a single-field formulation of governing equations is used to model two-phase flows. Constant density and dynamic viscosity of the two phases, namely ρ − , ρ + and µ − , µ + are combined into single-fields using the phase indicator:
ρ(x, t) = χ(x, t)ρ − + (1 − χ(x, t))ρ + ,(2)µ(x, t) = χ(x, t)µ − + (1 − χ(x, t))µ + .(3)
The Navier-Stokes equations in the single-field formulation model the incompressible two-phase flow read as
∇ · v = 0,(4)∂ t (ρv) + ∇ · (ρv ⊗ v − T) = ρg + f Σ ,(5)
where T is the stress tensor
T = − p + 2 3 µ∇ · v I + µ ∇v + (∇v) T .(6)
In the standard model, i.e. with a constant surface tension and without phase change, the surface tension per unit volume f Σ exerts a force on the interface and is modeled following a CSF (Continuum Surface Force) model [40] f Σ = σκn Σ δ Σ
in which σ is the surface tension coefficient and assumed constant in this work, κ denotes twice the local mean curvature of interface, and δ Σ is the interface Dirac distribution. Note that in case of variable surface tension coefficient and phase change, eqs. (4) and (7) would need to be modified.
3.
A solution algorithm for two-phase flows with high density ratios using the collocated unstructured Finite Volume method
The volume fraction α(x, t) is introduced as the phase indicator χ averaged over a fixed control volume Ω c , i.e.
α c (t) = 1 |Ω c | Ωc χ(x, t) dV.(8)
The single-field formulation of the Navier-Stokes equations associates with the centroid of the control volume Ω c the density
ρ c (t) = α c (t)ρ − + (1 − α c (t))ρ + ,(9)
and the viscosity
µ c (t) = α c (t)µ − + (1 − α c (t))µ + .(10)
Equations (9) and (10) and are used in eq. (5), discretized by the collocated unstructured finite volume method [41,42,43], implemented in the OpenFOAM open-source software [44,45,46]. The surface tension force given by eq. (7) is discretized as
f Σ,c ≈ σκ c ∇α c ,(11)
where cell curvature κ c is calculated as described in [2] and ∇α c denotes a discrete finite volume gradient of the volume fraction.
The unstructured Finite Volume Hybrid Level Set / Front Tracking method
Hybrid multiphase flow simulation methods combine the sub-algorithms of the Front Tracking, Level Set, or Volume-of-Fluid methods to achieve better overall results. The structured Hybrid Level Set / Front Tracking method ( [47,48,49,38,50]) has demonstrated remarkable capabilities for simulating a wide range of multiphase flows. The unstructured Level Set / Front Tracking method -the LENT method [1,2] -shows promising computational efficiency and accuracy for surface tension driven flows on unstructured meshes.
The LENT method computes the cell-centered volume fractions α c (t) from the surface mesh (i.e. the Front) that discretely approximates the fluid interface [2,51]. However, it turns out that this step introduces an inconsistency in the solution of the single-field momentum conservation equation, as outlined below.
Numerical consistency of the single-field conservative two-phase momentum convection term
Bussmann et al. [15] were the first to consider the problem of numerical consistency of the two-phase momentum convective term in the setting of the collocated unstructured Finite Volume method. We expand on their work by improving the accuracy of the face-centered density evaluation and employing a solution algorithm that allows for an implicit discretization of the convective term, thus removing the CFL condition.
Consider first the Euler explicit collocated unstructured finite volume discretization of the mass conservation equation,
∂ t ρ + ∇ · (ρv) = 0,(12)
namely,
ρ n+1 c − ρ n c ∆t + 1 |Ω c | f ∈Fc ρ n f F n f = 0, (13) ρ n+1 c = ρ n c − ∆t |Ω c | f ∈Fc ρ n f F n f ,(14)
where F f is the volumetric flux at the face defined by the index f from the set of all indexes of cell-faces F c that belong to Ω c , defined as
F f := v f · S f ,(15)
with v n f as the face-centered velocity and S f := |S f |Ŝ f as the face area-normal vector, with |S f | denoting the area of the face S f . Second, we focus on the two-phase momentum advection with a prescribed initial velocity, spatially constant throughout the solution domain and its inlet/outlet boundaries. Without forces on the r.h.s of eq. (5), the momentum conservation equation becomes
∂ t (ρv) + ∇ · (ρv ⊗ v) = 0.(16)
Without forces on the r.h.s. of eq. (16), eq. (16) keeps the initial velocity spatially constant and without acceleration/deceleration. Therefore, a numerically consistent unstructured collocated FVM discretization of the two-phase momentum convection equation (16) must ensure that no artificial acceleration or deceleration occurs. For example, just like eq. (14), the Euler explicit discretization of eq. (16) is
ρ n+1 c v n+1 c = ρ n c v n c − ∆t |Ω c | f ∈Fc ρ n f F n f v n f .(17)
If this discretization is numerically consistent, and the velocity field remains spatially constant,
then v n f = v n c ,(18)
which is, of course, ensured for the initial spatially constant velocity (v 0 f = v 0 c ). Equation (18), applied to eq. (17), results in
ρ n+1 c v n+1 c = v n c ρ n c − ∆t |Ω c | f ∈Fc ρ n f F n f ,(19)
and dividing by ρ n+1
c finally gives v n+1 c = v n c ρ n c − ∆t |Ω c | f ∈Fc ρ n f F n f ρ n+1 c .(20)
As there are no forces on the r.h.s. of eq. (16), the velocity should not be changed simply by advecting the two-phase momentum, i.e.
v n+1
c = v n c ,(21)
and this condition is ensured in eq. (20) if
ρ n c − ∆t |Ω c | f ∈Fc ρ n f F n f ρ n+1 c = 1,(22)
which is equivalent to eq. (14): the Euler explicit discretized form of the mass conservation equation.
Consequently, a numerically consistent discretization of the momentum convection equation requires the new cell-centered density ρ n+1 c to be computed by solving a mass conservation equation. This observation justifies theoretically the use of the auxiliary mass conservation equation, introduced by Ghods and Herrmann [3], and used by [4,5,6,7,8].
Modern unstructured geometric flux-based Volume-of-Fluid methods ( [52,25,53,54], see [55] However, numerical methods such as Front Tracking, Level Set, and their hybrids, update the cell-centered density ρ n+1 c from the approximated fluid interface that is not advected by solving the volume fraction advection equation. Bussmann et al. [15] rely on this consistency of the Volume-of-Fluid method and use the VOF fluxes to first solve eq. (16) explicitly in the first step, followed by the second step that includes volume and surface forces. The approach from Bussmann et al. [15] works very well for the VOF method, however it cannot be directly applied to the Level Set method, the Front Tracking method or their hybrids, when the unstructured collocated Finite Volume method is used to discretize the single-field Navier-Stokes equations. The solution algorithm for high density ratios that we propose avoids the CFL condition imposed by Bussmann et al. [15] and increases the accuracy of the face-centered density ρ f required by the mass flux, and it is applicable to any multiphase flow simulation method that utilizes the single-field formulation of the Navier-Stokes equations.
A semi-implicit solution algorithm for high-density ratios
Section 3.2 provides the formal reasoning behind solving the mass conservation equation (or its equivalent) for ρ n+1 c . Since Ghods and Herrmann [3] introduced the use of the "auxiliary" mass conservation equation, other researchers have adopted this approach, with the main difference in the way the face-centered (mass flux) density ρ f is evaluated both in the discretized mass conservation equation (12) and the discretized momentum equation (5).
Ω + (t) ρ + n Σ Ω c Σ(t) Ω − (t) ρ −
The condition given by eq. (22), derived from eqs. (20) and (21) can be fulfilled only if the same face-centered (mass flux) density is used when discretizing the auxiliary mass conservation and momentum equations. Going one step further, the volumetric flux F f must also be the same in the discretized auxiliary mass conservation and momentum equations. Put together, the mass flux in the auxiliary discretized mass conservation equation must be equal to the mass flux in the discretized momentum conservation equation: this is the requirement for the mass flux consistency, mentioned throughout the literature.
It is relevant to point out that the same model for the single-field density given by eq. (9) is used throughout the literature. The basis of this model is mass conservation, and this fundamental principle further leads to an interesting conclusion regarding the evaluation of the face-centered (mass flux) density ρ f in the discretized mass and momentum conservation equations. The face centered density is evaluated differently throughout scientific publications reviewed in section 1, and here we show that there is a strict relationship between the phase indicator and the face centered density ρ f .
Consider the fixed control volume Ω c in fig. 2, that is separated by the fluid interface Σ(t) into two parts, occupied by fluids Ω ∓ (t). The single-field density model given by eq. (2) is adopted in every publication reviewed in section 1, an in the rest of the scientific literature on two-phase flow simulations. The mass conservation principle together with the single-field density model (2) give
d dt Ωc ρ dV = − ∂Ωc ρv · ndS = − ∂Ωc [ρ − χ + ρ + (1 − χ)]v · n dS.(23)
The equality of surface integrals in eq. (23),
∂Ωc ρv · ndS = ∂Ωc [ρ − χ + ρ + (1 − χ)]v · n dS,
demonstrates that the mass flux of the single-field density over ∂Ω c is determined by the constant densities ρ ∓ and the phase indicator given by eq. (1), if eq. (2) is used to model the single-field density. In other words, the single-field density at ∂Ω c should be computed using the phase indicator as done on the r.h.s. of eq. (23), otherwise the mass conservation of the single-field density model given by eq. (2) will not be upheld. This relevant condition transfers to the discrete level, leading to an interesting consequence for the computation of the face-centered (mass flux) density, that has so far been computed in many ways throughout the literature. Specifically, when the surface integrals in eq. (23) are discretized using the unstructured collocated finite volume method, f ∈Fc
ρ f F f = f ∈Fc ρ − S f χ dS v f ·Ŝ f + ρ + S f dS v f ·Ŝ f − ρ + S f χ dS v f ·Ŝ f = f ∈Fc ρ − S f S f S f χ dS v f ·Ŝ f + ρ + S f dS v f ·Ŝ f −ρ + S f S f S f χ dS v f ·Ŝ f = f ∈Fc ρ − α f + ρ + (1 − α f ) F f ,(24)
where
α f := 1 |S f | S f χ dS ≡ |Ω − (t) ∩ S f | |S f |(25)
is the area fraction of the face S f ⊂ ∂Ω c , i.e. the ratio of the area of S f submerged in Ω − (t), and the total face-area |S f |. Further, S f ≡ |S f |, and F f is the volumetric flux v f · S f (eq. 15) in eq. (24). An important consequence of eq. (24) is the requirement for the evaluation of the face-centered (mass flux) density, necessary for ensuring the numerical consistency of the single-field two-phase momentum convection. Equation (24) requires all methods 3 that define ρ using eq. (9) to either compute ρ f using the area fractions or S f χ dS from eq. (24), or to achieve this equivalently when computing ρ n+1 c from the advected volume fractions α n+1 c , which is possible for the flux-based VOF methods [55].
Another important realization is that eq. (24) is valid at any time t -which is very relevant for the semi-implicit discretization developed within the ρLENT method, that applies eq. (24) at t n+1 .
It is true that many two-phase simulation methods do not advect the phase indicator when advecting the fluid interface, Σ(t) but this does not infer that eq. (24) cannot be applied. The idea of using an auxiliary mass conservation equation introduced by [3], made into a formal requirement by eqs. (20) and (21), allows the use of eq. (24): α f can be computed regardless of the approximation of the fluid interface Σ(t) and the method used to advect it.
Similar to other contemporary methods, the ρLENT method also first advects the interface using the velocity from the previous time step as shown in the left image of fig. 3a, resulting in the new position of the interface shown in the right image in fig. 3a, that is then used to geometrically calculate the face-centered density ρ n+1 is obtained, which is used to evolve the fluid interface in the next time step, from t n+1 to t n+2 . At this point, the numerically consistent cell-centered density ρ n+1 c has served its purpose and is reset using eq. (9), to make it consistent again with the fluid interface approximation.
Any two-phase flow simulation method has the possibility to compute the face-centered density ρ f (t) from the interface approximation in some way. The ρLENT method computes the face-centered density ρ f (t) by computing the face area fraction α f (t) (short: area fraction) of the face S f , submerged in the phase Ω − (t). The calculation of α f uses signed distances available in the unstructured LENT [1,2]. Other two-phase flow simulation methods rely on a different approximation of the fluid interface Σ(t), that can be used to geometrically approximate the area fraction α f without resorting to an interpolation of the field that abruptly changes in the interface-normal direction.
In the original Front-Tracking method, the density is updated utilizing the new position of marker points (the approximated interface) [56]. After the velocity field in the current step is computed, the position of marker points in the new time step can be updated immediately by
x n+1 p = x n p + ∆t v n p ,(26)
where x p ,v p indicate the position and interpolated velocity of marker points respectively, and ∆t is the time step length.The advection of marker points along Lagrangian trajectories eventually corrupts the triangular mesh, leading to discrepancies in the ratios of triangular angles and areas and self-intersections of the triangular mesh. The original Front Tracking method [57] deals with this by redistributing marker points based on quality criteria imposed on the triangular mesh, which involves manipulating the connectivity of the triangular mesh. Contrary to original Front Tracking [57], the LENT method reuses the principles from LCRM / LFRM methods [47,48,38,50] and reconstructs the interface using an iso-surface reconstruction algorithm. The iso-surface reconstruction does not add/delete marker points locally by changing the connectivity of the triangular surface mesh; it reconstructs the entire interface in the solution domain as an iso-surface. Following the strategy from LCRM / LFRM, the physics of the problem determines the iso-surface reconstruction frequency. The LENT method uses the marching tetrahedra [58] algorithm to enable the iso-surface reconstruction on unstructured meshes. However, the marching tetrahedra algorithm introduces many triangles per cell (even with regularization), causing instabilities in front tracking. We are developing an alternative iso-surface reconstruction that relies on a higher-order signed-distance interpolation and results in a favorable ratio of triangle-to-cell length scales.
Once the marker points are advected and redistributed, the cell density is updated depending on x n+1 p , namely
ρ n+1 = ρ(x n+1 p ).(27)
The face-centered density used for the mass flux is then interpolated from densities of two adjacent cells. Contrary to LENT, the face-centered density is updated by ρLENT using the phase indicator approximated at each cell-face. A 2D interface is depicted in fig. 3b, where α n+1 f is the area fraction at t n+1 : the ratio of the cell-face area submerged in the phaseΩ − (t n+1 ) ≈ Ω − (t n+1 ), and the total face area |S f |. More precisely, the area fraction α n+1 f is computed by the ρLENT method using a second-order accurate approximation from signed distances [59], used in [2] to equivalently approximate the volume fraction α c (see eq. (8)). The Level Set component of the LENT method [1] calculates signed distances from the triangular surface mesh that approximates the interfacẽ Σ(t n+1 ) ≈ Σ(t n+1 ) := ∂Ω − (t n+1 ). With the narrow band approach from [1], the signed distances can be computed efficiently at any point in a close vicinity ofΣ(t). The original LENT method [1] computes signed distances at cell-centers and cell corner-points, and the proposed ρLENT additionally computes signed distances at face centers. Each face S f is triangulated using its centroid x f , as shown in fig. 4. The face centroid x f , together with the two successive cell-corner points that belong to the face S f , x f,i , x f,i+1 , forms a triangle (x f , x f,i , x f,i+1 ). Face-triangles may be partially submerged in the phaseΩ − (t n+1 ), in which case the submerged area of the triangle is computed using the nearest signed distances toΣ(t n+1 ) from the triangle points ( fig. 4. The second-order approximation developed in [59] is used here for computing the area fraction of a triangle submerged inΩ − (t n+1 ). Any other secondorder method can be applied. For example, a linear interpolation of signed distances along the edges of the triangle may be used equivalently, or a geometrical intersection betweenΩ − (t n+1 ) and the triangle. The total submerged area of the face S f is then the sum of the submerged areas of face-triangles A n+1
x f , x f,i , x f,i+1 ), namely (φ f , φ f,i , φ f,i+1 ), as shown inf := |Ω − (t n+1 ) ∩ S f | = t∈T f |Ω − (t n+1 ) ∩ T t |,(28)
where T f is the set of indexes of the triangles in the triangulation of the face S f . As mentioned above, other two-phase flow simulation methods may compute |Ω − (t n+1 ) ∩ T t | differently. The area fraction α n+1 f is then computed as
α n+1 f = |Ω − (t n+1 )∩S f | |S f | φ n+1 f,i φ n+1 f,i+1 x f,i+1 x f,ĩ Ω + (t n+1 ) n Σ φ n+1 f x fΣ (t n+1 ) Ω − (t n+1 )α n+1 f := |Ω − (t n+1 ) ∩ S f | |S f | = A f |S f | ,(29)
as shown in fig. 4. Once the area fraction α n+1 f is approximated, it is used to compute the facecentered densities required by eq. (24), namely
ρ n+1 f = α n+1 f ρ − + (1 − α n+1 f )ρ + ,(30)
at the new time step, because the interface has been advected forward in time to t n+1 with the available velocity v n . The discretized continuity equation (14) then obtains the form
ρ n+1 c = ρ n c + ∆t |V Ωc | f ρ n+1 f F m f , F m f = v m f · S f .(31)
It is important to note that, although ρ n+1 f appears in eq. (31), ρLENT does not use an implicit discretization for eq. (31): ρ n+1 f is geometrically computed from the fluid interface approximatioñ Σ n+1 , so eq. (31) is solved explicitly (exactly). The exact (non-iterative) evaluation of ρ n+1 c from eq. (31), alongside eq. (20), further infers the possibility of exact numerical consistency for the discretized convective term in the single-field momentum equation, which is in fact achieved and supported by the results.
In addition to density, the viscosity is updated utilizing the area fraction α f . Note that there is no need to calculate the cell-centered viscosity for the unstructured FVM discretization, only the face-centered viscosity is updated as follows
µ n+1 f = α n+1 f ρ − ν − + (1 − α n+1 f )ρ + ν + .(32)
The non-linearity of the convective term in the momentum equation eq. (5), namely ρvv, is usually linearized when solving the single-field Navier-Stokes equations using the unstructured Finite Volume method. The convective term is discretized as
Ωc ∇ · (ρvv)dV ≈ f ∈Fc ρ n+1 f F m f v n+1 f .(33)
Numerical consistency also does not depend on the implicit / explicit discretization. As an analogue, observe the Euler implicit and Euler explicit discretization of the momentum convection equation: in the lim ∆x→0,∆t→0 , both implicit and explicit discretization must converge to eq. (5). Therefore, the requirement given by eqs. (20) and (21), is valid for an implicit discretization as well.
The volumetric flux F m f is initialized to F n f and iterated within the SAAMPLE [2] pressurevelocity coupling algorithm loop until m = n + 1 is reached. The ρLENT algorithm is outlined in algorithm 1 and it extends the SAAMPLE algorithm [2]. It is relevant to note that F m f is iterated from F n f to F n+1 f and p m is solved for from p n to p n+1 such that the discrete incompressibility condition f ∈Fc F n+1 f is ensured. Advect the interface toΣ n+1 .
[1].
3:
Compute the signed-distance field φ n+1 fromΣ n+1 at xc, x f , xp in the narrow-band. [2] and eq. (33).
9:
Make ρ n+1 c consistent withΣ n+1 , i.e. ρ n+1
c = α n+1 c ρ − + (1 − α n+1 c )ρ + .
10:
Make µ n+1 c consistent withΣ n+1 , i.e. µ n+1
c := α n+1 c ρ − ν − + (1 − α n+1 c )ρ + ν + . 11: end while
The p − v coupling -mentioned in the step 8 in algorithm 1 -requires some further explanation. The semi-implicit discretization (with the convective term linearized as an explicit mass-flux and implicit velocity) of the single-field momentum equation using the implicit collocated unstructured finite volume method [60,61], results in
a c v n+1 c + k∈Nc a k v n+1 k = −(∇p) n+1 c − [(∇ρ) · (g · h)] c + (f Σ ) n+1 c ,(34)
where N c is the index-set of cells that are face-adjacent to cell Ω c , and the total pressure is ex-
Verification and validation
The hybrid Level Set / Front Tracking method is not strictly volume conservative, and volume errors arise from three sources. First, the iso-surface reconstruction -that handles the topological changes of the fluid interface -introduces volume errors by interpolating the level-set function. This error source can be reduced using higher-order level set function interpolation. Second, the Front Tracking method approximates the fluid interface as a surface triangulation and advects the interface in a co-moving reference frame by displacing the triangulation points along Lagrangian trajectories. The volume errors introduced by Front Tracking are reducible significantly by a second (or higher)-order temporal integration of the Lagrangian displacements. The third source of volume conservation errors is the phase-indicator model: we approximate volume fractions from signed distances stored at cell centers and cell-corner points [2]; however, we are investigating a more accurate geometrical intersection between the Front and the volume mesh [65]. In conclusion, the volume conservation of the hybrid Level Set / Front Tracking method depends on the physics of the problem. For the verification problems, the ρLENT method recovers very low maximal relative volume conservation errors of 5.13 · 10 −4 for the coarsest resolution of only 6 cells per droplet diameter and 5.49 · 10 −5 for the finest resolution of 26 cells per droplet diameter. The volume conservation errors of such small magnitude have no effect on the numerical stability of the two-phase momentum convection term, so their detailed visualization is omitted for brevity.
Results presented in this section are publicly available [66], as well as the implementation of the ρLENT method as an OpenFOAM module [67].
Time step size
The time step size limit due to the CFL condition is given by
∆t ≤ ∆t CF L = ∆x U ,(37)
where ∆x is cell length and U is a characteristic velocity. In the cases, ∆x is the ratio of domain side length to resolution N , while U is equal to magnitude of the ambient flow velocity vector, i.e. U = |v a | = 1. Another restriction for the time step size arises from the propagation of capillary waves on interfaces between two fluids. This time step constraint is firstly introduced by Brackbill et al. [40], and afterwards revised by Denner and van Wachem [68]. It has the form
∆t ≤ ∆t cw = (ρ d + ρ a )∆x 3 2πσ ,(38)
in which ρ d and ρ a are density of droplet and ambient fluid, respectively, σ is the surface tension coefficient. In the case setup procedure, the method devised by Tolle et al. [2] is followed, i.e., using a compare function ∆t = min (k cw ∆t cw , k CF L ∆t CF L )
where k cw and k CF L are arbitrary scale factors between 0 and 1. In the following, k cw = 0.5 and k CF L = 0.2 are used.
mantle mantle outlet v R L y L x L z inlet C z C x C y x y z v = (0, 0, 1) v = 0 v = (0, 0, 1) inlet: mantle:
oulet:
R = 0.2, C x = C y = 0.5, C z = 0.4, L x = L y = 5R, L z = 6R t end = 0.41s ∇p = 0 ∇p = 0 p = 0
Translating droplet
Following the setup of Popinet [69], a sphere of radius R = 0.2 translates in a rectangular domain having side lengths L x = L y = 5R, L z = 6R. The initial position of the sphere's centroid is C x = C y = 0.5, C z = 0.4. One corner of the rectangular domain is located in the origin as shown in fig. 5. The boundary conditions of the rectangular domain are set as follows: ∇v · n = 0 and p = 0 for the outlet, v = v a and constant normal pressure gradient ∂p/∂n at the mantle and the inlet. The initial conditions for internal field is set to p(t 0 ) = 0 and v(t 0 ) = v a . The end time of simulation is set to t end = 0.41 s, which corresponds to a droplet displacement of one diameter.
Two groups of cases are tested to verify the ρLENT method, respectively. For the first group, only the advection of momentum is considered, and the ambient flow has a constant density ρ a = 1, while the density of the droplet ρ d varies between (1, 10 2 , 10 3 , 10 4 ), resulting in four density ratios. Three mesh resolutions N ∈ (16,32,64) are tested. For each mesh resolution N , the domain is discretized equidistantly into 1.2N 3 hexahedral cells, as shown in fig. 6. The exact solution is given by v n+1 c = v n c = v c (t 0 ) = v a and can be used to verify the numerically consistent discretization of the single-field conservative two-phase momentum convection.
Viscosity and surface tension forces are included in the second test case group. A smaller range of density ratios is simulated, ρ − /ρ + ∈ (1, 10, 10 2 , 10 3 ). The same kinematic viscosity is used for the ambient and the droplet phase, namely ν ∈ (0.057735, 0.018257, 0.0057735, 0.0). The surface tension coefficient is constant σ = 1.
Droplet translation without viscosity and surface tension forces
When the momentum is transported only by advection, no forces are exerted on the droplet body and surface. As a result, the velocity field in the overall domain should remain spatially constant and equal to v a = (0, 0, 1). The maximum norm L ∞ is employed to measure how much the numerical velocity deviates from the analytical one, i.e.,
L ∞ (v) = max i v i − v ∞ v ∞ ,(40)
where v i denotes velocity of all interface-intersected cells, and v ∞ = v a = (0, 0, 1). The numerically inconsistent solution can cause large nonphysical interface deformations leading to a complete deterioration of the solution, visible for a verification configuration in the left image in fig. 8. The deterioration is amplified by the p − v coupling algorithm that will calculate a pressure field p that enforces ∇ · v = 0. This, in turn, causes artificial acceleration in all cells where v n+1 c = v a . The consistent ρLENT method ensures the shape of the droplet is preserved, as shown on the right image in fig. 8. The fig. 7a contains the velocity error calculated with the old, inconsistent method. Every line in the diagram is labeled by the number of the case, mesh resolution N , and droplet density ρ − . The default ambient density is 1. Thus, the ρ − also represents the density ratio. As shown in fig. 7a, the cases with the density ratio ρ − = 1 run successfully until end time t end = 0.41s. All cases with a density ratio higher than 1, namely ρ − > 1, diverge and stop early. Cases with a very high density ratio of 10 4 (e.g., case 0011 and 0003) fail catastrophically.
When ρLENT is used, as shown in fig. 7b, the velocity error remains exactly 0 in all cases. This means that the interface velocity remains consistent with the ambient flow and is unaffected by the mesh resolution and density ratio. The results demonstrate the exact recovery of numerical consistency for the advection of the two-phase momentum, using the conservative formulation of single-field two-phase Navier-Stokes equations.
Droplet translation with viscosity and surface tension forces
Here, viscous and capillary forces are taken into account when solving the momentum equation. Since SAAMPLE is a well-balanced algorithm [2], surface tension force is balanced by the pressure gradient, if the curvature is exactly calculated and propagated as a constant in the normal direction with respect to the interface. In the absence of gravity, such a droplet does not accelerate or decelerate. In other words, the velocity error L ∞ stays the same, namely 0, in theory. The temporal evolution of L ∞ is shown in fig. 9. The inconsistent method remains stable only for ρ − /ρ + = 1.
For the results of all other cases, i.e., with ρ − /ρ + > 1, the velocity error increases exponentially, and the simulations crash. In contrast, as depicted in fig. 9b, the ρLENT demonstrates numerically stable results for all tested density ratios. Additional numerical errors are introduced compared with two-phase momentum advection, specifically when approximating the curvature [2]. Therefore, L ∞ cannot exactly be equal to zero, as shown in fig. 7b. However, as seen in fig. 9b, the final L ∞ error given by eq. (40) 10 −4 and 10 −2 , which is acceptable.
(a) Old inconsistent method: interface stable only for cases with density ratio ρ − /ρ + = 1 (b) New consistent method: interface stable for density ratios ρ − /ρ + ∈ [1, 10, 100, 1000] Table 1: Realistic fluid properties are combined into four tests: water droplet/air ambient, mercury droplet/air ambient, silicone oil droplet/air ambient, silicone oil droplet/water ambient. Table 1 contains the physical properties used for the test-case configuration of the translating sub-millimeter droplet with realistic physical properties. In terms of size, a spherical droplet of radius R = 0.25 mm is translating a distance of three diameters with velocity 0.01 m/s in z-direction of the rectangular solution domain (L x = L y = 5R, L z = 10R). The initial centroid position of the droplet is (2.5R, 2.5R, 2R). Surface tension and viscous forces are not considered for this setup. As depicted in fig. 10, it is obvious that L ∞ (v) remains stable over time when the droplet translates. Even in the cases with a density ratio of over 10000, as shown in fig. 10d, no matter how high the resolution is, the results from ρLENT the method can reach machine precision.
Apart from the observation mentioned above, table 2 reveals another advantage of ρLENT method, that is, the conspicuous computational efficiency. As shown in table 2, the execution time of the solver using ρLENT method to simulate a case is always short. cases resolution execution time (s) 16 6. 62
Conclusions
The proposed ρLENT method exactly ensures numerical consistency of the single-field incompressible two-phase momentum convection, discretized by the unstructured collocated Finite Volume Method. The ρLENT method is straightforward and can be applied directly to any two-phase flow simulation method that relies on the collocated FV method for equation discretization: the only difference is the computation of area fractions α n+1 f from the approximated fluid interfaceΣ n+1 . We show, by analyzing the two-phase momentum advection equation, that the numerical consistency requires the computation of the cell-centered density ρ n+1 c using a mass flux identical to the one used in the two-phase momentum convective term. This provides the theoretical reasoning behind the auxiliary mass conservation equation, originally introduced by Ghods and Herrmann [3]. Following the importance of the face-centered (mass flux) density pointed out by [7], we derive the expression for the mass flux density using the principle of mass conservation and connect the mass flux density with the phase indicator. We achieve this by avoiding the temporal integration of the conserved property as done very recently by Arrufat et al. [31], which allows us to express the mass fluxes using the phase indicator in a discrete setting. The consistent cell-centered density ρ n+1 c is used in the p − v coupling algorithm [2] to obtain the velocity v n+1 c , necessary to evolve the fluid interface in the next step from t n+1 to t n+2 . Once the velocity is obtained by p − v coupling, the cell-centered density ρ n+1 c is again made consistent with the fluid interface. Using the face-centered (mass-flux) density in the p − v coupling and advecting the interface first, enables ρLENT to discretize the momentum convection term implicitly, compared to the explicit convective term discretization that is used by Bussmann et al. [15], Ghods and Herrmann [3] in the collocated finite volume setting. The consistency of the mass flux in the auxiliary density equation with the mass flux computed using the phase indicator, justifies theoretically the use of the same schemes for these two fluxes by Ghods and Herrmann [3], Patel and Natarajan [28], Manik et al. [29].
Results demonstrate the recovery of an exact solution, with the error in the L ∞ norm exactly equaling 0, for the canonical droplet translation verification case studies [69]. Droplets with submillimeter diameters and with realistic fluid properties are also advected exactly. Validation cases with realistic surface tension forces and viscosity demonstrate numerical stability of ρLENT, resulting in the relative L ∞ norm for the parasitic currents between 10 −4 and 10 −2 for realistic density ratios.
Acknowledgments
Figure 1 :
1The different domains for phase 1 drop surrounded by a phase 2
for a recent review), potentially ensure this property, since they solve the conservative formulation of the volume fraction advection equation for α n+1 c by computing phase-specific fluxed volumes, and then use the cell-centered volume fraction α n+1 c to compute ρ n+1 c with eq. (9). However, the temporal discretization scheme used in the momentum equation for the convective term, must be consistent with the integration of the fluxed phase-specific volumes, used to obtain α n+1 c . Additionally, the α n+1 c ∈ [0, 1] must hold near machine epsilon. Any correction to α n+1 c performed after the numerical solution of the volume fraction advection equation, that bounds α n+1 c within [0, 1], results in a discrepancy between ρ n+1 c computed using the mass flux that gives unbounded α n+1 c , and the ρ n+1 c computed from the a-posteriori bounded α n+1 c using eq. (9). It is important to note that if the pressure gradient is included on the r.h.s of eq. (16), any error in v n+1 c will result in non-zero source terms on the r.h.s. of the resulting pressure equation, in the p − v coupling algorithm. Since the pressure gradient enforces ∇ · v = 0 ( f ∈Fc F f = 0 on the discrete level), this results in artificial velocities similar to parasitic currents caused by the surface tension force.
Figure 2 :
2A two-phase fixed control volume Ωc separated by the interface Σ(t).
Interface Σ at t n and t n+1 and the respective Ω − (t n ) and Ω − (t n+1 ) in gray color, used to compute α n c and α n+1 c , that are further used to compute ρ n c and ρ
Figure 3 :
3Updating the face-centered (mass flux) density in the ρLENT method. then used to update the cell-centered density ρ n+1 c by solving a mass conservation equation. The index m in the volumetric flux refers to the linearization of the convective term in the momentum equation. The same mass flux ρ n+1 f F m f is used in the implicitly discretized momentum conservation equation. The pressure-velocity coupling algorithm iterates the linearized volumetric flux F m f to F n+1 f . Finally, the cell-centered velocity v n+1 c
Figure 4 :
4Computing area fractions from signed distances in the method.
Algorithm 1
1The ρLENT solution algorithm.1: while simulation time ≤ end time do 2:
Figure 5 :
5Translating droplet case setup.
Figure 6 :
6Half section of mesh N = 64, droplet at initial position. (a) Old inconsistent method. (b) New consistent method.
Figure 7 :
7Temporal evolution of velocity error norm L∞(v): the left figure depicts the results from old inconsistent SAAMPLE algorithm, the right shows the results from new consistent method.
Figure 8 :
8Comparison of the strong interface deformation with the inconsistent LENT method (left) and the numerically consistent interface shape of the ρLENT method. Parameters: N = 64, ρ − /ρ + = 10 4 , t = 0.0008s.
Figure 9 :
9Temporal evolution of velocity error norm L∞(v) for the viscous flow with surface tension forces: the left diagram depicts the results from the old inconsistent method, and the right diagram contains the results from the ρLENT method. The legends of these diagrams are large, and the full information is available in Appendix A: fig. A.11 for fig. 9a, fig. A.12 for fig. 9b. 4.2.3. Translating sub-millimeter droplet with realistic physical properties materials/properties (25°C) density (kg m −3 ) kinematic viscosity (m 2 s −1 ) surface tension (N m
( a ) 37 Figure 10 :
a3710Silicone oil droplet in water, density ratio 0.96 (b) Silicone oil droplet in air, density ratio 788.92 (c) Water droplet in air, density ratio 842.17 (d) Mercury droplet in air, density ratio 11431.Temporal evolution of velocity error norm L∞(v) with pure advection: ρLENT method used in simulating two-phase flows with different density ratios, mesh resolution: N = 16, 32, 64.
Funded by the German Research Foundation (DFG) -Project-ID 265191195 -SFB 1194. Calculations for this research were conducted on the Lichtenberg high performance computer of the TU Darmstadt.
Figure A. 12 :
12Full figure of fig. 9b
Table 2 :
2Execution time for the ρLENT method.
In this publication ρ − denotes the density of the denser fluid, so that ρ − ≥ ρ + and ρ − /ρ + ≥ 1 holds.
The Level Set and VoF methods do not exactly preserve the shape of a translating droplet.
All two-phase flow simulation methods encountered by the authors use eq. (9).
pressed using the dynamic and the hydrostatic pressure. The diagonal coefficient a c corresponds to the cell Ω c , and k denotes the coefficients contributed from cells that are face-adjacent to Ω c . An interesting aspect of the semi-implicit discretization in eq.(34), is a potentially more stable discretization of the surface tension force term (f σ ) c at the new time step t n+1 . Namely, the Front Tracking[57]in the LENT method[1](also in LCRM[47], and LCRM[38]) discretizes eq. (26) in a Lagrangian reference frame. Once the new interface is obtained as {x n+1 p } p∈P in a Lagrangian reference frame, the new interface is mapped onto an Eulerian signed-distance field. Since the Lagrangian advection is unconditionally stable, the mapping is geometrical and therefore bounded, and f n+1does not introduce a CFL condition, contrary to other Eulerian two-phase flow methods. This is a potential benefit that will be investigated in the future, as it may allow for larger time steps, than those limited by hydrodynamic stability condition[62]or CFL. Here, the Euler explicit method has been used in eq. (26) for simplicity; other explicit higher-order temporal discretizations can be used instead of eq. (26).Equation eq. (34) is discretized semi-implicitly, because of the linearized convective term, that contributes the volumetric flux F f to the a c,k coefficients in eq.(34). Linearizing the convective term introduces a need for iteration. Iterations are also introduced by splitting eq. (34) into two equations: one for v n+1 c , and another for p n+1 c . Dividing the equation eq. (34) with a c and applying the discrete divergence ∇ c ·, results in the pressure equation , v n+1 c ) is standard in the context of collocated unstructured finite volume method[61]: the inner iterations and the assembly of the pressure equation originates from the PISO algorithm[63], the outer iterations originate from the SIMPLE algorithm[64], and the tolerance-based control of outer iterations is described in detail in[2]. In addition, the implementations of the LENT method[1], the SAAMPLE algorithm[2]and the ρLENT method are publicly available . This description, the details on the tolerance-based outer iteration control in[2], and the publicly available implementation in OpenFOAM, provide sufficient information for an interested reader willing to understand or further extend the methodology.Appendix A. Parameter study figures with legends
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"Task Driven Generative Modeling for Unsupervised Domain Adaptation: Application to X-ray Image Segmentation",
"Task Driven Generative Modeling for Unsupervised Domain Adaptation: Application to X-ray Image Segmentation"
]
| [
"Yue Zhang \nMedical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA\n\nDepartment of Mathematics, Applied Mathematics and Statistics\nCase Western Reserve University\n44106ClevelandOHUSA\n",
"Shun Miao \nMedical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA\n",
"Tommaso Mansi \nMedical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA\n",
"Rui Liao \nMedical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA\n"
]
| [
"Medical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA",
"Department of Mathematics, Applied Mathematics and Statistics\nCase Western Reserve University\n44106ClevelandOHUSA",
"Medical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA",
"Medical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA",
"Medical Imaging Technologies\nSiemens Healthineers Technology Center\n08540PrincetonNJUSA"
]
| []
| Automatic parsing of anatomical objects in X-ray images is critical to many clinical applications in particular towards image-guided invention and workflow automation. Existing deep network models require a large amount of labeled data. However, obtaining accurate pixelwise labeling in X-ray images relies heavily on skilled clinicians due to the large overlaps of anatomy and the complex texture patterns. On the other hand, organs in 3D CT scans preserve clearer structures as well as sharper boundaries and thus can be easily delineated. In this paper, we propose a novel model framework for learning automatic X-ray image parsing from labeled CT scans. Specifically, a Dense Image-to-Image network (DI2I) for multi-organ segmentation is first trained on Xray like Digitally Reconstructed Radiographs (DRRs) rendered from 3D CT volumes. Then we introduce a Task Driven Generative Adversarial Network (TD-GAN) architecture to achieve simultaneous style transfer and parsing for unseen real X-ray images. TD-GAN consists of a modified cycle-GAN substructure for pixel-to-pixel translation between DRRs and X-ray images and an added module leveraging the pre-trained DI2I to enforce segmentation consistency. The TD-GAN framework is general and can be easily adapted to other learning tasks. In the numerical experiments, we validate the proposed model on 815 DRRs and 153 topograms. While the vanilla DI2I without any adaptation fails completely on segmenting the topograms, the proposed model does not require any topogram labels and is able to provide a promising average dice of 85% which achieves the same level accuracy of supervised training (88%).Disclaimer: This feature is based on research, and is not commercially available. Due to regulatory reasons its future availability cannot be guaranteed. | 10.1007/978-3-030-00934-2_67 | [
"https://arxiv.org/pdf/1806.07201v1.pdf"
]
| 49,317,508 | 1806.07201 | 4032ea0e2d6d4276af69ad61de40d9456018d1df |
Task Driven Generative Modeling for Unsupervised Domain Adaptation: Application to X-ray Image Segmentation
Yue Zhang
Medical Imaging Technologies
Siemens Healthineers Technology Center
08540PrincetonNJUSA
Department of Mathematics, Applied Mathematics and Statistics
Case Western Reserve University
44106ClevelandOHUSA
Shun Miao
Medical Imaging Technologies
Siemens Healthineers Technology Center
08540PrincetonNJUSA
Tommaso Mansi
Medical Imaging Technologies
Siemens Healthineers Technology Center
08540PrincetonNJUSA
Rui Liao
Medical Imaging Technologies
Siemens Healthineers Technology Center
08540PrincetonNJUSA
Task Driven Generative Modeling for Unsupervised Domain Adaptation: Application to X-ray Image Segmentation
Unsupervised Domain Adaptation · Deep Learning · Image Parsing · Generative Adversarial Networks · Task Driven
Automatic parsing of anatomical objects in X-ray images is critical to many clinical applications in particular towards image-guided invention and workflow automation. Existing deep network models require a large amount of labeled data. However, obtaining accurate pixelwise labeling in X-ray images relies heavily on skilled clinicians due to the large overlaps of anatomy and the complex texture patterns. On the other hand, organs in 3D CT scans preserve clearer structures as well as sharper boundaries and thus can be easily delineated. In this paper, we propose a novel model framework for learning automatic X-ray image parsing from labeled CT scans. Specifically, a Dense Image-to-Image network (DI2I) for multi-organ segmentation is first trained on Xray like Digitally Reconstructed Radiographs (DRRs) rendered from 3D CT volumes. Then we introduce a Task Driven Generative Adversarial Network (TD-GAN) architecture to achieve simultaneous style transfer and parsing for unseen real X-ray images. TD-GAN consists of a modified cycle-GAN substructure for pixel-to-pixel translation between DRRs and X-ray images and an added module leveraging the pre-trained DI2I to enforce segmentation consistency. The TD-GAN framework is general and can be easily adapted to other learning tasks. In the numerical experiments, we validate the proposed model on 815 DRRs and 153 topograms. While the vanilla DI2I without any adaptation fails completely on segmenting the topograms, the proposed model does not require any topogram labels and is able to provide a promising average dice of 85% which achieves the same level accuracy of supervised training (88%).Disclaimer: This feature is based on research, and is not commercially available. Due to regulatory reasons its future availability cannot be guaranteed.
Introduction
Semantic understanding of anatomical objects in X-ray images is critical to many clinical applications, such as pathological diagnosis, treatment evaluation and surgical planning. It serves as a fundamental step for computer-aided diagnosis and can enable intelligent workflows including organ-based autocollimation, infinite-capture range registration, motion compensation and automatic reporting. In this paper, we study one of the most important problems in semantic understanding of X-ray image, i.e., multi-organ segmentation.
While X-ray understanding is of great clinical importance, it remains a very challenging task, mainly due to the projective nature of X-ray imaging, which causes large overlapping of anatomies, fuzzy object boundaries and complex texture patterns. Conventional methods rely on prior knowledge of the procedure (e.g., anatomical motion pattern from a sequence of images [1]) to delineate anatomical objects from X-ray images. Modern approaches utilize deep convolutional networks and have shown superior performance [2]. However, they typically require a large amount of pixel-level annotated training data. Due to the heterogeneous nature of X-ray images, accurate annotating becomes extremely difficult and time-consuming even for skilled clinicians. On the other hand, large pixel-level labeled CT data are more accessible. Thousands of X-ray like images, the so-called Digitally Reconstructed Radiographs (DRRs), are generated from labeled CTs and used in [3] to train an X-ray depth decomposition model. While using automatically generated DRRs/labels for training has merits, the trained model cannot be directly applied on X-ray images due to their appearance Generalization of image segmentation models trained on DRRs to X-ray images requires unsupervised domain adaptation. While many effective models [4,5] have been studied, most of them focus on feature adaptation which naturally suits for recognition and detection. However, segmentation task desires pixel-wise classification which requires delicate model design and is substantially different. Recently, pixel-level adaptation models [6,7] have been proposed which utilize generative adversarial networks and achieve promising results on image synthesis and recognition. Still, continuing study on image segmentation especially for medical applications remains blank.
In this paper, we present a novel model framework to address this challenge. Specifically, we first create DRRs and their pixel-level labeling from the segmented pre-operative CT scans. A Dense Image-to-Image network (DI2I) [8,9] is then trained for multi-organ (lung, heart, liver, bone) segmentation over these synthetic data. Next, inspired by the recent success of image style transfer by cycle generative adversarial network (cycle-GAN) [7], we introduce a task driven generative adversarial network (TD-GAN) to achieve simultaneous image synthesis and automatic segmentation on X-ray images, see Figure 1 for an overview. We emphasize that the training X-ray images are unpaired with previous DRRs and are totally unlabeled. TD-GAN consists of a modified cycle-GAN substructure for pixel-to-pixel translation between DRRs and X-ray images. Meanwhile, TD-GAN incorporates the pre-trained DI2I to obtain deep supervision and enforce consistent performance on segmentation. The intuition behind TD-GAN is indeed simple: we transfer X-ray images in the same appearance as DRRs and hence leverage the pre-trained DI2I model to segment them. Furthermore, the entire transfer is guided by the segmentation supervision network. The contributions of our work are: 1) We propose a novel model pipeline for X-ray image segmentation from unpaired synthetic data (DRRs). 2) We introduce an effective deep architecture TD-GAN for simultaneously image synthesis and segmentation without any labeling effort necessary from X-ray images. To our best knowledge, this is the first end-to-end framework for unsupervised medical image segmentation.
3) The entire model framework can be easily adjusted for unsupervised domain adaptation problem where labels from one domain is completely missing. 4) We conduct numerical experiments and demonstrated the effectiveness of the proposed model with over 800 DRRs and 150 X-ray images.
Methodology
Problem Overview
In this paper, our goal is to learn a multi-organ segmentation model on unlabeled X-ray images using pixel-wise annotated DRRs data. Furthermore, these images are not paired, which means they are not taken from the same group of patients. We collected X-ray images that contain targeted organs such lung, heart, liver and bone (or a subset of them). The DRRs of the same region of interest (ROI) are generated by placing 3D labeled CT volumes in a virtual imaging system that simulates the actual X-ray geometry. Meanwhile, the pixel-level labeling of DRRs are generated by projecting 3D CT labels along the same trajectories. While most public datasets for multi-organ segmentation only consists of tens of cases, our dataset covers a richer variety of scanning ranges, contrast phases as well as morphological differences. In the next subsections, we first train a DI2I on the DRRs and then adapt it to the X-ray images with TD-GAN to provide deep segmentation supervision during the image synthesis.
Dense Image to Image Network for Segmentation on DRRs
We train a Dense Image-to-Image network (DI2I) on the labeled DRRs data. As is depicted in Figure 2, the network employs an encoder-decoder UNet structure with dense blocks [2,8,9]. The network consists of dense blocks which are generalizations from ResNets [10] by iteratively concatenating all feature outputs in a feed-forward fashion. This helps alleviating the vanishing gradient problem and thus can obtain a deeper model with higher level feature extraction. Despite these appealing properties, empirically we found that it achieved superior performance than classical UNet. The final output feature map has five channels which consists of a background channel x 0 and four channels x 1 , ..., x 4 corresponding to the four organs. By doing so, we alleviate the challenge of segmenting overlapped organs and simplify the problem into binary classifications. We further use a customized loss term which is a weighted combination of binary cross entropies between each organ channel and background channel,
L seg = − 4 i=1 w i (y i log(p i ) + (1 − y i ) log(1 − p i ))(1)
where y i is the ground truth binary label map for each organ and p i is calculated as exp(x i )/(exp(x 0 ) + exp(x i )) for i = 1, 2, 3, 4.
Task Driven Generative Adversarial Networks (TD-GAN)
We now describe the main deep architecture of TD-GAN. The network has a modified cycle-GAN [7] sub-structure with add-on segmentation supervisions. As is depicted in Figure 3, the base structure consists of two generators G 1 , G 2 and two discriminators D 1 , D 2 to achieve pixel-to-pixel image synthesis. The generators try to synthesize images in the appearance of the other protocol while the discriminators need to distinguish the generated images (fake) against the reals. When the cyclic training converges to optimal, the network will be able to transfer X-ray images to DRRs through G 2 and transfer back through G 1 . This also holds for DRRs. However, image synthesis only serves for a general purpose of appearance transfer and is not segmentation focused. Important prior knowledge such as organ boundaries, shapes and local variations are not carefully treated and could possibly lose during the synthesis process. We therefore add supervision modules to enforce the segmentation consistency, that is, during the transfer process we require the X-ray images that not only have the appearance of DRRs but also can be segmented by the pre-trained DI2I network. This is done by introducing conditional adversarial training on the translation of X-ray images and cycle segmentation consistency on DRRs. We detail this later on. The proposed TD-GAN involves 4 different paths that transfer and reconstruct images between the two protocols: real DRR → fake X-ray, real X-ray → fake DRR, real X-ray → reconstructed X-ray and real DRR → reconstructed DRR. Different losses are proposed for each path. We discuss the meaning of these paths in a top-down topological order that is shown in Figure 3. We denote the data distribution d ∼ p d and x ∼ p x for DRRs and X-ray images. Our main contribution comes in the segmentation driven losses in path real X-ray → fake DRR and real DRR → reconstructed DRR.
Real DRR → Fake X-ray. Given a real DRR image, the generator G 1 tries to produce the corresponding image in the appearance of X-ray images. The discriminator D 1 will need to distinguish the generated fake X-ray image and the real. Since we do not have any paired X-ray images with the DRRs, this real X-ray image is randomly selected from the training dataset. A successful generation from G 1 will confuse D 1 to make the wrong prediction. The loss function involved is a standard GAN loss for image style transfer,
L DX := E t∼px {log [D 1 (x)]} + E d∼p d {log [1 − D 1 (G 1 (d))]} .
Real X-ray → Fake DRR. The other generator G 2 will produce a fake DRR to challenge D 2 . We could also randomly select a real DRR in the training of D 2 . However, this is suboptimal since the labels of DRRs will be unused which contain important organ information such as size, shape and location and are crucial to our final segmentation task. Inspired by the conditional GANs [11], we leverage the pre-trained DI2I to predict the organ labels on the fake DRRs. The fake DRRs combined with their predicted labels are then fed into D 2 to compare with the real pairs. Therefore D 2 needs to not only distinguish the fake DRRs and the reals but also determine whether the image-label pairs are realistic. To confuse D 2 , the generator G 2 will particularly focus on the organs of interest during the image transfer. We hence will obtain a more powerful generator in the task of segmentation. Finally, to make the involved loss function differentiable, we only obtain the predicted probability map from the DI2I and do not binarize them. Denote U (·) as the pre-trained DI2I, we have the following loss function for this path,
L XD := E d∼p d {log [D 2 (d|U (d))]} + E x∼px {log [1 − D 2 (G 2 (x)|U (G 2 (x)))]} .(2)
We remark that the pre-trained DI2I is frozen during the training of TD-GAN, otherwise the supervision will be disturbed by the fake DRRs. Furthermore, TD-GAN can be easily adapted to other tasks by replacing U (·).
Real X-ray → Reconstructed X-ray. The idea behind this path is that once we transferred a X-ray image to a fake DRR through G 2 , we should be able to transfer this DRR back to X-ray image through G 1 . The final (reconstructed) X-ray image should be the same as the original one. This is called cycle-consistency. The corresponding loss function is calculated by l 1 distance,
L XX := E x∼px { G 1 (G 2 (x)) − x 1 } Real DRR → Reconstructed DRR.
Same argument also applies to the DRR reconstruction. Moreover, we enforce the reconstructed DRR to maintain cycle segmentation consistency by adding a segmentation loss as a regularization term. The implies that we would like that the reconstructed DRR is not only close to the original but maintain the same segmentation performance under DI2I. The two losses involved in this path are as follows,
L DD := E d∼p d { G 2 (G 1 (d)) − d 1 } , L seg := L seg ,(3)
where the segmentation loss L seg is the same as in equation (1). The total loss for TD-GAN is then a weighted summation of all the losses in the above paths. We demonstrate the effectiveness of these two losses in the experiments. We employ the same parameter setting as is suggested in [7]. The generator G 1 and G 2 utilize a same Resnet20 structure. The discriminator D 1 and D 2 contains four consecutive convolutional layers with increasing number of filters from 64 to 512 and a final output layer with sigmoid activation.
Experiments and Results
In this section we validate our methodology on a dataset of 815 labeled DRRs and 153 topograms. The topograms are acquired before the CT scan for isocen-tering and therefore co-registered with the CT. The CT scans are labeled pixelwisely and the labels of topograms are generated in a same way as the aforementioned DRRs masks. The co-registered CT is not used in training and the labels on topograms are used only for evaluation purpose. Our model is among the first approaches to address unsupervised domain adaptation for segmentation with unpaired data. We compare with the state-of-the-art image synthesis model cycle-GAN [7] since both can be used in our problem except ours are task driven. To further show the effectiveness of the deep architecture and the proposed losses, we also compare our models with TD-GAN adversarial (TD-GAN-A) and TD-GAN reconstruction segmentation (TD-GAN-S) by enabling either one of the conditional adversarial loss (2) and the cycle segmentation consistency loss (3).
We first train a DI2I for multi-organ segmentation on the labeled DRRs. A standard 5 fold cross-validation scheme is used to find the best learned weights. We evaluate the dice score on the testing dataset, summarized as follows (mean ± std): lung 0.9417 ± 0.017, heart 0.923 ± 0.056, liver 0.894 ± 0.061 and bone 0.910 ± 0.020. All the experiments are run on an 12GB NVIDIA TITAN X GPU. Next we load the pretrained DI2I into TD-GAN with weights all frozen and train the model to segment the topograms. We use all the DRRs as well as 73 topograms for training, 20 topograms for validation and 60 for testing. We visualize one example and compare the proposed method against vanilla setting (where the DI2I is tested directly on topograms) in Figure 4. For better visualization purpose, we only show the ground the truth labeling for lung, heart and liver. The numerical results are summarized in table 1. To better understand the performance of our model, we also train the DI2I on the topograms using their labels under the same data splitting setting, listed as Supervised in table 1.
While the direct application of the learned DI2I on topograms fails completely, it can be seen that our model significantly improved the segmentation accuracy and even provided the same level of accuracy compared with the supervised training with labeled topograms. Compared with the cycle-GAN which only performs image style transfer, both the partially task driven nets TD-GAN-A and TD-GAN-S can improve the performance. Furthermore, the final TD-GAN combines the advantages of all the three models and achieves the best.
Fig. 1 .
1Overview of the proposed task driven generative model framework.
Fig. 3 .
3Proposed TD-GAN architecture. Real X-ray images and DRRs are passed through 4 different paths for simultaneously synthesis and segmentation.
Fig. 4 .
4Visualization of segmentation results on topograms (bottom) against direct application of DI2I (top). The red curves stands for the boundary of the ground truth. The colored fill-in parts are the predictions by TD-GAN and DI2I.
Fig. 2. Illustration of the DI2I for segmentation on DRRs.Dense
Block
Dense
Block
Dense
Block
Dense
Block
Dense
Block
3 × 3 conv, BN, ReLU
2 × 2 max pooling
deconv with 80 filters
skip connection
1 × 1 conv, BN, ReLU
DRRs
Outputs
48
80
80
160
160
5
Dense Block
48 64 80 96 112
Table 1 .
1Average Dice results of segmentation on topograms. Objects Vanilla CGAN TD-GAN-A TD-GAN-S TD-GAN SupervisedBone 0.401 0.808
0.800
0.831
0.835
0.871
Heart 0.233 0.816
0.846
0.860
0.870
0.880
Liver 0.285 0.781
0.797
0.804
0.817
0.841
Lung 0.312 0.825
0.853
0.879
0.894
0.939
mean 0.308 0.808
0.824
0.844
0.854
0.883
Discussions and ConclusionsIn this paper, we studied the problem on multi-organ segmentation over totally unlabeled X-ray images with labeled DRRs. Our model leverages a cycle-GAN substructure to achieve image style transfer and carefully designed add-on modules to simultaneously segment organs of interest. The proposed model framework is general. By replacing the DI2I with other types of supervision networks, it can be easily adapted to many scenarios in computer-aided diagnosis such as prostate lesion classification, anatomical landmark localization and abnormal motion detection. We leave this for the future direction.
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| []
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[
"Kink dynamics in a system of two coupled scalar fields in two space-time dimensions",
"Kink dynamics in a system of two coupled scalar fields in two space-time dimensions"
]
| [
"A Alonso Izquierdo \nDepartamento de Matematica Aplicada\nUniversidad de Salamanca\nSPAIN\n"
]
| [
"Departamento de Matematica Aplicada\nUniversidad de Salamanca\nSPAIN"
]
| []
| In this paper we examine the scattering processes among the members of a rich family of kinks which arise in a (1+1)-dimensional relativistic two scalar field theory. These kinks carry two different topological charges that determine the mutual interactions between the basic energy lumps (extended particles) described by these topological defects. Processes like topological charge exchange, kinkantikink bound state formation or kink repulsion emerge depending on the charges of the scattered particles. Two-bounce resonant windows have been found in the antikink-kink scattering processes, but not in the kink-antikink interactions. arXiv:1711.08784v1 [hep-th] | 10.1016/j.physd.2017.10.006 | [
"https://arxiv.org/pdf/1711.08784v1.pdf"
]
| 118,956,348 | 1711.08784 | cc1f29390d0530cd9ad9700730223d4ac63dd71e |
Kink dynamics in a system of two coupled scalar fields in two space-time dimensions
A Alonso Izquierdo
Departamento de Matematica Aplicada
Universidad de Salamanca
SPAIN
Kink dynamics in a system of two coupled scalar fields in two space-time dimensions
In this paper we examine the scattering processes among the members of a rich family of kinks which arise in a (1+1)-dimensional relativistic two scalar field theory. These kinks carry two different topological charges that determine the mutual interactions between the basic energy lumps (extended particles) described by these topological defects. Processes like topological charge exchange, kinkantikink bound state formation or kink repulsion emerge depending on the charges of the scattered particles. Two-bounce resonant windows have been found in the antikink-kink scattering processes, but not in the kink-antikink interactions. arXiv:1711.08784v1 [hep-th]
Introduction
Over the last decades topological defects have played a key role in the understanding of new phenomena in a large number of disciplines of non-linear science. For this reason the search for these types of solutions in some PDEs receives much attention in both Mathematics and Physics, see [1,2]. Among these equations the relativistic non-linear Klein-Gordon equation
∂ 2 φ i ∂t 2 − ∂ 2 φ i ∂x 2 = − ∂U ∂φ i , i = 1, . . . , D
is a prominent member, which has been extensively cited in the literature [3,4]. Here D is the number of fields needed to describe a certain phenomenon. It is also referred to as the dimension of the internal space (φ 1 , . . . , φ D ). For instance, in Condensed Matter Physics D is the number of order parameters in 1D materials, see [5], where the dispersion relations in the different phonon branches are all of them relativistic. Here, the study of kinks in relativistic (1+1)-dimensional D = 2 coupled scalar field theory is addressed. For these systems an associated energy-momentum tensor can be found, which can be used to introduce a formal definition of kink: A kink is a non singular solution of the nonlinear coupled field equations of finite energy whose energy density T 00 is localized at any point in time [2]. This last characteristic portrays these solutions as energy lumps, of which each can be interpreted as "extended particles" of the model, which differentiates the kinks from plane wave packets (radiation), that delocalize its energy over time. The celebrated soliton and kink of the sine-Gordon and φ 4 models [1,2] are included in this context. These models encompass only one scalar field and provide theoretical support to explain, for instance, the appearance of superconductivity in type II materials [6,7,8,9], electric charge fractionization in trans-polyacetylene (CH) x [10], the Josephson effect [5], stabilization of interbrane spacings in brane world scenarios which resolve the problems of the cosmological constant and the large hierarchy between the scales of weak and gravitational forces [11,12], among others. Promotion of these models to the quantum realm has led to studies on the one-loop corrections to the masses of these (1+1)dimensional topological defects, see [13,14,15,16]. A very interesting phenomenon, which deserves special attention, is the resonant kink-antikink interaction. In the seminal paper [17], Campbell, Schonfeld and Wingate thoroughly describe the dynamics of interacting kinks and antikinks in the φ 4 model. There exists a critical velocity v c which characterizes the behavior of these scattering processes. For kinkantikink collisions where the initial speed is v 0 > v c these single solutions collide, bounce back and escape. However, if v 0 < v c they are compelled to collide a second time. Campbell and his collaborators discovered that there exist certain initial velocity windows where the kink and the antikink escape after the second impact, while at other windows they form a bound state. Other resonant windows, where the kink and the antikink escape after colliding N ≥ 3 times, have also been found in this model. These authors also quantitatively explain this behavior by using collective coordinates for the kink solutions in this model. This effect can be explained by the resonant energy transfer mechanism, where an energy exchange between the kink translational mode and the internal vibrational mode takes place in each collision. At the first collision the internal vibrational eigenmode of the kink and antikink is excited, which decreases the kinetic energy. For particular initial velocities the energy of the vibrational mode is given back to the kinetic energy at subsequent collisions, which allows the kink and the antikink to escape in opposite directions. The application of the collective coordinate method to kink-antikink scattering in the φ 4 model was initially addressed in [18] and later corrected in [19,20]. Another novel property unveiled in this work is the fractal structure followed by the separation velocity versus collision velocity graph [21]. Goodman was able to provide a deep understanding of this feature in his illustrative papers [22,23,24]. He derives from the collective coordinate model for the resonant energy transfer mechanism a family of iterated maps which closely describe the chaotic behavior of the kink-antikink scattering. The previously described pattern is not specific to the φ 4 model and has been found in many other situations such as in the kink-antikink interactions in the modified sine-Gordon model [25] and in the φ 6 model [19,26], the kink-impurity interactions in the sine-Gordon and φ 4 models [27,28], the soliton-defect interaction in the sine-Gordon model [29,30,31], the interaction of kinks with local inhomogeneities [32,33] and the collision of vector solitons in the coupled nonlinear Schrödinger model [34,35,36]. It is of note that the single kink in the φ 6 model lacks internal vibrational modes [19] and the resonant energy transfer mechanism is triggered by an internal vibrational mode of the combined kink-antikink configuration [37]. In addition, the presence of many kink vibrational modes can lead to the suppression of bounce-windows in kink-antikink collisions [38] and the presence of quasiresonances [39,40]. An enrichment of the previously mentioned models is reached by increasing the internal space dimension. This involves the presence of several scalar fields in the model, which can be coupled by a potential function U (φ i ). Rajaraman's quotation "This already brings us to the stage where no general methods are available for obtaining all localized static solutions (kinks), given the field equations" [2] highlights the analytical difficulty in searching for kink solutions in this type of system. A huge amount of effort has been devoted to this issue in the last few decades, although most of this effort has been aimed at identifying static kink manifolds that exist in some field theory models. Here, the solutions depend only on the spatial coordinate and do not evolve in time. This static picture provides unchanging topological defects, where the forces acting on every point are balanced. It is assumed that an isolated basic "particle" described by a topological defect must be a member of this static kink manifold. This set may also include composite kinks, which consist of a distribution of basic kinks laid out in such a way that the total force exerted at every point vanishes. In other words, the static kink manifold provides us with a description of a set of solutions where the evolution of time is frozen.
In this paper we shall address the study of the kink dynamics in a two coupled scalar field theory model whose potential term is given by U (φ, ψ) = (4φ 2 +ψ 2 −1) 2 +4φ 2 ψ 2 . This model arises as a special member of the one-parameter model family with U (φ, ψ) = (4φ 2 + 2σψ 2 − 1) 2 + 16σ 2 φ 2 ψ 2 discussed by Bazeia and coworkers in the references [41,42], where the authors identify a pair of topological kinks. Shifman and Voloshin showed that this family of systems can be found as the dimensional reduction of a generalized Wess-Zumino model with two chiral super-fields [43,44]. They found that the static kink manifold in this model comprises a one-parameter family of energy degenerate composite kinks. These solutions are formed by two basic kinks which belong to different topological sectors and whose centers are placed at distinct points. An explicit demonstration of the stability of some of these solutions is presented in [45]. In addition to this, Sakai and Sugisaka analytically explore the existence of bound states of wall-antiwall pairs [46]. A supersymmetric version of this model compatible with local supersymmetry, where the kinks of the (1+1)-dimensional model promote to exact extended solutions of N = 1 (3+1)-dimensional supergravity was constructed in [47]. In this framework, the coupling of this scalar field theory model to gravity in (4+1)-dimensions in warped spacetime is considered by Bazeia in [48]. The formation of planar networks of topological defects is addressed in [49,50]. The breaking of the classical energy degeneracy for the static kink family by quantum-induced interactions has been studied in [51,52]. The distinctiveness of the case σ = 1 2 in the previously mentioned model family was first noted in [53,54] where it was shown that the analogue mechanical system derived from the static Klein-Gordon equation is Hamilton-Jacobi separable. This mechanical analogy underlies the fact that solving the static Klein-Gordon equation is tantamount to finding the solutions of a Lagrangian dynamical system in which x plays the role of time, the point-like particle position is determined by the fields φ i , and the potential energy of the particle is −U . This point allowed the authors to identify a second one-parameter family of energy degenerate static composite kinks for this particular member. These solutions are made of four basic particles (energy lumps). This means that an arrangement of four single kinks for which the mutual interactions are counterbalanced exists. A first attempt to study the kink dynamics for this model was accomplished in [55] in the adiabatic approximation, where it is assumed that the kink motion is very slow. Under these circumstances the evolution of the particles or energy lumps can be studied as geodesics on the static kink moduli space [56,3,57,58,59]. The goal of this paper is to study the interactions between the basic "extended particles" beyond the adiabatic approximation in the proposed model. It will be shown that in this two-scalar field theory model there exist eight single particles or energy lumps (described by four kinks and its corresponding antikinks) which carry two topological charges. Also, these particles will be scattered to uncover the nature of the interactions between them. Processes like topological charge exchange, kink-antikink bound state formation and kink repulsion emerge depending on the charges of the scattered particles.
The non-linearity of the evolution equations does not allow analytical tools to be employed to describe the behavior of the scattering solutions in detail. For this reason these processes will be studied by means of numerical simulations. Also, the modified algorithm described by Kassam and Trefethen in [60] will be used, which has been designed to solve the numerical instabilities of the exponential time-differencing Runge-Kutta method introduced in [61]. This explicit method is spectral in space and fourth order in time [62]. A scattering process can be confined to occur in a bounded spatial interval, but the collisions between the energy lumps can originate radiation which travels at relativistic speeds and arrives to the spatial frontiers of the simulation in a short period of time. Therefore, the previous numerical method must be implemented in a spatial interval large enough to avoid the radiation to corrupt the results. In order to gain more control over the radiation evolution, the previous scheme will be complemented with the use of the energy conservative second-order finite difference Strauss-Vazquez algorithm [63] implemented with Mur boundary conditions [64], which absorb the linear plane waves at the boundaries. The efficiency of this numerical scheme has been proved in [65], and its global stability and convergence were established in [66]. More general numerical schemes in this framework are studied in references [67,68,69].
The organization of this paper is as follows: in Section 2 the (1+1)-dimensional two coupled scalar field theory model, which is dealt with in this paper, will be introduced. The static kink manifold will be described where the basic topological defects of the model can be identified. These solutions involve just a single energy lump and can be interpreted as the "basic extended particles" of the system. These fundamental kinks are distinguished by the value of its topological charges and chirality, which endow them with different properties. The composite kink solutions included in the manifold are also explained in this section. In Section 3, the scattering processes between the basic particles will be analyzed. The strategy consists of colliding boosted kink solutions whose centers are initially placed far away. Four different scattering events arise, which depend on the topological charge values of the kinks involved in the collision. This allows the distinct interactions between the particles or lumps arising in the system to be described. In Section 4, the conclusions and final comments will be provided. 2 The scalar field theory model and the static kink manifold
We shall deal with a (1+1)-dimensional two-coupled scalar field theory model whose dynamics is governed by the action
S = d 2 x 1 2 ∂ µ φ ∂ µ φ + ∂ µ ψ ∂ µ ψ − U (φ, ψ) ,(1)
where Einstein summation convention is assumed for µ = 0, 1. Here φ : R 1,1 → R and ψ : R 1,1 → R are dimensionless real fields and the Minkowski metric g µν in the two-dimensional spacetime is chosen in the form g 00 = −g 11 = 1 and g 12 = g 21 = 0. We shall denote the spacetime coordinates as x 0 ≡ t and x 1 ≡ x from now on. The potential function U (φ, ψ) in (1) is given by the non-negative expression
U (φ, ψ) = (4φ 2 + ψ 2 − 1) 2 + 4φ 2 ψ 2(2)
for our model. The Euler-Lagrange equations derived from the action (1) lead to the coupled nonlinear Klein-Gordon equations
∂ 2 φ ∂t 2 − ∂ 2 φ ∂x 2 = − ∂U ∂φ = −16φ 4φ 2 + 3 2 ψ 2 − 1 ,(3)∂ 2 ψ ∂t 2 − ∂ 2 ψ ∂x 2 = − ∂U ∂ψ = −4ψ 6φ 2 + ψ 2 − 1 ,(4)+ U (φ, ψ) ,(5)P [φ, ψ] = dx ∂φ ∂x ∂φ ∂t + ∂ψ ∂x ∂ψ ∂t .(6)
In addition to the previous continuous symmetries, there exist discrete symmetries in our model. The action functional (1) remains invariant by the symmetry group G = Z 2 × Z 2 generated by the transformations π 1 : (φ, ψ) → (−φ, ψ) and π 2 : (φ, ψ) → (φ, −ψ) in the internal space. The mirror reflection in the space coordinate π x : x → −x does also play an important role in the study of topological defects because it relates kink and antikink solutions. The simplest stable solutions of equations (3) and (4) are the static homogenous solutions which correspond with the minima of the potential function U (φ, ψ). From (2), they constitute the set of zeroes of the potential function M = {(φ 0 , ψ 0 ) ∈ R 2 : U (φ 0 , ψ 0 ) = 0}:
M = A 1 = ( 1 2 , 0), A 2 = (− 1 2 , 0), B 1 = (1, 0), B 2 = (−1, 0) ,
whose total energy is zero. Notice that π 1 (A 1 ) = A 2 and π 1 (A 2 ) = A 1 such that the transformation π 1 links the constant solutions A 1 and A 2 . This allows us to define the vacuum orbit A = {A 1 , A 2 }. Likewise, π 2 (B 1 ) = B 2 and π 2 (B 2 ) = B 1 and B = {B 1 , B 2 } is the second vacuum orbit in this model. In general, the configuration space comprises the set of maps φ : R 1,1 → R and ψ : R 1,1 → R whose total energy (5) is finite,
C = {Φ(x, t) = (φ(x, t), ψ(x, t)) ∈ R × R : E[Φ(x, t)] < +∞}.
Owing to the total energy conservation law, the compliance of the previous condition in a single instant t 0 is sufficient for a map Φ(x, t) to belong to C. Based on the previous definition, every member of C must satisfy the following asymptotic conditions
lim x→±∞ ∂Φ(x, t) ∂t = lim x→±∞ ∂φ(x, t) ∂t = lim x→±∞ ∂ψ(x, t) ∂t = 0 ,(7)lim x→±∞ ∂Φ(x, t) ∂x = lim x→±∞ ∂φ(x, t) ∂x = lim x→±∞ ∂ψ(x, t) ∂x = 0 ,(8)lim x→±∞ Φ(x, t) = lim x→±∞ (φ(x, t), ψ(x, t)) ∈ M .(9)
The following step will be to identify static kinks, time-independent finite-energy solutions of the field equations (3) and (4) whose energy density is localized. This type of solutions usually lives in nonzero topological sectors of the configuration space C and its spatial dependence asymptotically links two different elements of M. This behavior allows two different topological charges to be introduced
q 1 = 2 · [φ(+∞, t 0 ) − φ(−∞, t 0 )] , q 2 = ψ(+∞, t 0 ) − ψ(−∞, t 0 ) ,
in our two scalar field theory model. These magnitudes are invariant because of the previous asymptotic conditions (7)- (9). Solutions carrying non-zero topological charges are unable to evolve in time to zeroenergy solutions (this would require infinite energy). In particular, the linear stability of a static solution Φ(x) is studied by means of the spectrum of the second order small fluctuation operator
H[Φ(x)] = − d 2 dx 2 + V 11 (x) V 12 (x) V 12 (x) − d 2 dx 2 + V 22 (x) = (10) = − d 2 dx 2 + ∂ 2 U ∂φ ∂φ [Φ(x)] ∂ 2 U ∂φ ∂ψ [Φ(x)] ∂ 2 U ∂φ ∂ψ [Φ(x)] − d 2 dx 2 + ∂ 2 U ∂ψ ∂ψ [Φ(x)]
The existence of negative eigenvalues in the spectrum of the operator (10) implies that the solution Φ(x) is unstable. Notice that (10) is a 2 × 2 matrix partial differential operator making the identification of its spectrum an arduous task. The identification of the static kink manifold in this model has been tackled in previous works from two points of view. The first procedure exploits the Hamilton-Jacobi separability of the analogue mechanical model derived from the static Klein-Gordon equations [53]. The second method makes use of the presence of two different superpotentials
W I = 4 √ 2( 1 3 φ 3 − 1 4 φ + 1 4 φψ 2 ) and W II = 1 3 √ 2 φ 2 + ψ 2 (4φ 2 + ψ 2 − 3)
, which lead to the same potential function (2) and reduce the problem (3) and (4) to first order differential equations, see [54]. The static kinks reported in [53,54] are listed below and will be organized and displayed in a way that facilitates the intelligence of the kink dynamics, an issue that will be researched in the next section and represents the main goal of this work.
(1) We shall start this description by introducing the basic energy lumps or particles of the model, which are characterized by the eight static kinks
K (q 1 ,q 2 ,λ) static (x) = q 1 4 λ + tanh( √ 2 x) , −λ q 2 1 2 1 − λ tanh[ √ 2 x](11)
where q i , λ = ±1. In addition the compact notation x = x − x 0 being x 0 ∈ R the position of the kink center in the real line has been used. The parameter x 0 arises as an integration constant when solving the differential equations and proves the spatial translational invariance of the model. The parameters q i (i = 1, 2) in the expression (11) are the topological charges associated with these kinks and take the values of ±1, see Figure 1. In any case these kinks asymptotically link elements in the vacuum orbit A = {A 1 , A 2 } (placed in the ψ = 0 axis) with elements of the vacuum orbit B = {B 1 , B 2 } (located in the φ = 0 axis), see Figure 1. Indeed, the parameter λ introduced in (11) (which will be called chirality for reasons which will be clear later on) determines the sense of this connection, if the kink comes from points of the orbit A at x = −∞ and arrives to a B type element at x = ∞ then λ = −1, while if the reverse sense takes place then λ = 1. All the eight basic kinks (11) share the same total energy,
E[K (q 1 ,q 2 ,λ) static (x)] = √ 2 3
although the kink energy density distribution
ε[K (q 1 ,q 2 ,λ) static (x)] = 1 8 sech 4 [ √ 2 x] 2 + cosh[2 √ 2 x] + λ sinh[2 √ 2 x](12)
distinguishes between kinks with different chirality λ. The energy density (12) is localized around only one point in such a way that the K (q 1 ,q 2 ,λ) static (x) kink can be interpreted as a basic particle (a single energy lump in the space line), see Figure 1 (right). The exponential decay of the energy density is stronger when x → ∞ than when x → −∞ for kinks with chirality λ = −1, as can observed in the energy density profile represented by a red dashed curve in Figure 1 (right). The opposite pattern is found for kinks with λ = 1, see the blue solid curve in Figure 1 (right). From a physical point of view, this fact is evident because this type of kinks asymptotically connects vacua with different mass matrices. In other words K (q 1 ,q 2 ,1) static (x) kinks are more energetic at the right side than at the left side of the energy density peak. The converse behavior occurs for kinks with chirality λ = −1. In a static scenario (where forces are absent), all the kink solutions with the same chirality are undistinguishable. static (x) kink centered at the origin (left), the basic kink orbits connecting type A and B vacua (middle) and energy density distribution for kinks with different chirality (right).
The action of the field reflection transformations π i (i = 1, 2) on the solutions (11) reverses the sign of the topological charge q i ,
π 1 [K (q 1 ,q 2 ,λ) static (x)] = K (−q 1 ,q 2 ,λ) static (x; x 0 ) , π 2 [K (q 1 ,q 2 ,λ) static (x)] = K (q 1 ,−q 2 ,λ) static (x; x 0 ) .
The mirror or spatial reflection symmetry π x changes all the properties (both topological charges and chirality) of the basic kinks
π x [K (q 1 ,q 2 ,λ) static (x)] = K (−q 1 ,−q 2 ,−λ) static (x) ,
although the kink orbit remains unchanged (which is traced in reverse order).
By convention we shall refer to solutions (11) with negative chirality K (q 1 ,q 2 ,−1) static (x) as kinks and those with positive chirality K (q 1 ,q 2 ,1) static (x) as antikinks where q i = ±1. In addition, the term kink and antikink of the same type will be used for those solutions which share the same orbit; that is, for the couple of kinks K (q 1 ,q 2 ,−1) static (x) and K (−q 1 ,−q 2 ,1) static (x), which are related by the transformation π x , otherwise it will be said that the kinks and antikinks are of different types. As shown later on, these distinctions simplify the language used in the scattering study.
The kink fluctuation operator H[K (q 1 ,q 2 ,λ) static (x)] follows the form (10) where the potential wells V ij (x), i, j = 1, 2 are given by
V 11 (x) = 4[2 + 3 tanh( √ 2x)(λ + tanh( √ 2x))] V 12 (x) = −6 √ 2q 1 q 2 (1 + λ tanh( √ 2x)) 1 − λ tanh( √ 2x) V 22 (x) = 1 2 [7 + 3 tanh( √ 2x)(−2λ + tanh( √ 2x))]
which have been depicted in the Figure 2 for the values λ = q 1 q 2 = −1. The zero mode ∂ ∂x K (q 1 ,q 2 ,λ) static (x) is part of the spectrum as a consequence of the translational symmetry of the model. Numerical studies point out that the rest of the spectrum is a continuous spectrum on the threshold value ω 2 = 2. No internal vibrational eigenmodes for the single kink appear in this model.
Figure 2: Potential well components V ij (x), i, j = 1, 2, of the kink fluctuation operator H[K (q1,q2,λ) static (x)].
The static picture of the kink manifold of the model does not finish here. Furthermore, together with the basic kinks, there is a pair of two-parameter families of static composite kinks which are described below.
(2) The first of the previously mentioned families, which links the points A 1 and A 2 , is determined by the expression
K (q 1 ,0) static (x, b) = q 1 4 sinh(2 √ 2 x) cosh(2 √ 2 x) + b 2 , b [b 2 + cosh(2 √ 2 x)] 1 2 , (q 1 = ±2, b ∈ R) .(13)
The magnitude q 1 inserted in (13) is the first topological charge associated with these kinks, which ranges the values 2 and −2. The second topological charge vanishes for these solutions. Every member of the kink family (13) can be interpreted as the concatenation of two basic static kinks following the form
K (q 1 ,q 2 ,−1) static (x − x 1 ) ∪ K (q 1 ,−q 2 ,1) static (x − x 2 ) with x 1 ≤ x 2 ,(14)
that is, a kink with topological charges (q 1 , q 2 ) followed by an antikink with the charges (q 1 , −q 2 ) in the spatial coordinate x. Notice that the involved kink and antikink are of different types because they share the same charge q 1 . The parameter b in (13) measures the distance between these two basic lumps. All of these features are illustrated in Figure 3. Take note that the energy density distribution for the solution (13) with b = 5 displayed in Figure 3 (middle) consists of two single energy lumps.
In particular, the solution K
(q 1 ,0) static (x, 0), obtained from (13) with b = 0, K (q 1 ,0) static (x, 0) = q 1 4 tanh(2 √ 2 x), 0(15)
describes a (q 1 , q 2 )-kink and a (q 1 , −q 2 )-antikink whose centers are placed at the same point. The energy density of this particular solution is represented in Figure 3 (right). The simplicity of the expression static (x, 0) to be studied. In this case, the small kink fluctuation operator
H[K (q 1 ,0) static (x, 0)] = − d 2 dx 2 + 32 − 48 sech 2 2 √ 2 x 0 0 − d 2 dx 2 + 2 − 6 sech 2 2 √ 2 x(16)
is a second order differential diagonal matrix operator. The discrete spectrum of (16) comprises two zero modes ξ T 0 (x) = (sech 2 2 √ 2x, 0) and ξ 0 T = (0, sech 1 2 2 √ 2x), which are in turn the ground states. Indeed, the presence of two zero modes occurs not just for the solution (15) but for every member of (13). This result underlies the fact that the kink family (13) depends on two real parameters x 0 and b. By changing the value of one of these parameters a shift is made from solutions to solutions of the model. If the change is infinitesimal, the difference between the new and the original solutions becomes a zero mode.
Therefore ∂ ∂x 0 K (q 1 ,0) static (x, b) ≡ ∂ ∂x K (q 1 ,0) static (x, b) and ∂ ∂b K (q 1 ,0)
static (x, b) correspond to the analytical expression of these eigenmodes. The first of these modes arises because of the spatial translational symmetry, and infinitesimally changes the kink center x 0 without changing the kink orbit. The second one deforms the orbit of the original solution, giving rise to a kink family member infinitesimally close in the b-parameter space. The final result of this change is that the basic kinks (which constitute this solution) separate infinitesimally. The excited state ξ T 1 = (sech 2 √ 2x tanh 2 √ 2x, 0) with eigenvalue ω 2 1 = 24 completes the discrete spectrum of (16). The excitation of this mode induces an internal vibration, the energy lump contracts (making its peak higher) and then stretches (lowing the energy peak) in a periodic sequence. The lack of negative eigenvalues in the spectrum of the fluctuation operator (16) implies that the particular kink (15) is stable. In references [53,54] it has been proven by applying the Morse theory on the kink orbit manifold that the previous conclusion is valid for every member of the family (13).
(3) There exists another two-parameter family of static composite kink solutions, determined by the expression
K (0,q 2 ) static (x, c) = sinh 2 √ 2c sinh 2 √ 2x cosh 2 2 √ 2x + 2 cosh 2 √ 2c cosh 2 √ 2x + 1 , q 2 2 sinh 2 √ 2x [cosh 2 2 √ 2x + 2 cosh 2 √ 2c cosh 2 √ 2x + 1] 1 2(17)
where the parameter c ∈ R and q 2 is the second topological charge associated with these solutions whose possible values are ±2. These kinks live in the topological sector which joins the members B 1 and B 2 of the vacuum orbit B. As a result the first topological charge q 1 vanishes. In Figure 4, the energy density of two members of this family is displayed. It can be observed that the solutions (17) consist of four basic lumps following the antikink-kink-antikink-kink arrangement
K (q 1 ,q 2 ,1) static (x − x 1 ) ∪ K (−q 1 ,q 2 ,−1) static (x) ∪ K (−q 1 ,−q 2 ,1) static (x) ∪ K (q 1 ,q 2 ,−1) static (x + x 1 ) ,(18)
where the kink and antikink in the middle of this sequence are exactly overlapped (giving rise to a K (q 1 ,0)
static (x, 0) configuration) and the rest ones are equidistant from this central lump. The distance between these constituents is set by the value of the family parameter c, see Figure 4. For the special value c = 0 the four basic kinks are located at the same point, and the solution (17) reduces to the expression
K (0,q 2 ) static (x, 0) = 0, q 2 4 tanh √ 2x(19)
whose density energy is displayed in Figure 4 (right). The evolution of the K (0,q 2 )
static (x, 0)-kink fluctuations is coded in the spectrum of the second order differential matrix operator
H[K (0,q 2 ) static (x, 0)] = − d 2 dx 2 + 8 − 24 sech 2 √ 2 x 0 0 − d 2 dx 2 + 8 − 12 sech 2 √ 2 x .(20)
The discrete spectrum of (20) begins with a negative eigenvalue ω 2 0 = −10, whose eigenfunction is given by η T 0 = (sech 3 √ 2x, 0). In addition, two degenerate zero modes η T 1 = (sech 2 √ 2x tanh √ 2x, 0) and η 1 T = (0, sech 2 √ 2x) with zero eigenvalue ω 2 1 = 0 are involved. The excitation of the longitudinal and transversal modes η 1 T and η T 1 infinitesimally moves the family parameters x 0 and c. The action of η 1 T is a simple translation of the kink solution, while the action of η 2 T consists of separating an antikink and a kink from the original energy lump formed by four merged basic particles. An eigenvalue ω 2 2 = 6 is also included in the discrete spectrum of (20) with the degenerate eigenfunctions η T 2 = (sech √ 2x(4 − 5 sech 2 √ 2x), 0) and η 2 T = (0, sech √ 2x tanh √ 2x). These correspond to internal vibrational modes of the kink. Finally a continuous spectrum emerges on the threshold value ω 2 = 8. The presence of the negative eigenvalue ω 2 0 implies that the kink (19) is unstable. This seems also to be supported by the fact that for this solution a kink K (q 1 ,q 2 ,−1) static (x) and its own antikink K (−q 1 ,−q 2 ,1) static (x) are placed at the same point. It should be noted that the evolution of the kink (19) excited by the mode η T 0 has not been described here because now the fluctuation grows indefinitely and the original kink changes into a completely different configuration. The application of the Morse theory indicates that all of the members of the family (17) are unstable due to the presence of a conjugate point in the kink orbit space, see [53,54].
In summary, the study of the static kink manifold in this model reveals that there are four basic particles of the model characterized by the value of the pair of topological charges (q 1 , q 2 ) with q i = ±1 and the chirality λ = −1. The corresponding antiparticles emerge by changing the sign of the topological charges and chirality. The static picture of the solutions described in this section points out that two basic particles of the type K (q 1 ,q 2 ,−1) static (x) and K (q 1 ,−q 2 ,1) static (x) can be placed at any location in the real line, giving rise to the family (13). If these lumps stand still the configuration remains unchanged over time because all of the forces are balanced. Additionally, there is another configuration with an analogous behavior described by the family (17). Here the composite kink K (q 1 ,0) static (x; x 0 , 0) is surrounded by an antikink and a kink carrying topological charge −q 1 and equidistant to this central lump. In this case, all the local forces are again counteracted; however, this is an unstable configuration which is spoilt if a little perturbation is introduced on this arrangement.
Kink dynamics: a study of two basic kink scattering
In this section the kink dynamics for the two-scalar field theory model introduced in Section 2 is investigated, such that complete intelligence of the effect of the nonlinearity on the kink evolution can be achieved in this model. In Section 2, the existence of four basic particles and the corresponding antiparticles are unveiled, which are distinguished by chirality, even in the static framework. In this section the aim is to uncover kink interactions by studying the scattering of the basic lumps or particles of the model. In particular, the results displayed in Section 2 present some of the following questions:
1. What is the evolution of the static configuration (14) if the basic lumps are pushed? In other words, how does a kink with charges (q 1 , q 2 ) and an antikink with charges (q 1 , −q 2 ) evolve when they are scattered each other?
2. Similarly, how does an antikink with charge (q 1 , q 2 ) and kink with charge (−q 1 , q 2 ) behave when they are propelled against each other? There are no static kink configurations with this arrangement, so the permanent presence of forces in this process is presumed.
3. The fate of the collision between a kink and its own antikink is also unknown. In some models collision speeds less than a critical velocity produce a bound state, where kink and antikink are trapped in an oscillatory movement, while in other models this event ends in mutual annihilation.
In addition, the resonant energy transfer mechanism can appear, giving rise to resonant initial velocity windows, which allow the kink and antikink ro escape after a finite number of collisions.
All of these questions will be addressed in this section by employing a numerical analysis. However, before tackling this task this subsection will end by introducing some comments regarding the boosted basic kinks. In a relativistic system, the static kinks (11) can be endowed with a time dependence by introducing a Lorentz boost
K (q 1 ,q 2 ,λ) (x, t; v 0 ) = K (q 1 ,q 2 ,λ) static x − v 0 t 1 − v 2 0 ,(21)
such that the lumps move with constant velocity v 0 . The total energy is now increased by the Lorentz factor
E[K (q 1 ,q 2 ,λ) (x, t; v 0 )] = √ 2 3 1 − v 2 0 .
Also, when the speed v 0 is increased, length contraction implies that the energy lumps are concentrated in a smaller region. This behavior is illustrated in Figure 5 for several velocities v 0 . In the following sections the results obtained by the numerical simulations, carried out with the purpose of describing the interactions between the basic particles in this model, will be discussed. The initial configuration must consist of a concatenation of alternating basic kinks and antikinks because of the topological constraints. The trajectory for this type of profiles asymptotically begins in the vacuum orbit of a given type, A or B, then travels towards the vacuum orbit of the other type, B or A, to later return to the vacuum orbit of the first type, see Figure 1(middle). The scattering processes related by the reflection transformations π i , i = 1, 2 and π x are equivalent. The evolution of the particles in a given event can be derived from an equivalent one. As a result, the present study will be restricted to non-equivalent processes. Furthermore, the analysis of the interactions between two of the basic kinks and antikinks, which can be considered as the fundamental events, will be discussed. These interactions are distinguished by the kink-antikink or antikink-kink arrangements, and by the relation between its types. We recall that a kink and an antikink are said to be of the same type if they share the same orbit, otherwise they are considered as different types.
3.1
The K (q 1 ,q 2 ,−1) -K (q 1 ,−q 2 ,1) scattering process: an exchange of the second topological charge
Here we shall deal with the scattering of the two basic lumps which comprise the static composite kink family (13). In this case, a basic kink is placed on the left of a basic antikink in the x-axis.
These are chosen to be of different type. In Section 2 it is shown that if these basic lumps stand motionless, the dynamics leaves this situation unchanged. Following on, the evolution of these kinks when they are obliged to collide with each other at speed v 0 is analyzed. If the mass center is fixed at the spatial coordinate origin the initial configuration can be represented by the kink-antikink concatenation (21) and x 1 is large enough to generate a continuous profile. The evolution of these basic lumps is displayed in Figure 6, where velocity v 0 has been chosen as v 0 = 0.2. It can be observed that the kink with charge (q 1 , q 2 ) and the antikink with charge (q 1 , −q 2 ) approach each other, coalesce, giving rise to an energy density sharper than the sum of the individual lump energy densities due to nonlinear interactions, finishing with the lumps bouncing back and exchanging the second topological charge q 2 . This scattering process is represented by the relation
K (q 1 ,q 2 ,−1) (x + x 1 , t; v 0 ) ∪ K (q 1 ,−q 2 ,1) (x − x 1 , t; −v 0 ) where K (q 1 ,q 2 ,±1) (x, t; v 0 ) is defined inK (q 1 ,q 2 ,−1) (v 0 ) ∪ K (q 1 ,−q 2 ,1) (−v 0 ) → K (q 1 ,−q 2 ,−1) (−v 0 ) ∪ K (q 1 ,q 2 ,1) (v 0 )
where the velocities of each lump are indicated. The final outcome consists of a kink with charge (q 1 , −q 2 ) that travels to the left and an antikink with charge (q 1 , q 2 ) that moves to the right. This phenomenon is quite an elastic event, because it takes place with a negligible emission of radiation. In reference [55] the above kink-antikink scattering process has been addressed under the adiabatic approximation where very slow motion is assumed for the lumps following the Manton's scheme [56]. In this context it is assumed that the kink family parameters x 0 and b, which arise in the expression (13), depend on time. This approach postulates that under the assumption of slowness of the process, a composite kink (13) evolves into a new configuration characterized by the same expression (13) although with different parameter values. If we plug this generalized form of (13) into the action functional (1) ordinary differential equations for the variables x 0 (t) and b(t) are obtained. The kink dynamics is now described by geodesics on the two dimensional static kink moduli space (x 0 , b). In other words, the dynamics is ruled by the excitation of the two zero modes associated with these composite kinks (13).
The results found in this subsection endorse the description of the scattering introduced in [55]. Indeed, they conclude that the validity of this analysis is applicable beyond the adiabatic approximation; that is, the description of the previous kink scattering is valid for an extensive range of collision velocities v 0 . Simulations with v 0 = 0.9 show that the previously mentioned pattern is maintained; however, a very small amount of kinetic energy is now converted to radiation.
3.2
The K (q 1 ,q 2 ,1) -K (−q 1 ,q 2 ,−1) scattering process: repulsive forces in action
In this subsection the scattering between an antikink with charge (q 1 , q 2 ) and a kink with charge (−q 1 , q 2 ) is analyzed. Now the basic lumps obliged to collide have the opposite first topological charge but the second topological charge is the same. The initial configuration consists of an antikink placed on the left of a different type of kink in the x-coordinate. We fix the mass center at the origin of the spatial axis such that the speed of each kink lump is v 0 in this reference frame. The initial arrangement can be represented by means of the antikink-kink concatenation K (q 1 ,q 2 ,1) (x+x 1 , t; v 0 )∪K (−q 1 ,q 2 ,−1) (x−x 1 , t; −v 0 ). In Figure 7 (left) the evolution of this profile, where v 0 = 0.2, is displayed. The particles approach each other but when they are close enough the lumps repel each other, avoiding collision. This involves the presence of repulsive forces between antikinks with charge (q 1 , q 2 ) and kinks with charge (−q 1 , q 2 ). The resulting configuration is similar to the original one although now the lumps move away from each other. Therefore this process can be represented as
K (q 1 ,q 2 ,1) (v 0 ) ∪ K (−q 1 ,q 2 ,−1) (−v 0 ) → K (q 1 ,q 2 ,1) (−v 0 ) ∪ K (−q 1 ,q 2 ,−1) (v 0 ) .
The previous pattern is general for any initial velocity v 0 . In Figure 8 (left) the final velocity of the lumps after the scattering process is represented with respect to v 0 . The dashed line in Figure 8 (left) characterizes the result if the process is elastic. Notice that these curves are indistinguishable for initial speeds v 0 up to 0.7. In these cases the radiation emission is negligible and the kink scattering is practically elastic. However, for higher initial velocities radiation phenomena are appreciable. The collision between the antikink and the kink is now so violent that a part of the kinetic energy of these extended particles is emitted in form of radiation, which decelerates the lumps. In Figure 7 (right), this type of events is illustrated for the case v 0 = 0.9 using a top view in order to enhance the visualization of the process. It can be observed that after impact each basic particle emits radiation in both directions, where part of this radiation is trapped between the lumps because a large amount of it is reflected upon reaching the kink cores. The rest of the radiation advances towards the simulation frontiers. In Figure 8 (right), the minimum distance d min between the lumps in the scattering process is plotted as a function of the initial velocity v 0 . Both of the graphs in Figure 8 have been generated by means of a discrete number of points with the initial velocity step ∆v 0 = 0.01.
3.3
The K (q 1 ,q 2 ,−1) -K (−q 1 ,−q 2 ,1) scattering process
In this subsection the scattering between a kink and an antikink of the same type is studied. In this case, the initial kink configuration is given by a kink with charge (q 1 , q 2 ) (this solution asymptotically starts at the point A i and arrives to the point B j following a given orbit) which is placed to the left of the antikink with charge (−q 1 , −q 2 ) (this solution returns from B j to the initial point A i retracing the previous kink trajectory in converse sense). Thus, this situation is dealing with the collision of a kink and its own antikink, which are pushed together with a velocity v 0 . As usual we fix the mass center at the origin of the x-axis. The initial configuration is represented by the kink-antikink concatenation K (q 1 ,q 2 ,−1) (x + x 1 , t; v 0 ) ∪ K (−q 1 ,−q 2 ,1) (x − x 1 , t; −v 0 ). Here we can distinguish two different types of scattering events, which are separated by the critical velocity v c ≈ 0.29703 in the initial velocity space. We find that:
• If v 0 < v c the involved particles are trapped in a bound state (bion) where the kink and the antikink are forced to approach and bounce back over and over again. In Figure 9 (left) this event has been depicted for v 0 = 0.2. In this process a part of the total energy is converted into radiation (observe the small ripples in the Figure). This process is represented by means of the relation
K (q 1 ,q 2 ,−1) (v 0 ) ∪ K (−q 1 ,−q 2 ,1) (−v 0 ) → K (q 1 ,q 2 ,−1) K (−q 1 ,−q 2 ,1) + radiation
where the use of the symbol emphasizes the formation of the kink-antikink bion.
An intriguing point related to the previous description is the fate of the kink-antikink coupling: does it become a stable oscillatory state or do the kink lumps succumb to mutual annihilation, leaving a radiation vestige? With the purpose of investigating this question, the evolution of the total energy in the simulation interval is plotted in Figure 10 together with the long term evolution of the period of the motion. The numerical simulations indicate that the energy loss after a period is very small and decreases over time. The total energy for t = 5000 is approximately 0.691125. From a numerical point of view it is not possible to guarantee that this configuration is completely stable although we can affirm that the survival time of this state is long. Therefore we can infer that kink and antikink form a very lasting bound state. The evolution of period T in this cyclic motion is shown in Figure 10. After a short interval of time the period T tends to the value 2.77 ± 0.01. This value seems to be independent of the shooting velocities v 0 in those cases where the bound state is formed. Notice however that the kink-antikink motion is not a pure oscillation because the frequency oscillates slightly. This behavior is pointed out in the framed plot inside Figure 10 (right) where a zoomed image of the period evolution curve is exhibited. • If the speed v 0 > v c then the attraction force is not strong enough to attach the kink to the antikink. They collide and bounce back with an escape velocity v f . This type of events has been illustrated in Figure 9 (right) for the case v 0 = 0.4. Notice that the kink-antikink collision is followed by radiation emission.
The global pattern is shown in the Figure 11 where the final velocity of the lumps is depicted with respect to its initial velocity. This graphic has been generated using a discrete number of points with step ∆v 0 = 0.001. This study has been refined near the critical velocity v c . Observe that if v 0 < v c the final velocity is zero, which implies the formation of a bion (the kink and the antikink form a bound state). As before the dashed line describes an elastic scattering process. Notice that in the kink-antikink scattering described in this Section the N -bounce reflection (N ≥ 2) does not arise. A qualitative explanation of this fact is that the resonant energy transfer mechanism becomes effective when internal vibrational eigenmodes are present. We recall that in two coupled scalar field theories the kink fluctuation operator (10) is in general a 2 × 2 non-diagonal matrix differential operator, whose spectrum must be usually identified numerically. As previously mentioned the basic kink fluctuation operator H[K (q 1 ,q 2 ,λ) static (x)] lacks discrete eigenmodes other than the zero modes. The resonant energy transfer mechanism could also be activated by internal vibrational modes associated with the combined kink-antikink configuration, see [37,40,19]. In Figure 12 the potential well components of the kink-antikink fluctuation operator have been depicted. This operator comprises a continuous spectrum on the threshold value 2 in addition to two discrete eigenvalues which are approximately zero and come from the zero modes of the kink and the antikink. Therefore the collision between these kinks excites continuous eigenmodes which are responsible for the radiation phenomena. This heuristically justifies the absence of resonant windows in these scattering processes. 3.4 The K (q 1 ,q 2 ,1) -K (−q 1 ,−q 2 ,−1) scattering process Finally we shall describe the last class of two-lump scattering events. This process is similar to the previous one in the sense that we study the collision between a kink and an antikink of the same type although now they are arranged in the reverse order. Therefore the initial configuration consists of the concatenation of an antikink with charge (q 1 , q 2 ) followed in the x-axis by its own kink with charge (−q 1 , −q 2 ). A first difference with respect to the kink-antikink scattering explained in the previous subsection is that now the attractive forces are much weaker than in that event. This can be seen by the fact that the antikink-kink bound state arises when the collision velocity is much smaller than in the previous case. Indeed the critical velocity v c which divides the initial velocity regimes where the single lumps escape and where they are forced to collide a second time is given by v c ≈ 0.04162
The general behavior of these scattering processes is illustrated in Figure 13 where the final velocity of the single lumps is represented as a function of the initial velocity v 0 .
We can observe that:
• For the regime v 0 < v c we find two possibilities. The first one is shown in Figure 14 (left) where the formation of a antikink-kink bion is depicted. In this simulation the basic lumps travel with a collision velocity v 0 = 0.02. After the first impact some vibrational modes are excited although radiation emission is negligible. Notice the small fluctuations in the energy density between the collisions. The single lumps do not manage to escape each other and remain trapped in a bound state. In the figure 13 this situation is characterized by a zero final velocity. The second possibility is described in the Figure 14 (middle) for the value v 0 = 0.03970. In this case the single lumps escape after the second collision. The energy accumulated in the internal vibrational mode is transferred back to the kinetic energy of the lumps, which breaks the bound estate. In this model this behavior arises for narrow initial velocity windows, which are called the 2-bounce resonance windows. In the Figure 13 (top) we plot the set of resonance windows which arise just below the critical velocity v c . The existence of resonance windows has been explored using a step ∆v 0 = 0.00001. The presence of other non-detected resonance windows, whose width is narrower than that value, is taken for granted due to the fractal nature of these windows sets, see [22,23,24]. • For collision velocities v 0 > v c the antikink-kink coupling is avoided. The simulation displayed in Figure 14 (right) for v 0 = 0.2 shows that the basic lumps attract each other, collide and bounce back. Finally the antikink and the kink move away. The global process is quite elastic even for high speeds v 0 because the radiation emission is small.
The explanation of the previous pattern can be understood by means of the resonant energy transfer mechanism although the internal vibrational eigenmode is associated with the combined antikinkkink configuration. In Figure 15 we have depicted the potential well components of the second order small fluctuation operator associated with a configuration following the form
φ KK (x) = φ (q 1 ,q 2 ,1) static (x − x 1 ) + φ (−q 1 ,−q 2 ,−1) static (x − x 1 ) − 1 2 ψ KK (x) = ψ (q 1 ,q 2 ,1) static (x − x 1 ) + ψ (−q 1 ,−q 2 ,−1) static (x − x 1 )
which represents an antikink followed by a kink separated by a distance equals to 2x 1 . The second diagonal component V KK 22 (x) is a potential well whose asymptotic behavior tends to the value 8 while the valley floor level reaches the value 2. The well width increases as the antikink-kink distance increases. A continuous spectrum emerges on the threshold value ω 2 = 8. In the Figure 15 (right) the discrete eigenvalues of the fluctuation operator associated to the previous configuration are plotted as a function of the magnitude x 1 . We can observe that the number of discrete eigenmodes grows as the separation between the antikink and the kink increases. The potential well V KK 22 (x) becomes narrower as the lumps approach each other and only a few discrete eigenmodes survive this process. The resonant energy transfer mechanism can be activated by some of these excited modes making the resonant windows to arise.
Conclusions and further comments
The study of the kink dynamics in a two coupled scalar field theory model in two space-time dimensions with potential term (2) has revealed a rich plethora of different interactions between the basic topological defects of the model. The static kink manifold unveils the existence of four basic particles together with its corresponding antiparticles described respectively by four single kinks and its antikinks. An important feature which rules the dynamics is the kink-antikink relation type. A kink and an antikink which live on the same trajectory are said to be of the type. As we have shown in the previous sections there exists an attractive force between this type of kinks. The kink ordering is also a property to be considered in these scattering processes. For instance, in the kink-antikink interaction there exists a critical velocity v c ≈ 0.29703 which distinguishes the initial velocity regimes where the bion formation and the lump reflection occur. In the antikink-kink interaction the velocity v c ≈ 0.04162 plays the same role but now two-bounce resonant windows arise just below this value. A qualitative explanation of these different behaviors underlies the small fluctuation operator spectrum valued on the combined kink-antikink or antikink-kink configurations. In the first case there is no internal vibrational eigenmodes and the continuous spectrum begins at the value 2. In the second case the continuous spectrum starts at the threshold value 8 and we can find vibrational modes in the range [2,8]. Therefore in the kinkantikink interaction the continuous eigenmodes are easily excited, which implies radiation emission while in the second case the vibrational modes play a predominant role involving the presence of the twobounce resonant windows. In addition the kink-antikink interaction is stronger at short distances than the antikink-kink force although this last one has a longer range.
On the other hand repulsive forces manage the antikink-kink interaction when the involved lumps are of different type. Radiation emission is relevant for high speeds. In contrast the kink-antikink interaction is almost absent, its effect is only appreciable when the lumps are merged and compel the particles to concentrate its energy in a small region, such that the energy density peak quadruple the value of the single lump one. These two-body scattering processes conform the fundamental events in this model and they constitute the blocks which allow to explain more complex scattering processes.
As a final comment, it would be interesting to investigate the kink dynamics in other two scalar field theory models in order to acquire a more global perspective of the problem. The celebrated MSTB model [53] or its generalizations [70] arise as natural candidates to this scrutiny. The study of the kink dynamics in massive nonlinear S 2 -sigma models [71] also constitutes a challenging problem.
A Numerical analysis for kink scattering processes
In this appendix we introduce the particular expressions obtained from the Strauss-Vazquez numerical scheme with Mur contour conditions adapted to study the evolution of a kink configuration ruled by the non-linear partial differential equations (3) and (4). Computational limitations compel us to restrict the space coordinate to the interval [x m , x M ] where we assume that the relevant kink scattering processes occur. The evolution of the phenomenon is studied in the time period [0, T ]. We construct a finite mesh with J space subintervals and N time subintervals for the spacetime. With this notation the space and time steps are respectively given by
δ = x M − x m J , τ = T N .
We shall denote φ n j = φ(x m + j δ, nτ ) and ψ n j = ψ(x m + j δ, nτ ), the values of the fields at the mesh points. We recall that the total energy (5) is an invariant magnitude for the scalar field theories which we are dealing with. This fact suggests the use of the energy conservative implicit second order numerical scheme
φ n+1 j − 2φ n j + φ n−1 j τ 2 − φ n j+1 − 2φ n j + φ n j−1 δ 2 + U [φ n+1 j , ψ n j ] − U [φ n−1 j , ψ n j ] φ n+1 j − φ n−1 j = 0,(22)ψ n+1 j − 2ψ n j + ψ n−1 j τ 2 − ψ n j+1 − 2ψ n j + ψ n j−1 δ 2 + U [φ n j , ψ n+1 j ] − U [φ n j , ψ n−1 j ] ψ n+1 j − ψ n−1 j = 0,(23)
where U stands for the potential term U (φ, ψ) given in (2). This is the adaptation of the Strauss-Vazquez scheme introduced in [63, 65, 66] to our two-scalar field theory context. By construction this algorithm preserves the discrete total energy
E n = j δ 1 2τ 2 (φ n+1 j − φ n j ) 2 + 1 2δ 2 (φ n+1 j+1 − φ n+1 j )(φ n j+1 − φ n j ) + + 1 2τ 2 (ψ n+1 j − ψ n j ) 2 + 1 2δ 2 (ψ n+1 j+1 − ψ n+1 j )(ψ n j+1 − ψ n j ) +(24)+ 1 2 U [φ n+1 j , ψ n j ] − U [φ n j , ψ n+1 j ]
which can be understood as a discretization of the total energy (5). One reason for the convenience of the numerical method introduced in (22) and (23) underlies the fact that some kink scattering processes involve radiation phenomena where linear plane waves are emitted and travel with large speeds in both spatial directions. The use of an energy conservative numerical method allows us to control the amount of energy which escapes through the frontiers of our finite interval [x m , x M ], the only possibility of energy change in our numerical scheme. This strategy must be complemented with the use of absorbing contour conditions. If the spatial interval is large enough to the kink scattering occurs far away from the boundaries the dynamics in these peripheral regions is described by the linear partial differential equations ∂ 2 φ ∂t 2 − ∂ 2 φ ∂x 2 = 0 , ∂ 2 ψ ∂t 2 − ∂ 2 ψ ∂x 2 = 0 . This fact suggests the use of second order absorbing Mur contour conditions for our problem [64], which are given by the relations
φ n+1 0 − φ n 1 − n c − 1 n c + 1 (φ n+1 1 − φ n 0 ) = 0 ,(25)
Figure 1 :
1Scalar field components for the K(1,1,1)
Figure 3 :
3Orbits of the kink family K (q1,0) static (x, b) for several values of the parameter b (left) and energy density distributions for the K
Figure 4 :
4Orbits of the kink family K (0,q2) static (x, c) for several values of the parameter c (left) and energy density distributions for the K
Figure 5 :
5Energy density for several boosted basic kinks. Take note that the higher the velocity is, the thinner and taller the kink energy density is.
Figure 6 :
6Energy density representation in the K (q1,q2,−1) -K (q1,−q2,1) scattering process for the initial velocity v 0 = 0.2.
Figure 7 :
7Energy density representation in the K (q1,q2,1) -K (−q1,q2,−1) scattering process for the initial velocities v 0 = 0.2 (left) and v 0 = 0.9 (right).
Figure 8 :
8Graphic representation of the final velocity (left) and the minimal distance (right) of the lumps in the K (q1,q2,1) -K (−q1,q2,−1) scattering as functions of the initial velocity v 0 .
Figure 9 :
9Energy density representation in the K (q1,q2,−1) -K (−q1,−q2,1) scattering process for the initial velocities v 0 = 0.2 (left) and v 0 = 0.4 (right)
Figure 10 :
10Evolution of the total energy of the K (q1,q2,−1) -K (−q1,−q2,1) scattering process for the velocity v 0 = 0.2 in the simulation interval (left) and evolution of the kink-antikink motion period (right) in the time interval [0, 5000].
Figure 11 :
11Graphical representation of the final velocity v f of the lumps as a function of the initial velocity v 0 in the K (q1,q2,−1) -K (−q1,−q2,1) scattering.
Figure 12 :
12Potential well components V ij (x), i, j = 1, 2 of the second order small fluctuation operator associated with the combined kink-antikink configuration.
Figure 13 :
13Graphical representation of the final velocity v f of the lumps as a function of the initial velocity v 0 in the K (q1,q2,1) -K (−q1,−q2,−1) scattering. The top figure is a zoomed image of the small box remarked in the bottom figure where the 2-bounce resonant windows arise.
Figure 14 :
14Energy density representation in the K (q1,q2,1) -K (−q1,−q2,−1) scattering process for the initial velocities v 0 = 0.02 (left), v 0 = 0.03970 (middle) and v 0 = 0.2 (right).
Figure 15 :
15Potential well components V ij (x), i, j = 1, 2 (left) and discrete eigenvalues as a function of the lump separation (right) of the second order small fluctuation operator associated with the combined antikink-kink configuration(22).
AcknowledgmentsThe authors acknowledge the Spanish Ministerio de Economía y Competitividad for financial support under grant MTM2014-57129-C2-1-P. They are also grateful to the Junta de Castilla y León for financial help under grant VA057U16.where n c = τ /δ. These contour conditions have the ability of absorbing the radiation which arrives to the frontiers of our simulations. The initial conditionslet start the algorithm by fixingwhich corresponds to a second order approximation consistent with the numerical scheme(22)and(23). Finally we have to implement a procedure for estimating the error derived from the numerical method. This error control will be accomplished firstly by monitoring the evolution of the model invariants. The algorithm has been constructed to keep the total energy E(t) constant but the total momentum M (t) given by(6)should also be a constant of motion. In this sense we shall supervise the evolution of the discrete version of the total momentumA significant variation of the magnitude (29) along the time would indicate that the algorithm fictitiously accelerates the particles. In addition to this protocol we shall also analyze the difference between the results obtained by two simulations where the second one halves both the space and time steps used in the first one. In this sense we construct the functionswhich give the maximum discrepancy between the values of the field components on the set of all the mesh points for every instant t. In (30) the notation φ n j (δ, τ ) stands for the value of the field obtained by the simulation with space and time steps δ and τ respectively. A standard choice of the parameters in our simulations is given by the values x m = −10, x M = 10, T = 50, J = 16000 and N = 160000. This involves a value of the steps δ = 0.00125 and τ = 0.003125. A upper bound for the error parameters |ξ i (t)| for this setting is 5 × 10 −5 , that is, |ξ i (t)| ≤ 5 × 10 −5 for most of the simulations. The error estimation of the numerical procedure is measured by the difference between the results of two simulations where the space and time steps have been halved. The evolution of the total momentum is also monitored. This magnitude must be constant over time because it is an invariant of the system.
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| []
|
[
"Asynchronous Distributed Optimization over Lossy Peer-to-Peer Networks via ADMM: Stability and Exponential Convergence",
"Asynchronous Distributed Optimization over Lossy Peer-to-Peer Networks via ADMM: Stability and Exponential Convergence"
]
| [
"N Bastianello ",
"R Carli ",
"L Schenato ",
"M Todescato "
]
| []
| []
| In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-topeer network. In particular we are interested in the case in which communications between nodes are lossy and the agents are not synchronized among themselves. We address this problem by proposing a modified version of the relaxed ADMM (R-ADMM), which corresponds to the generalized Douglas-Rachford operator applied to the dual of our problem. By exploiting results from operator theory we are then able to prove the almost sure convergence of the proposed algorithm under i.i.d. random packet losses and asynchronous operation of the agents. By further assuming the cost functions to be strongly convex, we are able to prove that the algorithm converges exponentially fast in mean square in a neighborhood of the optimal solution. Moreover, we provide an upper bound to the convergence rate. Finally, we present numerical simulations of the proposed algorithm over random geometric graphs in the aforementioned lossy and asynchronous scenario. | null | [
"https://arxiv.org/pdf/1901.09252v1.pdf"
]
| 119,729,625 | 1901.09252 | 737fc405b2c28c4ccf8fc12098e040adff7e7fc1 |
Asynchronous Distributed Optimization over Lossy Peer-to-Peer Networks via ADMM: Stability and Exponential Convergence
N Bastianello
R Carli
L Schenato
M Todescato
Asynchronous Distributed Optimization over Lossy Peer-to-Peer Networks via ADMM: Stability and Exponential Convergence
Index Terms distributed optimizationADMMasynchronous updatelossy communicationsoperator theoryDouglas-Rachford operator
In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-topeer network. In particular we are interested in the case in which communications between nodes are lossy and the agents are not synchronized among themselves. We address this problem by proposing a modified version of the relaxed ADMM (R-ADMM), which corresponds to the generalized Douglas-Rachford operator applied to the dual of our problem. By exploiting results from operator theory we are then able to prove the almost sure convergence of the proposed algorithm under i.i.d. random packet losses and asynchronous operation of the agents. By further assuming the cost functions to be strongly convex, we are able to prove that the algorithm converges exponentially fast in mean square in a neighborhood of the optimal solution. Moreover, we provide an upper bound to the convergence rate. Finally, we present numerical simulations of the proposed algorithm over random geometric graphs in the aforementioned lossy and asynchronous scenario.
I. INTRODUCTION
From classical control theory to more recent machine learning applications, many problems can be cast as optimization problems [1] and, in particular, as large-scale optimization problems, given the advent of Internet-of-Things and the everincreasing growth of large-scale cyber-physical systems. Hence, stemming from classical optimization theory, in order to break down the computational complexity, parallel and distributed optimization methods have been the focus of a wide branch of research [2]. Within this vast topic, typical applications foresee computing nodes to cooperate in order to achieve a desired common goal such as
min x N i=1 f i (x)(1)
where, usually, each f i is owned by one node only. While parallel optimization methods usually rely on a shared memory architecture to implement the communication among agents, in distributed systems a message passing architecture is employed, in which agents can exchange transmissions with a (subset) of the other agents. The message passing (or peer-to-peer) architecture however introduces some issues due to the implementation of the transmission protocols, which may suffer from faulty and time-varying communications, on top of the possible asynchronism of the agents' operations. A popular class of algorithms that have been proposed to solve (1) is that of distributed subgradient methods. These classes can handle non-differentiable convex cost functions, and require only the computation of local (sub-)gradients. However, they exhibit sub-linear converge rates even if the cost functions are smooth [3], [4]. The recent works [5], [6], [7], [8] derived convergence guarantees in the presence of time-varying communications. However, the need for a diminishing stepsize implicitly requires a synchronization mechanism between agents.
Another popular class of distributed optimization algorithms is the one of gradient consensus methods, which consists of strategies that exploit first and second-order derivatives for computing descent directions. For example in [9], [10] the authors apply quasi-Newton distributed descent schemes to general time-varying directed graphs. A different approach, based on computing Newton-Raphson directions through average consensus algorithms, has been proposed in [11]. Even if initially proposed for synchronous implementations, this scheme has been later extended to cope with asynchronous and faulty communication schemes [12]. However, in [12], the convergence is proved only locally and no characterization of the convergence rate is provided.
Finally, dual decomposition schemes have been widely employed to solve distributed problems, and we refer to [13] for a comprehensive tutorial. Among these algorithms, the Alternating Direction Multiplier Method (ADMM) has attracted the attention of the scientific community for its simple distributed implementation and good convergence speed. This algorithm was originally proposed in mid '70s as a general convex optimization strategy, then exploited in the context of networked optimization [14], and recently popularized by the survey [15]. Substantial research has been dedicated in optimizing the free parameters of ADMM in order to obtain faster convergence rates, but these are mainly restricted to synchronous implementations over undirected communication graphs [16], [17], [18], [19]. Some recent exceptions extend dual decomposition and ADMM to asynchronous scenarios, see [20], [21], [22], [23]. Other works have addressed the problem of random delays in the communication/update rounds in ADMM schemes [24], [25], [23]. However the strategies [24], [23] are restricted to networks with server-client communication topologies, i.e. based on a shared memory architecture, and all three do not explicitly address packet losses. To the best of our knowledge, the first explicit focus on the robustness of the ADMM to packet losses, in general peer-to-peer networks, was given in [26], [27], which are preliminary versions of this paper. In the present work we extend them to allow asynchronous operations of the agents, and further study the convergence rate of the proposed algorithm. The tools used to propose and analyze a robust version of the ADMM are drawn from the vast literature on operator theory. Briefly, the underlying idea is to convert optimization problems into the problem of finding the fixed points of suitable nonexpansive operators, that is operators with unitary Lipschitz constant [28]. Given an operator T , a simple idea to find its fixed pointsx = Tx would be that of iteratively applying the operator itself [28]. However, in many situations, this strategy does not provide the desirable results, mainly for two reasons: i) evaluating the operator might be computationally expensive, and ii) the iterates might not converge. To deal with issue i), the so-called splitting methods have been introduced, which exploit the problem's structure to break it down into smaller and more tractable subproblems. Examples of splitting methods are the Peaceman-Rachford (PRS) [29] and Douglas-Rachford (DRS) [30], [31] splitting. In particular, by applying the DRS to the dual of the problem at hand, it is possible to derive the updates of the standard relaxed-ADMM [32]. The second issue can be addressed by introducing the Krasnosel'skii-Mann iterate, which performs an averaging or relaxation of the operator. For further details on a variety of splitting operators and their application in asynchronous setups we refer to [33], [25] and [34].
In this paper we present and analyze a modified version of the relaxed ADMM which is amenable of distributed implementation in peer-to-peer networks with unreliable communications and asynchronous operations of the agents. In particular, by deriving the ADMM as an application of the Douglas-Rachford splitting method, we are able to exploit recent results on randomized nonexpansive operators to establish the almost sure convergence of the proposed algorithm, provided mild assumptions on the asynchronous and lossy nature of the network are satisfied. In addition, assuming the functional costs are strongly convex, we show that the convergence is locally exponential in mean-square, also providing an upper-bound on the convergence rate.
The remainder of the paper is organized as follows. In Section II we review some preliminary results and the standard relaxed ADMM. In Section III we introduce the consensus optimization problem of interest. In Section IV we describe the modified relaxed ADMM algorithm we propose to deal with distributed optimization, and in Section V we establish its local exponentially fast convergence for strongly convex functions. In Section VI we show how the algorithm introduced in Section IV is amenable of an implementation asynchronous and robust to packet losses which converges almost surely to the optimal solution; in addition, we show that for strongly convex costs the convergence rate is locally exponential in mean-square sense. Finally, Section VII reports some numerical results to complete the analysis of the proposed algorithm.
II. PRELIMINARIES
In this section we introduce some notation and briefly review the relaxed ADMM algorithm.
A. Notation and useful definitions
We denote a graph with G = (V, E), where V is the set of N vertices, labeled 1 through N , and E is the set of undirected edges. For i ∈ V, by N i we denote the set of neighbors of node i in G, namely,
N i = {j ∈ V : (i, j) ∈ E} .
Consider the scalar function f :
R n → R ∪ {+∞}. We say that f is closed if ∀a ∈ R the set {x ∈ dom(f ) | f (x) ≤ a} is closed. We say that f is proper if it does not attain −∞.
A function is said to be convex if ∀x, y ∈ R n and ∀λ ∈ [0, 1] it holds f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y). Moreover, we say that it is strongly convex if it is twice differentiable, f ∈ C 2 , and its Hessian is bounded from below, that is ∇ 2 f (x) cI for all x with c some positive scalar. We define the convex conjugate of f as f * (y) = sup x∈R n { y, x − f (x)}.
By operator on R n we mean a mapping T : R n → R n that assigns to each point x in R n the corresponding point T x ∈ R n . An operator T is nonexpansive if for any two x, y ∈ R n it holds T x − T y ≤ x − y , and averaged if there exist α ∈ R + and R nonexpansive such that T = (1 − α)I + αR.
In the following we will use the class of proximal operators, defined for a closed, proper and convex function f and penalty parameter ρ > 0 as prox ρf (x) = arg min
y f (y) + 1 2ρ y − x 2 .
Notice that proximal operators are nonexpansive.
B. The Relaxed-ADMM algorithm
Consider the following optimization problem, which we assume has a finite optimal solution:
min x∈R n ,y∈R m {f (x) + g(y)} s.t. Ax + By = c(2)
with f : R n → R ∪ {+∞} and g : R m → R ∪ {+∞} closed, proper and convex functions. Let us define the augmented Lagrangian for problem (2) as
L ρ (x, y; w) =f (x) + g(y) − w (Ax + By − c) + ρ 2 Ax + By − c 2(3)
where ρ > 0 is the penalty parameter and w is the vector of Lagrange multipliers. The Relaxed ADMM (R-ADMM) [33] consists in alternating the following three steps
y(k + 1) = arg min y {L ρ (x(k), y; w(k)) + ρ(2α − 1) By, (Ax(k) + By(k) − c) } (4) w(k + 1) = w(k) − ρ(Ax(k) + By(k + 1) − c) − ρ(2α − 1)(Ax(k) + By(k) − c) (5) x(k + 1) = arg min x L ρ (x, y(k + 1); w(k + 1)).(6)
A well known result is that the R-ADMM algorithm can be alternatively derived applying the relaxed Douglas-Rachford splitting (DRS) operator to the dual of problem (2), see [15], [33]. The following convergence result applies. Proposition 1: Assume that the functions f and g are closed, convex and proper. If 0 < α < 1 and ρ > 0, then the iterates of the R-ADMM algorithm converges to the optimal solution of (2). We conclude this section by observing that setting α = 1/2 returns the classical ADMM, thoroughly analyzed in [15].
III. DISTRIBUTED CONSENSUS PROBLEM FORMULATION
In this section we introduce the distributed optimization problem of interest. Consider the undirected, connected graph G = (V, E) with N nodes. We are interested in solving the following optimization problem
min x N i=1 f i (x)(7)
where f i : R n → R ∪ {+∞} are closed, proper and convex functions and each f i is known only to node i. In the following we denote by x * a solution of (7) and assume that such a solution exists and is finite. Assigning a local copy x i of x to each agent, (7) can be equivalently formulated as
min x1,...,x N N i=1 f i (x i ) s.t. x i = x j , ∀(i, j) ∈ E(8)
By introducing for each edge (i, j) ∈ E the two bridge variables y ij and y ji , the constraints in (8) can be further rewritten as
x i = y ij x j = y ji y ij = y ji ∀(i, j) ∈ E. Defining x = [x 1 , . . . , x N ] , f (x) = i f i (x i )
, and stacking all bridge variables in y ∈ R 2n|E| , we can reformulate the problem as 1
min x f (x)
s.t. Ax + y = 0 (9) y = Py for a suitable A ∈ R 2n|E|×nN matrix, and with P ∈ R 2n|E|×2n|E| being a permutation matrix that swaps y ij with y ji . Notice that P is symmetric.
Making use of the indicator function ι (I−P) (y) which is equal to 0 if (I − P)y = 0, and +∞ otherwise, we can finally rewrite problem (8) as
min x,y f (x) + ι (I−P) (y) s.t. Ax + y = 0.(10)
Clearly (10) conforms to problem (2) and therefore it can be addressed by applying the R-ADMM algorithm.
IV. R-ADMM FOR DISTRIBUTED OPTIMIZATION In this section we employ (4), (5) and (6) to solve problem (10). To do so we introduce the Lagrange multipliers w ij and w ji which are associated to the constraints x i = y ij and x j = y ji , respectively. It is possible to show that R-ADMM applied to (10) is amenable of a distributed implementation as algorithmically described in Algorithm 1. More precisely, observe that node i stores in memory only the variables x i , y ij , w ij , j ∈ N i , and updates them exchanging information only with its neighbors, i.e, with nodes in N i .
Algorithm 1 Distributed R-ADMM using (4)- (6).
Input: Set the termination condition K > 0. For each node i, initialize x i (0), {y ij (0), w ij (0)} j∈Ni . 1: k ← 0 2: while k < K each agent i do 3: broadcast x i (k), y ij (k) and w ij (k) to all neighbors j ∈ N i 4: collect {x j (k), y ji (k), w ji (k)} received from each neighbor j ∈ N i 5:
compute in order
y ij (k + 1) = 1 2ρ (w ij (k) + w ji (k)) + 2αρ(x i (k) + x j (k)) − ρ(2α − 1)(y ij (k) + y ji (k)) w ij (k + 1) = 1 2 (w ij (k) − w ji (k)) + 2αρ(x i (k) − x j (k)) − ρ(2α − 1)(y ij (k) − y ji (k)) x i (k + 1) = arg min xi f i (x i ) + ρ 2 |N i | x i 2 + j∈Ni ρ(2α − 1)y ji (k) − 2αρx j (k) − w ji (k) x i 6: k ← k + 1 7: end while
Notice that, to find the solution of (7), we are only interested in the trajectories of the x i variables in Algorithm 1, and not in those of the auxiliary variables y ij and w ij . Moreover, as we mentioned above, the R-ADMM can be derived by applying the relaxed DRS operator to the dual of problem (10). In the next Proposition, based on the previous observations and leveraging the particular structure of problem (10) 2 , we propose a simpler, but otherwise equivalent, implementation of the distributed R-ADMM in terms of both memory and communication requirements. This new formulation, which, beside the x i 's variables, involves only the auxiliary variables z ij , i ∈ V, j ∈ N i , will be instrumental for the derivation of a robust and asynchronous R-ADMM.
Proposition 2: The trajectories of the variables x i , i ∈ V, generated by Algorithm 1, starting from a given initial condition x(0), y(0), w(0), are identical to the trajectories generated by iterating, for k ≥ 1, the following two updates
x i (k) = arg min xi f i (x i ) − j∈Ni z ji (k) x i + ρ 2 |N i | x i 2 (11)
for all i ∈ V, and
z ij (k + 1) = (1 − α)z ij (k) − αz ji (k) + 2αρx i (k) z ji (k + 1) = (1 − α)z ji (k) − αz ij (k) + 2αρx j (k)(12)
for all (i, j) ∈ E, where the auxiliary variables are initialized as z ij (1) = w ij (1) + ρy ij (1).
Proof: See Appendix A. The previous proposition naturally suggests an alternative distributed implementation of the ADMM Algorithm 1, in which each node i stores and updates in its local memory the variables x i and z ij , j ∈ N i , and then exchanges them with its neighbors. An equivalent implementation, which reduces the number of transmissions required, can be obtained if the node i stores and updates the variables z ji , j ∈ N i instead of the variables z ij , j ∈ N i . This implementation is formally described in Algorithm 2.
Algorithm 2 Modified distributed R-ADMM.
Input: Set the termination condition K > 0. For each node i, initialize x i (0) and z ji (0), j ∈ N i . 1: k ← 0 2: while k < K each agent i do 3: compute x i (k) according to (11) 4:
compute, for j ∈ N i , the temporary variable q i→j as
q i→j = −z ji (k) + 2ρx i (k)(13) 5:
transmit, for j ∈ N i , q i→j to node j 6: gather q j→i from each neighbor j 7:
update the auxiliary variables as
z ji (k + 1) = (1 − α)z ji (k) + αq j→i(14)
8:
k ← k + 1 9: end while
Observe that, at the beginning of each iteration node i updates x i based only on local information according to (11). Then it computes the temporary variable q i→j which is sent to neighbor j. At the same time it receives the quantity q j→i from neighbor j and it uses this information to update z ji as in (14).
Remark 1: As we can see, both Algorithms 1 and 2 need a single round of transmissions at each time k. However, they differ for the number of variables that each node has to transmit and update. The comparison between the two algorithms is reported in Table I.
2|N i | + 1 |N i | + 1 Send 2|N i | + 1 |N i | Temporary 3|N i | |N i | Remark 2:
The R-ADMM formulation presented in Algorithms 1 and 2 is edge-based in the sense that there is a local auxiliary variable associated with each neighbor of a node i. Therefore the memory requirement scales in the worst case as O(N ). On the other hand, node-based implementations of the ADMM have been proposed [35], [36], for which the memory requirement scales as O(1). We observe that the distributed R-ADMM presented in this Section can be reformulated as a node-based algorithm if the nodes store and update, alongside x i , the auxiliary variables u i = j∈Ni z ij and v i = j∈Ni z ji instead of z ji , j ∈ N i . However, this modified version is not suitable for handling packet losses and asynchronism.
Remark 3: The formulation of the R-ADMM presented in Algorithm 2 is derived using the same idea employed in [25] of interpreting the R-ADMM as an application of the DRS to the dual problem. Also the R-ADMM algorithm proposed in [25] to solve problem in (1) (see Section 2.6.2), involves only the use of variables x i , z ij but the actual implementation differs from Algorithm 2. Additionally, it is worth mentioning that the authors of [25] have derived the R-ADMM within the framework of the ARock algorithm, introduced in the context of parallel computing where agents share a common memory. Interestingly, it is shown that the ARock framework successfully handles asynchronous updates and delayed information. However, due to the reliance of the convergence proof on the common memory, ARock it is not suitable to deal with arbitrary packet losses, which, instead, is the focus of the second part of the present paper.
Clearly, the equivalence between Algorithm 2 and Algorithm 1 and thus with the R-ADMM reviewed in Section II-B, allows us to state the following convergence result.
Corollary 1: Let (α, ρ) be such that 0 < α < 1 and ρ > 0. Then, for any initial conditions, the trajectories k → x i (k), i ∈ V, generated by Algorithm 2, converge to the optimal solution of (7), i.e.,
lim k→∞ x i (k) = x * , ∀i ∈ V,
for any x i (0) and z ji (0), j ∈ N i .
V. LOCALLY EXPONENTIAL CONVERGENCE
In this section we prove the locally exponential convergence of the distributed R-ADMM illustrated in Algorithm 2 under the assumption that the local costs are strongly convex.
We first derive a formulation of the update equations (11) and (12) as affine functions perturbed by locally vanishing functions; this derivation is inspired by the approach adopted in [19].
Thanks to the strong convexity assumption, f i is twice differentiable and hence it is possible to write the first order optimality condition for the minimization problem (11) as
∇f i (x i (k)) + [A ] i Pz(k) + ρ|N i |x i (k) = 0 (15)
where matrices A and P have been introduced in (9), [A ] i denotes the i-th row of matrix A and z ∈ R 2n|E| is the vector obtained stacking together the variables z ij , i ∈ V, j ∈ N i . Moreover, when x i is sufficiently close to the (unique) optimum x * the first order Taylor expansion for the gradient of f i can be computed as
∇f i (x i ) = ∇f i (x * ) + ∇ 2 f i (x * )(x i − x * ) + r i (x i − x * )(16)
where (15) and (16) we get
r i (x) / x → 0 when x → 0. Now, combining equations−[A ] i Pz(k)−ρ|N i |x i (k) = ∇f i (x * ) + ∇ 2 f i (x * )(x i (k) − x * ) + r i (x i (k) − x * ) which, solved for the x i (k) yields x i (k) = −H −1 i [A ] i Pz(k) − H −1 i (r i (x i (k) − x * ) + b i ) where H i = ∇ 2 f i (x * ) + ρ|N i |I is invertible and b i = ∇f i (x * ) − ∇ 2 f i (x * )x * . In compact form for the entire vector x(k), we can write x(k) = −H −1 A Pz(k) − H −1 (r(x(k) − x * ) + b)(17)
where
H = diag{H 1 , . . . , H N }, b = [b 1 · · · b N ] , r(x) = [r 1 (x 1 ) · · · r N (x N ) ] , and x * = [(x * ) · · · (x * ) ] . From (12) we have z(k + 1) = ((1 − α)I − αP)z(k) − 2αρAx(k)
where I is the identity matrix of size R 2n|E|×2n|E| . Combining the two equations above yields
z(k + 1) = Qz(k) − F r(x(k) − x * ) + c(18)
where Q = (1 − α)I − αP + 2αρAH −1 A P, F = 2αρAH −1 and c = −2αρAH −1 b.
The following Lemma characterizes the spectral properties of the matrix Q.
then A Pv i = 0, for i = 1, . . . , m.
Proof: See Appendix B.
Let now Γ (Q) be the spectrum of the matrix Q and let
γ M = max {|γ| : γ ∈ Γ (Q), γ = 1}
Observe that, from Lemma 1, it follows γ M < 1. We have the following result. Proposition 3: Assume that the cost functions f i are strongly convex. Then, there exists a neighborhood B x * of the optimal point x * such that, if x(0) ∈ B x * , then Algorithm 2 converges exponentially fast, i.e.,
x i (k) − x * ≤ Cγ k x i (0) − x * , ∀i ∈ V,
for suitable constants C > 0 and 0 ≤ γ ≤ γ M . Proof: See Appendix B.
Remark 4: Notice that the matrix Q depends both on the topology of the communication graph, via matrices A and P, and on the curvature of the cost functions f i , via the matrix of Hessians H. This result is different from the case of Newton algorithms whose convergence rate depends only on the structure of the graph, see [12].
VI. ASYNCHRONOUS DISTRIBUTED R-ADMM OVER LOSSY NETWORKS
The distributed algorithms illustrated in the previous section work under the standing assumption that the communication channels are reliable, that is, no packet losses occur, and that the nodes update all at the same time, i.e., they are synchronized with each other. The goal of this section is to relax these requirements and, in particular, to show how Algorithm 2 can be modified to still guarantee convergence, under probabilistic assumptions on communication failures and asynchronous updates.
A. Robust and Asynchronous R-ADMM
The following assumption will stand throughout the remainder of the paper. Assumption 1: During any iteration of Algorithm 2, a node i performs an update with probability q, and a packet that is sent from the node to one of its neighbors is lost with some probability p. In addition, an update or packet loss event is independent of any other update and loss events occurring at any k ∈ N.
Consider now Algorithm 2 and notice that node j updates the variable z ij if and only if it receives the information q i→j from node i. For this to happen, two events need to take place: node i performs an update, which happens with probability q; and the packet transmitted from i to j is not lost, which happens with probability 1 − p. To provide a formal description of this probabilistic framework, we introduce the set of random variables β i→j (k), k = 0, 1, 2, . . ., i ∈ V, j ∈ N i , such that β i→j (k) = 1 if node i updates at iteration k and the packet q i→j is received by node j, β i→j (k) = 0 otherwise. Under Assumption 1, by independence of the update and packet loss events it holds
P[β i→j (k) = 1] = q(1 − p) =: s.
In this scenario, Algorithm 2 is modified as shown in Algorithm 3. We refer to the robust and asynchronous ADMM as raR-ADMM.
Algorithm 3 Robust and asynchronous distributed R-ADMM.
Input: Set the termination condition K > 0. For each node i, initialize x i (0) and z ji (0), j ∈ N i . 1: k ← 0 2: while k < K every agent i do 3: if scheduled to update then 4: compute x i (k) according to (11) 5: for all j ∈ N i , compute the quantity q i→j as
q i→j = −z ji (k) + 2ρx i (k)(20)
6: for all j ∈ N i , transmit q i→j to node j 7:
end if 8: for j ∈ N i if q j→i was received do 9: update z ji as z ji (k + 1) = (1 − α)z ji (k) + αq j→i
10:
end for 11: k ← k + 1 12: end while
In the modified algorithm, at the k-th iteration node i updates x i according to Eq. (11) if it scheduled to do so. Then, for j ∈ N i , it computes q i→j as in (20) and transmits it to node j. If node j receives q i→j , then it updates z ij as z ij (k + 1) = (1 − α)z ij (k) + αq i→j , otherwise z ij remains unchanged, i.e., z ij (k + 1) = z ij (k). This last step can be compactly described as
z ij (k + 1) = (1 − β i→j (k)) z ij (k) + β i→j (k) ((1 − α)z ij (k) + αq i→j ) .(22)
B. Almost-sure convergence
The following Proposition characterizes the convergence properties of Algorithm 3. Proposition 4: Consider the raR-ADMM described in Algorithm 3 and suppose that Assumption 1 holds true. Let (α, ρ) be such that 0 < α < 1 and ρ > 0. Then, for any initial conditions, the trajectories k → x i (k), i ∈ V, generated by Algorithm 3, converge almost surely to the optimal solution of (7), i.e.,
P lim k→∞ x i (k) = x * = 1, ∀i ∈ V,
for any x i (0) and z ji (0), j ∈ N i .
Proof: See Appendix C. Notice that, while Algorithms 1 and 2 are characterized by the same convergent behavior in the synchronous and lossless scenario, we were able to prove convergence in the asynchronous and lossy case only for the latter. In particular, recall that we derived Algorithm 2 as the application of the DRS operator to the dual of problem (10). Similarly, by casting Algorithm 3 in the framework of operator theory, we are able to exploit recent results on randomized nonexpansive operators to establish the convergence of the raR-ADMM algorithm.
Remark 5: Observe that both Proposition 1, for the case of reliable communications, and Proposition 4, for the randomized updating scenario, establish convergence provided that 0 < α < 1 and ρ > 0. However, these conditions are only sufficient and not necessary and, in particular, the convergence might hold also for values of α ≥ 1. Indeed, in the simulation Section VII we show that, for the case of quadratic functions f i , i ∈ V, the region of attraction in parameter space is larger. Moreover, despite what the intuition would suggest, the larger the packet loss probability p (or, the smaller the node update probability q), the larger the region of convergence. However, this increased region of stability is counterbalanced by a slower convergence rate of the algorithm.
Remark 6: The R-ADMM algorithm proposed in [25] allows only a single agent to update at each iteration, while the proposed Algorithm 3 is fully parellel, i.e. guarantees convergence when an arbitrary number of agents updates simultaneously.
C. Mean Square Exponential Convergence
The results stated in Proposition 4 can be refined in case the functions f i are strongly convex; indeed, under this stronger assumption, it is possible to show that the mean-square convergence is locally exponential.
To do so, let us introduce the random diagonal matrix B(k) ∈ R 2n|E|×2n|E| such that
B(k) = diag{β i→j (k)I n , i ∈ V, j ∈ N i },(23)
and observe that, plugging the update (18) into the randomized update (22) it is possible to derive
z(k + 1) = (I − B(k))z(k) + B(k)[Qz(k) − F r(x(k) − x * ) + c] = G(k)z(k) + B(k)[−F r(x(k) − x * ) + c](24)
where
G(k) = I − B(k) + B(k)Q.
We are now interested in evaluating the behavior of the mean square error E[ x(k) − x * 2 ] as k → ∞. In particular, we show that it converges to zero exponentially fast also providing an upper bound to the rate of convergence.
We start observing that, from Lemma 3 reported in Appendix B, it follows
x(k) − x * 2 ≤ A P(z(k) − z * ) 2 .
Then, by iterating (24) and taking the expected value we get the following inequality
E x(k) − x * 2 ≤ E A PG G G(k; 0)(z(0) − z * ) 2 + k−1 =0 E A PG G G(k; + 1)B( )Fr(x( ) − x * ) 2(25)
where
G G G(k; ) = G(k − 1) · · · G( ) if ≤ k − 1, I if > k − 1.
The following Lemma outlines the properties of G G G(k; 0) when k → ∞. Proof: See Appendix C. Consider now the first term in the right-hand side of (25), or equivalently, the term
(z(0) − z * ) E G G G (k; 0)PAA PG G G(k; 0) (z(0) − z * ).(27)
Next we show that this term converges exponentially to zero as k → ∞. Notice that (27) can be rewritten as
(z(0) − z * ) ∆ ∆ ∆(k)(z(0) − z * ), where ∆ ∆ ∆(k) = E G G G (k; 0)PAA PG G G(k; 0) ,
if k ≥ 1 and where ∆ ∆ ∆(0) = PAA P. A simple recursive argument shows that
∆ ∆ ∆(k + 1) = E G (0)∆ ∆ ∆(k)G(0) = L(∆ ∆ ∆(k)).
This shows that ∆ ∆ ∆(k) is the evolution of a linear dynamical system which can be written in the form
∆ ∆ ∆(k + 1) = L(∆ ∆ ∆(k)) where L : R N ×N → R N ×N is given by L(M) = E G (0)MG(0) .
By iterating it is clear that ∆ ∆ ∆(k) = L k (∆ ∆ ∆(0)) and by using Lemma 1 combined with Lemma 2 we obtain
lim k→∞ L k (∆ ∆ ∆(0)) = E m i=1 m j=1 i v i ∆ ∆ ∆(0)v j j = 0.
Now let R be the convergence rate to zero of ∆ ∆ ∆(k). If we consider the reachable subspace R of the pair (L, P AA P), namely the smallest L-invariant subspace of R N ×N containing P AA P, then we have that R = sr(L |R ), where sr denotes the spectral radius. Next we characterize the eigenspace of L relative to the eigenvalue 1.
Proposition 5: Consider the raR-ADMM described in Algorithm 3. Suppose Assumption 1 holds true. Moreover let 1 , . . . , m be the random vectors such that (26) holds. Then the eigenspace of L relative to 1 is m 2 dimensional and it is generated by E i j , i, j = 1, . . . , m.
Proof: See Appendix C.
Clearly the reachability subspace R will be contained in the subspace generated by the remaining eigenvectors; this implies that R will be upper-bounded by the second dominant eigenvalue of L. Precisely, let Γ (L) be the spectrum of the operator L and letγ
M = max {|γ| : γ ∈ Γ (L), γ = 1} (28) then R ≤γ M < 1.
The following result establishes that, locally, E[ x(k) − x * 2 ] converges exponentially fast to zero and that R and, in turn, γ M are suitable upper-bounds for the convergence rate.
Proposition 6: Assume functions f i are strongly convex. Then, there exists a neighborhood B x * of the optimal point x * such that, if x(0) ∈ B x * , then Algorithm 3 converges exponentially fast -in mean-square sense -to the optimal solution, i.e.,
E x i (k) − x * 2 ≤ Cγ k x i (0) − x * 2 , ∀i ∈ V,
for suitable constants C > 0 and 0 ≤γ ≤ R ≤γ M . Proof: See Appendix C.
D. Computing the upper-boundγ M
Next, we provide a numerical way to computeγ M . To do so, it is useful to introduce a matrix representation of the linear operator L. In general, given a matrix M ∈ R N ×N , we define vect(M ) to be the N 2 column vector having M i,j in position (i − 1)N + j. It is known that vect(XY Z) = (Z ⊗ X)vect(Y ), where ⊗ is the Kronecker product of matrices. Using the properties of the Kronecker product we can derive that the linear operator L is described by the matrix
L := E G (0) ⊗ G (0) Indeed, if we vectorize the matrix ∆ ∆ ∆(k), it holds vect(∆ ∆ ∆(k + 1)) = E G (0) ⊗ G (0) vect(∆ ∆ ∆(k)) = L vect(∆ ∆ ∆(k)).
Let Γ (L) be the spectrum of the matrix L, then
γ M = max {|γ| : γ ∈ Γ (L), γ = 1} .
The next Proposition gives an explicit formula for computing the matrix L which can be then exploited to provide an upper bound to the convergence rate by evaluating its spectrum. Before stating the Proposition, we introduce the following matrix Observe that, from the definition of the matrix B given in (23), it turns out that S is a diagonal matrix whose diagonal elements are of the form E[β i→j (0)β l→m (0)] and, more specifically,
S = E[B(0) ⊗ B(0)].(29)E[β i→j (0)β l→m (0)] = s 2 if i = l s 2 /q if i = l, j = m s if i = l, j = m.
We have the following result.
Proposition 7: The matrix L = E[G (0) ⊗ G (0)] is given by L =(1 − 2s)I ⊗ I + sI ⊗ Q + S − (I ⊗ Q )S + sQ ⊗ I − (Q ⊗ I)S + (Q ⊗ Q )S ,(30)
where S is as in (29).
Proof: See Appendix C.
Remark 7:
The derivation employed in this section extends the results reported in [37] which were tailored to randomized consensus problem.
Remark 8: Note that Propositions 3, 6 and 7 hold globally in the case of quadratic cost functions. This is a consequence of the fact that the update equation (18) holds with r(x) = 0 for any x.
VII. SIMULATIONS
In this section we discuss some numerical results obtained applying Algorithm 3 to solve distributed consensus optimization problem (7) in lossy and asynchronous scenarios. In order to simplify the numerical analysis, we restrict to the case of quadratic cost functions of the form
f i (x i ) = a i x 2 i + b i x i + c i(31)
where, in general, the quantities a i , b i , c i ∈ R are different for each node i. In this case the update of the primal variables becomes linear and, in particular, Eq. (11) reduces to
x i (k) = j∈Ni z ji (k) − b i 2a i + ρ|N i | .
Moreover, we consider the case of random geometric graphs with communication radius r = 0.1[p.u.] in which two nodes are connected if and only if their relative distance is less that r. Figure 1 shows an example of such graphs. First of all, in order to evaluate the effect of packet losses on the algorithm, Figure 2 shows the evolution of the relative error
log x(k) − x * x *
computed with respect to the unique minimizer x * and averaged over 100 Monte Carlo runs, with different values of packet loss probability p and for fixed values of step size α = 0.75, penalty ρ = 3 and update probability q = 0.8. As expected, the higher the packet loss probability, the smaller the rate of convergence. Figure 3 represents instead the stability boundaries of the algorithm as a function of the step-size α and penalty ρ, for different packet losses and with synchronous updates. In particular, each curve in Figure 3 represents the numerical boundary below which the algorithm is found to be convergent and above which it diverges. As predicted by Proposition 4, the convergence is guaranteed for all values of the penalty ρ, but the algorithm is seen to converge even for values of the step size α larger than one. Even more interesting is the fact that the stability boundary enlarges as the packet loss probability increases, a phenomenon that, however, is compensated by a lower convergence rate, as shown by Figure 2. The raR-ADMM has been also tested for different values of the step-size α and with fixed penalty ρ = 3, packet loss probability p = 0.4 and update probability q = 0.6, and the relative error behavior is depicted in Figure 4. Notice that for α = 1/2, which corresponds to the case of the classical ADMM, the convergence speed is slower than that for the cases of α = 0.75 and 1 3 . Therefore it is possible to exploit the relaxation of the ADMM in order to achieve faster convergence rates, which motivates the choice of using the slightly more complex raR-ADMM over the simpler classic ADMM.
Finally, in Figure 5 we compare, for different values of α and ρ, the upper-boundγ M with the empirical convergence rate of the raR-ADMM algorithm obtained averaging over 100 Monte Carlo trials. In particular, the values ofγ M are plotted in the trasparent surface on top. Notice that in particular for smaller values of the parameter ρ the upper bound is not tight, suggesting that further work could be done to improve the tightness of the bound.
VIII. CONCLUSIONS AND FUTURE DIRECTIONS
In this paper we addressed distributed convex optimization problems over a peer-to-peer network with both unreliable communications and asynchronous updates of the nodes. We proposed a modified version of the relaxed ADMM that, exploiting operator theoretical results, can be shown to converge almost surely under the same assumptions as the lossless and synchronous
version.
Moreover, by further assuming the local costs to be strongly convex, we proved local exponential mean-square convergence of the proposed algorithm. Finally, numerical results complement the theoretical analysis and show the performance of the algorithm.
APPENDIX A PROOF OF PROPOSITION 2
The following derivation of the ADMM is standard, however we report it in its entirety because it is instrumental for proving the equivalence of the formulation in Algorithm 1 and Equations (11)- (12).
We start by showing how the updates (11)- (12) can be derived applying the Douglas-Rachford splitting operator to the dual of the distributed problem (10). The following derivation resembles the one proposed in [25]. The dual of (10) is defined as min
w {d f (w) + d g (w)}(32)
with d f (w) = f * (A w), and d g (w) = ι * (I−P) (w). Applying the relaxed Douglas-Rachford splitting to (32) consists in iterating the following equations
w w w(k) = prox ρdg (z(k)) (33) ξ ξ ξ(k) = prox ρd f (2w(k) − z(k))(34)z(k + 1) = z(k) + 2α(ξ ξ ξ(k) − w(k)),(35)
where ξ ξ ξ and z are two auxiliary variables. In particular we can compute (33) by performing the following two updates
y(k) = arg min y ι (I−P) (y) − z (k)y + ρ 2 y 2 (36) w(k) = z(k) − ρy(k).(37)
This fact can be shown by observing that
prox ρdg (z) = arg min u ι * (I−P) (u) + 1 2ρ u − z 2
where the minimum on the right-hand side can be computed as
min u ι * (I−P) (u) + 1 2ρ u − z 2 = max v min u v u + 1 2ρ u − z 2 − ι (I−P) (v) ,
where this equality follows from standard algebraic manipulations. The solution to min u {v u+ 1 2ρ u − z 2 } is u * = z −ρv, therefore it follows that
min u ι * (I−P) (u) + 1 2ρ u − z 2 = − min v ι (I−P) (v) − z v + ρ 2 v 2
and by the definition of the proximal operator we can derive (36)- (37). A similar derivation can be applied to the update (34) to prove that it is equivalent to
x(k) = arg min x f (x) − (2w(k) − z(k)) Ax + ρ 2 Ax 2 (38) ξ ξ ξ(k) = 2w(k) − z(k) − ρAx(k).(39)
Therefore the ADMM is characterized by the equations (36)-(37), (38)-(39) and (35). The next goal is to show that the formulation of Algorithm 1 is equivalent to the formulation we derived above. We show this fact by proving that the updates derived above are equivalent to the equations (4)- (6), which particularized for the distributed problem (10) give Algorithm 1. By rewriting (37) as z(k) = w(k) + ρy(k) and substituting it into (39) we obtain ξ ξ ξ(k) = w(k) − ρ(Ax(k) + y(k)).
Rearranging (39) as 2w(k) − z(k) = ξ ξ ξ(k) + ρAx(k) and substituting (40) into it yields 2w(k) − z(k) = w(k) − ρy(k). Plugging this result into (38) gives (6):
x(k) = arg min x f (x) − (w(k) − ρy(k)) Ax + ρ 2 Ax 2 = arg min x f (x) + ι (I−P) (y(k)) − w (k)(Ax + y(k)) + ρ 2 Ax + y(k) 2 = arg min x L(x, y(k); w(k))
where the second equality was derived adding the following terms independent of x: ι (I−P) (y(k)), w(k) y(k) and (ρ/2) y(k) 2 ;
and the last using the definition of augmented Lagrangian (3).
Consider again (40), we can rewrite ξ ξ ξ(k) − w(k) = −ρ(Ax(k) + y(k)) and plugging it into (35) we get
z(k + 1) = z(k) − 2αρ(Ax(k) + y(k)) = w(k) + ρy(k) − 2αρ(Ax(k) + y(k)) = w(k) − ρ(2α − 1)(Ax(k) + y(k)) − ρAx(k)(41)
where the second equality was derived using z(k) = w(k)+ρy(k) and the third by adding and subtracting ρAx(k). Evaluating now (37) at time k + 1 we obtain w(k + 1) = z(k + 1) − ρy(k + 1), and combining it with (41) we obtain (5):
w(k + 1) =w(k) − ρ(Ax(k) + y(k + 1)) − ρ(2α − 1)(Ax(k) + y(k)).
Using (41) into (36) finally we derive (4):
y(k + 1) = arg min y f (x(k)) + ι (I−P) (y) − w(k) (Ax(k) + y) + ρ 2 Ax(k) + y 2 + ρ(2α − 1)y (Ax(k) + y(k))
where we added the terms -independent on yf (x(k)), −w(k) Ax(k) and ρ Ax(k) 2 .
Now that we have shown the equivalence of the two ADMM formulations (36)-(37), (38)-(39) and (35) and (4)- (6), we particularize the former to the distributed problem of interest, which yields (11) and thus (38) becomes x(k) = arg min
x f (x) + (Pz(k)) Ax + ρ 2 Ax 2 ,(44)
and equation (35)
z(k + 1) = (1 − α)z(k) − αPz(k) − 2αρAx(k).(45)
Consider the inner product (Pz(k)) Ax, exploiting the particular structure of the two matrices we can compute (Pz(k)) Ax = − N i=1 j∈Ni z ji (k) x i . Moreover, since each primal variable x i appears in exactly |N i | constraints, we have that
Ax 2 = N i=1 |N i | x i 2 .
Substituting these results into (44) gives (11), while using the structure of A and P in (45) yields (12).
Finally, with the derivation above we have shown that the variables x in both algorithms are equivalent. Thus if they start from the same initial condition, the trajectories k → x(k) that they generate are equal; this can be ensured by requiring that (37) hold at time k = 1, that is z(1) = w(1) + ρy(1).
APPENDIX B PROOFS OF SECTION V A. Proof Lemma 1
Let us start by assuming that functions f i are quadratic and, more specifically, that Eq. 16 holds true with the residual equal to 0, i.e.,
∇f i (x i ) = ∇f i (x * ) + ∇ 2 f i (x * )(x i − x * ).(46)
In this case (18) simplifies to z(k + 1) = Qz(k) + c
We characterize now the properties of matrix Q, which will extend to the more general case of strongly convex functions thus proving the Lemma. By Proposition 1, we know that the sequence {z(k)} k∈N converges to a fixed point z * for the DRS operator, which thus must satisfy z * = Qz * + c. Therefore, the eigenvalues of Q must lie either in one or strictly inside the unitary circle, and all eigenvalues in one must be semisimple.
We are now interested in proving point 2) in the Lemma. Since the solution is unique by strong convexity, we know using (17) that
x * = −H −1 A Pz * − H −1 b.(48)
Moreover, the vector z * + c i v i , c i ∈ R, must satisfy
x * = −H −1 A P(z * + c i v i ) − H −1 b(49)
since (z * + c i v i ) = Q(z * + c i v i ) + c and hence (z * + c i v i ) is a fixed point. Imposing equivalence between the right-hand sides of both (48) and (49) yields
c i H −1 A Pv i = 0
which, by invertibility of H, implies v i ∈ ker(A P). Since this result does not depend on the particular eigenvector v i chosen, then 2) is true and the Lemma is proved.
B. Proof Proposition 3
We start by stating the following Lemma which will be instrumental for concluding the proof. Lemma 3: The updates (11) can be rewritten as where the i-th row of A selects the auxiliary variables relative to neighbors of i only. Therefore we can write
x i (k) = ρ|N i | prox hi 1 ρ|N i | [A ] i Pz(k) where h i (x i ) = f (x i )x i (k) = arg min xi f i (x i ) − [A ] i Pz(k), x i + ρ|N i | x i 2 2 = arg min xi f i (x i ) + ρ|N i | 2 x i − [A ] i Pz(k) ρ|N i | 2 = ρ|N i | prox hi 1 ρ|N i | [A ] i Pz(k)
where h i (x i ) = f i (x i )/(ρ|N i |) and the term [A ] i Pz(k)/(ρ|N i ) 2 -independent from x i -was added to derive the second equality.
We provide now the proof of Proposition 3, which has been inspired by the one presented in [19] for a different formulation of the ADMM.
We can exploit Lemma 1 to give a formula for Q k . In particular, using the spectral decomposition theorem [38, p. 603-604] we can write
Q = m i=1 v i w i w i v i + r i=m+1 (γ i V i + N i )
where w i is a left eigenvector of the i-th eigenvalue in one, while for the remaining m − r distinct eigenvalues, V i is the spectral projector and N i = (Q − γ i I)V i are nilpotent matrices. Using the properties of the V i and N i matrices, it holds that
Q k = m i=1 v i w i w i v i k + r i=m+1 (γ i V i + N i ) k
and moreover
(γ i V i + N i ) k = γ k i V i + ki j=1 k j γ k−j i V i N j i
where k i is the index of γ i , that is the smallest integer such that rk((Q − γ i I) ) = rk((Q − γ i I) +1 ).
By Lemma 3 and the nonexpansiveness of the proximal operator it follows that
x(k) − x * ≤ A P(z(k) − z * ) .
Substituting the explicit formula for z(k) − z * and using the triangle and Cauchy-Schwarz inequalities we obtain
x(k) − x * ≤ A PQ k z(0) − z * + k−1 l=0 A PQ k−l−1 F r(x(l) − x * ) .(50)
Notice that A PQ k = r i=m+1 A P(γ i V i + N i ) k by property 2) in Lemma 1, and thus depends only on the eigenvalues inside the unitary circle. Therefore there exists C M > 0 such that A PQ k ≤ C M γ k M . Since the Douglas-Rachford operator is nonnexpansive, it follows that z(k + 1) − z * ≤ z(k) − z * and, in turn, that also the sequence x(k) − x * is bounded.
Since lim k→∞ x(k) − x * = 0, from the definition of residual, we can argue that there exists a sequence of positive numbers δ 0 , δ 1 , . . . such that δ k+1 ≤ δ k , lim k→∞ δ k = 0 and
r(x(k) − x * ) ≤ δ k x(k) − x * .
Hence we can bound the error at time k as from which, since |a + b h | < 1 for all h, we get that lim k→∞ e(k) = 0. We conclude the proof by observing that, for any > 0 it holds lim k→∞ k h=1 (a + b h ) (a + ) k = 0.
x(k) − x * ≤ C M γ k M + k−1 =0 C M γ k− −1 M F δ x( ) − x * ,(51)
Lemma 1 :
1Let 0 < α < 1 and ρ > 0. Let γ be an eigenvalue of Q. Then, either γ = 1 or |γ| < 1. Moreover, if m is the algebraic multiplicity of the eigenvalue 1, then the following two properties hold true: 1) the geometric multiplicity of 1 is m, that is, dim (Ker(Q − I)) = m; 2) if the vectors v 1 , . . . , v m are such that v 1 , . . . , v m = Ker(Q − I),
Lemma 2 :
2Let v 1 , . . . , v m be defined as in Lemma 1. Then, it holdsG G G(∞; 0) = lim k→∞ G G G(k; 0) = v 1 1 + . . . + v m m(26)where 1 , . . . , m are random variables taking values in R 2n|E| .
Fig. 1 :
1Example of a random graph for N = 10.
Fig. 2 :
2Evolution, in log-scale, of the relative error of Alg. 3 computed w.r.t. the unique optimal solution x * as function of different values of packet loss probability p, for step size α = 0.75, penalty ρ = 3 and update probability q = 0.8. Average over 100 Monte Carlo runs for the case of cycle graphs.
Fig. 3 :
3Evolution, in log-scale, of the relative error of Alg. 3 computed w.r.t. the unique optimal solution x * as function of different values of packet loss probability p for step size α = 0.75 and penalty ρ = 3. Average over 100 Monte Carlo runs for the case of cycle graphs.
Fig. 4 :Fig. 5 :
45Evolution, in log-scale, of the relative error of Alg. 3 computed w.r.t. the unique optimal solution x * as function of different values of the step-size α, for penalty ρ = 3, packet loss probability p = 0.4 and update probability q = 0.6. Average over 100 Monte Carlo runs for the case of cycle graphs. Empirical and theoretical convergence rates as a function of the tunable parameters α and ρ, averaged over 100 Monte Carlo iterations, with packet loss p = 0.4 and update probability q = 0.6.
-(12); a similar derivation can be done to compute Algorithm 1. The KKT conditions for (36) are (I − P)y = 0 and − z(k) + ρy + (I − P)ν ν ν = 0, with ν ν ν the vector of Lagrange multipliers. Rearranging the second yields y = (1/ρ)(z(k) − (I − P)ν ν ν) and substituting it into the right-hand side of y = Py gives y = P 1 ρ (z(k) − (I + P)ν ν ν) = 1 ρ (Pz(k) + (I − P)ν ν ν) (42) since by the properties of P it holds P(I − P) = −(I − P). Finally, summing y = (1/ρ)(z(k) − (I − P)ν ν ν) with (43) into (37) and the resulting equation for w(k) into (39) gives now ξ(k) = −Pz(k) − ρAx(k)
/(ρ|N i |) and where [A ] i denotes the i-th row of the matrix A . Proof: Recalling the definition of A and P, we can see that j∈Ni z ji (k) = [A ] i Pz(k)
where C M = C M z(0) − z * .Now let us consider the sequence e(0), e(1), e(2), . . ., such thate(0) = x(0) − x * , e(1) = C M γ M +C M F δ 0 x(0) − x * and e(k + 1) = (a + b k )e(k), k ≥ 1, where a = γ M , b k = C M F δ k .Recalling the definition of r, we know that there exists a ball centered on x * , such that, ifx(0) belongs to it then r(x(0) − x * ) ≤ δ 0 x(0) − x * with δ 0 such that a + b 0 < 1, i.e., δ 0 < (1 − γ M )/(C M F ).In this case, by a standard inductive argument, one can show that x(k) − x * ≤ e(k) for all k.
TABLE I :
IComparison of R-ADMM implementations.Alg. 1
Alg. 2
Store and Update
Hereafter we will use bold letters to denote the vectors and matrices stacking local quantities.
With the notation of (2): matrix B is the identity matrix and function g is the indicator function.
Even though the convergence is guaranteed only for 0 < α < 1.
APPENDIX C PROOFS OF SECTION VI A. Proof Proposition 4We prove the convergence of Algorithm 3 by resorting to the stochastic Krasnosel'skii-Mann (KM) iteration studied in[22]. In particular, first we briefly review the stochastic KM and then we show that the proposed algorithm conforms to it. Consider the classic KM iterationwhere T is an averaged operator and where at each iteration all the components of z are updated. The idea underlying the stochastic KM is to allow for only a random subset of coordinates to be updated at each iteration.To formalize this, let I = {1, . . . , M } be the set of indices enumerating the coordinates of z, and let ξ ⊂ I. We introduce the operatorT (ξ) : X → X whose i-th element is given byT. Therefore picking at each time k a random subset ξ k of I we allow the stochastic KM to update only some of the coordinates in z. On a probability space (Ω, F, P), we define the random i.i.d. sequence {ξ k } k∈N , with ξ k : Ω → 2 I , to keep track of which coordinates are updated at each instant. The stochastic KM iteration is finally defined asand consists of the α-averaging of a stochastic operator. The stochastic iteration satisfies the following convergence result, which is a particular case of[22,Theorem 3]. Proposition 8: Let T be a nonexpansive operator with at least a fixed point, and let the step size be α ∈ (0, 1). Let {ξ k } k∈N be a random i.i.d. sequence on 2 I such that ∀i ∈ I, ∃ I ∈ 2 I s.t. i ∈ I and P[ξ 1 = I] > 0.Then for any deterministic initial condition z(0) the stochastic KM iteration (52) converges almost surely to a random variable with support in the set of fixed points of T .Recall from Section VI that the update equation of Algorithm 3 can be compactly written aswhere T DRS denotes the Douglas-Rachford operator applied to the problem in hand. Equivalently we haveTherefore Proposition 8 applies withTB. Proof Lemma 2As done in Lemma 1, suppose the functions f i are quadratic and, in turn, the update rule we consider isLet z * be a fixed point of the synchronous iteration z(k + 1) = Qz(k) + c, that is z * = Qz * + c. Observe that it holds also thatConsider now the update z(k + 1) − z * = G(k)z(k) − z * + B(k)c, and, notice that, from (53) it follows z(k) − z * = G(k)(z(k) − z * ), and, in turn, z(k) − z * = G G G(k; 0)(z(0) − z * ).Since we know that z(k) converges almost surely to some fixed pointz * ,z * = Qz * + c , then we havēwhere α i depends on the particular sequence of matrices B(0), B(1), B(2), . . ., and on the initial condition z(0). This concludes the proof.C. Proof Proposition 5Letz(t) be such thatz (k + 1) = G(k)z(k).Hencez(k) = G G G(k; 0)z(0). and, in turn, from Lemma 2, lim k→∞z (k) = m h=1
If i = l then β i→j and β →k are independent and E[β i→j β →k ] = s 2 . If i = and j = k then E[β 2 i→j ] = s, while if i = but j = k it is E. • S = E[b ⊗ B, The diagonal elements of each block are of the type E. β i→j β i→k ] = s 2 /q. This last result can be derived terms can be computed using the property of the following Kronecker product (AC ⊗ BD) = (A ⊗ B)(C ⊗ D) for matrices A, B, C, D of suitable dimensions• S = E[B ⊗ B] is a block diagonal matrix with the block relative to edge (i, j) equal to E[β i→j diag{β β β}]. The diagonal elements of each block are of the type E[β i→j β →k ]. If i = l then β i→j and β →k are independent and E[β i→j β →k ] = s 2 . If i = and j = k then E[β 2 i→j ] = s, while if i = but j = k it is E[β i→j β i→k ] = s 2 /q. This last result can be derived terms can be computed using the property of the following Kronecker product (AC ⊗ BD) = (A ⊗ B)(C ⊗ D) for matrices A, B, C, D of suitable dimensions.
I ⊗ B)(I ⊗ Q)] = s(I ⊗ I)(I ⊗ Q) = sI ⊗ Q. • E[i ⊗ Bq] = E, • E[I ⊗ BQ] = E[(I ⊗ B)(I ⊗ Q)] = s(I ⊗ I)(I ⊗ Q) = sI ⊗ Q.
B ⊗ I)(Q ⊗ I)] = s(I ⊗ I)(Q ⊗ I) = sQ ⊗ I. • E[ Bq ⊗ I] = E, • E[BQ ⊗ I] = E[(B ⊗ I)(Q ⊗ I)] = s(I ⊗ I)(Q ⊗ I) = sQ ⊗ I.
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| []
|
[
"Ab initio instanton rate theory made efficient using Gaussian process regression",
"Ab initio instanton rate theory made efficient using Gaussian process regression"
]
| [
"Gabriel Laude ",
"Danilo Calderini ",
"David P Tew ",
"Jeremy O Richardson "
]
| []
| []
| Ab initio instanton rate theory is a computational method for rigorously including tunnelling effects into the calculations of chemical reaction rates based on a potentialenergy surface computed on the fly from electronic-structure theory. This approach is necessary to extend conventional transition-state theory into the deep-tunnelling regime, but it is also more computationally expensive as it requires many more ab initio calculations. We propose an approach which uses Gaussian process regression to fit the potential-energy surface locally around the dominant tunnelling pathway. The method can be converged to give the same result as from an on-the-fly ab initio instanton calculation but it requires far fewer electronic-structure calculations. This makes it a practical approach for obtaining accurate rate constants based on high-level electronic-structure methods. We show fast convergence to reproduce benchmark H + CH 4 results and evaluate new low-temperature rates of H + C 2 H 6 in full dimensionality at a UCCSD(T)-F12b/cc-pVTZ-F12 level.This journal is | 10.1039/c8fd00085a | [
"https://pubs.rsc.org/en/content/articlepdf/2018/fd/c8fd00085a"
]
| 52,298,275 | 1805.02589 | 3163acf2a76079fe989c8d2209e417c54e11bceb |
Ab initio instanton rate theory made efficient using Gaussian process regression
Gabriel Laude
Danilo Calderini
David P Tew
Jeremy O Richardson
Ab initio instanton rate theory made efficient using Gaussian process regression
10.1039/c8fd00085aReceived 27th April 2018, Accepted 21st May 2018
Ab initio instanton rate theory is a computational method for rigorously including tunnelling effects into the calculations of chemical reaction rates based on a potentialenergy surface computed on the fly from electronic-structure theory. This approach is necessary to extend conventional transition-state theory into the deep-tunnelling regime, but it is also more computationally expensive as it requires many more ab initio calculations. We propose an approach which uses Gaussian process regression to fit the potential-energy surface locally around the dominant tunnelling pathway. The method can be converged to give the same result as from an on-the-fly ab initio instanton calculation but it requires far fewer electronic-structure calculations. This makes it a practical approach for obtaining accurate rate constants based on high-level electronic-structure methods. We show fast convergence to reproduce benchmark H + CH 4 results and evaluate new low-temperature rates of H + C 2 H 6 in full dimensionality at a UCCSD(T)-F12b/cc-pVTZ-F12 level.This journal is
Introduction
Transition-state theory (TST) has surely become the most popular method for evaluating reaction rates in gas-phase chemistry. 1 It has achieved this status due to its simplicity and the fact that it can be evaluated with efficient computational algorithms. Two geometry optimisations are needed, for the reactant and transition states and two Hessian calculations, one at each stationary point. As only a small number of electronic-structure calculations are needed to evaluate the TST rate, expensive high-level ab initio methods can be used. This is necessary to achieve a good prediction, as small errors in the PES lead to exponential errors in the rate. TST however is based on classical dynamics and neglects important quantum effects such as tunnelling, 2 which can dominate the mechanism of certain chemical reactions of interest. 3 Ring-polymer instanton theory has proved itself to be a useful and accurate method for computing the rate of a chemical reaction dominated by tunnelling. 7 The method is based on a rst-principles derivation from the path-integral representation of the quantum rate [8][9][10] and can be thought of as a quantummechanical generalisation of TST. A ring-polymer discretisation of the path integral allows a simple optimisation algorithm to be used for locating the dominant tunnelling pathway, known as the "instanton". [10][11][12][13] As with TST, it is possible to combine the instanton method with ab initio electronic-structure calculations to evaluate the potential-energy surface (PES) on the y. 14-18 When compared with benchmark quantum dynamics approaches applied to polyatomic reactions, the instanton method typically gives low-temperature rates within about 20-30% of an exact calculation on the same PES. 15,19 This is, in many cases, less than the error in the rate which can be expected to result from the best achievable convergence of the electronic Schrödinger equation, implying that the accuracy of instanton theory itself is not the major issue.
The ab initio instanton method is very efficient when compared with other quantum dynamics approaches, including path-integral molecular dynamics or wave-function propagation. However, it remains considerably more computationally expensive than a TST calculation. The major reason for this expense is that energies, gradients and Hessians of the PES are required, not just at the transition state, but for each ring-polymer bead along the instanton, of which about 100 may be required. For high-accuracy electronic-structure methods, such as those provided by coupled-cluster theory, gradients and Hessians are typically evaluated using nite-differences, and can thus consume a lot of computational power. If the ring-polymer instanton method is to become widely applied in place of TST, the number of ab initio points will need to be reduced to bring the computational expense down, closer to that of a TST calculation.
It is important that high-quality electronic-structure calculations are employed as results can be strongly-dependent on the PES and give signicant errors when using cheaper and less-accurate surfaces. 19,20 One suggestion for decreasing the computational effort required is to run the instanton calculation using a low-level surface and partially correct the result using a few high-level single-point calculations along the optimised pathway. 14,21,22 This approach (termed the 'dual-level instanton approach') certainly improves results, but it cannot always been relied upon as, in certain cases, the location of the instanton pathway may vary considerably depending on the quality of the PES. One can also use Taylor series expansions around the stationary points to obtain an approximate instanton solution analytically. [23][24][25][26][27] These approaches also have the potential to break down when the instanton pathway exhibits strong corner-cutting behaviour and deviates signicantly from the transition state.
The procedure which has generally been followed for ring-polymer molecular dynamics rate theory 28,29 or wave-function propagation methods 30,31 has been to use an analytical function for the PES which is tted to approximately reproduce ab initio points on the surface. In particular much attention has been given to water potentials, 32-34 on which instanton calculations have also been carried out for comparison with high-resolution spectroscopy. 35,36 Despite improvements and automation of this procedure, it remains a difficult task to t a global potential, and is oen based on tens of thousands of ab initio points, 37 computations which we wish to avoid.
The reason why these tting procedures are typically difficult to carry out in practice is because a PES is a complex high-dimensional function. For many applications, including molecular dynamics or wave-function propagation, it is important to have a globally-accurate PES. In particular, if non-physical minima exist in the PES, the dynamics could be attracted there and give nonsensical results. Instanton theory has a particular advantage in that it only requires knowledge of a small region of the PES, located along a line representing the dominant tunnelling pathway. This implies that it might be possible to t a locally-accurate surface around this small region in an efficient manner, as represented in Fig. 1. In this way we ensure that no extrapolation is used, but only interpolation, which is expected to be well behaved.
In this paper, we describe how we use Gaussian process regression (GPR) 38 to t a local representation of the PES and thereby obtain the instanton rate using only a small number of ab initio calculations. By converging the rate with respect to the number of electronic-structure calculations, it is possible to obtain the same results as ab initio instanton theory, for a fraction of the cost. In this way, our GPR approach is almost as efficient as a TST calculation, but has the accuracy of a fully-converged ab initio instanton calculation. We are then able to take advantage of recent developments in high-accuracy electronic-structure methods, 39 which might otherwise be too expensive for an on-the-y calculation. A similar combination of GPR and path-optimisation has been used successfully by the group of Jónsson. 40,41 A number of new developments are Fig. 1 The only areas of the PES which need to be accurately known are those around the instanton pathway or the reactant minimum (in order to obtain their partition functions). In this image, they are represented by the coloured areas, whereas those that are not built into the GPR are unshaded. The blue points represent the beads along the instanton path, while the black points represent the reactant and the transition state. Note that the tunnelling pathway cuts the corner to explore a space far from the transition state. necessary for our implementation, as instanton theory also requires accurate knowledge of Hessians along the path, and because we apply the approach to gasphase reactions, we must account for rotational invariance.
In the following, we describe the background theory as well as the particulars of our implementation of the approach. The results are then presented for two applications and the convergence properties discussed.
Theory
The results in this paper are computed by combining together a number of different approaches. Ring-polymer instanton theory is used to evaluate the rate based on a GPR t to the PES, which has a training set composed of coupledcluster electronic-structure calculations. It will be necessary to transform some data between different coordinate systems to use an appropriate set for each part of the calculation. The instanton equations are dened with Cartesians, as are the inputs and outputs of the electronic-structure calculations, but the GPR is best built using internal coordinates to ensure that it is rotationally invariant. In this way we formally make no further approximations to the instanton theory and also avoid having to construct a kinetic-energy operator in curvilinear coordinates.
Ring-polymer instanton theory
In the ring-polymer version of instanton theory, 10 the dominant tunnelling pathway is represented by a path discretised into N segments. The points where the segments begin and end are given by Cartesian coordinates, x i , called "beads". Because the instanton pathway folds back on itself, only one half of the path need be specied. 12,13 A path dened by a set of N/2 beads, {x 1 ,.,x N/2 }, has the associated half-ring-polymer potential
U N=2 À x 1 ; .; x N=2 Á ¼ X N=2À1 i¼1 X 3n j¼1 m j 2b N 2 ħ 2 À x i;j À x ðiÀ1Þ;j Á 2 þ X N=2 i¼1 V ðx i Þ;(1)
where x i,j is the Cartesian coordinate of the ith bead in the jth nuclear degree of freedom with associated mass m j . The number of degrees of freedom is 3n, where n is the number of atoms. The spring constants are dened by the temperature, T,
such that b N ¼ b/N and b ¼ (k B T) À1 .
The instanton conguration is dened as the saddle point of eqn (1) and in practical applications it can be located using quasi-Newton geometry optimisers. [11][12][13] These require gradients of the target function at each iteration but use update formulae to avoid recomputing the Hessians. 42 The gradient of the ringpolymer potential depends on the gradients of the underlying PES at each bead geometry. In the on-the-y implementation, these are obtained directly from an electronic-structure package, but here they are derivatives of the GPR tted potential.
Once the instanton pathway is optimised, the theory accounts for uctuations up to second order. Thus in order to evaluate the rate, we require Hessians of each bead. Again, these can be computed by an electronic-structure package or from the GPR. The calculation of a Hessian is usually carried out using second-order nite-differences and is therefore on the order of 3n-times more expensive than a gradient calculation. Under the instanton approximation, the rate is given by
k inst Q r ¼ 1 2pbħ Q trans Q rot Q vib expðÀS=ħÞ;(2)
where the action is S ¼ 2b N ħU N/2 and explicit expressions for the instanton vibrational, rotational and translational partition functions are given in ref. 10. The result should be converged with respect to the number of beads, N. Typically on the order of N ¼ 100 beads are used to obtain a rate converged to two signicant gures.
CCSD(T)-F12 theory
For electronic structures where the independent particle model is qualitatively correct, electronic energies computed at the basis set limit CCSD(T) level of theory are expected to be accurate to better than 1 kcal mol À1 for reaction barriers, 0.1 pm for structures and 5 cm À1 for harmonic vibrational wavenumbers. 43 Until relatively recently, the cost associated with using the large basis sets traditionally required to access the basis set limit has prevented this high level of theory from being routinely used in quantum dynamics simulations, which typically require many thousands of energy evaluations. With the maturation of modern F12 explicitly correlated theory, 44 near basis set limit CCSD(T) energies can now be computed using small (triple-zeta) orbital basis, at a cost only 15% larger than a traditional CCSD(T)/TZ calculation, and quantum dynamics studies can be performed using near basis set limit CCSD(T) Born-Oppenheimer potentialenergy surfaces on a routine basis. In F12 theory the standard manifold of correlating orbitals |abi that parameterise two-body correlation functions in pair theories is supplemented with one geminal basis function per occupied orbital pair ij, chosen to directly model the Coulomb hole in the rst-order pair correlation function m ij ¼ X a\b t ij ab jabi þ X k\l c ij klQ f ðr 12 Þjkli:
The correlation factor f(r 12 ) is chosen to be a linear combination of Gaussians 45 t to an exponential function 46 with a length-scale of 1a 0 , appropriate for valence electrons, and the many electron integrals that arise due to the explicit dependence on the interelectronic distance, r 12 , and the presence of the strong orthogonality projector,Q, are decomposed into one-and two-electron components by inserting in approximate resolutions of the identity. 47 The coefficients t ij ab are optimised in the presence of xed geminal contributions, 48 to reduce the geminal basis-set superposition error, 49 with coefficients chosen to satisfy the rst-order singlet and triplet cusp conditions. 50 Small but numerically expensive geminal contributions to the energy Lagrangian function are neglected if they rank higher than third order in perturbation theory, 51 resulting in the CCSD(T)(F12*) approximation. 39 In this work we use the Molpro electronic structure package 52 and are restricted to using the slightly less accurate CCSD(T)-F12b approximation 53 where the geminal contributions from third order ring diagrams are also neglected. Nevertheless, the CCSD(T)-F12b energies computed in a TZ basis set are within 0.2 kJ mol À1 per valence electron of the CCSD(T) basis set limit and retain the intrinsic accuracy of the wavefunction ansatz. 54
Gaussian process regression (GPR)
Gaussian process regression is a machine learning algorithm which can be used to efficiently generate complex hypersurfaces with limited data. 38 Recent work has applied this technique for the construction of potential-energy surfaces 55-57 and determining minimum energy paths 40,41 at a much lower computational cost. In this paper, a local representation of the PES is constructed around the instanton pathway and used to evaluate reaction rate constants. Before carrying out the construction of a local PES with GPR, we rst note that we have to utilise an internal coordinate system that accounts for rotational invariance. We dene this internal coordinate system as q ¼ q(x), where x is a set of Cartesian coordinates. This transformation to a rotationally invariant coordinate system is dened in Section 2.4.
In the simplest case, the training set is composed of known values of the potential, V(q j ), at the M reference points {q 1 ,.,q M }. This denes the column vector y with elements
y j ¼ 3 + V(q j ),(4)
where 3 is an energy shi chosen such that the average of these elements is approximately zero. Noting that the derivative of a Gaussian process is also a Gaussian process, 38,40,41 it is also possible to include gradients and Hessians into the training set as described in ref. 40.
The potential for an unknown point q* can be predicted from GPR as
V ðq*Þ ¼ À3 þ X M j¼1 k À q*; q j Á w j ;(5)
where k(q i , q j ) is a covariance function for the prior. We chose a squaredexponential covariance function with length-scale g and a prefactor f:
k À q i ; q j Á ¼ f 2 exp À 1 2g 2 q i À q j 2 :(6)
The elements, w j , of the vector, w, are determined by solving the linear equations
(K + s 2 I)w ¼ y,(7)
where the covariance matrix is dened by K ij ¼ k(q i , q j ). By differentiating eqn (5), one obtains expressions for the gradient and Hessian of the PES. Because the covariance function is smooth, the PES is guaranteed to be differentiable to all orders. This is in contrast to the otherwise similar method of Shepard interpolation which produces surfaces which are not smooth enough to carry out instanton optimizations. 19 s is a noise term, which is introduced to avoid overtting, and should be chosen to be the expected self-consistent error in the reference data. Together, f, s and g are known as hyperparameters. Their values can be optimised by maximising the log marginal likelihood, Alternatively, one can also optimise the hyperparameters through the minimisation of errors by cross-validation. 38 The method above allows us to construct a local PES from a training set of M points. In our implementation, the general idea is as follows. Firstly, we construct an approximate PES with GPR using a small number of points, and then optimise the ring polymer based on this PES. Aer this, we rene the PES by adding new ab initio evaluations of points along the previously predicted ring-polymer conguration. Using the rened PES, we obtain a new ring-polymer conguration and then compare it with the previous one to check if the pathway has converged to the true instanton pathway. If this is not satised, the PES is rened again through the addition of more ab initio evaluations; this is continued iteratively until the convergence is achieved. The abovementioned scheme is further elaborated in Section 3.
Q ¼ À 1 2 y T w À 1 2 log K þ s 2 I À M 2 log 2p:(8)
The general scheme described above is similar to that done by the group of Jónsson, 40,41 wherein they obtain the minimum energy path using a GPR-aided nudged elastic band (NEB) method. This appears to have been highly successful, effectively reducing the number of ab initio evaluations required by an order of magnitude in comparison to a conventional NEB calculation. In this paper, we intend to emulate this drastic reduction in the computational effort for locating the instanton pathway and evaluating rates. As mentioned before, there are some differences in our implementation, such as the need for rotational invariance and accurate knowledge of the Hessians. We have found that the accuracy of the Hessians returned by GPR is signicantly improved by explicitly providing Hessian data into the training set.
Non-redundant internal coordinate system
We would like to build the GPR representation of the PES using an internal coordinate system which is rotationally and translationally invariant. This is necessary as the relative rotational orientation of individual beads along the instanton pathway is not known a priori. However, we will need to be able to convert the information obtained from the GPR-based PES back into a Cartesian coordinate system in order to evaluate the instanton rate. Also note that the data available from electronic-structure packages are in Cartesian coordinates, which will need to be converted into internal coordinates in order to build the GPRbased PES.
Much recent work into machine-learning algorithms for describing intermolecular forces has further required permutational invariance. [58][59][60][61] Such advanced approaches could also be applied to our problem. However, as we only need to t the potential locally, it is an unnecessary complication and thus we choose to neglect permutational symmetry. For our studies here, this is no inconvenience as we only need to compute the instanton rate for one of the equivalent reaction pathways and multiply the rate by the degeneracy.
A simple translationally and rotationally-invariant coordinate system for representing molecular geometries is provided by the n  n distance matrix, 62 dened as wherer i is a three-dimensional vector of the Cartesian coordinates of atom i, such
D ij ¼ & kr i Àr j k À1 ; i . j 0; i # j;(9)thatr 1 ¼ (x 1 , x 2 , x 3 ),r 2 ¼ (x 4 , x 5 , x 6 ), etc.
Although it is possible to convert data from a Cartesian coordinate system into this set, 63 the back transformation is not well dened, as the internal coordinates are redundant. In order to obtain a non-redundant set of internal coordinates, we follow the approach of Baker et al. 64 Firstly, we unravel the matrix D to give the coordinates as a vector of length n 2 ,
d ¼ [D 11 D 12 . D 21 D 22 . D nn ] T .(10)
The B matrix is dened to describe how changes in the Cartesian coordinates affect these redundant coordinates as
B ¼ vd vx ¼ 2 6 6 6 6 6 4 vD 11 vr 11 . . . vD 11 vr n3 « « « « « vD nn vr 11 . . . vD nn vr n3 3 7 7 7 7 7 5 :(11)
The elements of this n 2 Â 3n matrix are given explicitly by
vD ij vr ka ¼ 8 < : À À r ia À r ja Á kr i Àr j k À3 ; k ¼ i . j À r ia À r ja Á kr i Àr j k À3 ; i . j ¼ k 0; otherwise(12)
where a runs over the indices of three-dimensional space. A square matrix, G ¼ BB T , is formed and then diagonalised to obtain the eigenvalues and eigenvectors. The non-redundant eigenvectors are those corresponding to the nonzero eigenvalues (of which there will be 3n À 6 for a nonlinear isolated molecule), whereas the redundant eigenvectors have zero eigenvalues. The non-redundant eigenvectors are collected into the columns of a matrix, U. With this, we can now transform d into a non-redundant coordinate system, dened by
q ¼ U T d.(13)
It is this internal coordinate system which is used to build the GPR representation.
Note that the matrix U is built only once at a reference geometry and is used to dene the transformation to q at all other geometries. The reference geometry used in our studies was the transition state, although this is not a requirement. The same U matrix is then used for new geometries to give a consistent denition of the internal coordinates q ¼ q(x).
Therefore the required relationship between the internal coordinates and Cartesians is given by dq
¼ B q dx, where B q ¼ vq vx ¼ U T B:(14)
The gradient and Hessian in the non-redundant internal coordinate system are dened as This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
g q ¼ vV vq H q ¼ v 2 V vqvq ;(15)
and similarly, in Cartesian coordinates,
g x ¼ vV vx H x ¼ v 2 V vxvx :(16)
Given a geometry x to dene the appropriate orientation, the gradients and Hessians obtained from the GPR in internal coordinates can be transformed back to Cartesian coordinates. Obtained using the chain rule, the transformations are dened by
g x ¼ B q T g q(17)H x ¼ B q T H q B q þ g q T vB q vx ;(18)where vB q vx ¼ U T vB vx .
In order to transform the gradients and Hessians obtained from electronicstructure calculations into the q coordinate system, these equations need to be inverted. However, as B q is not a square matrix, we need to dene the generalised inverse as
(B q T ) À1 ¼ (B q B q T ) À1 B q .(19)
The required transformations are
g q ¼ (B q T ) À1 g x(20)H q ¼ B q T À1 H x À g q T vB q vx ! B q T À1 ! T :(21)
These equations dene all of the necessary transformations needed for converting the ab initio data into reduced coordinates, and for converting it back to a Cartesian system at a given orientation.
Method
Our aim is to reproduce the same result as an ab initio instanton calculation performed on the y. As with these calculations, we must therefore consider convergence with respect to N. For our new approach based on GPR, we must also simultaneously converge the result with respect to the number of points in the training set.
Here we outline our standard protocol for computing converged instanton rates using GPR. This is made up of two parts: rst, in which the instanton pathway is located, and second, in which the uctuation terms are converged to yield the nal instanton rate. We have attempted to design this protocol to be stable and efficient. In our study, we have found that this protocol posed no signicant problems for the systems tested here. In future studies, one could consider improvements which may increase the efficiency further. In a realistic working environment, a researcher has the freedom to add information to the GPR however they like until the result is converged. Our protocol is designed for the case that single-point ab initio calculations are by far the most expensive part of the calculation. We also assume that the Hessian calculations are orders of magnitude slower than potential or gradient evaluations. This is commonly the case for many electronic-structure methods, especially if the Hessians are computed using nite differences. The efficiency of our protocol should thus be measured in terms of the number of ab initio calculations required, and in particular the number of Hessians. We present these numbers for specic examples in the next section.
The protocol described below is intended for a calculation of a single instanton rate at a given temperature, as is the approach used in the H + CH 4 benchmarks we present in Section 4.1. If, as is common, one needs the rate at multiple temperatures, it is recommended to start just below the crossover temperature, T c . The optimised instanton can be used as the initial guess and GPR training set for a calculation at a lower temperature. We use this more efficient approach for our H + C 2 H 6 calculations in Section 4.2.
Protocol
(1) Optimise the reactants and transition state (using a standard Quantum Chemistry package), and obtain gradients and Hessians for the optimised geometries. The optimised transition-state geometry in Cartesian coordinates is notated x ‡ .
(2) By diagonalising the mass-weighted Hessian at the transition-state, calculate the cross-over temperature,
T c ¼ ħu b 2pk B ;(22)
where u b is the magnitude of the imaginary frequency.
(3) An initial guess for the instanton conguration is obtained using 13
x i ¼ x ‡ þ D cos 2pi N z i ¼ f1; .; N=2g;(23)
where z is the normalised non-mass-weighted eigenvector corresponding to the imaginary mode at the transition state and D is a user-dened spread of points. Typically we choose D $ 0.1Å and N ¼ 16 for an initial guess. However, if previous instanton optimisations at a higher temperature have been performed successfully, these congurations usually provide a better initial guess.
(4) Calculate ab initio potentials and gradients for the points obtained in step 23. (5) Repeat until convergence: (a) Optimise hyperparameters using methods dened previously under Section 2.3.
(b) Starting with a low number of beads N, locate the ring-polymer path by increasing N until the action S/ħ is converged to 2 decimal places.
(c) Check if the mean bead displacement
Dx ¼ 1 N X N i¼1 kx new i À x old i k\PC,
where PC corresponds to the path convergence limit. Also check that the convergence of the action |S new À S old |/ħ # 10 À2 . If this is satised, this means that the ring-polymer path has converged. Continue to step 6.
Otherwise provide new inputs (ab initio energies and gradients) to the GPR training set along the current ring polymer (i.e. increase the number of training points M) and then go back to step 5a.
(6) Repeat until convergence is achieved: (a) Provide a couple of new points along the converged instanton pathway to the GPR training set, this time also including the Hessians.
(b) Optimise the hyperparameters using the methods dened previously under Section 2.3.
(c) Locate the instanton pathway and calculate the rate, k, increasing N until this converges.
(d) Test if |k new À k old |/k new # RC, where RC corresponds to the rate convergence limit.
If this is satised, the iterative algorithm is terminated, and the current value of k is taken as the converged instanton rate.
Otherwise, return to step 6a. In the following calculations, we built the GPR using energies in hartrees (E h ) and Cartesian coordinates inångströms (Å). In these units, the typical values used for the length-scale were g $ 0.3-0.4, and for the prefactor, f ¼ 0.09. We specied the noise term differently for the potentials, gradients and Hessians, as s V $ 10 À6 , s G $ 10 À4 and s H $ 10 À3 . The convergence limits of PC ¼ 10 À2Å and RC ¼ 10 À2 were used.
We have outlined the simplest protocol which has the desired properties of converging the instanton rate without needing a large number of ab initio calculations. However, it is not necessarily the optimal choice for all problems. In particular, it should be noted that in this work, new information is provided to the GPR training set at the positions of beads chosen by hand. This was done in a systematic way, wherein during the path convergence step, the beads were chosen such that they are evenly distributed along the current ring polymer. Once the path is converged, beads where the Hessians are to be included were chosen in a similar manner, i.e. evenly distributed along the converged pathway. There may be better ways of providing new information to the GPR training data; for instance one can evaluate the expected tting error along the current pathway and then provide points at the areas with high variance. By being more selective, one can potentially further reduce the number of ab initio calculations required.
Results
The method described above was applied to the following two systems:
H + CH 4 / H 2 + CH 3 H + C 2 H 6 / H 2 + C 2 H 5 .
The rst is a standard benchmark reaction for testing quantum rate theories and has been studied with various methods including MCTDH, 65,66 ring-polymer molecular dynamics, 67 the quantum instanton, 68 as well as ring-polymer instanton theory. 12,15,19 The second reaction is beyond the current limits of exact quantum mechanics unless reduced dimensionality models are used. Using the GPR formalism, we are able to present a converged ab initio instanton rate for the rst time. We compare these results with those predicted by other semiclassical methods.
H + CH 4
An on-the-y ab initio instanton calculation has been done by one of us for this polyatomic reactive system. 15 Here, we use this reaction as a benchmark case for our GPR-aided instanton calculation and show that we are able to obtain the same result as an on-the-y calculation with a signicant reduction in the number of potentials, gradients, and most importantly, Hessians required.
The electronic-structure method used in ref. 15 was RCCSD(T)-F12a/cc-pVTZ and we use exactly the same method for the training set for the GPR. Note that in this paper, as well as in ref. 15, this method is also used to obtain the properties of the isolated reactants (including the H atom). To account for the indistinguishability of the H atoms, the instanton rate formula is multiplied by 4.
The results in ref. 15 were computed using N ¼ 128, which we also use here. This required the calculation of 64 ab initio potentials and gradients per iteration of the instanton optimisation scheme. Because approximately 10 iteration steps are usually required for an instanton optimisation, about 640 gradients were computed in addition to the 64 Hessians once the instanton had been optimised. Here, we followed the protocol outlined in the previous section independently for three different temperatures. This allows us to accurately determine the computational effort required for a converged rate.
In Table 1, the rows correspond to the iterations of step 5 of the protocol, in which the pathway is optimised by adding more potentials and gradients to the GPR training set. The action is seen to converge to two decimal places aer only a few iterations. Here, this was done with fewer than 50 potentials and gradients for all three temperatures. This means that a reduction in the number of gradient evaluations by an order of magnitude has been achieved. This fast convergence is also observed in Fig. 2, where it is seen that, at the lowest temperature studied, the pathway already has the correct shape aer the second iteration. In this gure, the potential along the pathway is plotted as a function of a cumulative mass-weighted path length,
l i ¼ X i i 0 ¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 3n j¼1 m j À x i 0 þ1;j À x i 0 ;j Á 2 v u u t :(24)
It should be noted that the plots are shied such that they are centred around l ¼ 0.
In Table 2, the GPR model is further rened, as described in step 6 of the protocol, by providing more observations (i.e. more ab initio potentials, gradients Fig. 2 Convergence of the ring-polymer instanton at 200 K for H + CH 4 . The initial GPR training set was defined by eqn (23). The ring-polymer beads are plotted as a function of their potential energy and the path length, l, as defined by eqn (24). Table 2 The rates obtained from the GPR-based instanton calculations are given as the information provided to the GPR training set is increased. The error is measured relative to the on-the-fly ab initio results of ref. 15. Note that for the rate calculation, one further Hessian is needed at the reactant geometry, but that this is not included in the GPR training set and Hessians) to the GPR training set. Our ndings show that it is necessary to include a few Hessians directly in the GPR training set, and that the transitionstate Hessian alone is not sufficient to describe the uctuation terms of the instanton. Note that at low temperatures, the GPR requires a few more Hessians to converge the rate. This is due to the fact that the instanton stretches out more at lower temperatures, thus meaning that the GPR needs more information as the instanton covers a larger area of the PES. The convergence is fast and it takes no more than 8 Hessians to converge the rates for all of the temperatures to less than 1% of that of the ab initio calculation. This is a remarkable improvement in terms of the computational effort required over the ab initio instanton calculations as the Hessian calculations account for a huge percentage of the computational effort required. Having reduced the number of Hessians required from 64 to 8, the reduction in computational power needed would allow us to investigate problems involving larger molecules and to also use higher-level electronic-structure methods.
H + C 2 H 6
The H abstraction reaction from ethane follows the same mechanism as abstraction from methane. From a theoretical point of view, it is of interest as the number of degrees of freedom is signicantly higher such that full-dimensional exact quantum methods are not applicable and approximations must be made. There are two types of approximations which can be used to make the simulation tractable. One makes use of semiclassical dynamics, and the other involves reducing the dimensionality of the system. The instanton method is an example of the former, as are other semiclassical extensions of the transition-state theory 25,69 and ring-polymer molecular dynamics. 70 Reduced-dimensionality models allow the quantum scattering theory to be applied 71 and can also be combined with semiclassical approaches. 25,26,72 Experimental results are available at 300 K, 73,74 but unfortunately not at lower temperatures, where the tunnelling effect is more important. Here, we compare the results of our instanton rate calculations with other theoretical calculations, and discuss the relative efficiency of the various methods.
4.2.1 Ab initio calculations. Due to the efficiency of the GPR-aided instanton approach seen in our benchmark tests, we are able to use high-accuracy and computationally expensive electronic-structure methods. The method we choose is UCCSD(T)-F12b as discussed in Section 2.2. Table 3 shows the predictions for barrier heights, V ‡ , and imaginary frequencies, u b , with increasingly large basis sets. Hessians with cc-pVQZ and cc-pV5Z basis sets were not evaluated due to the large amount of computational resources that would be required. However, we can see that cc-pVTZ-F12 reproduces almost the same barrier height as cc-pV5Z, in accordance with the study by Spackman et al. 75 which suggested that the cc-pVnZ-F12 basis sets have similar performance to the cc-pV(n + 2)Z basis sets (where n ¼ D, T, Q, etc.) in terms of results when using CCSD(T)-F12. Hence in the following calculations, we will use the cc-pVTZ-F12 basis set.
With our chosen method, the crossover temperature is predicted to be 337 K. We ran three instanton calculations, rst at 300 K, and then used this as a starting point for a calculation at 250 K, and in the same way for 200 K. This approach may slightly reduce the number of iterations needed for convergence. For instance, it can be seen in Fig. 3 that the optimisation of the 200 K instanton is obtained in only a few iterations and that the path is almost correct even aer the rst. The convergence criteria used for this system were similar to those used in the H + CH 4 system. The Cartesian representation of the optimised path for H abstraction from ethane is shown in Fig. 4. It is seen that the mechanism is similar to that of H + CH 4 , shown in ref. 15, in that the abstracted hydrogen does most of the tunnelling, and is accompanied by a small movement of its neighbouring hydrogens. The atoms at the far end of the ethane molecule hardly participate in the instanton at all. Note, however, that they still make a contribution to the uctuations, and thus cannot be neglected. 76 The results of our GPR-based instanton calculations are presented in Table 4. These rates account for the degeneracy of the reaction by multiplying the formula in eqn (2) by a factor of 6. The 300 K result was obtained with a training set including 33 potentials and gradients, and 6 Hessians. The calculations at the lower temperatures of 250 K and 200 K added an additional 6 Hessians to the training set (i.e. at 250 K, the training data includes 6 Hessians from 300 K and 6 Hessians from 250 K) in order to converge the rates. This represents a reduction in the computational effort by an order of magnitude, similar to what has been observed for H + CH 4 . Table 3 Barrier heights and imaginary frequencies for H + C 2 H 6 using increasingly larger basis sets at the UCCSD(T)-F12b level It is clear from our calculation that the tunnelling effect makes a large contribution to the rate, even at 300 K. This is conrmed by experimental results at this temperature, which in various setups have been measured to be 3.13 Â 10 À17 cm 3 s À1 (ref. 73) or 7.47 Â 10 À17 cm 3 s À1 , 74 and which both lie in the same order of magnitude as our prediction. Note that we expect the instanton approach to slightly overpredict the rate (by up to a factor of 2) at 300 K as this lies close to the value of T c . 77 Unfortunately, no experimental results are available for comparison at lower temperatures where the tunnelling effect is predicted to increase dramatically. Table 4 also compares our predicted rate with those of the reduceddimensionality quantum scattering (RD-QS) calculations by Horsten et al. 71 and a full-dimensional semiclassical transition-state theory (SCTST) rate calculation by Greene et al. 25 The RD-QS calculations utilised a similar electronic structure method to our calculations, albeit with F12a rather than F12b, which gives a barrier height only 0.1 kJ mol À1 lower. The SCTST calculations employed the CCSD(T)/cc-pVTZ method for the energies at the stationary points, which gives a barrier height 0.4 kJ mol À1 lower. We expect these differences to lead to only a minor deviation.
The instanton results are in quite close agreement with RD-QS, where the rates differ by no more than 25%. This is what is typically expected when comparing results obtained with the instanton method and those obtained with exact quantum methods. 19 This conrms that, at least for this system, the reduceddimensionality approach is not causing an appreciable error in the tunnelling effect.
There is a slightly larger discrepancy between the instanton and SCTST results, 25 which increases at lower temperatures. The SCTST rate calculation involved a total of 118 ab initio Hessians at the MP2/cc-pVTZ level, with energies at stationary points evaluated with CCSD(T)/cc-pVTZ. The GPR-instanton method required only 6 Hessians to converge the rate at each temperature and thus a high-level of theory for the Hessian calculations can be used as well. There are two reasons for the discrepancy in the SCTST rates. One is that lower-level electronic-structure theory was used in ref. 25 for the Hessian calculations. The second is that at low temperatures, the instanton pathway stretches far from the transition state and the PES cannot therefore be well represented by a Taylor series around the transition state. In this case, there are no dramatic differences between the theoretical predictions. It seems that the H + C 2 H 6 reaction follows a simple pathway for which reduced-dimensionality models are applicable. However, we expect that for more complex reactions there will be a larger discrepancy and that, in many cases, the full-dimensional instanton theory will be the most accurate.
4.2.2 Results on a tted PES. In order to get an idea of the accuracy of the instanton approach for this reaction, we compare instanton rates with those of other semiclassical approaches based on the tted, global CVBMM potentialenergy surface. 69 This PES was constructed by dividing the system into a reactive part which would be treated with semiempirical valence bond theory and a non-reactive part treated with molecular mechanics. It was parameterised against density functional theory, of which more details can be found in ref. 69. The barrier height obtained with the CVBMM PES is 47.90 kJ mol À1 and has a predicted crossover temperature of 352 K. Table 5 presents the rates of three methods, the instanton theory (this work), the quantum instanton theory (QI) 78 and the small curvature tunnelling correction to canonical variational TST (CVT/SCT). 69 The tunnelling factors are seen to be about a factor of 2 larger than those from the ab initio method, mainly due to the fact that the CVBMM barrier is too narrow and thus overpredicts the tunnelling factors
The CVT/SCT rate is in close agreement with that of the instanton theory, which at least in this cases implies that the dominant tunnelling pathway is well approximated by the minimum-energy pathway used by CVT/SCT. It is expected that, in general for more complex reactions, the instanton method, which denes the tunnelling pathway in a rigorous manner, will give a more accurate result.
Unlike the ring-polymer instanton approach, the QI method does not use a steepest-descent approximation and thus includes anharmonic vibrational effects in full dimensionality. In order to do this, it samples over a statistically large number of path-integral congurations and would therefore not be a practical computational method when combined with high-level ab initio potentials. Nonetheless, these anharmonic effects only change the rate by less than 50% at the lowest temperature studied. This is in agreement with the ndings of ref. 78 which showed that, at low temperatures, a small increase in the rate resulted from making a harmonic approximation to the internal rotation. This conrms that instanton theory gives a reliable prediction of the order-of-magnitude of the rate. The real advantage of the instanton approach over this method is that it can be applied to new reactions without needing to build a global PES at all.
Conclusions
We have demonstrated how ab initio instanton theory can be made efficient using GPR to t the PES locally around the dominant tunnelling path. This was demonstrated rst using the H + CH 4 reaction as a benchmark, for which we have shown that the number of electronic-structure calculations can be reduced by an order of magnitude, while converging the rate to within 1% of the benchmark result. We then proceeded to evaluate the instanton rates for H + C 2 H 6 , based on UCCSD(T)-F12b/cc-pVTZ-F12 electronic-structure calculations. Most importantly, the number of Hessians needed for all of these calculations is about 6, which makes the method more efficient than full-dimensional SCTST calculations and almost as efficient as a classical TST calculation.
When studying a complex network of reactions, TST is commonly used to obtain a rate for the many possible reaction steps. 79 By evaluating the crossover temperature for each step, it can be easily determined whether tunnelling is likely to play a role, and instanton calculations can be run for these steps only. As there are typically many more steps for which tunnelling is not important than those for which it is, the number of ab initio calculations needed for the instanton calculations would be small in comparison to the overall total. In this way, tunnelling can be rigorously accounted for without signicantly increasing the computational effort.
In this work, we suggested a simple protocol which, in our tests, showed no particular problems. We note, however, that it could still be improved in a number of ways which would further increase the efficiency. For instance, when using estimates of the GPR tting error, we could select new points to be added to the training set in a more systematic way. These could also be used to estimate the tting error in the rate constant in a similar way to what has been done for TST calculations. 80 Other techniques might allow us to reduce the number of high-level calculations by including low-level ab initio information into the GPR training set. One possibility would be to use this low-level information only for the initial iterations to locate the region of space where the instanton is likely to exist on the high-level surface. The nal iteration could be done using only the high-level information to ensure convergence to the correct result. However, one could also consider combining the high-and low-level information in the training set, as in the duallevel approach. 22 By using a larger value of the noise term for the low-level points, the GPR would then t itself accurately to the high-level points, and the low-level information could be used as a rough guide for the shape. Typically the frequency calculations from the low-level calculations are a good approximation even if the absolute energies are not, and so most Hessians could be derived from the lowlevel calculations. One could imagine systematically converging to the correct result by adding more high-level ab initio points such that the accuracy would not be compromised. We have shown in this paper that we can converge the rate with respect to the number of ring-polymer beads, N, as well as with respect to the number of points included in the GPR training set. However, the accuracy of our method is still limited by the computational expense of electronic-structure methods, which are rarely possible to fully converge. Methods such as F12 have been very useful for increasing this efficiency 44 as well as linear-scaling methods 81 and the use of graphical processing units. 82 Our GPR-aided instanton theory can take advantage of such improvements to ab initio electronic-structure calculations in an efficient way.
We did not nd particularly large differences in the rate predictions for the H + C 2 H 6 reaction between the instanton approach and other theories. This is due to the rather simple mechanism exhibited by the H abstraction reaction, which follows a pathway close to the minimum-energy path, making the CVT/SCT and reduced-dimensionality models valid. The advantage of the instanton theory is that no a priori choice of reduced coordinates, or tunnelling coordinate is made. This makes the approach applicable also to more complex reactions as well as tunnelling splitting calculations. 35 In these cases it is expected that the instanton path will deviate more strongly from the minimum-energy path, and the fulldimensional instanton theory will be required to obtain an accurate prediction. The proof of principle outlined in this work for combining GPR with the instanton theory will then be exploited in future studies of new reactions.
Note added in proof
It has come to our attention that in a recent paper, Kästner and coworkers apply a similar scheme to compute ab initio instanton rates using a neural network t of the PES. 83 Their ndings are similar to our own, in that a huge computational saving results from the approach and they also stress the importance of including ab initio Hessians in the training data. Neural networks have been found to be one of the best methods for tting global PESs. However, our simpler GPR scheme may have some advantages over the neural networks in this case where only a local description is required based on a small training set. In particular, GPR has a limited number of hyperparameters which can be easily optimised, whereas neural networks appear to have a strong dependence on randomly chosen initial weights and require an averaging procedure to obtain a prediction for the rate.
Conflicts of interest
There are no conicts to declare.
- 6 a
6Laboratory of Physical Chemistry, ETH Zurich, Switzerland. E-mail: [email protected] b On exchange from School of Chemistry, University of Edinburgh, UK c Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
Faraday Discuss., 2018, 212, 237-258 This journal is © The Royal Society of Chemistry 2018 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Faraday Discuss., 2018, 212, 237-258 This journal is © The Royal Society of Chemistry 2018 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
is © The Royal Society of Chemistry 2018 Faraday Discuss., 2018, 212, 237-258 | 243 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Faraday Discuss., 2018, 212, 237-258 This journal is © The Royal Society of Chemistry 2018 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM.
is © The Royal Society of Chemistry 2018 Faraday Discuss., 2018, 212, 237-258 | 245 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Faraday Discuss., 2018, 212, 237-258 This journal is © The Royal Society of Chemistry 2018 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Faraday Discuss., 2018, 212, 237-258 This journal is © The Royal Society of Chemistry 2018 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Fig. 3
3Convergence of the ring-polymer instanton at 200 K for H + C 2 H 6 . The initial GPR training set was given by points along the 250 K instanton path. The path length, l, is defined by eqn(24).Paper Faraday DiscussionsThisjournal is © The Royal Society of Chemistry 2018 Faraday Discuss., 2018, 212, 237-258 | 251 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
Fig. 4
4Representation of the ring-polymer instanton for H + C 2 H 6 at 200 K.
Faraday Discuss., 2018, 212, 237-258 This journal is © The Royal Society of Chemistry 2018 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
This journal is © The Royal Society of Chemistry 2018 Faraday Discuss., 2018, 212, 237-258 | 239 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.Paper
Faraday Discussions
Table 1
1Convergence of the instanton path with the iteration of protocol step 5. The number of potentials (V), gradients (G), and Hessians (H) included in the GPR training set is explicitly noted. Here the single Hessian in the training set corresponds to that of the transition stateT (K)
Iteration
Training set
S/ħ
Dx (10 À3Å )
300
1
10V, 10G, 1H
25.167
-
2
19V, 19G, 1H
25.243
41.7
3
28V, 28G, 1H
25.249
1.78
250
1
10V, 10G, 1H
28.957
-
2
19V, 19G, 1H
29.244
37.3
3
28V, 28G, 1H
29.301
3.17
4
37V, 37G, 1H
29.274
1.77
5
46V, 46G, 1H
29.278
0.13
200
1
10V, 10G, 1H
32.586
-
2
19V, 19G, 1H
33.871
70.1
3
28V, 28G, 1H
33.945
2.54
4
37V, 37G, 1H
33.904
1.90
5
46V, 46G, 1H
33.907
0.70
Table 4
4Calculated rates (in cm 3 s À1 ) for H + C 2 H 6 obtained by the GPR-aided instanton method and other direct dynamics methods. The tunnelling factor, k tun , is defined as the ratio between the instanton rate and Eyring TSTT/K
GPR-aided instanton
SCTST 25
RD-QS 71
k tun
Rate
300
15
7.0(À17)
3.88(À17)
6.23(À17)
250
38
6.4(À18)
9.51(À18)
7.97(À18)
200
623
5.7(À19)
2.50(À19)
6.69(À19)
Faraday Discussions
Paper
Table 5
5Rate comparison between different methods using the CVBMM PES. All of the rates are in cm 3 s À1 T/K This journal is © The Royal Society of Chemistry 2018 Faraday Discuss., 2018, 212, 237-258 | 253 Open Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.Instanton
CVT/SCT 69
QI 78
k tun
Rate
300
24
1.25(À16)
1.44(À16)
1.15(À16)
250
80
1.46(À17)
-
1.40(À17)
200
1296
1.61(À18)
1.90(À18)
1.16(À18)
AcknowledgementsThis work has been nancially supported by the Swiss National Science Foundation (project no. 175696).View Article OnlineNotes and references
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Open Access Article. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported LicenceOpen Access Article. Published on 27 June 2018. Downloaded on 1/9/2021 8:57:08 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.
| []
|
[
"Robustness of the optical-conductivity sum rule in Bilayer Graphene",
"Robustness of the optical-conductivity sum rule in Bilayer Graphene"
]
| [
"L Benfatto \nCentro Studi e Ricerche \"Enrico Fermi\"\nvia Panisperna 89/AI-00184RomeItaly\n\nDepartment of Physics\nCNR-SMC-INFM\nUniversity of Rome \"La Sapienza\"\nP.le Aldo Moro 5I-00185RomeItaly\n",
"S G Sharapov \nDepartment of Physics\nWestern Illinois University\n61455MacombILUSA\n\nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n",
"J P Carbotte \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n"
]
| [
"Centro Studi e Ricerche \"Enrico Fermi\"\nvia Panisperna 89/AI-00184RomeItaly",
"Department of Physics\nCNR-SMC-INFM\nUniversity of Rome \"La Sapienza\"\nP.le Aldo Moro 5I-00185RomeItaly",
"Department of Physics\nWestern Illinois University\n61455MacombILUSA",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada"
]
| []
| We calculate the optical sum associated with the in-plane conductivity of a graphene bilayer. A bilayer asymmetry gap generated in a field-effect device can split apart valence and conduction bands, which otherwise would meet at two K points in the Brillouin zone. In this way one can go from a compensated semimetal to a semiconductor with a tunable gap. However, the sum rule turns out to be 'protected' against the opening of this semiconducting gap, in contrast to the large variations observed in other systems where the gap is induced by strong correlation effects. | 10.1103/physrevb.77.125422 | [
"https://arxiv.org/pdf/0712.1885v2.pdf"
]
| 119,271,975 | 0712.1885 | cd29cad5fd754c47559179fdd3ee10a6797bffa5 |
Robustness of the optical-conductivity sum rule in Bilayer Graphene
8 Apr 2008 (Dated: April 8, 2008)
L Benfatto
Centro Studi e Ricerche "Enrico Fermi"
via Panisperna 89/AI-00184RomeItaly
Department of Physics
CNR-SMC-INFM
University of Rome "La Sapienza"
P.le Aldo Moro 5I-00185RomeItaly
S G Sharapov
Department of Physics
Western Illinois University
61455MacombILUSA
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
J P Carbotte
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
Robustness of the optical-conductivity sum rule in Bilayer Graphene
8 Apr 2008 (Dated: April 8, 2008)
We calculate the optical sum associated with the in-plane conductivity of a graphene bilayer. A bilayer asymmetry gap generated in a field-effect device can split apart valence and conduction bands, which otherwise would meet at two K points in the Brillouin zone. In this way one can go from a compensated semimetal to a semiconductor with a tunable gap. However, the sum rule turns out to be 'protected' against the opening of this semiconducting gap, in contrast to the large variations observed in other systems where the gap is induced by strong correlation effects.
I. INTRODUCTION
Sum rules on the conductivity have played an important role in the analysis of optical conductivity data, which give information on electron dynamics. In its simplest form, for an infinite free electron band, the sum rule gives the plasma frequency which is independent of temperature, interactions and impurity scattering. 1,2 In tight-binding models it is related to the second derivative with respect to the momentum k x,y of the band dispersion ε k , times the probability of occupation of the state |k . 1,2 In the special case where only nearest-neighbor hopping is present on a square lattice, the sum-rule integral reduces to minus one half the kinetic energy in appropriate units. As a consequence, in general it is affected by the interactions present in the system, and these determine both its absolute value and its variations with temperature. This fact has been investigated recently both experimentally and theoretically in the context of high-T c cuprate superconductors (see Ref. [1,2] for a review). It follows from the Sommerfeld expansion that the relative temperature variation of the sum-rule integral in non-interacting tight-binding models is proportional to (T /t) 2 , where t is the nearest-neighbor hopping. In the experiments done in the normal state of cuprates, the temperature variations of the the sum rule are much larger than this estimate, and in some cases deviate from a simple T 2 law. The discrepancy can be attributed to correlation effects. For example, in the studies of the Hubbard or t − J model 3,4 , a new energy scale emerges associated with the reduced width of the renormalized band. This parameter replaces the hopping parameter t in the (T /t) 2 dependence leading to its enhancement. It is shown in other theoretical approaches that the sum rule integral measures instead a specific average of the quasiparticle lifetime [5][6][7][8] normalized to the Fermi energy ε F , and hence such optical experiments ultimately probe correlation effects through lifetime broadening. These examples illustrate that optical data can provide important insight into correlation effects.
Graphene, which is a single layer of graphite, has recently been isolated and its properties investigated (see Ref. 9 for a review). Bilayer graphene 10 and thicker graphite films are also now widely produced. Since graphene possesses truly remarkable properties both from the technological and theoretical points of view, there has already been considerable work done on monolayer and bilayer graphene and also on related materials. In particular, recent data are available on the AC conductivity in the infrared region of graphite 11,12 , several layer epitaxial graphite 13 , and on monolayer 14 and bilayer 15 graphene. There has also been much theoretical work on the microwave and infrared conductivity of graphene [16][17][18] (see Ref. 19 for a review) and on a bilayer 20-22 and multilayer. 18,23 Thus, it is of interest to anticipate some general results on the optical-conductivity sum rule behavior of these systems, which are likely to be tested experimentally in the near future. The optical sum rules for the (in-plane) longitudinal and Hall conductivities were studied for monolayer graphene in Ref. [24], where the unusual dependences of the sum rules on temperature and chemical potential were revealed. For example, at the neutral Dirac point the linear dispersion law of quasiparticles leads to a T 3 dependence of the sum-rule, instead of the T 2 law found in tight-binding bands 1,2 .
A specific and very useful feature of graphene, not available in ordinary metals, is that it is possible to change the chemical potential µ and thus the number of carriers by tuning the gate voltage V g (V g ∝ µ 2 for monolayer graphene) in a field-effect device. It was shown 24 that in the limit T = 0 the concentration-dependent part of the sum rule goes like (|µ|/t) 3 . For a finite temperature T ≪ |µ|, there is a temperature dependent correction to the sum of order (|µ|/t)(T /t) 2 which is very much as in the more ordinary T 2 -law case discussed above.
In this paper we generalize the previous work to the case of bilayer graphene. In the case considered most often, a bilayer graphene consists of two coupled hexagonal lattices. The inequivalent sites A1, B1 and A2, B2 on the bottom and top graphene sheets are arranged according to Bernal (A2−B1) stacking: every B1 site in the bottom layer lies directly below an A2 site in the upper layer. The strongest inter-layer coupling between pairs of A2 − B1 orbitals, γ 1 , changes the electron dispersion from a linear to a quadratic form 21,25 . In addition to this effect, bilayer graphene is the only known material in which the elec-tronic band structure can be changed significantly simply by applying an electric field perpendicular to the layers. 9 Indeed, while the unbiased bilayer system is a semi-metal, with parabolic valence and conduction band touching each other at the neutrality point, the system becomes a semiconductor when the two layers are electrostatically inequivalent. The corresponding semiconducting gap ∆ between valence and conduction bands is tunable continuously from zero to ≈ 0.2 eV. 9,26,27 As shown experimentally, the asymmetry gap ∆ and the carrier concentration can be controlled independently through an applied gate voltage and chemical deposition of potassium or other atoms on the vacuum side of the structure. 26,27 Recently, similar results have also been obtained in a double-gated structure made of bilayer graphene sandwiched in between two gate electrodes 28 . This effect has been understood by means of both tight-binding calculations 26,29,30 and ab initio calculations 31 .
Beyond the clear potential impact that this effect has on device applications, it is extremely interesting from the theoretical point of view to understand how this field-induced band transition will affect the conduction properties of the system. In light of the previous discussion concerning the physics of cuprates, one could ask if the opening of the gap due to the metal-insulator transition in Hubbard-like models, or across the metalsuperconductor transition in superconducting models, share commonality with the physics of bilayer graphene. As we shall demonstrate below, these phenomena are drastically different. Indeed, we show that the sum rule is essentially protected against the opening of the semiconductor gap in this bilayer system. Despite the fact that ∆ can be as big as a significant fraction of the Fermi energy, the relative sum-rule changes from the unbiased to the biased system are of order of 10 −3 . Thus, even though a general redistribution of spectral weight is expected due to the opening of the gap, the total sum rule is practically constant. One would have expected the large band structure changes involved to lead to corresponding significant changes in the optical sum. Here, however, we show that this is not the case. The relative changes in optical spectral weight induced by the opening of the gap must largely compensate each other, leaving the total weight almost unchanged.
II. DERIVATION OF THE SUM RULE
For a generic electronic model described by the Hamiltonian H the optical-conductivity sum rule is given by
: 1,2 W (T ) = ∞ −∞ Reσ xx (ω)dω = πe 2 V τ xx ,(1)
where V is the unit-cell volume and τ xx is the so-called mass tensor, which appears in the second-order expansion of H evaluated at finite vector potential A:
H(A) ≈ H(0)− n,α=x,y e c A α (n)j α (n) − e 2 2c 2 A 2 α (n)τ αα (n) ,(2)
and j α is the current density in the α direction. The vector potential A is introduced in the tight-binding Hamiltonian (3) written below by means of the Peierls sub-
stitution, a † 2n b 2n+δ δ δ ′ → a † 2n exp − ie c n n+δ δ δ ′ Adr b 2n+δ δ δ ′ .
Here a 2n and b 2n+δ δ δ ′ are the Fermi operators of electrons on A2, B2 sublattices of the second layer. (Note that this Peierls substitution corresponds to the 2nd layer, for the 1st layer it is similar. The spin label is omitted.) The positions of A2 and B1 atoms are denoted as n and they are connected to their nearest neighbors on B2 (A1) sites by the three vectors δ δ δ ′ (δ δ δ). Accordingly, the general tight-binding Hamiltonian for a bilayer reads
H = −t n,δ δ δ (a + 1n+δ δ δ b 1n + h.c.) − t n,δ δ δ ′ (a + 2n b 2n+δ δ δ ′ + h.c.) + γ 1 n (a + 2n b 1n + h.c.) + γ 3 n,δ δ δ ′ (a + 1n b 2n+δ δ δ ′ + h.c.), − 1 2 ∆ n (a + 1n+δ δ δ a 1n+δ δ δ + b + 1n b 1n − (1 → 2)),(3)
where t is the in-plane hopping parameter, and γ 1 the strongest inter-layer coupling. When ∆ = 0 this Hamiltonian is equivalent to the Slonzewski-Weiss-McClure (SWM) model 32,33 for the bulk graphite, provided that one retains only in-plane and γ 1 out-of-plane hopping, and assumes no dispersion along the c axis (perpendicular to the planes) 34 . As mentioned above, the connected sites A2 and B1 lie directly below and above each other, so that the magnetic field perpendicular to the bilayer does not affect this coupling. This is not the case for the weaker A1 − B2 coupling γ 3 included in the Hamiltonian (3) for completeness. In what follows, however, we will neglect this term, whose role is negligible as far as the longitudinal optical sum rule is concerned. Finally, the parameter ∆ represents the asymmetry of the on-site energies on the two layers, and it is induced by the electric field applied perpendicular to the graphene bilayer. As we shall see below, it is responsible of the splitting of the valence and conduction bands. The Hamiltonian (3)
± (k), α = 1, 2 with ε (α) ± (k) = ± γ 2 1 2 + ∆ 2 4 + |φ(k)| 2 + (−1) α Γ, Γ = γ 4 1 4 + |φ(k)| 2 (γ 2 1 + ∆ 2 ),(4)
where the function φ(k) = −t δi e ikδi = −t δ ′ i e −ikδ ′ i . By expanding φ(k) around the two inequivalent K, K ′ points in the Brillouin zone one gets the usual linear dependence, |φ(k)| ≈ v F |k|, where the wave-vector k is measured from K points and v F = √ 3ta/(2 ) is the Fermi velocity, with a the lattice constant (V = √ 3a 2 /2). Using this form, one can easily see from Eq. (4) that at large momenta the four bands reproduce the two Dirac cones |ε (α) ± | = v F k ≡ |p| of each uncoupled layer. The low-energy band dispersion is, however, drastically affected by the interplane hopping γ 1 and the asymmetry parameter ∆. Indeed, at ∆ = 0 for |ε (1) ± | < γ 1 /4 the two low-energy bands are parabolic with the dispersion ε (1) ± ≈ ± 2 k 2 /2m, where the effective mass m = γ 1 /(2v 2 F ), and touch each other as in a semimetal (see Fig. 1a, where we used t = 3.1 eV, γ 1 = 0.4 eV 26,30 ). However, when ∆ = 0 the system becomes a semiconductor with a gap∆ = ∆γ 1 / γ 2 1 + ∆ 2 at the momentum p m / v F , that corresponds to the energy p 2 m = (∆ 2 /2)(γ 2 1 + ∆ 2 /2)/(γ 2 1 + ∆ 2 ), see Fig. 1b. Although the linear approximation for φ(k) is sufficient for the analysis of the low-energy band structure and for the numerical study of the mass tensor (9) below, the derivation of this tensor must be done using an unexpanded expression for φ(k). 24 It follows from the definition (2) that (the factor 2 accounts for the spin)
τ xx = 2 2 N k (a + 1k b 1k + a + 2k b 2k ) ∂ 2 φ * (k) ∂k 2 x + h.c. ,(5)
where we kept the full momentum dependence of φ(k) (N is the number of unit cells). The thermal average τ xx is calculated from the the imaginary time Green's function (GF) G = − T τ ΨΨ † , where we introduced Ψ + = (a + 1k , b + 2k , a + 2k , b + 1k ). Then the averages a + 1k b 1k and a + 2k b 2k are
a + 1k b 1k = T iωn e −iωn0 +Ĝ 41 (iω n , k), a + 2k b 2k = T iωn e −iωn0 +Ĝ 23 (iω n , k),(6)
where iω n is the fermionic Matsubara frequency. The corresponding elementsĜ 41 andĜ 23 of the GFĜ can be found from the inverse GF 21
G −1 (iω n , k) = z + 1 2 ∆ 0 0 −φ * (k) 0 z − 1 2 ∆ −φ(k) 0 0 −φ * (k) z − 1 2 ∆ −γ 1 −φ(k) 0 −γ 1 z + 1 2 ∆ (7) with z = iω n + µ. Then G 41 (iω n , k) = φ(k)[−|φ(k)| 2 + (∆/2 − z) 2 ] [z 2 − (ε (1) ) 2 ][z 2 − (ε (2) ) 2 ] ,(8)
andĜ 23 =Ĝ 41 (−∆). Calculating the Matsubara sum in Eq. (6) and using the identity φ(k)(∂ 2 φ * (k)/∂k 2 α )+ c.c = −(a 2 /3)|φ(k)| 2 , we finally obtain
τ xx V = − 2a 2 3 2 VN k,λ=±,α |φ(k)| 2 f (ε (α) λ (k)−µ)M (ε (α) λ (k)), (9) where M (ε (α) λ (k)) = 1 2 + (−1) α γ 2 1 + ∆ 2 4Γ 1 ε (α) λ (k)(10)
and f (ε) = 1/[exp(ε/T ) + 1] is the Fermi distribution. One can verify that for γ 1 = ∆ = 0 Eq. (9) reduces to the doubled one-layer sum of Ref. [24]. The expression (9) does not show any clear formal resemblance to the kinetic-energy density of the bilayer system, that one would define using the bands (4) as:
K = 2 VN k,λ=±,α ε (α) λ (k)f (ε (α) λ (k) − µ).(11)
Nonetheless, as we shall see these two quantities show approximately the same doping dependence, satisfying again the general relation between sum rule and kinetic energy, despite the complicated band evolution with doping of the bilayer system.
III. SUM RULE AS A FUNCTION OF DOPING
As we already mentioned, it was shown experimentally 26-28 that one can tune independently the asymmetry gap ∆ between the two layers and the total carrier density (the value of µ). This corresponds to controlling the excess carrier density n 1,2 (difference between the densities of electrons and holes) in each layer as schematically shown in Fig. 1c. The charge on the surfaces below the bottom layer and above the top layer is n g and n 0 , respectively. In the experimental configuration of Ref. 26 n g is varied by changing the gate voltage and n 0 by chemical deposition of dopants on top of the upper layer (see also Ref. 27), while in the device of Ref. 28 both n 0 and n g are varied, by using two independent gates. The resulting asymmetry ∆ between on-site energies in the two layers can be determined by equating the voltage difference ∆/e between the plates of the capacitor to its value evaluated from the charge density e(n 0 + n 2 − n g − n 1 ): 21,26,30
∆ = e 2 d 2ε 0 [n 0 + n 2 (∆) − n g − n 1 (∆)] = n 0 ed ε 0 + n 2 (∆)ed ε 0 .
(12) Here d is the bilayer thickness, ε 0 is the permittivity of free space and in the last equality we used that the total carrier excess n = n 1 + n 2 = −(n 0 + n g ). Since n 1 (∆ = 0) − n 2 (∆ = 0) = 0, we obtain that ∆ = 0 at n = −2n 0 . The gap closes when the system is doped away from half-filling, as indicated by the ARPES measurements of Ref. 27. However, if the value of n 0 is tuned to be exactly at zero 28 , the gap closes exactly at the neutrality point, where the semimetal band structure is reproduced. For finite ∆ the excess carrier densities n 1,2 are given by
n 1 (∆, µ) = 1 N V k,α [f (ε (α) + (k) − µ) − f (ε (α) + (k) + µ) +g (α) (k) [f (ε (α) − (k) − µ) − f (ε (α) + (k) − µ)] ,(13)
where g (α) (k) = (∆/ε (α) + (k))(1/2 + (−1) α |φ(k)| 2 /Γ), and n 2 (∆, µ) = n 1 (−∆, µ). To determine selfconsistently the dependence ∆ on n, one has to solve simultaneously Eq. (12) and the equation n 1 (∆, µ) + n 2 (∆, µ) = n for the chemical potential µ. Once ∆(n) and µ(n) are determined, one can compute τ xx and K as a function of n.
Following Ref. 27 we use d = 3.4Å, and show results for n 0 = 0, n 0 = ±12 × 10 12 cm −2 . This choice is suggested by the fact that in the measurements of Ref. 27 the gap closes at n = −2n 0 given approximately by 24 × 10 12 cm −2 (notice that n expressed in units of 10 12 cm −2 corresponds to 1.9 × 10 3 n uc , where n uc is the number of electrons per unit cell n uc ). The doping dependence of the τ xx at T = 0 is shown in Fig. 2. As one can see, even though the charging gap ∆ changes consistently with doping (see inset of Fig. 2), due to the screening effects of the bilayer device, nonetheless the sum rule attains overall variations of ∼ 0.1%, which are undetectable from an experimental point of view. In other words, despite the large optical-conductivity spectralweight redistribution associated with the opening of the gap through the semimetal-semiconductor transition induced by doping, the sum rule is not affected. It follows the same behavior that one would obtain in (electrostatically) uncoupled layers where ∆ = 0 at all doping (see the dashed-dotted line in Fig. 2). In the regime where the gap is relatively small (for example at negative doping for the n 0 = 12 × 10 12 cm −2 curve in Fig. 2) the sum rule follows approximately the same doping dependence found in the single-layer case, i.e. [τ xx (n) − τ xx (0)]/V ≃ −2a 2 |µ| 3 /(9π 2 v 2 F ). Since 2a 2 /(9π 2 v 2 F ) ≈ 10 −2 eV −2 , and |µ| 3 has an overall variation of at most 0.1(eV) 3 in the considered doping range, one can easily get the small sum-rule variation with doping. At larger gap values, τ xx (n) − τ xx (0) is slightly larger than expected in the single-layer case, but is still too small to be detected experimentally. As far as the temperature dependence is concerned we always found a (T /t) 2 variation, as expected due to the parabolic structure of the low-energy bands. Including other hoppings such as γ 3 in the Hamiltonian (3) can change the dispersion curve as discussed in 29 (see also Ref. 34) for energies less than 2 meV. While this can slightly affect the overall value of the optical integral and of the self-consistent ∆, it does not change the main conclusion of our calculation, which focus on the relative doping changes of the sum rule when a large semiconducting gap ∆ of order of a few hundred meV is opened. Having established that the relatively large band-structure changes caused by the opening of the gap ∆ lead to a negligible changes in the optical sum, we expect that the above mentioned hopping terms introduce negligible corrections as well. Analogously, the kinetic energy, shown in Fig. 3, is not much affected by the opening of the gap, and it attains a value which is approximately 6 times the sum rule, τ /V ∼ (1/6)|K|. There are however two differences in the doping dependence of the sum rule with respect to the kinetic energy. First, we notice that the very small changes in the sum rule and kinetic energy have the opposite trend: indeed, they both decrease when the gap opens, in contrast to what is found for example at the transition between a normal metal and a superconductor, where a kinetic-energy increase corresponds to a sum-rule decrease and vice-versa. This effect disappears when we set γ 1 = ∆ = 0 in our numerical work: indeed, in this case τ xx reproduces exactly twice the single-layer value computed in Ref. [24], and the sum rule and the kinetic energy track each other exactly as the doping is varied. Second, it is clear from Fig. 3 that the chemical-potential jump at the neutrality point in the biased case (see inset (b) of Fig. 3) is much more effective on the kineticenergy doping dependence. The two curves for |K| at n 0 = ±12 × 10 12 cm −2 in Fig. 3 show a kink at n = 0 which is clearly due to the µ(n) discontinuity at the neutrality point. Indeed, in the case where no gap is present in the system µ(n) goes smoothly across n = 0, as does the kinetic energy. Instead, the sum-rule curves in Fig. 2 are not much affected by the µ discontinuity at n = 0.
IV. CONCLUSIONS
In summary we investigated the behavior of the optical sum in bilayer graphene as a function of charge imbalance carrier density n and temperature. While a small temperature variation could be expected because of the relatively small effect of temperature on the elec-tronic excitations, the negligible dependence of the sum rule on the carrier density was not a priori expected. Indeed, despite the profound band-structure changes induced by the change of doping, no significant signature appears on the overall spectral weight. This implies that the spectral weight lost in the gap must appear above it, and the two must largely compensate for each other. Observe that in order to be able to test experimentally such a prediction one would need to define an "experimental" cut-off. Indeed, even though the relation (1) is theoretically established by integrating the optical conductivity to all frequencies, in practice an intrinsic cut-off is provided by the frequency ω c above which transitions to other electronic bands not considered in the present study would develop. 2 According to optical studies on graphite 35 , the optical sum rule saturates to the value corresponding to one electron per atom (as due to the π band considered here) around 8 eV, while optical transitions coming from the σ bands appear around 15 eV. However, in the present case a much smaller cut-off could be used, if one wants just to compare the spectral-weight variations induced by the opening of the gap. Indeed, the spectral-weight redistribution is expected to fall in a range of frequencies of order 2∆, as confirmed recently in Ref. 22 . Thus, an upper cut-off of order of ω c ≃ 1 − 2 eV should be enough to account for the gap-opening effects, and to test experimentally the predicted robustness of the optical sum rule.
It is worth noting that in our calculation only the electrostatic interaction between layers was included, since this is believed to be the most important effect in the system. However, as in the case of cuprates superconductors other mechanisms (electron repulsion or electronphonon interactions) could be at play and modify the sum rule. Thus, the experimental verification of our prediction would help understanding if other interactions need or need not be taken into account in dealing with these systems.
FIG. 1 :
1(Color online) Top: bilayer band dispersion for ∆ = 0 (a) and ∆ = 0.4 eV (b) as a function of p ≡ vF k. Bottom: general scheme of the experimental setup for a tunable-gap bilayer device (notation defined in the text).
FIG. 2 :
2(Color online) Doping dependence of the T = 0 sum rule for a bilayer system with a doping-dependent gap ∆(n), shown in the inset. We show results for different values of n0, given in units of 10 12 cm −2 . For comparison, we show the equivalent doping dependence of the sum rule when no gap is present between the two layers (dashed-dotted line).
FIG. 3 :
3(Color online) (a) Doping dependence of the T = 0 kinetic energy for a bilayer system with a doping-dependent gap ∆(n) and without it. Observe the small variations of K in the considered doping range. Insets: doping dependence of the chemical potential when a gap is present (b) or not (c), for parameters as in the main frame.
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| []
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[
"A New Proposed Dynamic Quantum with Re-Adjusted Round Robin Scheduling Algorithm and Its Performance Analysis Debashree Nayak",
"A New Proposed Dynamic Quantum with Re-Adjusted Round Robin Scheduling Algorithm and Its Performance Analysis Debashree Nayak"
]
| [
"ProfessorH S Behera \nVeer Surendra Sai University of Technology\nBurla SambalpurIndia\n",
"ProfessorR Mohanty \nVeer Surendra Sai University of Technology\nBurla SambalpurIndia\n",
"Research Associate \nVeer Surendra Sai University of Technology\nBurla SambalpurIndia\n"
]
| [
"Veer Surendra Sai University of Technology\nBurla SambalpurIndia",
"Veer Surendra Sai University of Technology\nBurla SambalpurIndia",
"Veer Surendra Sai University of Technology\nBurla SambalpurIndia"
]
| [
"International Journal of Computer Applications"
]
| Scheduling is the central concept used frequently in Operating System. It helps in choosing the processes for execution. Round Robin (RR) is one of the most widely used CPU scheduling algorithm. But, its performance degrades with respect to context switching, which is an overhead and it occurs during each scheduling. Overall performance of the system depends on choice of an optimal time quantum, so that context switching can be reduced. In this paper, we have proposed a new variant of RR scheduling algorithm, known as Dynamic Quantum with Readjusted Round Robin (DQRRR) algorithm.We have experimentally shown that performance of DQRRR is better than RR by reducing number of context switching, average waiting time and average turnaround time.General TermsScheduling, Round Robin Scheduling. | 10.5120/913-1291 | [
"https://arxiv.org/pdf/1103.3831v1.pdf"
]
| 5,146,391 | 1103.3831 | 7837916a4e8976bbd9b8959276f4cbfaf627b020 |
A New Proposed Dynamic Quantum with Re-Adjusted Round Robin Scheduling Algorithm and Its Performance Analysis Debashree Nayak
August 2010
ProfessorH S Behera
Veer Surendra Sai University of Technology
Burla SambalpurIndia
ProfessorR Mohanty
Veer Surendra Sai University of Technology
Burla SambalpurIndia
Research Associate
Veer Surendra Sai University of Technology
Burla SambalpurIndia
A New Proposed Dynamic Quantum with Re-Adjusted Round Robin Scheduling Algorithm and Its Performance Analysis Debashree Nayak
International Journal of Computer Applications
55August 201010Round Robin SchedulingContext SwitchingWaiting TimeTurnaround Time
Scheduling is the central concept used frequently in Operating System. It helps in choosing the processes for execution. Round Robin (RR) is one of the most widely used CPU scheduling algorithm. But, its performance degrades with respect to context switching, which is an overhead and it occurs during each scheduling. Overall performance of the system depends on choice of an optimal time quantum, so that context switching can be reduced. In this paper, we have proposed a new variant of RR scheduling algorithm, known as Dynamic Quantum with Readjusted Round Robin (DQRRR) algorithm.We have experimentally shown that performance of DQRRR is better than RR by reducing number of context switching, average waiting time and average turnaround time.General TermsScheduling, Round Robin Scheduling.
INTRODUCTION
Operating system is an interface between end user and system hardware, so that the user can handle the system in a convenient manner. In a single user environment, there was no need to choose any task because task execution continues one after another, but in multitasking environment, it becomes necessary for the processor to choose a task from the ready queue. Operating system follows a predefined procedure for selecting process among number of processes from the ready queue, known as Scheduling. Scheduler selects the ready processes from memory and allocates resource/CPU as per their requirement. Whenever one process waits for some other resource, scheduler selects next process and allocates CPU to it. This process continues till the system request for termination of execution and then the last CPU burst ends up with it.
Scheduling Algorithms
In the First-Come-First-Serve (FCFS) algorithm, process that arrives first is immediately allocated to the CPU based on FIFO policy. In Shortest Job First (SJF) algorithm, process having shortest CPU burst time will execute first. If two processes having same burst time and arrive simultaneously, then FCFS procedure is applied. Priority scheduling algorithm, provides priority (internally or externally) to each process and selects the highest priority process from the ready queue. In case of Round Robin (RR) algorithm, time interval of one time quantum is given to each process present in the circular queue emphasizing on the fairness factor.
Motivation
In RR scheduling fairness is given to each process, i.e. processes get fair share of CPU because of given time slice. So, it is better than other scheduling algorithms. Number of context switching incase of RR Scheduling is n in one round only, i.e. high in comparison to other scheduling algorithms. It gives low turnaround time and average waiting time. RR scheduling uses static time quantum that gives large waiting time and turnaround time in case of variable burst time which degrades the overall performance. This factor motivates us to design an improved algorithm which can overcome the above limitation.
Related Work
SARR algorithm [1] is based on a new approach called dynamictime-quantum, in which time quantum is repeatedly adjusted according to the burst time of the running processes. Mixed Scheduling (A New Scheduling Policy) [2], uses the job mix order for non preemptive scheduling FCFS and SJF. According to job mix order, from a list of N processes, the process which needs minimum CPU time is executed first and then the highest from the list and so on till the nth process. In Burst Round Robin (BRR) [3], a new weighting technique is introduced for CPU Schedulers. Here shorter jobs are given more time, so that processes having shorter jobs are cleared from the ready queue in a short time span.
Our Contribution
In this paper, the principal objective is to reduce context switching occur in RR scheduling. For that purpose, we have developed a method that drastically reduces context switching.
Organization of the Paper
This paper presents the method for reducing context switch, average waiting time and average turnaround time using random sorting and dynamic time quantum. Section 2 discusses background preliminaries. Section 3 presents the proposed approach. Section 4 shows experimental analysis. In Section 5 conclusion and future work towards our method is given.
BACKGROUND WORK 2.1 Terminologies
Burst time (b t ) is the time needed by the process to hold the control of CPU. Time Quantum (q t ) is a particular slice of time given to each process to have CPU for that time period only. Average Waiting Time (a wt ) is the time gap between arrival of one process and its response by the CPU. To achieve good result, a wt should be less. Average Turnaround Time (a tat ) is the time gap between the instant of process arrival and the instant of its completion. For getting good result, it should be less. Context Switch (CS) is the number of time CPU switches from one process to another. For better performance of the algorithm, it should be less.
RR Scheduling Algorithm
RR Scheduling Algorithm is the simplest and widely used algorithm as it gives fairness to each process. Newly arrived processes are kept in the rear part of the queue. Scheduler chooses each process from front of the queue and allocates the CPU for one time quantum. The performance of RR algorithm depends heavily on the size of the time quantum [1]. For smaller time quantum, the context switching is more and for larger time quantum, response time is more. Overall performance of RR may decrease for weak time quantum selection. Therefore, choice of an optimal time quantum is necessary.
PROPOSED APPROACH
In proposed approach, we have to arrange the processes in ascending order according to their burst time present in the ready queue. Then time quantum is calculated. For finding an optimal time quantum, median method is followed. The median can be found out using the following formulae [1].
Median x =
Where, x = median y = number located in the middle of a group of numbers arranged in ascending order n = number of processes Here, the time quantum is assigned to the processes. This time quantum is recalculated taking the remaining burst time in account after each cycle. In the next step we have to rearrange the sorted processes, i.e. among n processes, the process which needs minimum CPU burst time will be replaced as the first process and then the process with highest CPU burst time from the queue, will be replaced as the second process and so on.
Proposed Algorithm
Illustration
To demonstrate the above algorithm we have considered the following example. Arrival time is considered to be zero for the given processes P1, P2, P3, P4 and corresponding burst times are 21, 105, 12, 55 respectively. In first step the processes in the ready queue are sorted in ascending order. Then the time quantum is calculated in the second step. Here q t = 38. In third step sorted processes are rearranged as described in the 3 rd section, i.e. P3 with b t =12, P2 having b t = 105, P1 with b t =21 and P4 with b t = 55. After assigning q t to each process the remaining burst time of all process are P3=0, P2=67, P1=0 and P4=17. When a process completes its execution, it is deleted from ready queue automatically. Further the next time quantum is calculated from remaining burst times as per the 3 rd step in the algorithm. Here q t =42. Then the remaining burst times are P2=25 and P4 =0. According to the algorithm the next q t will be 25 and in the last step the process P2 will complete its execution and will be deleted from the ready queue.
EXPERIMENTAL ALANALYSIS 4.1 Assumptions
Our experiments are performed in a uni-processor environment and the processes taken are CPU bound processes only. Here we have taken n processes, i.e. P1, P2… Pn and all these processes are independent from each other. For all the processes, corresponding burst time (b t ) and arrival time (a t ) are known before submitting the processes to the processor.
Experimental Frame Work
The input parameters taken are as follows. P n is the number of processes. a t , b t , q t are the arrival time, burst time and quantum time respectively. The output parameters are context switch(CS), average waiting time(a wt ) and average turnaround time(a tat ). We have taken two cases, i.e. case 1 is for processes with zero arrival time (here each process arrive at same time) and case 2 is for processes without zero arrival time (here processes are arrived at different time). Under these two cases we have performed three different experiments taking three different types of data sets (data sets in increasing order, decreasing order and random order).
Results Obtained
This algorithm can work effectively with large number of data. In each case we have compared our proposed algorithm's results with Round Robin scheduling algorithm's result. For RR Scheduling Algorithm we have taken 25 as the fixed time quantum.
Case 1: With Zero Arrival Time
Increasing Order
We consider five processes P1, P2, P3, P4 and P5 arriving at time 0 with burst time 30, 42, 50, 85, 97 respectively shown in
Decreasing Order
We consider five processes P1, P2, P3, P4 and P5 arriving at time 0 with burst time 105, 90, 60, 45, 35 respectively shown in Table 4.3. Table 4.4 shows the comparing result of RR algorithm and our proposed algorithm (DQRRR).
Random Order
We consider five processes P1, P2, P3, P4 and P5 arriving at time 0 with burst time 92, 70,35,40,80 respectively shown in Table 4.5. Table 4.6 shows the comparing result of RR algorithm and our proposed algorithm (DQRRR).
CONCLUSION AND FUTURE WORK
The proposed variant of RR algorithm drastically decreases context switching. The proposed algorithm performs better than RR scheduling algorithm with respect to average waiting time, turnaround time and context switching. Our proposed algorithm can be further investigated to be useful in providing more and more task-oriented results in future along with developing adaptive algorithms to fit the varying situations in today's multifaceted complex working of operating system.
Fig.4. 13 :
13Context Switching(DQRRR vs. RR) Fig.4.14: Average Waiting Time(DQRRR vs. RR) Fig.4.15: Average Turnaround Time(DQRRRvs.RR) Fig.4.16: Context Switching(DQRRR vs. RR) Fig.17: Average Waiting Time(DQRRR vs. RR)
Table 4 .
41. Table 4.2 shows the comparing result of RR algorithm and our proposed algorithm (DQRRR).
Table 4 .
41.Data in Increasing Order
Table 4.2.Comparison between RR and DQRRR
Fig.4.2: Gantt chart for RR in Table 4.2
Fig.4.2: Gantt chart for DQRRR in Table 4.2
Table 4 .
4Fig.4.4: Gantt chart for DQRRR inTable 4.43.Data in Decreasing Order
Table 4.4.Comparison between RR and DQRRR
Fig.4.3: Gantt chart for RR in Table 4.4
Table 4 .
45. Data in Random Order
Table 4 .
46.Comparison between RR and DQRRR
Fig.4.5: Gantt chart for RR in Table 4.6
Fig.4.6: Gantt chart for DQRRR in Table 4.6
Case 2: Without Zero Arrival Time
Increasing Order
We consider five processes P1, P2, P3, P4 and P5 arriving at time
0,2,6,6,8 and burst time28,35,50,82,110 respectively shown in
Table 4.7. Table 4.8 shows the comparing result of RR algorithm
and our proposed algorithm (DQRRR).
Table 4 .
47.Data in Increasing Order
Table 4.8.Comparison between RR and DQRRR
Fig.4.7: Gantt chart for RR in Table 4.8
Table 4 .
49.Data in Decreasing OrderTable 4.10.Comparison between RR and DQRRR Fig.4.9: Gantt chart for RR in Table 4.10 Fig.4.10: Gantt chart for DQRRR in Table 4.10 .11. Table 4.12 shows the comparing result of RR algorithm and our proposed algorithm (DQRRR).Table 4.11.Data in Random Order Table 4.12.Comparison between RR and DQRRR Fig.4.11: Gantt chart for RR in Table 4.12 Fig.4.12: Gantt chart for DQRRR in Table 4.12Random Order
We consider five processes P1, P2, P3, P4 & P5 arriving at time
0, 1,2,5,7 and burst time 26,82,70,31,40 respectively shown in
Table 4
Fig.18: Average Turnaround Time (DQRRR vs. RR)0
26
P2
1
82
P3
2
70
P4
5
31
P5
7
40
algorithms
RR
DQRRR
q t
25
26,55,21,6
CS
12
7
a wt
149.4
95.6
a tat
199.2
145.4
Self-Adjustment Time Quantum in Round Robin Algorithm Depending on Burst Time of Now Running Processes. J Rami, Matarneh, American J. of Applied Sciences. 610Rami J. Matarneh."Self-Adjustment Time Quantum in Round Robin Algorithm Depending on Burst Time of Now Running Processes", American J. of Applied Sciences 6(10): 1831-1837,2009.
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| []
|
[
"Half-Metallic Ferromagnetism in the Heusler Compound Co 2 FeSi revealed by Resistivity, Magnetoresistance, and Anomalous Hall Effect measurements",
"Half-Metallic Ferromagnetism in the Heusler Compound Co 2 FeSi revealed by Resistivity, Magnetoresistance, and Anomalous Hall Effect measurements"
]
| [
"Dirk Bombor \nIFW Dresden\n01171DresdenGermany\n",
"Christian G F Blum \nIFW Dresden\n01171DresdenGermany\n",
"Oleg Volkonskiy \nIFW Dresden\n01171DresdenGermany\n",
"Steven Rodan \nIFW Dresden\n01171DresdenGermany\n",
"Sabine Wurmehl \nIFW Dresden\n01171DresdenGermany\n\nInstitut für Festkörperphysik\nTechnische Universität Dresden\nD-01062DresdenGermany\n",
"Christian Hess \nIFW Dresden\n01171DresdenGermany\n",
"Bernd Büchner \nIFW Dresden\n01171DresdenGermany\n\nInstitut für Festkörperphysik\nTechnische Universität Dresden\nD-01062DresdenGermany\n"
]
| [
"IFW Dresden\n01171DresdenGermany",
"IFW Dresden\n01171DresdenGermany",
"IFW Dresden\n01171DresdenGermany",
"IFW Dresden\n01171DresdenGermany",
"IFW Dresden\n01171DresdenGermany",
"Institut für Festkörperphysik\nTechnische Universität Dresden\nD-01062DresdenGermany",
"IFW Dresden\n01171DresdenGermany",
"IFW Dresden\n01171DresdenGermany",
"Institut für Festkörperphysik\nTechnische Universität Dresden\nD-01062DresdenGermany"
]
| []
| We present electrical transport data for single-crystalline Co2FeSi which provide clear-cut evidence that this Heusler compound is truly a half-metallic ferromagnet, i.e. it possesses perfect spinpolarization. More specifically, the temperature dependence of ρ is governed by electron scattering off magnons which are thermally excited over a sizeable gap ∆ ≈ 100 K (∼ 9 meV) separating the electronic majority states at the Fermi level from the unoccupied minority states. As a consequence, electron-magnon scattering is only relevant at T ∆ but freezes out at lower temperatures, i.e., the spin-polarization of the electrons at the Fermi level remains practically perfect for T ∆. The gapped magnon population has a decisive influence on the magnetoresistance and the anomalous Hall effect (AHE): i) The magnetoresistance changes its sign at T ∼ 100 K, ii) the anomalous Hall coefficient is strongly temperature dependent at T 100 K and compatible with Berry phase related and/or side-jump electronic deflection, whereas it is practically temperature-independent at lower temperatures. | 10.1103/physrevlett.110.066601 | [
"https://arxiv.org/pdf/1207.6611v1.pdf"
]
| 118,781,604 | 1207.6611 | 543fe0a249e637a6850db30771dd264a95ee955e |
Half-Metallic Ferromagnetism in the Heusler Compound Co 2 FeSi revealed by Resistivity, Magnetoresistance, and Anomalous Hall Effect measurements
(Dated: May 2, 2014)
Dirk Bombor
IFW Dresden
01171DresdenGermany
Christian G F Blum
IFW Dresden
01171DresdenGermany
Oleg Volkonskiy
IFW Dresden
01171DresdenGermany
Steven Rodan
IFW Dresden
01171DresdenGermany
Sabine Wurmehl
IFW Dresden
01171DresdenGermany
Institut für Festkörperphysik
Technische Universität Dresden
D-01062DresdenGermany
Christian Hess
IFW Dresden
01171DresdenGermany
Bernd Büchner
IFW Dresden
01171DresdenGermany
Institut für Festkörperphysik
Technische Universität Dresden
D-01062DresdenGermany
Half-Metallic Ferromagnetism in the Heusler Compound Co 2 FeSi revealed by Resistivity, Magnetoresistance, and Anomalous Hall Effect measurements
(Dated: May 2, 2014)BomborPACS numbers:
We present electrical transport data for single-crystalline Co2FeSi which provide clear-cut evidence that this Heusler compound is truly a half-metallic ferromagnet, i.e. it possesses perfect spinpolarization. More specifically, the temperature dependence of ρ is governed by electron scattering off magnons which are thermally excited over a sizeable gap ∆ ≈ 100 K (∼ 9 meV) separating the electronic majority states at the Fermi level from the unoccupied minority states. As a consequence, electron-magnon scattering is only relevant at T ∆ but freezes out at lower temperatures, i.e., the spin-polarization of the electrons at the Fermi level remains practically perfect for T ∆. The gapped magnon population has a decisive influence on the magnetoresistance and the anomalous Hall effect (AHE): i) The magnetoresistance changes its sign at T ∼ 100 K, ii) the anomalous Hall coefficient is strongly temperature dependent at T 100 K and compatible with Berry phase related and/or side-jump electronic deflection, whereas it is practically temperature-independent at lower temperatures.
We present electrical transport data for single-crystalline Co2FeSi which provide clear-cut evidence that this Heusler compound is truly a half-metallic ferromagnet, i.e. it possesses perfect spinpolarization. More specifically, the temperature dependence of ρ is governed by electron scattering off magnons which are thermally excited over a sizeable gap ∆ ≈ 100 K (∼ 9 meV) separating the electronic majority states at the Fermi level from the unoccupied minority states. As a consequence, electron-magnon scattering is only relevant at T ∆ but freezes out at lower temperatures, i.e., the spin-polarization of the electrons at the Fermi level remains practically perfect for T ∆. The gapped magnon population has a decisive influence on the magnetoresistance and the anomalous Hall effect (AHE): i) The magnetoresistance changes its sign at T ∼ 100 K, ii) the anomalous Hall coefficient is strongly temperature dependent at T 100 K and compatible with Berry phase related and/or side-jump electronic deflection, whereas it is practically temperature-independent at lower temperatures.
PACS numbers:
The class of Heusler compounds offers a tool box for studying various materials funtionalities and interesting physical ground states [1][2][3]. For example, for some Co 2based Heusler compounds full spin polarization at the Fermi level referred to as half-metallic ferromagnetism [3][4][5][6] has been predicted [see fig. 1a for an illustration of the density of states (DOS) of a half-metallic ferromagnet (HMF)], which renders these materials prime candidates for spintronics applications. Most prominent among these materials is the Heusler compound Co 2 FeSi, since this ferromagnetic metal possesses the highest Curie temperature ever found in Heusler compounds (T C = 1100 K) together with a large magnetic moment of m = 6 µ B /f.u. [7].
In this letter we present the electronic transport properties, viz., resistivity, magnetoresistance and Hall effect, of high quality Co 2 FeSi single crystals [8]. Our data provide fresh evidence of Co 2 FeSi being a true HMF: At high temperatures (T 100 K), the resistivity ρ is governed by scattering off ferromagnetic magnons. This magnon scattering process is exponentially suppressed at lower temperatures which implies a complete suppression of the minority electron density of states at the Fermi level where the excitation gap for these levels is extracted from resistivity as ∆ ∼ 100 K. This crucial finding is further corrobarated by a sign change of magnetoresistance from negative at high temperatures (T 110 K) to positive at lower temperatures. Furthermore, we observe that the activation of ferromagnetic magnons across ∆ has a profound impact on the anomalous Hall effect (AHE). While the anomalous Hall coefficient is practically temperature independent at T 100 K, consistent with dominating extrinsic skew scattering or the ordinary Lorentz-force in- duced Hall effect, it strongly increases at higher T , where it follows ∼ ρ 2 , suggestive of dominating Berry phase and/or side-jump contributions caused by activated ferromagnetic magnons.
Experimental details -Large single crystals of Co 2 FeSi, which crystallizes in the L2 1 structure with a unit cell size of a = 5.64Å [9] (fig. 1b), were prepared using the floating zone technique [8]. Magnetization measurements confirm the predicted magnetization of m = 6 µ B /f.u. and yield a saturation field of µ o H s = 0.4 T (see fig. 1c). The susceptibility below H s is χ = 5.4. Due to the high Curie temperature, the temperature dependences of the saturation field and susceptibility are negligible up to room temperature. For resistivity measurements in magnetic field, a single crystal with dimensions of a = b = c = 3 mm was used. Resistivity was measured using a standard 4-probe alternating current dctechnique. Hall measurements were done simultaneously using 2 additional Hall voltage contacts.
Resistivity - Fig. 2 shows the resistivity ρ as a function of temperature T for zero magnetic field and for µ 0 H = 15 T. In zero field the resistivity shows a clear metallic temperature dependence with a monotonic increase with increasing temperature. A residual resistivity of ρ R ≈ 4 µΩcm and a residual resistivity ratio of ρ(300K)/ρ R = 6.5 shows the extraordinary quality of our samples.
The resistivity of metals is usually dominated by electron-phonon scattering. This typically leads to a linear temperature dependence at elevated temperatures (T 50 K) and a saturation towards residual resistivity at lower temperatures. Such a linear part is completely absent in Co 2 FeSi. However, for ferromagnetic materials like the compound under scrutiny, another scattering process may be found, i.e., electron-magnon scattering, which often leads to a quadratic temperature dependence of the resistivity (see e.g. [10][11][12][13][14][15]).
However, the attempt to fit ρ(T ) of Co 2 FeSi with a quadratic and a quadratic plus linear temperature dependence (for pure magnon and magnon/phonon scattering of the electrons, respectively) fails completely (see fig. 3). Thus, the conventional electron-magnon scatter- ing of band ferromagnets cannot account for the observed temperature dependence of the resitivity. It can, however, be rationalized much better if the predicted halfmetallic character of the material is taken into account. In HMF, the minority spin electronic states are completely gapped at the Fermi level, where k B ∆ is the minimum excitation energy of majority charge carriers to occupy empty minority states involving a spin flip (see fig. 1b). As a consequence, at low temperatures (T ∆) the generation of magnons (which involves a spin flip) and in particular electron-magnon scattering must be exponentially supressed. At higher temperatures (T ∆) spin flips and magnon scattering would become possible when the Fermi distribution smears out the occupation of energy states around the Fermi level.
In order to test against this scenario we fitted the resistivity with combined scattering of a residual (ρ R ), magnonic (ρ M ) and phononic (ρ P ) contribution:
ρ(T ) = ρ R + ρ M (T ) + ρ P (T ) .(1)
The residual resistivity ρ R is temperature independent and caused by defects of the ideal crystal lattice. The usually quadratic magnonic term (ρ M ) was weighted by a Boltzmann factor:
ρ M (T ) = CT 2 · e −∆/T .(2)
The parameter C is a measure of the strength of the magnon scattering process. The phononic part of the scattering process (ρ P ) is described by the Bloch- Grüneisen formula:
ρ P (T ) = A T Θ D 5 ΘD/T 0 x 5 (e x − 1)(1 − e −x ) dx . (3)
With the Debye temperature Θ D = 332 K, determined from specific heat measurements, the resistivity fit 1 yields an energy gap of ∆ = 103 K , (k B ∆ = 8.9 meV) 2 .
It is worth to point out that the substantial size of the gap implies that the spin-polarization of the electrons at the Fermi level remains practically perfect for T ∆.
Magnetoresistance -An independent verification of the afore derived scenario of gapped-out magnon scattering can be obtained from the corresponding magnetoresistance of Co 2 FeSi, because the influence of a magnetic field on the magnons should be reflected by the resistivity. As can be seen in fig. 2, a magnetic field of 15 T yields a relatively small (∼ 2 %) but well resolvable change of the resistivity as compared to the zero-field data. The inset highlights the sign and the temperature dependence of ∆ρ = ρ(15T) − ρ(0). Apparently ∆ρ > 0 (∆ρ < 0) for T 110 K (T 110 K).
To get a more detailed view of this dependence on the magnetic field we measured the resistivity while sweeping the magnetic field and maintaining the temperatures ( fig. 4). The inset of fig. 4 shows the slope of the resistivity for fields higher than the saturation field. Up to the saturation field of µ 0 H s = 0.4 T a small decrease of the resistivity can be seen, which can naturally be explained by alignment and joining of magnetic domains which decreases the probability of domain wall scattering. For high fields H > H s , the resistivity decreases with increasing field at high temperatures; at low temperatures it increases with increasing field.
This temperature dependence of the magnetoresistance can perfectly be explained by the afore presented scenario of the resistivity being dominated by scattering of spin-polarized electrons off gapped magnons. The negative magnetoresistance at high temperatures signals a decrease of the electronic scattering with increasing field. This is consistent with ferromagnetic magnons being the main scattering centers for electrons since a magnetic field enlarges the magnon energy and thus lowers the number of magnons. Upon lowering the temperature, magnons in a HMF are expected to freeze out exponentially since only one spin state is present at the Fermi level. Hence electron-defect scattering must become important at lower temperatures and one expects a more conventional positive magnetoresistance. Indeed, such a behaviour is observed at T 110 K: ∆ρ > 0 and ∆ρ(B) approaches a positive quadratic field dependence, which is characteristic for conventional multi-band metals.
Hall effect -Having established Co 2 FeSi being a HMF and activated magnon generation for T 100 K, we now move on to the anomalous Hall effect (AHE) of Co 2 FeSi. The Hall resistivity of this compound is shown in fig. 5 along with its corresponding Hall coefficients in the inset. The Hall resistivity ρ xy (µ 0 H) is characteristic for the AHE in ferromagnetic materials: it has a linear dependence on the applied magnetic field but with a strong kink and a changing slope just at the saturation field of µ 0 H s = 0.4 T. Since the Hall effect depends on the magnetic field H as well as on its magnetization M , one considers two Hall coefficients, viz. the ordinary Hall coefficient R and the so-called anomalous Hall coefficient R A , which connects the applied magnetic field H and the magnetization M of the sample, respectively, with the Hall resistivity [15,16]:
ρ xy = µ 0 (R · H + R A · M ) .(5)
Thus, from our Hall data in high magnetic fields the ordinary Hall coefficient R can be obtained, and with the knowledge of this coefficient and χ = 5.4, the anomalous Hall coefficient R A can be extracted from the lowfield data, where M = χ · H. The results are shown in the inset of fig. 5. The ordinary Hall coefficient R is small and does not change much with temperature as expected for a metal. It slightly decreases with increasing temperature which indicates a change of the mobility of the charge carriers consistent with the multi-band nature of this material [7]. At low temperatures the anomalous Hall coefficient R A is comparable to the ordinary one, while at T 100 K it strongly increases with T . The huge increase of R A cannot be explained by Lorentz force deflection, and implies the relevance of fundamental electronic deflection mechanisms known in ferromagnetic materials. More specifically, following the classification given by Nagaosa et al. [16], one expects on the one hand R A ∼ ρ 2 for the intrinsic Berry phase-related contribution to the AHE as well for the so-called side-jump scattering contribution. On the other hand, for the extrinsic skew scattering, R A ∼ ρ is expected. In order to test the AHE in Co 2 FeSi against one of these scenarios, we plot R A as a function of the measured ρ(T ) ( fig. 6). Indeed, for ρ larger than the residual resistivity ρ R , the data very well follow R A ∝ (ρ − ρ R ) 2 + const, which provides clear evidence that the AHE is dominated by the intrinsic Berry phase and/or the side-jump scattering contributions. Interestingly, this resistivity regime corresponds to temperatures T 100 K, i.e., to high temperatures where magnons are significantly populated, which may affect both the Berry phase as well as the side-jump scattering. At lower resistivity values, which correspond to the low-temperature regime where magnons apparently freeze out, the observed ∼ ρ 2 behavior breaks down and R A becomes independent of ρ, implying that a qualitatively different deflection mechanism becomes dominant. Indeed the comparable magnitude of the anomalous (R A ) and ordiniary (R) Hall coefficients for T 100 K suggests a contribution of the Lorentz force induced Hall effect (see inset fig. 5), but also scew scattering contributions cannot be excluded. Note, that the temperature dependent change of ρ in this regime is very small, and thus R A cannot be tested against the expected linear dependence of ρ. Impurity dependent investigations of R A should shed more light on this issue, but are beyond the scope of this study. It is, however, worth to mention that Imort et al., in agreement with the conjectured skew scattering, reported R A ∼ ρ for Co 2 FeSi thin films, which naturally contain more defects than our high quality crystal [17]. Finally, it is worth to point out that the easily accessible and by temperature well distinct regimes of the AHE, suggests Co 2 FeSi as an ideal test bed for theoretical treatments of the AHE.
In summary, we have investigated the electronic transport properties of Co 2 FeSi in magnetic field. The temperature dependence of the resistivity is governed by electron-magnon scattering which exponentially freezes out over the substantial energy gap ∆ ≈ 100 K at low temperatures und thus provides clear-cut evidence of Co 2 FeSi being a HMF. This finding is corroborated by the magnetoresistance which is negative at high temperatures and positive at low temperatures. Furthermore, we observe that activation of ferromagnetic magnons at high temperatures T ∆ has a strong impact on the anomalous Hall effect. In this high-temperature regime it is strongly temperature dependent and governed by Berry phase and side-jump deflection whereas at T ∆ it becomes temperature independent and is consistent with dominating skew scattering or intrinsic Lorentz force induced Hall effect.
FIG. 1
1: a) L21 crystal structure of Co2FeSi. b) Sketch of the density of states of a HMF in dependence of energy for minority and majority electrons with energy gap at Fermi level EF for minority electrons. c) Magnetic moment of Co2FeSi single crystal.
FIG. 2 :
2Temperature dependence of the resistivity in zero and 15 T magnetic field. Upper left inset: Difference (∆ρ = ρ(15T) − ρ(0T), ρ0 = ρ(0T)) of these measurements. Lower right insets: Close-up to low and high temperatures.
FIG. 3 :
3Resistivity in dependence of the temperature. Measured values are displayed as black dots. Blue chopped lines show fits with magnonic (∼ T 2 ) and magnonic/phononic temperature dependence. The red continuous line shows a fit where an additional Boltzmann factor was added to the magnonic scattering, its fitting parameters are displayed (see text for formulas).
FIG. 4 :
4Magnetic field dependence of resitivity at different temperatures. Inset: slope of ρ(B) in the applied field range of µ0H = (3 . . . 15) T.
FIG. 5 :
5Hall resistivity in dependence of the applied magnetic field for different temperatures. Inset: ordinary Hall coefficient R and anomalous Hall coefficient RA in dependence of the temperature.
FIG. 6 :
6Anomalous Hall coefficient RA in dependence of the zero field resistivity ρ (black circles). Line: quadratic fit for temperatures above 100 K.
Note that electron-phonon scattering can be left out if one fits the data only up to 200 K.2 We estimate the uncertainty to less than 30 %.
AcknowledgementsThis work has been supported by the Deutsche Forschungsgemeinschaft through the Priority Programme SPP1538 (Grant No. HE 3439/9), and through the Emmy Noether Programme WU595/3-1 (S.W.).
* Electronic address: [email protected] † Electronic address: c.hess@ifw-dresden. de* Electronic address: [email protected] † Electronic address: [email protected]
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| []
|
[
"A Classical Algorithm for Quantum SU(2) Schur Sampling",
"A Classical Algorithm for Quantum SU(2) Schur Sampling"
]
| [
"Vojtěch Havlíček ",
"Sergii Strelchuk ",
"Kristan Temme ",
"\nDepartment of Computer Science\nDepartment of Applied Mathematics and Theoretical Physics\nUniversity of Oxford\nWolfson Building, Parks RoadOX1 3QDOxfordUK\n",
"\nIBM T.J. Watson Research Center\nUniversity of Cambridge\nWilberforce RoadCB2 3HU, 10598Cambridge, Yorktown HeightsNYUK, USA\n"
]
| [
"Department of Computer Science\nDepartment of Applied Mathematics and Theoretical Physics\nUniversity of Oxford\nWolfson Building, Parks RoadOX1 3QDOxfordUK",
"IBM T.J. Watson Research Center\nUniversity of Cambridge\nWilberforce RoadCB2 3HU, 10598Cambridge, Yorktown HeightsNYUK, USA"
]
| []
| Many quantum algorithms can be represented in a form of a classical circuit positioned between quantum Fourier transformations. Motivated by the search for new quantum algorithms, we turn to circuits where the latter transformation is replaced by the SU(2) quantum Schur Transform -a global transformation which maps the computational basis to a basis defined by angular momenta. We show that the output distributions of these circuits can be approximately classically sampled in polynomial time if they are sufficiently close to being sparse, thus isolating a regime in which these Quantum SU(2) Schur Circuits could lead to algorithms with exponential computational advantage. Our work is primarily motivated by a conjecture that underpinned the hardness of Permutational Quantum Computing, a restricted quantum computational model that has the above circuit structure in one of its computationally interesting regimes. The conjecture stated that approximating transition amplitudes of Permutational Quantum Computing model to inverse polynomial precision on a classical computer is computationally hard. We disprove the extended version of this conjecture -even in the case when the hardness of approximation originated from a difficulty of finding the large elements in the output probability distributions. Finally, we present some evidence that output of the above Permutational Quantum Computing circuits could be efficiently approximately sampled from on a classical computer. * [email protected] C. QFT † QST Classical C. QST † arXiv:1809.05171v3 [quant-ph] | 10.1103/physreva.99.062336 | [
"https://arxiv.org/pdf/1809.05171v3.pdf"
]
| 118,930,572 | 1809.05171 | 8849c10e4fba948c65aa08142ef3a8a365773c8a |
A Classical Algorithm for Quantum SU(2) Schur Sampling
Vojtěch Havlíček
Sergii Strelchuk
Kristan Temme
Department of Computer Science
Department of Applied Mathematics and Theoretical Physics
University of Oxford
Wolfson Building, Parks RoadOX1 3QDOxfordUK
IBM T.J. Watson Research Center
University of Cambridge
Wilberforce RoadCB2 3HU, 10598Cambridge, Yorktown HeightsNYUK, USA
A Classical Algorithm for Quantum SU(2) Schur Sampling
Many quantum algorithms can be represented in a form of a classical circuit positioned between quantum Fourier transformations. Motivated by the search for new quantum algorithms, we turn to circuits where the latter transformation is replaced by the SU(2) quantum Schur Transform -a global transformation which maps the computational basis to a basis defined by angular momenta. We show that the output distributions of these circuits can be approximately classically sampled in polynomial time if they are sufficiently close to being sparse, thus isolating a regime in which these Quantum SU(2) Schur Circuits could lead to algorithms with exponential computational advantage. Our work is primarily motivated by a conjecture that underpinned the hardness of Permutational Quantum Computing, a restricted quantum computational model that has the above circuit structure in one of its computationally interesting regimes. The conjecture stated that approximating transition amplitudes of Permutational Quantum Computing model to inverse polynomial precision on a classical computer is computationally hard. We disprove the extended version of this conjecture -even in the case when the hardness of approximation originated from a difficulty of finding the large elements in the output probability distributions. Finally, we present some evidence that output of the above Permutational Quantum Computing circuits could be efficiently approximately sampled from on a classical computer. * [email protected] C. QFT † QST Classical C. QST † arXiv:1809.05171v3 [quant-ph]
I. INTRODUCTION
Charaterizing the power of quantum computers is one of the two major challenges in quantum computation, with the other being their scalable implementation. A seminal approach to the former problem is the study of conditions which make quantum algorithms amenable to methods of efficient classical simulation. A number of important quantum algorithms can be cast in a form of classical circuit positioned between a pair of circuits which implement quantum Fourier transformation. These are, for example, algorithms for the Hidden Subgroup Problem which in particular include the Shor's factoring algorithm [1,2]. While the latter provides strong evidence that quantum computers outperform the classical ones, Schwarz and van den Nest [3] showed that the respective quantum circuit could be efficiently classically simulated if its output distribution was sufficiently close to being sparse.
In our current work, we aim to characterize a different class of circuits that instead of the quantum Fourier transform contain the quantum Schur transform (QST) as depicted on Fig. 1. QST is a map from the computational basis to a basis defined by angular momentum [4][5][6] and it underpins a variety of quantum information processing tasks, including spectrum estima- tion [7,8], hypothesis testing [9][10][11][12], quantum computing using decoherence-free subspaces [13], communication without a shared reference frame [14,15], and quantum color coding [16]. A quantum circuit that efficiently implements this transform was first described in [4][5][6] and recently improved by Kirby and Strauch [17,18]. The extent to which circuits using QST could be used to devise new quantum algorithms is, to our knowledge, largely unexplored -possibly with the exception of [19] and [20].
QST is a centerpiece in the analysis of Permutational Quantum Computing (PQC) [21] -a restricted quantum computational model based on recoupling of angular momenta [20,22]. It has been conjectured that PQC has supra-classical computational power. One of the conjec- tures supporting this belief stated that an approximation of its transition amplitudes in the regime where they encode matrix elements of the symmetric group irreps in the Young's orthogonal form [20,23] is hard to compute classically if we require inverse polynomial precision (in the number of input qubits). While in our previous work we presented an efficient classical algorithm for approximating such transition amplitudes [21], an intriguing question remained: Is it also possible to identify all PQC transition amplitudes that can be approximated using classical methods with the inverse polynomial precision? Since the expected output probability of an n-qubit quantum circuit C with an input state |y is given by:
E x | x|C|y | 2 = 1 2 n x | x|C|y | 2 = 1 2 n ,
approximating these values with an inverse polynomial precision cannot distinguish the majority of x|C|y amplitudes from zeroes (see Fig. 2). Could we exploit the difficulty that arises from finding large matrix elements encoded in the output of the algorithm and thus demonstrate the (exponential) quantum computational advantage?
We show that this is not the case by describing a classical method that finds all large output probabilities in polynomial time. Our proof technique uses the simulation technique of Schwarz and van den Nest [3] where the authors studied analogous problem in the context of the quantum Fourier transform. This approach uses a variant of the Kushilevitz-Mansour algorithm used in classical learning theory [24,25]. We adapt it for distributions arising in the class of circuits using the QST, which include the relevant regime of Permutational Quantum Computing. We then show how to classically approximately sample their output distributions. The sampling algorithm becomes efficient for output distributions that are sufficiently close to sparse.
1 3 n − 1 n 2 S 2 [2] S 2 [3] S 2 [n−1] S 2
[n] , Z Figure 3. Sequentially coupled basis on n qubits. The numbers at the leaf nodes label qubits. Every vertex • carries a total spin operator S 2 A , that forces qubits in set A to one of its eigenstates. Similar diagrams can be used to label basis states and are shown in Appendix C or [20].
Our results additionally imply that sampling from the quantum Schur circuits can only lead to exponential computational advantage if the individual elements of the output distribution cannot be resolved by polynomial approximation with the quantum device by taking polynomially many samples. A way to circumvent this restriction, similarly to the case of circuits that use the quantum Fourier transform, could be to use a technique utilized in the Shor's algorithm that reconstructs group generators by sampling log |G| group elements for a superpolynomially large |G|. There is no meaningful counterpart to this approach for the QST as of now.
II. QUANTUM SU(2) SCHUR SAMPLING
The studied circuits are derived from the Permutational Quantum Computing -a computational model based on recoupling of angular momenta [20,22]. We hence review the basics of the angular momentum the-
S k = 1 2 (X k , Y k , Z k ) ,
where X k , Y k , Z k denote the Pauli X, Y, Z operators on the k-th qubit. The total spin operator on a qubit subset A ⊆ [n] is given by:
S 2 A := k∈A S k · k ∈A S k .
We write S 2 := S 2 [n] . The operators S 2 A and S 2 B commute if and only if the sets A and B are disjoint or one is contained in the other. Let:
Z A := 1 2 k∈A Z k ,
denote the azimuthal spin operator on a qubit subset A.
We again use Z [n] := Z. The operators Z A and S 2 A commute for any A ⊆ [n] and share an eigenspace labeled by quantum numbers j A and m A . The quantum number j A is the total spin of qubits in A and m A is the azimuthal spin. Both spin numbers are subject to constraints: the azimuthal spin m A only takes values in integer steps between −j A and j A , while the total spin numbers are either integer or half-integer and combine according to the angular momentum addition rules [26,27]:
j A∪B ∈ {|j A − j B |, |j A − j B | + 1, . . . , j A + j B } . (1)
Sets of commuting spin operators can be used to define complete orthonormal bases [20]. A particular basis is given by coupling a qubit at a time; that is by the joint eigenstates of:
S 2 [2] , S 2 [3] , . . . S 2 , Z.
We call it the sequentially coupled basis. The basis states are labeled by eigenstates j [2] , j [3] . . . , j [n−1] , J and M of the spin operators. By Eq. 1, these are subject to:
j [1] = 1 2 , j [k+1] = j [k] ± 1 2 ,(2)
which can be expressed diagrammatically by a branching diagram (Fig. 4). Up to the quantum number M , the sequential basis states correspond to paths in this diagram that start at j [1] = 1 2 . Let A k be the set of all such paths on k qubits. Any path j ∈ A k can be labelled by a bitstring by writing 1 for any edge of the path and 0 for an edge of the path j in the branching diagram. For example:
1 2 → 1 → 1 2 → 1 → 101.
Any prefix of length m ≤ k − 1 in such a bitstring contains at most m 2 zeroes, since the path never goes below the horizontal axis of the branching diagram. These bitstrings play a role in the representation theory of the symmetric group and are called Yamanouchi symbols [28,29]. The sets of Yamanouchi symbols with the same Hamming weight correspond to Young diagrams on two rows, which can be seen in Fig. 4. This is underpinned by the SU(2) Schur-Weyl duality, that states that the n-qubit Hilbert space decomposes into the tensor product of the symmetric group S n modules (isomorphic to the Young diagrams on two rows) and the special unitary group SU(2) under their joint action.
See Appendix E for additional details of this correspondence and [4,18] for detailed discussion of the underlying representation theory.
For the sequentially coupled basis, the SU(2) Schur-Weyl duality gives the SU(2) Quantum Schur Transform as described in [4-6, 17, 18, 30]. It is a sequence of the Clebsch-Gordan transformations, that couple j and j eigenspaces into a |J, M, j, j state by:
|J, M, j, j = m,m C J,M j,m; j ,m |j, m |j , m ,
where the summation over m runs from −j to j in integer steps (and similarly for m ) and the C J,M j,m; j ,m are the Clebsch-Gordan coefficients. The transform between the computational and the sequentially coupled basis is given by a cascade of the Clebsch-Gordan transforms [4,17]. For example on 3 qubits:
|J, M, j [2] = m1,m2 m [2] ,m3
C J,M j [2] ,m [2] ; 1 2 ,m3 C j [2] ,m [2] 1 2 ,m1; 1 2 ,m2 |m 1 m 2 m 3 = m1m2m3 [U Sch ] J,M,j [2] m1m2m3 |m 1 m 2 m 3 .
where we omitted the j = 1 2 numbers for qubits for brevity. The extension to the n ≥ 3 qubit case is straightforward. We label the sequentially coupled basis states on n qubits by |J , M , where J is a path in A n .
Permutational Quantum Computing in the sequentially coupled basis uses the permutation gate between two sequentially coupled basis states. Its transition amplitudes are:
J , M |U π |J , M ,
where the permutation gate U π is defined by its action on a computational basis state |x 1 . . . x n as:
U π |x 1 x 2 x 3 . . . x n = |x π(1) x π(2) x π(3) . . . x π(n) .
Both Z and S 2 operators commute with U π and consequently, M = M and J = J . The matrix J , M |U π |J , M block-diagonalizes to J, M blocks; each of which corresponds the an irreducible representation of the symmetric group in the Young's orthogonal form. The transition amplitudes are then the matrix elements of these matrices [20]. Approximating them to polynomial precision was conjectured hard classically in [20,23] but an efficient classical algorithm was found in [21].
The methods we present here work for a broader family of quantum circuits we call the SU(2) Quantum Schur Sampling circuits. These have transition amplitudes:
J , M |W |J , M ,
where W is defined by its action on a computational basis state |x , x ∈ {0, 1} n :
W |x = |w(x) ,
with w : {0, 1} n → {0, 1} n being a classical function given by a sequence of Toffoli gates -we consider only such W where this sequence is poly(n) long [31].
The circuits become similar in structure to Shor's algorithm in a sense of Fig. 1 if we allow for ancillary qubits. The simulation results apply also to these circuits, which we discuss in Section VI. j [2] j [3] j [4] J, M
1 3 4 5 2 j [2]
j [3] j [4] J , M
III. FINDING LARGE PROBABILITIES
We now describe an algorithm for finding large probabilities in the output of the circuits (see Fig. 2). Our approach is built on the concept of computational tractability introduced in [32]: Definition 1. An n-qubit state |ψ is computationally tractable (CT) if it is possible to classically sample from the distribution:
p(x) = {| x|φ | 2 : x ∈ {0, 1} n },
in polynomial time and the overlaps x|φ can be computed to exponential precision for a computational basis state |x in polynomial time.
We proved in [21] that the sequentially coupled basis states are CT. As a corollary, we show that |φ = W |J , M is also CT:
Lemma 1. |φ = W |J , M is CT. Proof. Since x|J , M can be efficiently computed be- cause |J , M is CT, so can be x|W |J , M = w −1 (x)|J , M . The distribution: p(x) = | x|W |J , M | 2 ,
can be efficiently sampled by applying the inverse of w(x) to the samples drawn from | x|J , M | 2 :
p(w −1 (x)) = | w −1 (x)|J , M | 2 = | x|W |J , M | 2 .
Since W is made of polynomially-many Toffoli gates, the inverse is obtained by applying the circuit in reverse to the bitstring x. Figure 6. The summation J⊇j runs over all paths J ∈ An that contain j ∈ A k . As an example, in the diagram above j = 1 2 → 0 → 1 2 → 1 and k = 4, n = 6. The summation runs over the paths within the shaded region. It follows that (in terms of the Yamanouchi symbols) J ⊇ j = {01111, 01110, 01101, 01100}.
We also state Lemma 3 of [32], which is an application of the Chernoff-Hoeffding bound.
Lemma 2 (CT state overlap ( , δ)-approximation [32]). An overlap φ|ψ between two CT states can be approximated byã, such that:
|ã − φ|ψ | ≤ ,
with probability 1 − δ in poly( 1 , n, log 1 δ ) time. We say that the overlap φ|ψ is ( , δ)-approximated byã.
We now show how to approximate a set of output probability marginals, an enabling result for extension of the techniques used by Schwarz and van den Nest in [3]. Given a path j ∈ A k for k ≤ n, define the output marginal p(j):
p (j) := J⊇j M p (J , M ) = φ| J⊇j;M |J , M J , M | φ := φ| Π (j) |φ ,
where the summation J⊇j sums all paths J ∈ A n that contain j ∈ A k (see Fig. 6). The summation M runs from −J to J in integer steps. We use where the sum m runs over m ∈ {−j, −j + 1, . . . j}. Lemma 3. For j ∈ A k , p(j) can be classically ( , δ)approximated byp(j) in poly 1 , n, log 1 δ time. Proof. We first show that the marginal p(j) on k qubits can be written as a transition amplitude of a larger, (n + k)-qubit circuit as:
φ|j, m j, m|φ = ( j, m| φ|) U SWAPS (|φ |j, m ) ,
where U SWAPS is a permutation gate on k + n qubits. Write symbolically |j, m = |ψ = |ψ 1 ψ 2 . . . ψ k and |φ = |φ 1 φ 2 . . . φ n , so that:
φ|j, m j, m|φ = φ|ψ ψ|φ = φ 1 . . . φ n |ψ 1 . . . ψ k ψ 1 . . . ψ k |φ 1 . . . φ n .
Let U SWAPS swap the (n + i)-th and (n − k + i)-th qubits for all 1 ≤ i ≤ k:
U SWAPS |ψ 1 . . . ψ k |φ 1 . . . φ n = |φ 1 . . . φ k |ψ 1 . . . ψ k , φ k+1 . . . φ n .
This gives: Since m sums 2j + 1 ≤ n + 1 terms, it follows that p(j) can be also ( , δ)-approximated.
φ| ψ| U SWAPS |ψ |φ = φ 1 . . . φ n | ψ 1 . . . ψ k |φ 1 . . . φ k |ψ 1 . . . ψ k , φ k+1 . . . φ n = φ 1 . . . φ n |ψ 1 . . . ψ k ψ 1 . . . ψ k |φ 1 . . . φ n = φ|ψ ψ|φ ,
We now combine Lemmas 1, 2 and 3 to describe a classical algorithm that finds large elements in the output distribution of quantum Schur circuits. It is an adaptation of the Kushilevitz-Mansour algorithm [24]. Theorem 1. Let p(J ) : A n → [0, 1] be a probability distribution on paths. There is a classical algorithm that outputs a set L ⊆ A n in poly n, 1 θ , log 1 γ time, such that for some θ > 0:
∀ J ∈ L : p(J ) ≥ θ 2 , ∀ J ∈ A n : p(J ) > θ =⇒ J ∈ L,(3)
with probability at least 1 − γ.
|p(j2) −p(j2)| ≤ θ 4 .
Add j to L2 ifp(j2) ≥ 3 4 θ. 2. Continue for k = 3, . . . , n. Assume L k−1 has been found. For any path j k−1 ∈ L k−1 , take all possible steps in the branching diagram. This gives paths j k ∈ A k that end at j k = |j k−1 ± 1 2 |. For every such path, compute the approximatioñ p(j k ) such that:
|p(j k ) − p(j k )| ≤ θ 4 .
Ifp(j k ) ≥ 3 4 θ, add the path j k to L k .
In every step of the computation, check if
|L k | > 2 θ . If true, halt and output ∅. The algorithm never halts if all approximation steps succeed.
Output L = Ln.
Proof. See Algorithm 1.
The algorithm runs in n steps, each of which succeeds with probability at least (1 − δ) |L k | . Since |L k | ≤ 2 θ , the success probability is at least:
(1 − δ) 2n/θ ≥ 1 − 2δn θ := 1 − γ.
Thanks to δ < θ 2n , it follows that 1 − γ > 0. The algorithms terminates in poly n, 1 θ , log 1 γ time as it halts whenever the number of elements in any list exceeds 2 θ . Since p(j k ) ≥ θ 2 for each j k ∈ L k , the final list L contains at most 2 θ elements by normalization. So if all approximation steps succeed, the algorithm does not halt before it outputs L.
Algorithm 1 has an interesting consequence: since it runs in polynomial time whenever θ = 1/poly(n), paths with polynomially small p(J ) = M p(J , M ) can be found in polynomial time. When such path J is found, it is possible to approximate p(J , M ) for all M , since there are 2J + 1 ≤ n + 1 distinct values of M it has to be approximated for by Lemma 2. Such approximation of transition amplitudes has the same precision as if when polynomially many samples were taken with a quantum computer. The SU(2) Quantum Schur sampling circuits therefore cannot encode classically hard-to-approximate quantities in amplitudes that could be resolved by sam-pling, because any such quantity could be found by the presented algorithm and then approximated by Lemma 2.
IV. APPROXIMATE SAMPLING
Following Schwarz and van den Nest [3], we use the above algorithm to approximately sample the quantum Schur circuits under additional sparsity constraint on their output distribution:
p −p 1 ≤ .
We also adapt a technical lemma from [24]. By the triangle inequality:
pI S∩T − p 1 ≤ p − pI T 1 + pI S∩T − pI T 1 .
Since p is -approximate t-sparse, it follows that:
p − pI T 1 ≤ p − p t 1 ≤ .
We also have that: (4) because all elements in T /S are ≤ t and there is at most t of them. This gives:
pIJ ∈S;M p(J , M ) ≤ 2 .
Theorem 1 and Lemma 4 can be combined to define a probability distribution close to the quantum output that can be sampled from in poly(t, 1 , n) time. Assume that p(J , M ) is -approximately t-sparse. Let L ⊆ A n be the set of paths generated by the Kushilevitz-Mansour algorithm with threshold θ = t . Choose: Check if M ≤ 2J +1. If yes, define M = (M −J −1) and output (J , M ). Otherwise repeat. A valid Yamanouchi symbol will be found in poly(n) trials by a dimensionality argument (Appendix D). This procedure samples the probability distributionp defined above, which has been shown 6 close in the total variational distance to p.
= min (n + 1)|L| , 4t ,(5)
While the above algorithm runs inp in poly( 1 , n, t) time, it discards significant amount of paths during the uniform sampling of paths which may be an unnecessary bottleneck for the eventual implementation. We explain how to avoid this problem by an alternative algorithm for sampling the paths, based on the Greene-Nijenhuis-Wilf algorithm [33] in Appendix E.
V. HOW SPARSE IS THE OUTPUT?
We consider the range of applicability of the outlined algorithm. Since the set of classical gates W is large, we limit this analysis to Permutational Quantum Computing in the sequentially coupled basis and study the output distributions for n ≤ 10 qubits. We randomly chose 5 paths and consider 10 random permutations for each. This gives 50 sets of output distributions with dimension d determined by J of each path. Recall that d can be exponentially large in n.
All chosen distributions contained an element greater than 1 2n . As a sufficient condition for 1 n -approximate 2n 2 -sparsity by Lemma 4, we checked if the sum of all elements less than 1 2n 2 is less than 1 2n . Distributions for permutations on 4 to 9 qubits all have this property, while the fraction that do not have it for n = 10 qubits was estimated to be less than 0.1%. Being a sufficient condition, some of these distributions are nevertheless very far from flat -an example is shown in Fig. 7.
We also consider a stricter sufficient condition: for all J-blocks with dimension d > n, we computed the fraction of output distributions for which the sum of all elements except for the largest C (log 2 d) D ones is less than 1/ log 2 d for some constants C and D. Since d < 2 n , this condition suffices for 2n-approximate (Cn D )-sparsity of the output. Almost all of the distributions, with the exception of about 0.4% of those for n = 9, were 2 log(d)approximate log(d) 2 -sparse. While we were not able to prove that a significant fraction of the output distributions are -approximate t-sparse for some t = poly(n) and = 1/poly(n), the results give some indication that closeto-sparse output distributions could be common for the relevant regime of Permutational Quantum Computing.
VI. CIRCUITS WITH ANCILLAS
The proposed simulation technique extends to quantum Schur sampling circuits with ancilla qubits, with transition amplitudes given by:
0| k J , M | W |J , M |0 k ,
for J ∈ A n and J ∈ A n , such that k + n = k + n. Note that W |J , M |0 k is CT. Since the marginal approximation of Lemma 3 relies only on approximating overlaps of the form:
( j, m| φ|) U SWAPS (|φ |j, m ) ,
where |φ is a computationally tractable state, it also extends to marginals:
0| k j, m| φ| U SWAPS |φ |j, m |0 k .
since |φ |j, m |0 k is CT (see [32] for details). We give some evidence that these circuits can give rise to computationally interesting structures, largely inspired by [23]. Prepare:
U Sch H ⊗n |0 ⊗n = 1 √ 2 n J,M |J , M ,
and consider a classical circuit W that encodes the path J a the Yamanouchi symbol x into an ancilla register. This should done before the Schur transform as the W gate is generally controlled in the computational basis. A way to implement this is to use the form of QST which encodes the information about irreps explicitly into the computational basis input at the expense of logarithmic overhead in number of qubits [4,18]. Additionally, compute the value of J to another ancilla register of log n−1 qubits, giving the state:
1 √ 2 n J,M |J , M |x(J ) |J .
Apply the permutation gate U π to the first register. After applying the gate sequence H ⊗n U † Sch and measuring the first n qubits and the J register, we have that:
p(0 . . . 0 n , J) = 1 4 n J J |U π |J 2 .
where J runs over all paths that end at J. Here T (π) = J J | U π |J is the trace of U π over the Jblock, which is (up to a sign) the square of the character of the conjugacy class of π of the S n irrep. defined by J. This quantity is known to be #P-hard by [34], so we know that there exist π ∈ S n for which exact computation of T (π) becomes intractable under the standard complexity theoretic assumptions. Despite the fact that an efficient classical method for computing additive approximations to this quantity was given by [23] (its existence is in fact a consequence of Theorem 1), it is still possible that its multiplicative approximation retains hardness. This could lead to another class of probability distributions unlikely to be sampled from classically akin to [35][36][37]. On the discouraging side, limitations of the quantum 'Fourier-Schur' sampling in the context of addressing the hidden subgroup problem were identified in [19].
VII. DISCUSSION
Circuits using the QST underpin a diverse range of protocols in quantum information processing, from state discrimination to computational models such as Permutational Quantum Computing. While studying the computational power of the transform, we singled out a class of circuits with QST blocks that extend a computationally interesting regime of Permutational Quantum Computing. The key result that enabled this analysis was the efficient approximation of quantum Schur sampling circuits studied in [21] as means to characterize its com-putational power. Building on the work of Schwarz and Van den Nest [3,32], we showed that large elements of the output distributions can be efficiently found, which precludes the possibility that the circuits could encode quantities that would be hard to classically approximate by taking polynomial number of samples. We subsequently proved that these circuits can be classically efficiently approximately sampled from if their output distribution becomes sufficiently close to a sparse one.
Our algorithm is a random walk on the angular momentum branching diagram associated with the computation. One distinctive feature of the algorithm is then that it is not limited to the angular momentum and can be extended to other branching diagrams. It will remain efficient as long as the counterparts of the Clebsch-Gordan coefficients remain efficiently computable to high precision and the out-degree of any vertex of the branching diagram will be bounded by a constant (see also the discussion in [21]). One of the interesting cases where our techniques could apply with little adaptation is the case of q-deformations of the SU(2) branching diagrams, applied in the study of topological phases of matter [38,39].
Circuits using similar structure but using an SU(d) Schur-Weyl transformation for d > 2 were recently applied in study of Boson Sampling with partially distinguishable bosons in the first quantization [40]. The possibility of leveraging the simulation techniques proposed here in this context remains open.
The J eigenspaces are 2J + 1-degenerate due to possible values of the M number. We then have that:
It follows that eigenstates of S n span the n-qubit Hilbert space. This also implies that there exist exponentially large blocks for fixed J that asymptotically scale as 2 n , since the summation in Eq. D1 runs only over polynomially many J. In particular, this makes the sampling algorithm of Theorem 3 run in polynomial time.
Tableaux of this shape -every such tableau is sampled with probability 1 d(J) and the sampling algorithm runs in O(n 2 ) time. Convert the Young tableau to the Yamanouchi symbol and the corresponding path J using Appendix E. Lastly, choose M ∈ {−J, −J + 1, . . . J} uniformly randomly. The probability of choosing a specific (J , M ) is then given by: Π(J) 1 (2J + 1)d(J) = 1 2 n , as wanted. The sampling procedure is then the following: s Theorem 3. Assume that p(J , M ) is -approximate t-sparse. It can be sampled classically in poly(n, 1 , t) time to 6 error in the total variational distance.
Proof. Use the Kushilevitz-Mansour algorithm in Theorem 1 with threshold θ = t to find L and compute b = J∈L;Mp (J , M ). Flip a coin with a bias b.
• With probability b, output a sample drawn fromp(J , M )/b for J ∈ L and corresponding M .
• With probability 1 − b, output (J , M ) for J ∈ L uniformly randomly.
To sample (J , M ) uniformly randomly, use the above algorithm to uniformly randomly generate a (J , M ) and check if J ∈ L. If yes, output. If no, sample again.
Figure 1 .
1Schematic diagrams of the quantum circuit used in Shor's factoring algorithm (Left) and the circuits we consider here (Right). QST denotes the SU(2) Quantum Schur Transformation. The classical circuits between the transforms can represent, for example a polynomially-long sequence of Toffoli gates.
Figure 2 .
2(Left) A part of the | x|C|y | 2 matrix for a typical PQC instance. After normalization, most matrix elements are indistinguishable from zeros within the polynomially small approximation window. We show how to classically find the large elements. (Right) The output matrix with sorted output, demonstrating that an overwhelming fraction of the probabilities are usually small.
Figure 5 .
5Schematic representation of Permutational Quantum Computing in the sequentially coupled basis. The applied permutation is (1, 2, 3)(4, 5).
, m j, m| .
as desired. Since both |j, m and |φ states are CT and U SWAP is a permutation on up to 2n objects,( j, m| φ|) U SWAPS (|φ |j, m ) , can be ( , δ)-approximated by Lemma 2. Therefore: p(j) = φ| Π(j) |φ = m φ|j, m j, m|φ = m ( j, m| φ|) U SWAPS (|φ |j, m ) .
Definition 2 (
2-approximate t-sparsity). A probability distribution p(J , M ) is t-sparse if it has at most t non-zero elements p(J , M ). A probability distributioñ p(J , M ) is -approximately t-sparse if there exists a tsparse distribution p(J , M ) such that:
Lemma 4 .
4Let p(J , M ) be an -approximate t-sparse distribution. Let S be the set of all (J , M ) for which p(J , M ) is greater than t . Then: J ∈S;M p(J , M ) ≤ 2 . Proof. Let p t (J , M ) be a t-sparse distribution that isclose to p(J , M ). Define T to be the set of all (J , M ) for which p t is nonzero, i.e. the support of p t . Trivially, S ∩ T ⊆ S, which implies that: J ∈S;M p(J , M ) ≤ J ∈S∩T ;M p(J , M ). Define the indicator I A : A → {0, 1} on the set A as follows: J , M ) (I S∩T (J , M ) − 1) = pI S∩T − p 1 .
S∩T − pI T 1 = J∈T ;M p(J , M )I S∩T (J , M ) = J∈T /S;M p(J , M ) ≤ t t = .
JJJ
and compute approximationsp(J , M ) for all J ∈ L and M by Lemma 2. Define a normalization factor α as: α = 1 2 n − J∈L (2J + 1) , such that J ∈L;M α = 1 (see Appendix D). Use theapproximationsp(J , M ) to define: p = p(J , M ) for J ∈ L, p • := α(1 − J∈L;Mp (J , M )) otherwise, , so thatp becomes uniform on all J outside L. The constantp • is chosen so thatp is normalized. Then: p − p 1 = J∈L;M |p(J , M ) − p(J , M )| + J ∈L;M |p(J , M ) − p(J , M )| ≤ + j ∈L;M |p(J , M ) − p(J , M )|, ∈L;M |p • −p • | ≤ J∈L;M p(J , M ) − J∈L;Mp (J , M ) ≤ J∈L;M |p(J , M ) −p(J , M )| ≤ . ∈L;M |p • − p(J , M )|. We now use the set S from from Lemma 4. S ⊆ L by the defining property of L. It follows that: ∈L;M |p(J , M ) − p(J , M )| ≤ 5 , and: p − p 1 ≤ 6 . We now show how to classically samplep. Theorem 2. Assume that p(J , M ) is -approximate tsparse. It can be sampled classically in poly(n, 1 , t) time to 6 error in the total variational distance. Proof. Use the Kushilevitz-Mansour algorithm in Theorem 1 with threshold θ = t to find L. Compute b = J∈L;Mp (J , M ). Flip a coin with a bias b.• With probability b, output a sample drawn from p(J , M )/b for J ∈ L and corresponding M .• With probability 1 − b, output (J , M ) for J ∈ L uniformly randomly.To sample (J , M ) uniformly randomly, generate a random bitstring with n − 1 bits and check if it encodes a Yamanouchi symbol. This can be verified by checking that any prefix of m ≤ n − 1 bits has at most m 2 zeroes. Once found, generate a random integer M from [n + 1].
Figure 7 .
7Output distribution of PQC-SEQ that does not satisfy the sufficient sparsity condition for n = 10. The horizontal line labels the (2n 2 ) −1 threshold. The distribution is actually 1/10-approximate 21-sparse and 'fools' the proxy criteria by having a single overwhelmingly large element. The p-axis is logarithmic.
This can be seen by:Appendix C: Diagrammatic representation of the spin basis statesAppendix D: Completeness of the sequentially coupled basisThe argument comes from[28]. Denote the number of paths in A k that end at j [k] by d(j[k]). It follows from Eq. 2 that such j[k]can be reached by taking a step in a path j k−1 ∈ A k−1 that ends either at j [k−1] + 1 2 or j [k−1] − 1 2 . This gives a recurrence:which is solved by:Appendix E: Paths to Young TableauxHere we show that the paths are one to one with the standard Young tableaux on two rows, which we use to give an improved sampling method in Appendix F. Let J ∈ A n be a path and let:be its Yamanouchi symbol. The shape of the corresponding standard Young tableau is determined by J and n -it will have n 2 + J boxes in the first row and n 2 − J boxes in the second row. Write 1 to the first box in the upper row, then read the Yamanouchi symbol x from left to right. If the i-th bit x i = 0, add an element i + 1 to the leftmost empty box in the lower row. Conversely, if x i = 1, add i + 1 to the leftmost empty box in the upper row. The resulting Young tableau is in the standard form (its elements are increasing both along its rows and columns). The elements in each row are increasing by construction. The elements in each column also increase, which can be seen from the property that any prefix of length m ≤ n − 1 of the Yamanouchi bitstring contains at most m 2 zeroes -in other words, the upper row will be always filled faster than the lower one. Paths are also onto the standard two-row Young tableaux, which can be proved by converting the tableaux to bitstrings by reversing the above algorithm and checking the defining property of the Yamanouchi symbol.As an example, take the sequentially coupled basis state on n = 3 qubits:The path ends at J = 1 2 , which means that the corresponding Young diagram will have 2 boxes in the upper and 1 in the lower rows:The path for this state is 1 2 → 0 → 1 2 , which gives a Yamanouchi symbol = 01. It also gives a prescription to fill the Young diagram by the above algorithm, giving the tableau:so that the quantum state can be equivalently labeled as:There is a one-to-one correspondence between the semi-standard Young tableaux of the same shape filled with ↑, ↓ and M -see[18]for discussion of this. However, since M and n completely determine the filling in this case, there is no need to use this here.Appendix F: Sampling with the Greene-Nijenhuis-Wilf algorithmWe now describe how to sample the paths with n steps uniformly randomly using the algorithm proposed by Greene, Nijenhuis and Wilf in[33]. First, fix an endpoint of the path by sampling J from the distribution Π(J) = 2J+1 2 n d(J) where d(J) is the number of paths that end at J, as defined in Appendix D. Take a two-row Young diagram with n 2 +J boxes in the upper and n 2 − J in the lower row and use the GNW algorithm to uniformly generate a standard YoungAppendix G: Simplification of the marginal projectorThe aim of this section is to simplify the marginal projector expression as:for j ∈ A k and J⊇j runs over all paths J ∈ A n that contain j. The sum m runs over m ∈ {−j, −j + 1, . . . j}.To do so, we repeatedly use the Clebsch-Gordan orthogonality: JM C JM jm,j m2 C JM jm ,j m 2 = δ m2,m 2 δ mm .As we study coupling in the sequential basis, we have that:We have for the projector Π (j) that: This has the same form as the initial expression, but the projector is now defined by summing over paths with n − 1 steps. It is possible to continue recursively and write:for any integer 0 ≤ i ≤ n − k. For i = n − k, one obtains that:Since j ∈ A k , there is only one path contributing to J k ⊇j , the path j itself. It follows that:We can write:where the final summation runs over m ∈ {−j, −j + 1, . . . j}.
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| []
|
[
"Numerical Reconstruction of the Spatial Component in the Source Term of a Time-Fractional Diffusion Equation",
"Numerical Reconstruction of the Spatial Component in the Source Term of a Time-Fractional Diffusion Equation"
]
| [
"Daijun Jiang ",
"Yikan Liu ",
"Dongling Wang "
]
| []
| []
| In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H 1 -norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm. | 10.1007/s10444-020-09754-6 | [
"https://arxiv.org/pdf/1812.04235v2.pdf"
]
| 119,272,664 | 1812.04235 | 42bd02f8a3ad374f90e2da93b1078fff583f8de3 |
Numerical Reconstruction of the Spatial Component in the Source Term of a Time-Fractional Diffusion Equation
May 2020
Daijun Jiang
Yikan Liu
Dongling Wang
Numerical Reconstruction of the Spatial Component in the Source Term of a Time-Fractional Diffusion Equation
May 2020Time-fractional diffusion equationInverse source problemFinite ele- ment methodIterative thresholding algorithm Mathematics subject classification (2010) 35R1165M3241A35
In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H 1 -norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm.
Introduction
Let T > 0 and Ω ⊂ R d (d = 1, 2, 3) be a bounded convex polygonal domain. For 0 < α < 1, by ∂ α t we denote the Caputo derivative defined as (see, e.g., [10, p. where Γ( · ) is the usual Gamma function. In this paper, we consider the following initial-boundary value problem for a time-fractional diffusion equation (TFDE) with the homogeneous Neumann boundary condition (∂ α t u − △ + 1)u(x, t) = f (x)µ(t), (x, t) ∈ Q := Ω × (0, T ), u(x, 0) = 0,
x ∈ Ω, ∂ ν u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), (1.1) where ∂ ν u := ∇u · ν and ν = (ν 1 , . . . , ν d ) denotes the unit outward normal derivative on the boundary ∂Ω. The source term in (1.1) takes the form of separated variables, in which the spatial component f (x) models the spatial distribution e.g. of the contaminant source, and the temporal component µ(t) describes the time evolution pattern. The governing equation in (1.1) is a typical representative of a wide range of TFDEs, which were proposed as a powerful candidate for describing anomalous diffusion phenomena in heterogenous media. In the past decades, TFDEs have been investigated thoroughly from both theoretical and numerical aspects. It reveals from [5,16,19] and other literature that TFDEs resemble their integer counterpart (i.e., α = 1) in such qualitative aspects like analyticity and maximum principle, but show certain difference in the senses of limited smoothing effect, short-/long-time asymptotic behavior and weak unique continuation. For the numerical simulation of TFDEs, we refer e.g. to [6,9,11,14,21].
Simultaneously, various kinds of inverse problems for TFDEs have also gathered consistent popularity within the last decade. Owing to the limited smoothing property of the forward problem, usually the ill-posedness of inverse problems for TFDEs is less severe than that for the case of α = 1, which can be witnessed from the backward problems. Nevertheless, due to the lack of techniques for analysis, for some inverse problems, one should relate TFDEs with other types of equations, so that one can indirectly obtain uniqueness and sometimes stability. For the inverse source problems on determining the spatial component, it turns out that the majority of existing literature dealt with the final observation data u( · , T ) except for [29]. We refer to Sakamoto and Yamamoto [20] for the generic well-posedness, and [24,26] etc. for the numerical reconstruction by various regularization methods. Recently, by a newly established unique continuation property for TFDEs, Jiang et al. [5] proved the uniqueness for the same problem by the partial interior observation and developed an iterative thresholding algorithm. On reconstructing the temporal component in the source term, we refer e.g. to [17,18,25]. For a complete bibliography on inverse problems for TFDEs, see the topical review articles [8,12,13,15].
In the same direction of [5], in this paper we mainly focus on the numerical aspect of the following inverse source problem. In fact, here we can also consider a more general formulation such as
∂ α t u(x, t) − d i,j=1 ∂ ∂x i a ij (x) ∂u(x, t) ∂x j + c(x)u(x, t) = f (x)µ(t),
where we assume certain regularity of a ij and c, and (a ij ) d×d is a strictly positive-definite matrix on Ω . However, we restrict ourselves to the formulation (1.1) not only for its simplicity in the numerical simulation, but also due to the belief that the underlying ill-posedness are essentially the same.
On the other hand, we notice that the iteration method proposed in [5] lacks detailed analysis especially on the convergence issue, although the numerical performance was discussed. From the practical viewpoint, it is obligatory to reformulate Problem 1.1 in a discrete setting and investigate whether the resulting system inherits the corresponding properties of the continuous one. Moreover, we should verify the convergence of the discretized solution to the continuous one in some sense. This motivates the central topic of the present article, which turns out to be a very important and necessary supplementation to the investigation of Problem 1.1.
To fully discretize system (1.1), we shall employ the standard Galarkin method with piecewise linear finite element in space and the L1 scheme in time. This method is one of the most popular and successful numerical scheme for discretizing the subdiffusion problem, and it has been analyzed from various aspects. The optimal error estimate in L 2 (Ω) norm with respect to the regularity of the problem solution is established in [6] for uniform time step size. In order to deal with the weak singularity of time-fractional diffusion equations, the L1-type scheme on graded time meshes is employed in the discrete scheme, and the optimal error estimate in L ∞ (Ω) norm is established for d = 1 in [21] and d = 2, 3 in [9]. For the convergence analysis, in this paper we further establish the uniform H 1 (Ω) error estimate on [0, T ] by using the Poincaré-Friedrichs inequality and the discrete fractional Grönwall's inequality for the L1 method, which turns out to be novel to the best of our knowledge.
The remainder of this paper is organized as follows. Recalling key ingredients in theory, in Section 2 we interpret Problem 1.1 as an optimization problem with Tikhonov regularization, and demonstrate its regularizing effects. In Section 3, we discretize the optimization problem by a fully discrete finite element approximation and study its basic properties. Next, Section 4 is devoted to the convergence analysis of the solution to the discrete optimization problem. Finally, we propose the iterative thresholding algorithm to solve the discrete problem in Section 5, and implement several numerical examples to show the efficiency and accuracy of the algorithm.
Preliminary and Tikhonov regularization
In this section, we shall make general preparations and formulate Problem 1.1 stated in Section 1 as a stabilized minimization system, and establish the unique existence of the solution as well as the stability of the minimization formulation.
Let L 2 (Ω) be a usual L 2 -space equipped with the inner product ( · , · ), and H α (0, T ), H γ (Ω) (γ ∈ R) etc. denote Sobolev spaces (e.g., Adams [1]). Throughout this paper, C > 0 stands for generic constants which may change line by line. In a Banach space X, we denote the weak convergence of a sequence {z n } to z by z n ⇀ z in X as n → ∞.
We first revisit some basic facts concerning the initial-boundary value problem (1.1). For the solution regularity, we define the fractional Sobolev spaces 0 H α (0, T ) as
0 H α (0, T ) := H α (0, T ), 0 < α < 1 2 , g ∈ H 1 2 (0, T ); T 0 |g(t)| 2 t dt < ∞ , α = 1 2 , {g ∈ H α (0, T ); g(0) = 0}, 1 2 < α < 1.
We also define the Riemann-Liouville integral operator J α 0+ and the backward Riemann-Liouville integral operator J α T − respectively as
J α 0+ g(t) := 1 Γ(α) t 0 g(τ ) (t − τ ) 1−α dτ, J α T − g(t) = 1 Γ(α) T t g(τ ) (τ − t) 1−α dτ
for g ∈ L 2 (0, T ). We know that the Caputo derivative can be rewritten as ∂ α t g = J 1−α 0+ d dt g, and the following relation holds (see [5,Lemma 4
.1]) T 0 (J α 0+ g 1 ) g 2 dt = T 0 g 1 J α T − g 2 dt, ∀ g 1 , g 2 ∈ L 2 (0, T ). (2.1)
For later use, we show a technical lemma on the basis of (2.1)
Lemma 2.1. Let g ∈ 0 H α (0, T ) ∩ C[0, T ] and h ∈ C 1 [0, T ] with J 1−α T − h(T ) = 0. Then T 0 (∂ α t g) h dt = −g(0) J 1−α T − h(0) − T 0 g (J 1−α T − h) ′ dt.
Proof. Pick a sequence {g n } ⊂ C ∞ [0, T ] such that g n → g in 0 H α (0, T ) ∩ C[0, T ]. Then for each n = 1, 2, . . ., we utilize (2.1) and calculate
T 0 (∂ α t g n ) h dt = T 0 (J 1−α 0+ g ′ n ) h dt = T 0 g ′ n J 1−α T − h dt = g n J 1−α T − h t=T t=0 − T 0 g n (J 1−α T − h) ′ dt = −g n (0) J 1−α T − h(0) − T 0 g n (J 1−α T − h) ′ dt,
where the last equality follows from the assumption J 1−α T − h(T ) = 0. Then the proof is completed by passing n → ∞ on both sides of the above identity.
On the basis of [5, Lemma 2.4] and [19, Theorem 2.2(i)], we summarize the well-posedness of the forward problem (1.1) as follows.
Lemma 2.2. Let f ∈ L 2 (Ω) and µ ∈ L ∞ (0, T ). Then the initial-boundary value problem (1.1) admits a unique solution u(f ) ∈ 0 H α (0, T ; L 2 (Ω)) ∩ C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)). Moreover, there exists a constant C > 0 depending on Ω, T, α and µ such that
u(f ) H α (0,T ;L 2 (Ω)) + u(f ) C([0,T ];L 2 (Ω)) + u(f ) L 2 (0,T ;H 2 (Ω)) ≤ C f L 2 (Ω) .
In [19, Theorem 2.2(i)], it was proved that the solution u ∈ C([0, T ]; H −2γ (Ω)) for any γ > d 4 − 1 provided that the source term belongs to L ∞ (0, T ; L 2 (Ω)). Since we restrict the spatial dimensions to d = 1, 2, 3 throughout this paper, one can choose negative γ and thus u(f ) makes sense in C([0, T ]; L 2 (Ω)) under the assumptions of Lemma 2.2.
For the theoretical aspect of Problem 1.1, we recall the uniqueness result stated in [5, Theorem 2.6]. Suppose that we are given the noisy observation data u δ ∈ L 2 (ω × (0, T )) in practice. Usually, u δ satisfies
u δ − u(f * ) L 2 (ω×(0,T )) ≤ δ,
where f * ∈ L 2 (Ω) and δ > 0 stand for the true source term and the noise level, respectively. To deal with the ill-posedness of Problem 1.1, we still adopt a classical Tikhonov regularization methodology as that in [5] to consider the following optimization problem with Tikhonov regularization
min f ∈L 2 (Ω) J(f ), J(f ) := u(f ) − u δ 2 L 2 (ω×(0,T )) + β f 2 L 2 (Ω) ,(2.2)
where β > 0 is the regularization parameter, and u = u(f ) satisfies the initial condition u(x, 0) = 0 and the variational system associated with the equation (1.1): for any ϕ ∈ H α (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)),
T 0 Ω (∂ α t u(f ) ϕ + ∇u(f ) · ∇ϕ + u(f ) ϕ) dxdt = T 0 Ω f µ ϕ dxdt. (2.3)
Regarding the minimizer of (2.2), we first show the following unique existence result.
Theorem 2.1. For any u δ ∈ L 2 (ω × (0, T )), there exists a unique minimizer f * ∈ L 2 (Ω) to the optimization problem (2.2).
Proof. Since J(f ) is nonnegative, we know that inf f ∈L 2 (Ω) J(f ) is finite. Thus there exists a minimizing sequence {f n } ⊂ L 2 (Ω) such that lim n→∞ J(f n ) = inf f ∈L 2 (Ω) J(f ).
Then by the definition of J(f n ), it is obvious that {f n } is uniformly bounded in L 2 (Ω). Therefore, there exist f * ∈ L 2 (Ω) and a subsequence of {f n }, still denoted by {f n }, such that
f n ⇀ f * in L 2 (Ω) as n → ∞.
We shall prove that f * is indeed the unique minimizer of (2.2).
Since each f n corresponds with a solution u(f n ) to (1.1) with f = f n , it follows immediately from Lemma 2.2 that the sequence {u(f n )} is also uniformly bounded in 0 H α (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)).
This, together with the compact embedding H 2 (Ω) ⊂⊂ L 2 (Ω), indicates the existence of some u * ∈ 0 H α (0, T ; L 2 (Ω) ∩ L 2 (0, T ; H 2 (Ω)) and a subsequence of {u(f n )}, again still denoted by {u(f n )}, such that
u(f n ) ⇀ u * in 0 H α (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)). (2.4)
We claim u * = u(f * ). Actually, we utilize the fact that
T 0 Ω (∂ α t u(f n ) ϕ + ∇u(f n ) · ∇ϕ + u(f n ) ϕ) dxdt = T 0 Ω f n µ ϕ dxdt (2.5) for all ϕ ∈ L 2 (0, T ; H 1 (Ω)). Since (2.4) implies ∂ α t u(f n ) ⇀ ∂ α t u * , ∇u(f n ) ⇀ ∇u * in L 2 (Q), we pass n → ∞ in (2.5) to obtain T 0 Ω (∂ α t u * ϕ + ∇u * · ∇ϕ + u * ϕ) dxdt = T 0 Ω f * µ ϕ dxdt (2.6)
for all ϕ ∈ L 2 (0, T ; H 1 (Ω)). Next, we shall prove u * ( · , 0) = 0, which with (2.6) and the definition (2.3) of weak solutions implies u * = u(f * ). Indeed, it follows from Lemma 2.2 that u(f n ) ∈ 0 H α (0, T ; L 2 (Ω))∩C([0, T ]; L 2 (Ω)), which satisfies the assumption of Lemma 2.1. Then by taking ψ ∈ C 1 [0, T ] with J 1−α T − ψ(T ) = 0 and v ∈ L 2 (Ω) arbitrarily, we employ Lemma 2.1 to deduce
Ω T 0 ∂ α t u(f n ) ψ v dtdx = − Ω u(f n )( · , 0) J 1−α T − ψ(0) v dx − Ω T 0 u(f n ) d dt J 1−α T − ψ v dtdx.
Noting that u(f n )(x, 0) = 0 and letting n → ∞ in the above equation, we obtain
Ω T 0 ∂ α t u * ψ v dtdx = − Ω T 0 u * d dt J 1−α T − ψ v dtdx. (2.7)
In a parallel manner, we also have
Ω T 0 ∂ α t u * ψ v dtdx = − Ω u * ( · , 0) J 1−α T − ψ(0) v dx − Ω T 0 u * d dt J 1−α T − ψ v dtdx for any ψ ∈ C 1 [0, T ] with J 1−α T − ψ(T ) = 0 and v ∈ L 2 (Ω)
, which, together with (2.7), implies that u * ( · , 0) = 0.
By f n ⇀ f * in L 2 (Ω) and (2.4), we employ the lower semi-continuity of the L 2 -norm to conclude
J(f * ) = u(f * ) − u δ 2 L 2 (ω×(0,T )) + β f * 2 L 2 (Ω) ≤ lim inf n→∞ u(f n ) − u δ 2 L 2 (ω×(0,T )) + β lim inf n→∞ f n 2 L 2 (Ω) ≤ lim inf n→∞ J(f n ) = inf f ∈L 2 (Ω) J(f ),
indicating that f * is indeed a minimizer to the optimization problem (2.2). Furthermore, the uniqueness of f * is readily seen from the convexity of J(f ).
Next, we justify the stability of (2.2), namely, the minimization system (2.2) is indeed a stabilization for Problem 1.1 with respect to the perturbation in observation data.
Theorem 2.2. Let {u δ ℓ } ⊂ L 2 (ω × (0, T )) be a sequence such that u δ ℓ → u δ in L 2 (0, T ; L 2 (ω)) as ℓ → ∞, (2.8)
and {f ℓ } be a sequence of minimizers of problems
min f ∈L 2 (Ω) J ℓ (f ), J ℓ (f ) := u(f ) − u δ ℓ 2 L 2 (ω×(0,T )) + β f 2 L 2 (Ω) , ℓ = 1, 2, . . . .
Then {f ℓ } converges strongly in L 2 (Ω) to the minimizer of (2.2).
Proof. The unique existence of each f ℓ is guaranteed by Theorem 2.1. By definition, we have
J ℓ (f ℓ ) ≤ J ℓ (f ), ∀ f ∈ L 2 (Ω),
which implies the uniform boundedness of f ℓ in L 2 (Ω). Hence, there exist f * ∈ L 2 (Ω) and a subsequence of {f ℓ }, still denoted by {f ℓ }, such that
f ℓ ⇀ f * in L 2 (Ω) as ℓ → ∞.
Now it suffices to show that f * is indeed the unique minimizer of (2.2). Actually, repeating the same argument as that in the proof of Theorem 2.1, we can derive
u(f ℓ ) ⇀ u(f * ) in 0 H α (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) as ℓ → ∞,
up to taking a further subsequence. Combining the above convergence with (2.8), we obtain
u(f ℓ ) − u δ ℓ ⇀ u(f * ) − u δ in L 2 (0, T ; L 2 (ω)) as ℓ → ∞.
Therefore, we get
u(f * ) − u δ 2 L 2 (ω×(0,T )) ≤ lim inf ℓ→∞ u(f ℓ ) − u δ ℓ 2 L 2 (ω×(0,T )) . (2.9)
For any f ∈ L 2 (Ω), again we take advantage of the lower semi-continuity of the L 2 -norm to deduce
J(f * ) = u(f * ) − u δ 2 L 2 (ω×(0,T )) + β f * 2 L 2 (Ω) ≤ lim inf ℓ→∞ u(f ℓ ) − u δ ℓ 2 L 2 (ω×(0,T )) + β lim inf ℓ→∞ f ℓ 2 L 2 (Ω) ≤ lim inf ℓ→∞ u(f ℓ ) − u δ ℓ 2 L 2 (ω×(0,T )) + β f ℓ 2 L 2 (Ω) ≤ lim ℓ→∞ u(f ) − u δ ℓ 2 L 2 (ω×(0,T )) + β f 2 L 2 (Ω) = u(f ) − u δ 2 L 2 (ω×(0,T )) + β f 2 L 2 (Ω) = J(f ), ∀ f ∈ L 2 (Ω),(2.10)
which verifies that f * is the minimizer of (2.2). Next, we shall prove {f ℓ } converges to f * strongly in L 2 (Ω) by contradiction. Assuming that it is not true, then we know that { f ℓ L 2 (Ω) } does not converge to f * L 2 (Ω) . As f ℓ ⇀ f * in L 2 (Ω), by the weak lower semi-continuity of the norm, we have
f * L 2 (Ω) ≤ lim inf ℓ→∞ f ℓ L 2 (Ω) .
Hence, setting A := lim sup ℓ→∞ f ℓ L 2 (Ω) , we get
A = lim sup ℓ→∞ f ℓ L 2 (Ω) > lim inf ℓ→∞ f ℓ L 2 (Ω) ≥ f * L 2 (Ω) ,(2.11)
and there exists a subsequence {f m } of {f ℓ } such that
A = lim m→∞ f m L 2 (Ω)
.
Now taking f = f * in (2.10), we find that u(f * ) − u δ 2 L 2 (ω×(0,T )) + β f * 2 L 2 (Ω) = lim inf m→∞ u(f m ) − u δ m 2 L 2 (ω×(0,T )) + β f m 2 L 2 (Ω) = lim inf m→∞ u(f m ) − u δ m 2 L 2 (ω×(0,T )) + βA 2 .
This, together with (2.11), implies that
u(f * ) − u δ 2 L 2 (ω×(0,T )) = lim inf m→∞ u(f m ) − u δ m 2 L 2 (ω×(0,T )) + β A 2 − f * 2 L 2 (Ω) > lim inf m→∞ u(f m ) − u δ m 2 L 2 (ω×(0,T )) ,
which contradicts with (2.9). The proof of Theorem 2.2 is completed.
Fully Discrete Finite Element Approximation
In this section we propose a fully discrete finite element method for approximating the nonlinear optimization problem (2.2). We first introduce some appropriate time and space discretization. For the space discretization, we consider a shape regular triangulation T h of Ω with a mesh size h, consisting of tetrahedral elements (see [2]). Then we introduce the standard nodal finite element spaces of piecewise linear functions
V h = v h ∈ H 1 (Ω); v h | A ∈ P 1 (A), ∀ A ∈ T h , S h = f h ∈ L 2 (Ω); f h | A ∈ P 1 (A), ∀ A ∈ T h , where P 1 (A) is the space of linear polynomials on A. It is obvious that V h ⊂ S h .
To fully discretize the minimization problem (2.2), we also need the time discretization. To this end, we divide the time interval [0, T ] into M subintervals by the equidistant nodal points
0 = t 0 < t 1 < · · · < t M = T with t m = m τ , where τ = T /M is the step length.
Hereinafter, we additionally assume that the noisy observation data u δ satisfies u δ ∈ C([0, T ]; L 2 (ω)), so that each u δ ( · , t m ) makes sense in L 2 (ω).
To discretize the Caputo derivative in time, we refer to [7,14] and use the following approximation:
∂ α t u(x, t m+1 ) = 1 Γ(1 − α) m j=0 tj+1 tj ∂ s u(x, s) (t m+1 − s) α ds = 1 Γ(1 − α) m j=0 u(x, t j+1 ) − u(x, t j ) s tj+1 tj (t m+1 − s) −α ds + r m+1 τ = 1 Γ(2 − α) m j=0 d j u(x, t m+1−j ) − u(x, t m−j ) τ α + r m+1 τ = 1 Γ(2 − α) τ α m+1 j=0 γ j u(x, t m+1−j ) + r m+1 τ :=∂ α τ u(x, t m+1 ) + r m+1 τ , (3.1)
where d j := (j + 1) 1−α − j 1−α for j = 0, 1, . . . , m, and∂ α τ u(x, t m+1 ) and r m+1 τ denote the L1 numerical approximation 1 and local truncation error respectively. The coefficients γ j (j = 0, 1, . . . , m + 1) in the discrete convolution quadrature (3.1) are given by
γ j = 1, j = 0, (j + 1) 1−α − 2j 1−α + (j − 1) 1−α , j = 1, . . . , m, m 1−α − (m + 1) 1−α , j = m + 1.
Following the same line as that in [7], now we arrive at a fully discrete scheme for the forward problem (1.1):
given u 0 h = 0, find u m+1 h ∈ V h (m = 0, . . . , M − 1) such that (1 + b 0 ) u m+1 h , χ h + b 0 ∇u m+1 h , ∇χ h = m−1 j=0 (d j − d j+1 ) u m−j h , χ h + b 0 f h µ m+1 , χ h (3.2) or equivalently, find u m h ∈ V h (m = 1, . . . , M ) such that ∂ α τ u m h , χ h + (∇u m h , ∇χ h ) + (u m h , χ h ) = (f h µ m , χ h ) (3.3) holds for all χ h ∈ V h , where b 0 := Γ(2 − α) τ α .
Here and hereinafter, we understand f h ∈ S h as some approximation of f and µ m = µ(t m ). To emphasize the dependency, we also denote the solution to (3.2) or (3.3) as u m h (f h ). On the basis of (3.2), now we are well prepared to propose the fully discrete finite element approximation of the nonlinear optimization (2.2): To do so, we need the following lemmas which allows us to use the energy-like methods in the analysis. Lemma 3.1 (see [23]). For the L1 numerical approximation (3.1) of the Caputo derivative, there holds
min f h ∈S h J h,τ (f h ), J h,τ (f h ) := τ M m=0 c m u m h (f h ) − u δ ( · , t m ) 2 L 2 (ω) + β f h 2 L 2 (Ω) .∂ α τ u m h 2 L 2 (Ω) ≤ 2u m h ,∂ α τ u m h , m ≥ 1. (3.5)
Lemma 3.2 (see [11]). Suppose that the nonnegative sequences {v n } and {g n } satisfȳ
∂ α τ v n ≤ λ 1 v n + λ 2 v n−1 + g n , n ≥ 1,
where λ 1 , λ 2 are given positive constants independent of the time step τ . Then there exists a positive constant τ * > 0 such that
v n ≤ 2 v 0 + t α n Γ(1 + α) max 0≤j≤n g j E α (2λ t α n ) for 0 < τ < τ * , where E α (z) = ∞ k=0 z k Γ(1+kα)
is the Mittag-Leffler function and λ = λ 1 + λ2 2−2 1−α .
Just as the exponential function plays a fundamental role in the integral order equation, the Mittag-Leffler function naturally appears in the fractional order calculus, and it has many nice properties similar to the exponential function. In fact, we have E 1 (±z) = e ±z for all z ∈ C. Let us mention a couple of properties that we will use later, and refer to the recent monograph [4] for further details.
For α > 0, the Mittag-Leffler function E α (z) is an entire function. In particular, for real variable ξ ∈ R and α ∈ (0, 1), E α (ξ) is strictly positive and monotone increasing. Moreover, we have E α (ξ) > 0,
E α (0) = 1, lim ξ→−∞ E α (ξ) = 0, lim ξ→+∞ E α (ξ) = +∞, d dξ E α (ξ) = 1 α E α,α (ξ) > 0, where E α,α (ξ) := ∞ k=0 ξ k Γ(α+kα)
is the two parameter Mittag-Leffler function. These properties can be derived from the definition of E α (ξ) and the basic properties of gamma function Γ(ξ) for ξ > 0. In view of the above properties for Mittag-Leffler functions, we know that the function E α (2λ t α n ) involved in Lemma 3.2 is greater than one. Now we proceed to the proof the uniform H 1 -norm estimate for the solution u m h (f h ) to the fully discrete scheme (3.2) or (3.3).
(i) u m h (f h ) L 2 (Ω) ≤ C 0 f h L 2 (Ω) , (ii) ∇u m h (f h ) 2 L 2 (Ω) ≤ C 1 f h 2 L 2 (Ω) ,(3.6)
where C 0 , C 1 > 0 are constants independent of h and τ .
Proof. We adopt an inductive argument to prove (i) in (3.6). First, for m = 1, it follows immediately from (3.2) and u 0 h = 0 that
(1 + b 0 ) u 1 h (f h ), χ h + b 0 ∇u 1 h (f h ), ∇χ h = b 0 f h µ 1 , χ h , ∀ χ ∈ V h . Substituting χ h = u 1 h (f h ) into the above equation, we obtain b 0 u 1 h (f h ) 2 H 1 (Ω) ≤ (1 + b 0 ) u 1 h (f h ) 2 L 2 (Ω) + b 0 ∇u 1 h (f h ) 2 L 2 (Ω) = b 0 f h µ 1 , u 1 h (f h ) ≤ C b 0 f h L 2 (Ω) u 1 h (f h ) H 1 (Ω) , which implies u 1 h (f h ) H 1 (Ω) ≤ C 0 f h L 2 (Ω) . (3.7)
Next, assuming that (i) in (3.6) holds for some m ≥ 1, we shall prove that it also holds for m + 1. Choosing χ h = u m+1 h (f h ) in (3.2) and employing the monotone decreasing property of the sequence {d j }, we obtain
(1 + b 0 ) u m+1 h (f h ) 2 L 2 (Ω) + b 0 ∇u m+1 h (f h ) 2 L 2 (Ω) ≤ m−1 j=0 (d j − d j+1 ) u m−j h (f h ) L 2 (Ω) + C b 0 f h L 2 (Ω) u m+1 h (f h ) L 2 (Ω) ≤ (C 0 (1 − d m ) + C b 0 ) f h L 2 (Ω) u m+1 h (f h ) L 2 (Ω) .
By the same argument as that in the proof of [7, Lemma 4.1], we can suitably choose C 0 > 0 such that
C 0 (1 − d m ) + C b 0 ≤ C 0
holds uniformly for all m = 1, 2, . . .. Then we immediately obtain u m+1
h (f h ) L 2 (Ω) ≤ C 0 f h L 2 (Ω) .
We now prove the assertion (ii) in (3.6). Taking
χ h = 2∂ α τ u m h in (3.3) yields ∂ α τ u m h , 2∂ α τ u m h + ∇u m h , ∇(2∂ α τ u m h ) + u m h , 2∂ α τ u m h = f h µ m , 2∂ α τ u m h , (3.8) where ∇u m h , ∇(2∂ α τ u m h ) = ∇u m h , 2∂ α τ ∇u m h ≥∂ α τ ∇u m h 2 L 2 (Ω)
by (3.5). Then (3.8) leads to
2 ∂ α τ u m h 2 L 2 (Ω) +∂ α τ ∇u m h 2 L 2 (Ω) ≤ 2 ∂ α τ u m h 2 L 2 (Ω) + u m h 2 L 2 (Ω) + f h µ m 2 L 2 (Ω) .
Hence, we have that
∂ α τ ∇u m h 2 L 2 (Ω) ≤ u m h 2 L 2 (Ω) + f h µ m 2 L 2 (Ω) ≤ C 2 ∇u m h 2 L 2 (Ω) + Ω u m h dx 2 + f h µ m 2 L 2 (Ω) ≤ C 2 ∇u m h 2 L 2 (Ω) + C u m h 2 L 2 (Ω) + f h 2 L 2 (Ω) ≤ C 2 ∇u m h 2 L 2 (Ω) + C 3 f h 2 L 2 (Ω) , where we made use of the Poincaré-Friedrichs inequality u m h L 2 (Ω) ≤ C ∇u m h L 2 (Ω) + Ω u m h dx .
Now applying Lemma 3.2 and (3.7), we obtain that
∇u m h 2 L 2 (Ω) ≤ 2 ∇u 1 h 2 L 2 (Ω) + t α m Γ(1 + α) C 3 f h 2 L 2 (Ω) E α (2C 2 t α m ) ≤ 2 C 0 f h 2 L 2 (Ω) + t α m Γ(1 + α) C 3 f h 2 L 2 (Ω) E α (2C 2 t α m ) ≤ 2 C 0 + T α Γ(1 + α) C 3 E α (2C 2 T α ) f h 2 L 2 (Ω) ≤ C 1 f h 2 L 2 (Ω) .
This completes the proof.
The next theorem provides the existence of a solution to the discrete system (3.4).
Theorem 3.1. For each fixed τ > 0 and h > 0, there exists at least a minimizer to the discrete system (3.4).
Proof. It is readily seen from the non-negativity of
J h,τ (f h ) that inf J h,τ (f h ) is finite. Then there exists a minimizing sequence {f n h } ⊂ S h such that lim n→∞ J h,τ (f n h ) = inf f h ∈S h J h,τ (f h ).
Then {f n h } is uniformly bounded in S h ⊂ L 2 (Ω). Therefore, there exist f * h ∈ S h and a subsequence of
{f n h }, still denoted by {f n h }, such that f n h ⇀ f * h in L 2 (Ω) as n → ∞.
Similarly to the proof of Theorem 2.1, we shall prove that f * h is a minimizer of (3.4). Since τ > 0 is now a fixed constant, if follows from
(1 + b 0 ) u m+1 h (f n h ), χ h + b 0 ∇u m+1 h (f n h ), ∇χ h = m−1 j=0 (d j − d j+1 ) u m−j h (f n h ), χ h + b 0 f n h µ m+1 h , χ h for all χ h ∈ V h .
Passing n → ∞ in the above equation and using (3.9), we obtain
(1 + b 0 ) u m+1 h, * , χ h + b 0 ∇u m+1 h, * , ∇χ h = m−1 j=0 (d j − d j+1 ) u m−j h, * , χ h + b 0 f * h µ m+1 h , χ h for all χ h ∈ V h . By definition (3.
2) again, we conclude u m h, * = u m h (f * h ) by setting u 0 h, * = 0 artificially. Finally, collecting the above results, we again employ the lower semi-continuity to deduce
J h,τ (f * h ) = τ M m=0 c m u m h (f * h ) − u δ ( · , t m ) 2 L 2 (ω) + β f * h 2 L 2 (Ω) = lim n→∞ τ M m=0 c m u m h (f n h ) − u δ ( · , t m ) 2 L 2 (ω) + β lim inf n→∞ f n h 2 L 2 (Ω) ≤ lim inf n→∞ J h,τ (f n h ) = inf f h ∈S h J h,τ (f h )
.
Consequently, f * h is indeed a minimizer of J h,τ (f h ) over S h .
Convergence of the Fully Discrete Approximation
This section is devoted to the convergence analysis of the fully discrete finite element approximation (3.4).
In order to relate (3.4) with the continuous minimization problem (2.2), we start with the introduction of a few auxiliary tools and useful results. First, we recall the standard L 2 projection P h : L 2 (Ω) → V h and Ritz projection R h : H 1 (Ω) → V h , defined respectively by
(P h ϕ, χ h ) = (ϕ, χ h ), ∀ χ h ∈ V h , (∇R h ϕ, χ h ) = (∇ϕ, χ h ), ∀ χ h ∈ V h .
The operators P h and R h satisfy the following approximation properties (see [2,22]):
(I − P h )ϕ H γ (Ω) ≤ C h 1−γ ϕ H 1 (Ω) , ∀ ϕ ∈ H 1 (Ω), γ ∈ [0, 1], (4.1) (I − R h )ϕ L 2 (Ω) + h ∇(ϕ − R h ϕ) L 2 (Ω) ≤ C h γ ϕ H γ (Ω) , ∀ ϕ ∈ H γ (Ω), γ ∈ [1, 2]. (4.2)
The next lemma is a useful classical approximation result (see [27,28]).
Lemma 4.1. For a Banach space X and a function g ∈ C([0, T ]; X), define a step function approximation by
S ∆ g(x, t) = M m=1 1 (tm−1,tm] (t) g(x, t m ),
where 1 (tm−1,tm] is the characteristic function of (t m−1 , t m ]. Then we have the convergence
lim τ →0 T 0 S ∆ g( · , t) − g( · , t) 2 X dt = 0.
In order to demonstrate the convergence of the finite element approximation (3.4) to the continuous minimization problem (2.2), we shall turn to the following key error estimate, which is a straightforward consequence of [6, Theorem 3.6].
u(f )( · , t m ) − u m h (P h f ) L 2 (Ω) ≤ C f L 2 (Ω) µ C 1 [0,T ] h 2 | ln h| 2 + t α−1 m τ . (4.3)
Remark 4.1. According to [6,Theorem 3.6], the source term f µ should satisfy
t 0 (t − s) α−1 f µ ′ (s) L 2 (Ω) ds < ∞ (4.4)
for 0 < t ≤ T . This is automatically guaranteed by the assumption f ∈ L 2 (Ω) and µ ∈ C 1 [0, T ]. Therefore, the term including (4.4) with t = t m in [6, Theorem 3.6] is of order O(τ ), which is absorbed in the last term in (4.3). On the other hand, since
t α−1 m τ ≤ τ α → 0 as τ → 0, (4.3) immediately implies lim h,τ →0 u(f )( · , t m ) − u m h (P h f ) L 2 (Ω) = 0, m = 1, . . . , M.
The following lemma provides a crucial strong convergence.
lim h,τ →0 τ M m=0 c m u m h (f h ) − u δ ( · , t m ) 2 L 2 (ω) = u(f ) − u δ 2 L 2 (ω×(0,T )) . (4.5)
Proof. For simplicity, in this proof we abbreviate u m := u(f )( · , t m ). Since we have u(f ) ∈ C([0, T ]; L 2 (Ω)) by Lemma 2.2, first we can directly apply Lemma 4.1 to obtain ω×(0,T )) .
lim τ →0 τ M m=0 c m u m − u δ ( · , t m ) 2 L 2 (ω) = u(f ) − u δ 2 L 2 (
Using this convergence, (4.5) follows immediately if we can demonstrate
lim h,τ →0 τ M m=0 c m u m h (f h ) − u δ ( · , t m ) 2 L 2 (ω) = lim τ →0 τ M m=0 c m u m − u δ ( · , t m ) 2 L 2 (ω) . (4.6)
In order to show (4.6), we utilize the a priori estimates of u m h (f h ) and u m to deduce
τ M m=0 c m u m h (f h ) − u δ ( · , t m ) 2 L 2 (ω) − τ M m=0 c m u m − u δ ( · , t m ) 2 L 2 (ω) ≤ τ M m=0 u m h (f h ) − u m 2 L 2 (ω) 1 2 τ M m=0 u m h (f h ) + u m − 2 u δ ( · , t m ) 2 L 2 (ω) 1 2 ≤ C τ M m=0 u m h (f h ) − u m 2 L 2 (Ω) 1 2 .
Clearly, the convergence in (4.6) follows if there holds
lim h,τ →0 τ M m=0 u m h (f h ) − u m 2 L 2 (Ω) = 0. (4.7)
To this end, we split the error u m h (f h ) − u m into two parts:
u m h (f h ) − u m = (u m h (f h ) − u m h (P h f )) + (u m h (P h f ) − u m ) =: η m h + θ m h .
From the estimate (4.3) and Remark 4.1, we can easily see that θ m h L 2 (Ω) → 0 as h, τ → 0. Now we concentrate on the estimate of η m h . By the definitions of u m h (f h ) and u m h (P h f ), it is readily seen that η m h satisfies
(1 + b 0 ) η m+1 h , χ h + b 0 ∇η m+1 h , ∇χ h = m−1 j=0 (d j − d j+1 ) η m−j h , χ h + b 0 (f h − P h f )µ m+1 h , χ h (4.8)
for all χ h ∈ V h . By the assumption of Lemma 4.3 and (4.1), we have Picking any χ ∈ H 1 (Ω), we take χ h = R h χ ∈ V h as the test function in (4.8). Then it follows from (4.2) that χ h → χ in H 1 (Ω) as h → 0. Together with (4.9), we derive
f h ⇀ f, P h f → f in L 2 (Ω) as h → 0, which indicates f h − P h f ⇀ 0 in L 2 (Ω) as h → 0. Meanwhile,∇η m+1 h , ∇χ h = ∇η m+1 h , ∇χ + ∇η m+1 h , ∇(χ h − χ) → ∇η m+1 * , ∇χ as h → 0.
Using (4.9) again, we pass h → 0 in equation (4.8) to deduce
(1 + b 0 ) η m+1 * , χ + b 0 ∇η m+1 * , ∇χ = m−1 j=0 (d j − d j+1 ) η m−j * , χ ,
which turns out to be the semidiscrete scheme in time for the original problem (1.1) with f = 0. Then it follows immediately from [14] that η m * L 2 (Ω) = 0 or equivalently η m * = 0. Consequently, we obtain η m h L 2 (Ω) → 0 as h → 0, and finally conclude
u m h (f h ) − u m L 2 (Ω) ≤ θ m h L 2 (Ω) + η m h L 2 (Ω) → 0
as h, τ → 0. This completes the proof of (4.7).
We conclude this section with the convergence of the fully discrete finite element approximation (3.4) to the continuous minimization problem (2.2). Proof. It is routine to get the boundedness of {f * h } in L 2 (Ω). Thus there exist f * ∈ L 2 (Ω) and a subsequence of {f * h }, still denoted by {f * h }, such that f * h ⇀ f * in L 2 (Ω) as h → 0. Now it suffices to show that f * is the minimizer of the continuous problem (2.2). For any f ∈ L 2 (Ω), we have from (4.1) that
P h f ∈ S h , lim h→0 f − P h f L 2 (Ω) = 0. (4.10) Since f * h is the minimizer of J h,τ (f h ) over S h , we obtain J h,τ (f * h ) ≤ J h,τ (P h f ). (4.11)
Then using Lemma 4.3, (4.10), (4.11) and the lower semi-continuity of the L 2 -norm, we deduce
J(f * ) = u(f * ) − u δ 2 L 2 (ω×(0,T )) + β f * 2 L 2 (Ω) ≤ lim h,τ →0 τ M m=0 c m u m h (f * h ) − u δ ( · , t m ) 2 L 2 (ω) + β lim inf h→0 f * h 2 L 2 (Ω) ≤ lim inf h,τ →0 J h,τ (f * h ) ≤ lim inf h,τ →0 J h,τ (P h f ) ≤ lim h,τ →0 τ M m=0 c m u m h (P h f ) − u δ ( · , t m ) 2 L 2 (ω) + β P h f 2 L 2 (Ω) = u(f ) − u δ 2 L 2 (ω×(0,T )) + β f 2 L 2 (Ω) = J(f ),
which implies that f * is indeed a minimizer of the continuous problem (2.2).
Iterative Thresholding Algorithm and Numerical Examples
In this section, we develop an efficient iterative thresholding algorithm to solve the discrete minimization problem (3.2)-(3.4) and present several numerical experiments to show its efficiency and accuracy. Nearly all effective iterative methods for solving nonlinear optimization problems need the information of the derivatives of the concerned objective functional. We shall first derive the derivative of the discrete nonlinear functional J h,τ (f h ). Since u m h (f h ) is linear with respect to f h in view of (3.2), we have
u m h (f h ) ′ p h = u m h (p h ) for any p h ∈ S h . Hence, it is straightforward to obtain J ′ h,τ (f h )p h = 2τ M m=0 c m ω u m h (f h ) − u δ u m h (p h ) dx + 2β (f h , p h ). (5.1)
Needless to say, it is extremely expensive to use the above formula directly to evaluate the derivatives, since computing the derivative at one fixed point f h needs to solve equation (3.2) once for every direction p h ∈ S h . In order to reduce the computational costs for computing the derivatives, we recall the backward Riemann-Liouville derivative
D α t g(t) := − 1 Γ(1 − α) d dt T t g(s) (s − t) α ds = − d dt J 1−α T − g(t)
and introduce the following adjoint system
(D α t − △ + 1)v = 1 ω u(f ) − u δ in Q, J 1−α T − v = 0 in Ω × {T }, ∂ ν v = 0 on ∂Ω × (0, T ),(5.2)
where 1 ω denotes the characterization function of ω. As before, we still denote the solution of (5.2) as v(f ) to emphasize its dependency upon f .
Since p h ∈ S h was chosen arbitrarily, the above equality gives the following necessary condition for some element f * h ∈ S h to be a minimizer of the discrete optimization problem (3.4):
τ M m=0 c m µ m h v m h (f * h ) + β f * h = 0.
Following the same line as that in [3,5], this variational equation can be solved by the following iterative thresholding algorithm
f k+1 h = L L + β f k h − τ L + β M m=0 c m µ m h v m h (f k h ), k = 0, 1, . . . ,(5.6)
where L > 0 is a tuning parameter for the convergence. The choice of L should be larger than the operator norm of the forward operator
A : L 2 (Ω) −→ L 2 (ω × (0, T )), f −→ u(f ) ω×(0,T ) ,
and we refer to [5] for details. It is obvious that at each step of (5.6), we only need to solve equation (3.2) for u m h (f k h ) and then the discrete formulation of system (5.2) for v m h (f k h ) subsequently, which does not involve heavy computational costs. By the way, although there appears the backward Riemann-Liouville derivative D α T − in (5.2), we know that the solution v(f ) coincides with the following problem with a backward Caputo derivative
(∂ α T − − △ + 1)v = 1 ω u(f ) − u δ in Q, v = 0 in Ω × {T }, ∂ ν v = 0 on ∂Ω × (0, T ), (5.7) where ∂ α T − g(t) := − 1 Γ(1 − α) T t g ′ (s) (s − t) α ds.
Therefore, instead of (5.2) we just discretize system (5.7) to compute v m h (f h ) by almost the same discrete method for u m h (f h ). We are now ready to propose the iterative thresholding algorithm for the reconstruction.
If
f k+1 h − f k h L 2 (Ω) / f k h L 2 (Ω) < ǫ, stop the iteration.
Otherwise, update k ← k + 1 and return to Step 1.
Next we shall present several numerical experiments by applying Algorithm 5.1 to solve Problem 1.1 for d = 1, 2. In general, we set Ω = (0, 1) d , T = 1 and specify various parameters involved in Algorithm 5.1 as follows. Without loss of generality, we always choose the initial guess f 0 h as a constant, e.g., f 0 h ≡ 2. With the true source term f * and thus the noiseless data u(f * ), the noisy data u δ is generated as
u δ (x, t) = (1 + δ rand(−1, 1)) u(f * )(x, t), x ∈ ω, 0 < t ≤ T,err := f K h − f * L 2 (Ω) f * L 2 (Ω) ,
where we understand f K h as the result of the numerical reconstruction. In the one-dimensional case, we divide the space-time region Ω×[0, T ] = [0, 1] 2 into 40×40 equidistant meshes. We set the tuning parameter L, the stopping criteria ǫ and the known component µ(t) as L = 1, ǫ = 2 × 10 −3 and µ(t) = 5 + 10 t respectively in Algorithm 5.1. Except for the factors mentioned above, we shall test the numerical performance with different choices of exact source terms f * , fractional orders α, noise levels δ and observation subdomains ω.
Example 5.1. In this example, we fix the observation subdomain ω and the noise level δ as ω = Ω \ [1/20, 19/20] and δ = 1%, respectively. We test Algorithm 5.1 with different fractional orders α and exact source terms f * as follows: Table 1. We can see from Figures 1 that with different fractional orders α and 1% noise in the observation data, the numerical reconstructions f K h appear to be quite satisfactory in view of the ill-posedness of Problem 1.1, regardless of the casual choice of the initial guess f 0 h . In addition, we can observe from Table 1 that Algorithm 5.1 has two important advantages, namely, its strong robustness against the oscillating noise in observation data, and its insensitivity to the smallness of the observation subdomain ω.
(a) α = 0.3, f * (x) = sin(πx/2) + x 2 + 1. (b) α = 0.5, f * (x) = sin(πx) − 2.
Now we consider the more challenging two-dimensional case, where we divide the space-time region Ω×[0, T ] = [0, 1] 3 into 40×40×20 equidistant meshes. Here we set the tuning parameter L and the known component µ(t) in Algorithm 5.1 as L = 2 and µ(t) = 1 + 10π t 2 respectively. Analogously to the onedimensional counterpart, we evaluate the numerical performance of Algorithm 5.1 from various aspects, including different combinations of the true source terms, noise levels and observation subdomains.
Example 5.3. Parallel to Example 5.1, first we fix the observation subdomain ω and the noise level δ as ω = Ω \ [1/10, 9/10] 2 and δ = 1%, respectively. We take the stopping criteria ǫ = δ/3 and specify two pairs of fractional orders and true source terms as follows:
(a) α = 0.3, f * (x) = sin(x 1 ) + sin(x 2 ) + 1.
(b) α = 0.5, f * (x) = cos(πx 1 ) cos(πx 2 ) + 2.
In Figure 2, we illustrate the true source terms f * with the corresponding reconstructions f K h , and the iteration steps K and the relative errors are shown in the caption.
Example 5.4. In this example we fix α = 0.8, f * (x) = exp((x 1 + x 2 )/4) + 1 and test Algorithm 5.1 with various choices of noise levels δ and observation subdomains ω in a similar manner as that in Example 5.2. Here we set ǫ = δ/5 as the stopping criteria. For the choice of ω, we not only adjust its size, but also change its coverage of the boundary ∂Ω. The resulting numerical performance of the reconstructions is listed in Table 2.
It is readily seen from Examples 5.3-5.4 that Algorithm 5.1 also works efficiently and accurately in the two-dimensional case. It inherits almost all advantages witnessed in the one-dimensional tests in view of its strong robustness against oscillating noise as well as its insensitivity to the smallness of the observation subdomain.
91])
− s) α ds,
Problem 1 . 1 .
11Let ω ⊂ Ω be a nonempty open subdomain and u(f ) be the solution to (1.1). Provided that µ is known on [0, T ], determine the spatial component f in Ω by the partial interior observation of u(f ) in ω × (0, T ).
Lemma 2. 3 .
3Let ω ⊂ Ω be an arbitrarily chosen open subdomain and u(f ) be the solution to (1.1). Assume that f ∈ L 2 (Ω) and µ ∈ C 1 [0, T ] with µ(0) = 0. Then u = 0 in ω × (0, T ) implies f = 0 in Ω.
m (m = 0, 1, . . . , M ) are the coefficients of the composite trapezoidal rule for the time integration over [0, T ], i.e., c 0 = c M = 1 2 and c m = 1 for m = 1, . . . , M − 1. Before verifying the existence of a minimizer to the discrete minimization problem (3.4), we shall derive some useful a priori estimates for the discrete solution u m h (f h ) to (3.2) or (3.3).
Lemma 3. 3 .
3The fully discrete scheme (3.2) or (3.3) is unconditional stable. Moreover, the following estimates hold for m = 1, . . . , M :
Lemma 3.3 that for each m = 1, . . . , M , the sequence {u m h (f n h )} is uniformly bounded in V h ⊂ L 2 (Ω) with respect to n. Again, for each m = 1, . . . , M , this indicates the existence of u m h, * ∈ V h and a subsequence of {u m h (f n h )}, still denoted by {u m h (f n h )}, such that u m h (f n h ) ⇀ u m h, * in L 2 (Ω) as n → ∞, m = 1, . . . , M. In view of the norm equivalence in finite-dimensional spaces, the above two weak convergence results are actually strong, i.e., u m h (f n h ) → u m h, * in H 1 (Ω), m = 1, . . . , M, f n h → f * h in H 1 (Ω) as n → ∞. (3.9) Now it suffices to show u m h, * = u m h (f * h ). In fact, by definition (3.2) and up to taking a further subsequence, we see that {u m h (f n h )} M m=1 satisfies
Lemma 4 . 2 .
42Let u(f ) be the solution to system (1.1), where f ∈ L 2 (Ω) and µ ∈ C 1 [0, T ]. Then for the solution {u m h (P h f )} to the fully discrete scheme (3.2) with f h = P h f , we have the following error estimate: for any m = 1, . . . , M , there holds
Lemma 4. 3 .
3Let {f h } h>0 be a sequence in S h which converges weakly to some f ∈ L 2 (Ω) as h → 0. Let {u m h (f h )}and u(f ) be the solutions of (3.2) and (1.1), respectively. Then the following convergence holds:
by Lemma 3.3 and f h , P h f ∈ S h , we estimate η m h H 1 (Ω) ≤ C f h − P h f L 2 (Ω) ≤ C, m = 1, . . . , M. Then for each m = 1, . . . , M , there exist some element η m * ∈ H 1 (Ω) and a subsequence of {η m h }, still denoted by {η m h }, such that η m h ⇀ η m * in H 1 (Ω), η m h → η m * in L 2 (Ω) as h → 0, m = 1, . . . , M. (4.9)
Algorithm 5 . 1 .
51Choose a tolerance ǫ > 0, a regularization parameter β > 0 and a tuning parameter L > 0. Give an initial guess f 0 h , and set k = 0. 1. Compute f k+1 h according to the iterative update (5.6).
where rand(−1, 1) denotes the random number uniformly distributed on [−1, 1]. For simplicity, in all examples we set the regularization parameter β = 10 −4 . Besides the illustrative figures, we mainly evaluate the numerical performance of Algorithm 5.1 by the number K of iterations and the relative L 2 error
Figure 1
1compares the true source terms f * with the corresponding reconstructions f K h , and also shows the iteration steps K and the relative errors in the caption.
Example 5 . 2 .
52In this example, we fix α = 0.8 and the true source term f * (x) = − sin(πx) + x + 4, and test different combinations of the observation subdomain ω and the noise level δ to observe their influence on the numerical performance. We first fix the noise level δ = 1% and shrink the size of ω from Ω \ [1/10, 9/10], Ω \ [1/20, 19/20] to Ω \ [1/40, 39/40]. Next, we keep ω = Ω \ [1/20, 19/20] and enlarge the noise level δ from 0.5%, 1%, 2% to 4%. The resulting numerical performance of the reconstructions is listed in
Figure 1 :
1True source terms f * and their reconstructions f K h obtained in Example 5.1. Left: Case (a), K = 29, err = 2.53%. Right: Case (b), K = 4, err = 2.51%.
Figure 2 :
2True source terms f * and their reconstructions f K h obtained in Example 5.3. Left: Case (a), K = 20, err = 6.26%. Right: Case (b), K = 36, err = 7.17%.
Table 1 :
1Numerical performance in Example 5.2 under various combinations of the observation subdomain ω and the noise level δ. 5% Ω \ [1/20, 19/20] 2.96% 16 2% Ω \ [1/20, 19/20] 5.53% 20 4% Ω \ [1/20, 19/20] 11.17% 24δ
ω
err
K
1%
Ω \ [1/10, 9/10]
3.75%
13
1%
Ω \ [1/20, 19/20] 3.38%
18
1%
Ω \ [1/40, 39/40] 4.17%
27
0.
Table 2 :
2Numerical performance in Example 5.4 under various combinations of the observation subdomain ω and the noise level δ. 5% Ω \ [1/10, 9/10] 2δ
ω
err
K
1%
Ω \ [1/10, 9/10] 2
2.63% 13
1%
Ω \ [1/20, 19/20] 2
4.94% 27
1%
Ω \ [0, 9/10] × [1/10, 9/10] 4.36% 9
0.2.47% 24
1%
Ω \ [1/10, 9/10] 2
3.61% 15
2%
Ω \ [1/10, 9/10] 2
5.06% 7
4%
Ω \ [1/10, 9/10] 2
6.58% 5
For historical reasons, the numerical scheme in (3.1) is often referred to as L1 numerical approximation for Caputo fractional derivative. One possible explanation is that in this scheme we replace the first derivative with a first order linear interpolation in each cell [t j , t j+1 ], and is called L1 scheme. This name has nothing to do with the usual notation L 1 (Ω) for integrable function in Ω.
Acknowledgement The authors thank Professor Bangti Jin (University College London) for his constructive discussions, and appreciate the valuable comments by the anonymous referees.Lemma 5.1. Let f, p ∈ L 2 (Ω) and u, v be the solutions of systems (1.1) and (5.2) respectively. Then there holdsProof. From the variational system (2.3), we have for any ϕ ∈ H α (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)) thatOn the other hand, it is routine to get the variational system associated with (5.2) that for any ψ ∈Further, by the integration by parts, the relation (2.1) and noting thatSubstituting the above equality into (5.5), we arrive at the desired equality.Writing the above equality into its discrete counterpart as that in previous sections, we haveTherefore, we can rewrite (5.1) as
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| []
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[
"Two-Loop master integrals for heavy-to-light form factors of two different massive fermions",
"Two-Loop master integrals for heavy-to-light form factors of two different massive fermions"
]
| [
"Long-Bin Chen [email protected] \nSchool of Physics & Electronic Engineering\nGuangzhou University\n230 GuangZhou University City Outer Ring Road510006GuangzhouChina\n"
]
| [
"School of Physics & Electronic Engineering\nGuangzhou University\n230 GuangZhou University City Outer Ring Road510006GuangzhouChina"
]
| []
| We calculate the full set of the two-loop master integrals for heavy-to-light form factors of two different massive fermions for arbitrary momentum transfer in NNLO QCD or QED corrections. These integrals allow to determine the two-loop QCD or QED corrections to the amplitudes for heavy-to-light form factors of two massive fermions in a full analytical way, without any approximations. The analytical results of the master integrals are derived using the method of differential equations, along with a proper choosing of canonical basis for the master integrals. All the results of master integrals are expressed in terms of Goncharov polylogarithms. | 10.1007/jhep02(2018)066 | [
"https://arxiv.org/pdf/1801.01033v3.pdf"
]
| 118,859,397 | 1801.01033 | 2684ed1998bf47a8b8c7008512d01c7153075930 |
Two-Loop master integrals for heavy-to-light form factors of two different massive fermions
6 Feb 2018
Long-Bin Chen [email protected]
School of Physics & Electronic Engineering
Guangzhou University
230 GuangZhou University City Outer Ring Road510006GuangzhouChina
Two-Loop master integrals for heavy-to-light form factors of two different massive fermions
6 Feb 2018Prepared for submission to JHEPFeynman integralsMulti-loop calculationsGoncharov PolylogarithmsDi- mensional regularizationForm factors
We calculate the full set of the two-loop master integrals for heavy-to-light form factors of two different massive fermions for arbitrary momentum transfer in NNLO QCD or QED corrections. These integrals allow to determine the two-loop QCD or QED corrections to the amplitudes for heavy-to-light form factors of two massive fermions in a full analytical way, without any approximations. The analytical results of the master integrals are derived using the method of differential equations, along with a proper choosing of canonical basis for the master integrals. All the results of master integrals are expressed in terms of Goncharov polylogarithms.
Introduction
The calculation of heavy-to-light form factors have a number of applications in particle physics. For instance, the semileptonic decay of a heavy quark to a light massive quark (t → b + W + (l +ν), b → c + l +ν), the decay of a massive lepton to anther massive lepton (µ → e+ν µ +ν e , τ → µ+ν τ +ν µ ) , and the decay of W bosons into two massive quarks. The NNLO QCD or QED corrections to the decay of a heavy fermion to a light fermion were computed in a number of papers [1][2][3][4][5][6][7][8][9], and the masses of the light fermions were neglected to simplify the calculation. In Ref. [10][11][12], the semileptonic decay b → c + l +ν have been calculated with the mass of charm quark taken into account, an expansion in powers and logarithms of mass ratio mc m b was performed there. It has been shown that in the calculation of similar NNLO corrections to light fermion energy spectrum in heavy fermion decay, the mass of the light fermion can not be neglected since there exist the large logarithm terms ln( m heavy m light ) [13]. These large logarithm terms cancel out in the calculation of the total rate which make the calculation simpler. At order O(α 2 ) or O(α 2 s ), double-logarithmic ln 2 ( m heavy m light ) and single-logarithmic ln( m heavy m light ) enhanced terms will appear, which make it impossible to compute the radiative corrections to quantities such as the light fermion energy spectrum by neglecting the mass of the light fermion from the very beginning. Having the masses as regulators can simplify the treatment of real emission processes, however the computation of virtual corrections is more complicate compared to a purely massless case. The full dependence on the muon and electron masses for electron energy spectrum in muon decay at NNLO QED corrections have first been calculated in a numerical way in [14]. However, the appearance of m 2 light m 2 heavy ≪ 1 will cause numerical instability for the numerical evaluation of multi-dimension integrals, which make the analytical calculations highly desirable. Moreover, the analytic expressions are obviously desirable in order to have control over errors in approximations, especially when the convergence of expansion in powers of mass ratio works slowly.
On the theoretical side, unravelling the mathematical structure of Feynman integrals will be important to handle the complexity of their calculation and may help us to obtain a better understanding of the perturbative quantum field theory. The study of the mathematical properties of Feynman integrals has attracted increasing attention both by the physics and the mathematics communities. Significant progresses were achieved in understanding the analytical computations of multi-loop Feynman integrals in the last years .
One of the powerful methods to evaluate the master integrals analytically is the method of differential equations [15][16][17][18][19]. Along with the recent years' development [20][21][22][23][24], this method is becoming more and more powerful. It is pointed out in Ref. [20] that for multiloop Feynman integrals calculations, a suitable basis of master integrals can be chosen, so that the corresponding differential equations are greatly simplified, and their iterative solutions become straightforward in terms of dimensional regularization parameter ǫ = 4−D 2 . The choice of canonical basis will also simplify the determination of boundary conditions considerably. Following this proposal, substantive analytical computations of various phenomenology processes have been completed [25][26][27][28][29][30][31][32][33][34][35][36].
The two-loop master integrals of QED electron form factors have been calculated in [37], for on shell electrons of finite squared mass and arbitrary momentum transfer. The calculations of those master integrals have been refined by a suitable choice of basis [23]. The analytical results of two-loop master integrals for form factors of heavy fermion to massless fermion have been obtained in [6, 38? ? ]. All the master integrals for these two processes can be expressed in terms of Harmonic polylogarithms. However, for twoloop vertex integrals with two different type of massive fermions, the master integrals will contain one more scale, to the best of our knowledge, the integrals have not been calculated analytically in the literature. Furthermore, understanding the structure of loop integrals more generally is an interesting and important challenge. In this work, employing the method of differential equations, along with a proper choice of canonical basis, we calculate all the master integrals for two-loop heavy-to-light form factors of two massive fermions, the results are expressed in terms of Goncharov polylogarithms.
The outline of the paper is as follows. In section 2, the kinematics and notations are introduced for the processes we concern. We also present the generic form of the differential equations with respect to the kinematics variables in terms of the derivatives of the external momentum. In section 3, the Goncharov polylogarithms as well as Harmonic polylogarithms are introduced. In section 4, the canonical basis are explicitly presented, followed by the discussion of their solutions. In sections 5, the determination of the boundary conditions, as well as the analytical continuation are explained. Discussions and conclusions are made in section 6. In Appendix A, we present all the rational matrices of the system of differential equations in canonical form. All the analytical results up to weight four from our computation as well as the rational matrices in electronic form are collected in ancillary files accompanying the arXiv version of this publication.
Notations and Kinematics
We consider the process of a heavy fermion decay into a light massive fermion, or the decay of W boson into two massive fermions,
f 1 (p 1 ) → V * (q) + f 2 (p 2 ), (2.1) V (q) → f 1 (p 1 ) + f 2 (p 2 ),(2.2)
with p 2 1 = m 2 1 and p 2 2 = m 2 2 . For the decay of a heavy fermion to a light massive fermion, the squared momentum transfer have the following relations
q 2 = (p 1 − p 2 ) 2 = s < (m 1 − m 2 ) 2 . (2.3)
While for the decay of W boson into two different massive fermions, we have
q 2 = (p 1 + p 2 ) 2 = s > (m 1 + m 2 ) 2 . (2.4)
We take the dimensionless variables x and y to express the analytical results, they are defined by
s = m 2 1 (x − y)(1 − x y)
x , and m 2 = m 1 y.
(2.5)
In the above equations m 1 is treated as constants, s and m 2 are considered as variable. The derivatives of s and m 2 2 can be written in terms of the derivatives of the external momenta and expressed as
∂ ∂s = 1 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) ((s − m 2 1 − m 2 2 )p 1 − 2m 2 2 p 2 ) · ∂ ∂p 1 , ∂ ∂m 2 2 = 1 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) ((−s − m 2 1 + m 2 2 )p 1 + (s − m 2 1 + m 2 2 )p 2 ) · ∂ ∂p 1 .
(2.6)
The corrections to the processes (2.1) and (2.2) could be calculated using Feynman diagram approach. All the amplitudes can be expressed in terms of a set of 40 scalar integrals. The calculation of these scalar integrals always turns out to be the toughest parts in the whole work. We use packages FIRE [39][40][41] to reduce the group of scalar integrals into a minimum set of independent master integrals. FIRE is also adopted in the derivations of differential equations for master integrals.
Goncharov polylogarithms and Harmonic polylogarithms
The Goncharov polylogarithms (GPLs) [42] we use to express the analytical results are defined as follow
G a 1 ,a 2 ,...,an (x) ≡ x 0 dt t − a 1 G a 2 ,...,an (x) , (3.1) G− → 0 n (x) ≡ 1 n! log n x . (3.2)
They can be viewed as a special case belonging to a more general type of functions named Chen-iterated integrals [43]. If all the index a i belong to the set {0, ±1}, the Goncharov polylogarithms turn into the well-known Harmonic polylogarithms (HPLs) [44]
H− → 0 n (x) = G− → 0 n (x) ,(3.3)
H a 1 ,a 2 ,...,an (x) = (−1) k G a 1 ,a 2 ,...,an (x), (3.4) where k equals to the times of element (+1) taken in (a 1 , a 2 , . . . , a n ) . The Goncharov polylogarithms fulfil the following shuffle rules
G a 1 ,...,am (x)G b 1 ,...,bn (x) = c∈aXb G c 1 ,c 2 ,...,c m+n (x) . (3.5)
Here, aXb is composed of the shuffle products of lists a and b. It is defined as the set of the lists containing all the elements of a and b, with the ordering of the elements of a and b preserved. Both the GPLs and HPLs can be numerically evaluated within the GINAC implementation [45,46]. A Mathematica package HPL [47,48] is also available to reduce and evaluate the HPLs. Both the GPLs and HPLs can be transformed to the functions of ln, Li n and Li 22 up to weight four, with the algorithms and packages described in [49].
The canonical basis
All the amplitudes of two-loop QCD or QED corrections for the processes we concern can be reduced to a set of 40 master integrals, including two non-planar integrals. The master integrals M i (i = 1 . . . 40) are shown in Fig. 1. The vector of canonical basis F is built up with 40 functions F i (s, m 1 , m 2 , ǫ)(i = 1 . . . 40), defined in terms of the linear combinations of master integrals M i .
F 1 = ǫ 2 M 1 , (4.1) F 2 = ǫ 2 M 2 , (4.2) F 3 = ǫ 2 M 3 , (4.3) F 4 = ǫ 2 m 2 2 M 4 ,(4.4)F 5 = ǫ 2 m 2 1 M 5 ,(4.5)F 6 = ǫ 2 m 2 2 M 6 ,(4.6)F 7 = ǫ 2 m 1 m 2 (2M 6 + M 7 ) ,(4.7)F 8 = ǫ 2 m 2 1 M 8 , (4.8) F 9 = ǫ 2 m 1 m 2 (2M 8 + M 9 ) ,(4.9)
F 10 = ǫ 2 s M 10 , (4.10)
F 11 = ǫ 2 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 (M 10 + M 11 + M 12 ) ,(4.11)F 12 = ǫ 2 (m 2 1 − m 2 2 ) (M 10 + M 11 + M 12 ) + s (M 11 − M 12 ) , (4.12) M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24 M25 M26 M27 M28 M29 M30 M31 M32 M33 M34 M35 M36 M37 M38 M39 M40 −(q1 + q2 − p1 + p2) 2 Figure 1.
All the NNLO master integrals for heavy-to-light form factors with different type of fermion masses. The thin lines denote massless propagators; the thick dashed lines denote massive fermion with mass m 1 ; and the thick solid lines denote massive fermion with mass m 2 . A dot on a propagator indicates that the power of the propagator is raised to 2. Two dots represents that the propagator is raised to power 3.
F 13 = ǫ 2 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 2(s − m 2 1 + m 2 2 ) (2s M 13 − M 2 + M 3 ) , (4.13) F 14 = ǫ 2 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 2(s + m 2 1 − m 2 2 ) (2s M 14 − M 1 + M 3 )
, (4.14)
F 15 = ǫ 2 s 2 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) (s + m 2 1 − m 2 2 ) 2 M 15 + s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 s + m 2 1 − m 2 2 (F 13 − F 14 ) ǫ 2 + s m 2 1 (s + m 2 1 − m 2 2 ) 2 (M 1 + M 2 − 2M 3 ) ,F 16 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 16 , (4.16) F 17 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 17 , (4.17) F 18 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 18 , (4.18) F 19 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 19 , (4.19) F 20 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 20 , (4.20) F 21 = ǫ 2 2s (ǫ(M 19 + 2M 20 ) − m 2 2 M 21 − 2m 2 1 M 22 ) +2(m 2 2 − m 2 1 )(ǫ(M 19 + 2M 20 ) + m 2 2 M 21 − 2m 2 1 M 22 ) + 2 m 2 m 1 F 9 , F 22 = ǫ 2 2(m 2 1 − m 2 2 )(2ǫ(M 19 + 2M 20 ) + (m 2 1 + m 2 2 − s)M 21 − 4m 2 1 M 22 ) +2 m 1 m 2 F 7 − 2 m 2 m 1 F 9 , (4.21) F 23 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 23 , (4.22) F 24 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 24 , (4.23) F 25 = ǫ 2 s m 2 2 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) (s − m 2 1 + m 2 2 ) 2 M 25 +ǫ 3 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) 2(s − m 2 1 + m 2 2 ) (M 23 − M 24 ) − m 2 s (s − m 2 1 − m 2 2 ) m 1 (s − m 2 1 + m 2 2 ) 2 F 9 − s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 4(s − m 2 1 + m 2 2 ) F 11 + s m 2 2 (s − m 2 1 + m 2 2 ) 2 (F 2 − F 3 − 6F 8 + 2F 12 ) ,(4.24)F 26 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 26 ,(4.25)F 27 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 27 ,(4.26)F 28 = ǫ 2 s m 2 1 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) (s + m 2 1 − m 2 2 ) 2 M 28 +ǫ 3 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) 2(s + m 2 1 − m 2 2 ) (M 26 − M 27 ) − m 1 s (s − m 2 1 − m 2 2 ) m 2 (s + m 2 1 − m 2 2 ) 2 F 7 − s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 4(s + m 2 1 − m 2 2 ) F 11 + s m 2 1 (s + m 2 1 − m 2 2 ) 2 (F 1 − F 3 − 6F 6 − 2F 12 ) ,(4.27)F 29 = ǫ 4 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 29 ,
(4.28) For our choice of master integrals above, the differential equations for F = (F 1 . . . F 34 ) have the following canonical form
F 30 = ǫ 3 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) M 30 , (4.29) F 31 = ǫ 4 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 31 , (4.30) F 32 = ǫ 3 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) M 32 , (4.31) F 33 = ǫ 4 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) M 33 , (4.32) F 34 = ǫ 4 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 34 ,(4.F 37 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 37 , (4.36) F 38 = ǫ 2 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) M 38 + s − m 2 1 − m 2 2 2m 1 m 2 F 7 , (4.37) F 39 = ǫ 3 s − (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 M 39 , (4.38) F 40 = ǫ 2 (s − (m 1 + m 2 ) 2 )(s − (m 1 − m 2 ) 2 ) M 40 + s − m 2 1 − m 2 2 2m 1 m 2 F 9 .dF(x, y; ǫ) = ǫ dÃ(x, y) F(x, y; ǫ),(4.41)
with A(x, y) = A 1 ln(x) + A 2 ln(x + 1) + A 3 ln(x − 1) + A 4 ln(x + y) + A 5 ln(x − y) +A 6 ln(x y + 1) + A 7 ln(x y − 1) + A 8 ln(y) + A 9 ln(y + 1) + A 10 ln(y − 1) +A 11 ln(x 2 y − 2x + y) + A 12 ln(x 2 − 2y x + 1).
(4.42)
The notations A i (i = 1 · · · 12) are 40×40 matrices with rational numbers, they are presented in appendix. A. For reader's convenience, the rational matrices in electronic form are also provided in ancillary file accompanying the arXiv version of this paper. We can see in equation (4.42) that in equal masses case with y = 1, the alphabet turns into {x, x+1, x−1}, and the results of all the canonical basis can simply be reexpressed in terms of Harmonic polylogarithms.
The integral M 1 is defined as follow
M 1 = D D q 1 D D q 2 1 (−q 2 1 + m 2 1 ) 2 1 (−q 2 2 + m 2 1 ) 2 = 1 ǫ 2 ,(4.43)
with the measure of the integration is defined as
D D q i = 1 π D/2 Γ(1 + ǫ) m 2 1 µ 2 ǫ d D q i . (4.44)
For master integrals without numerators, their definition can readily be read off from Fig. 1, with the normalization defined above. For master integrals with numerator, we first define a series of propagators
P 1 = m 2 2 − q 2 1 , P 2 = m 2 1 − q 2 2 , P 3 = −(q 1 − p 1 ) 2 , P 4 = −(q 2 + p 2 ) 2 , P 5 = m 2 2 − (q 1 + q 2 + p 2 ) 2 , P 6 = m 2 1 − (q 1 + q 2 − p 1 ) 2 , P 7 = −(q 1 + q 2 − p 1 + p 2 ) 2 .
(4.45)
Then, the master integral with numerator (M 34 ) can be expressed as
M 34 = D D q 1 D D q 2 P 7 P 1 P 2 P 3 P 4 P 5 P 6 .
(4.46)
Boundary conditions and analytic continuation
Now, we are ready to perform the calculations of the differential equations. The first step is to specify all the boundary conditions that will completely fix the solutions of the differential equations. The results of basis (F 1 . . . F 5 ) and (F 35 , F 36 ) are already known in the literature [25,[50][51][52]. They can be recalculated with the assistance of Mathematica packages MB [53] and AMBRE [54][55][56]. The integrals (F 6 . . . F 9 ) have been calculated in [50,51], their boundary conditions could be determined by setting y = m 2 m 1 = 1, with the masses of two fermions equaling to each other, and their boundaries are proportional to the results of F 35 .
The integral M 10 does not have singularity at s = 0. Thanks to the normalization factor s that multiplying with M 10 in F 10 , we can readily know that F 10 = 0 at s = 0. Considering the fact that M (11,13,14,16...20,23,24,26,27,29...34,37,39) are regular at s = (m 1 − m 2 ) 2 as well as s = (m 1 +m 2 ) 2 , and their normalization factor to be s
− (m 1 + m 2 ) 2 s − (m 1 − m 2 ) 2 or (s−(m 1 +m 2 ) 2 )(s−(m 1 −m 2 ) 2 )
, the boundaries of basis F (11,13,14,16...20,23,24,26,27,29...34,37,39) are 0 at s = (m 1 − m 2 ) 2 and s = (m 1 + m 2 ) 2 . The boundary of F 15 at s = (m 1 − m 2 ) 2 and s = (m 1 + m 2 ) 2 can also be determined similarly and then expressed as
F 15 | s={(m 1 −m 2 ) 2 ,(m 1 +m 2 ) 2 } = 1 4 (F 1 + F 2 − 2F 3 )| s={(m 1 −m 2 ) 2 ,(m 1 +m 2 ) 2 } . (5.1)
Considering the fact that M 12 does not have singularity at s = 0, i.e. (x = y) or (x = 1 y ) , the boundary condition of F 12 can be determined from the differential equation of F 12 . To further illustrate it, we consider the differential of F 12 with respect to variable x and find that
∂F 12 ∂x = ǫ − F 11 + F 12 x − y + y F 11 − F 12 x y − 1 + F 12 x . (5.2)
Both F 11 and F 12 have finite limit at x = y and x = 1 y , this consistency leads to two relations between F 11 and F 12
F 11 | x=y = −F 12 | x=y . F 11 | x= 1 y = F 12 | x= 1 y . (5.3) -8 -
The boundary of (F 21 , F 22 , F 25 , F 28 , F 38 , F 40 ) can also be determined with the discussion above. By now, all the boundary conditions are fixed.
We then proceed to determinate the analytic continuation of all the master integrals. When considering the processes (2.1,2.2), the variables of the master integrals lie either in Euclidean region or in Minkowski region. In order to obtain the correct numerical results, their analytic continuation should be considered carefully. The proper analytic continuation can be achieved by the replacement s → s+i0 at fixed m 2 1 and m 2 2 . This transfer corresponds to x → x + i0.
After the determination of the boundary conditions and analytical continuation, we can readily obtain the analytical results of all the master integrals using the differential equations we obtain. The results of the master integrals can be written in terms of Goncharov polylogarithms introduced in section 3. The results for F i (i = 1 . . . 40) are calculated up to weight four in this work. All the analytic results up to weight four are collected in the ancillary file "results.m" which is supplied with the paper. Here, for illustration, we present the results for integrals (F 6 , F 10 , F 33 ) up to weight three.
F 6 = ǫ 2 (G −1,0 (y) + G 1,0 (y)) − 2ǫ 3 (G 1,1,0 (y) + G −1,−1,0 (y) + G 1,0,0 (y) + G −1,0,0 (y) −G 0,1,0 (y) − G 0,−1,0 (y) + 2G 1,−1,0 (y) + 2G −1,1,0 (y)) + O(ǫ 4 ), F 10 = 2ǫ 2 G 0,0 (y) − G 0,0 (x) + ǫ 3 G 0,0 (y) 2 (G 1 y (x) + G y (x) + 4G y (1) − 4G 1 y (1) − G 1 y (y)) + 4G 0 (y)(G 0,y (x) − G 0, 1 y (x) + G 0,0 (x) + G 0, 1 y (y) − G 1 y ,0 (1) − G y,0 (1) + π 2 3 ) + G 0 (x)( G 0,0 (y) 2 − 4(G 1 y (1) − G y (1))G 0 (y) − 4G 1 y ,0 (1) − 4G y,0 (1) + π 2 ) − 6G 0,0,0 (x) + 12(G 0,−1,0 (x) + G 0,1,0 (x) − G 0,−1,0 (y) − G 0,1,0 (y)) − 2(2G 0, 1 y ,0 (x) + 2G 0,y,0 (x) + G 1 y ,0,0 (x) + G y,0,0 (x)) + 2G 1 y ,0,0 (y) + 4G 0, 1 y ,0 (y) + 6ζ(3) + O(ǫ 4 ), F 33 = ǫ 3 2G 0,0,0 (x) − 2G 0,1,0 (x) − 2G 0,−1,0 (x) + 1 6 π 2 G 0 (x) + ζ(3) + O(ǫ 4 ). (5.4)
Note that the weight three results of F 33 depend only on variable x, the dependence of F 33 on variable y will start at O(ǫ 4 ).
The calculations are performed with our in house Mathematica code. It is desirable to perform an independent check on the analytical expressions of the master integrals. We check all the analytical results against the numerical results obtained from packages Fiesta [57,58] and SecDec [59,60]. Perfect agreement has been achieved between the analytical and numerical approaches with kinematics in both Euclidean region and Minkowski region. To be more specific, we show the numerical results of the master integrals M 33 (s, m 1 , m 2 ) obtained from packages Fiesta and SecDec and our analytical expressions in two different kinematics,
M SecDec 33 (5.4, 1.0, 0.2) = 0.890536 + i1.307051 ± (0.000010 + i0.000012) ǫ −5.989791 + i7.770764 ± (0.000306 + i0.000385), M FIESTA 33 (5.4, 1.0, 0.2) = 0.890545 + i1.307058 ± (0.000036 + i0.000039) ǫ −5.989754 + i7.772445 ± (0.00064 + i0.000639), M Ours 33 (5.4, 1.0, 0.2) = 0.8905387485 . . . + i1.307054048 . . . ǫ −5.989494850 . . . + i7.772448208 . . . , M SecDec 33 (−5.4, 1.0, 0.2) = −0.4466129 ± 0.0000004 ǫ − 0.507366 ± 0.000006, M FIESTA 33 (−5.4, 1.0, 0.2) = −0.446613 ± 0.000005 ǫ − 0.507387 ± 0.000049, M Ours 33 (−5.4, 1.0, 0.2) = −0.4466129967 . . . ǫ − 0.5073683817 . . . . (5.5)
We can see from the equations above that the numerical results obtained from packages Fiesta and SecDec with error estimates are perfectly agree to that precision with the numerical results obtained from our analytical answer. Note that it took Fiesta and SecDec packages each more than ten hours on a workstation with 16 cores (2.3 GHz) processor to reach the above precision for kinematic (s = 5.4, m 1 = 1.0, m 2 = 0.2) in Minkowski region.
Discussions and Conclusions
In summary, applying the method of differential equations, we calculate the full set of the two-loop master integrals for heavy-to-light form factors of two different types of massive fermions and arbitrary momentum transfer in NNLO QCD or QED corrections. With a proper choice of master integrals, it turns out that we can cast all the differential equations into the canonical form, which can straightforwardly be integrated order by order in dimensional regularization parameter ǫ. Under the determination of all boundary conditions, we express the results of all the basis in terms of Goncharov polylogarithms. The integrals allow to determine the two-loop amplitudes for heavy-to-light form factors of two massive fermions in a full analytical way, and thus to compute the NNLO corrections to the decay of heavy massive fermion into light massive fermion, or the decay of W boson into two massive quarks without any approximations. The integrals are also applicable to the calculations of NNLO QCD corrections to the inclusive decay of B meson into charmonium, and two-loop form factors of H + tb coupling vertex.
A The Matrices R
The rational matrices A i (i = 1 . . . 10) are expressed as
A 1 = , (A.1) A 2 = , (A.2) A 3 = , (A.3) A 4 = , (A.4) A 5 = ,
(A.5)
A 6 = , (A.6) A 7 = ,
(A.7)
A 8 = ,
(A.8)
A 9 = , (A.9) A 10 = .
(A.10)
Acknowledgments
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| []
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[
"Pulse Index Modulation",
"Pulse Index Modulation"
]
| [
"Member, IEEESultan Aldirmaz-Colak ",
"Erdogan Aydin ",
"Yasin Celik ",
"Yusuf Acar ",
"Senior Member, IEEEErtugrul Basar "
]
| []
| []
| Emerging systems such as Internet-of-things (IoT) and machine-to-machine (M2M) communications have strict requirements on the power consumption of used equipments and associated complexity in the transceiver design. As a result, multiple-input multiple-output (MIMO) solutions might not be directly suitable for these system due to their high complexity, inter-antenna synchronization (IAS) requirement, and high interantenna interference (IAI) problems. In order to overcome these problems, we propose two novel index modulation (IM) schemes, namely pulse index modulation (PIM) and generalized PIM (GPIM) for single-input single-output (SISO) schemes. The proposed models use well-localized and orthogonal Hermite-Gaussian pulses for data transmission and provide high spectral efficiency owing to the Hermite-Gaussian pulse indices. Besides, it has been shown via analytical derivations and computer simulations that the proposed PIM and GPIM systems have better error performance and considerable signal-to-noise ratio (SNR) gain compared to existing spatial modulation (SM), quadrature SM (QSM), and traditional M -ary systems.Index Terms-Hermite-Gaussian pulses, Internet-of-things (IoT), index modulation (IM), machine-to-machine (M2M), single-input single-output (SISO). | 10.1109/lcomm.2021.3073753 | [
"https://arxiv.org/pdf/2101.09340v1.pdf"
]
| 231,699,229 | 2101.09340 | 5f17f20dec4778c0de2122aed2759984141c2a96 |
Pulse Index Modulation
22 Jan 2021
Member, IEEESultan Aldirmaz-Colak
Erdogan Aydin
Yasin Celik
Yusuf Acar
Senior Member, IEEEErtugrul Basar
Pulse Index Modulation
22 Jan 20211Index Terms-Hermite-Gaussian pulsesInternet-of-things (IoT)index modulation (IM)machine-to-machine (M2M)single-input single-output (SISO)
Emerging systems such as Internet-of-things (IoT) and machine-to-machine (M2M) communications have strict requirements on the power consumption of used equipments and associated complexity in the transceiver design. As a result, multiple-input multiple-output (MIMO) solutions might not be directly suitable for these system due to their high complexity, inter-antenna synchronization (IAS) requirement, and high interantenna interference (IAI) problems. In order to overcome these problems, we propose two novel index modulation (IM) schemes, namely pulse index modulation (PIM) and generalized PIM (GPIM) for single-input single-output (SISO) schemes. The proposed models use well-localized and orthogonal Hermite-Gaussian pulses for data transmission and provide high spectral efficiency owing to the Hermite-Gaussian pulse indices. Besides, it has been shown via analytical derivations and computer simulations that the proposed PIM and GPIM systems have better error performance and considerable signal-to-noise ratio (SNR) gain compared to existing spatial modulation (SM), quadrature SM (QSM), and traditional M -ary systems.Index Terms-Hermite-Gaussian pulses, Internet-of-things (IoT), index modulation (IM), machine-to-machine (M2M), single-input single-output (SISO).
I. INTRODUCTION
Internet-of-things (IoT) and machine-to-machine (M2M) communications are emerging technologies which are supported by 5G. According to recent reports in 2020, the number of IoT connections exceeded the number of non-IoT ones. By 2025, it is expected that there will be more than 30 billion IoT connections, thus the efficient use of the spectrum comes to the fore in the system design [1]. IoT and M2M systems should have low-power equipments and low complexity. Therefore, single-input single-output (SISO) solutions are one step ahead of their multiple-input multiple-output (MIMO) counterparts.
MIMO transmission provides transmitter (Tx) and receiver (Rx) diversity gain and increases the data rate [2]. However, inter-antenna synchronization (IAS) and inter-antenna interference (IAI) are big problems of MIMO schemes. In 2008, Mesleh [7] which utilizes entire subcarriers for symbol transmission, unlike OFDM-IM. These techniques become very popular, then their variants have been proposed in a very short time [8], [9]. However, in the recent past, IM has been utilized to select orthogonal codes in codedivision multiple access (CDMA) communication [10]. Both OFDM and CDMA are utilized for wideband communication however, their complexities are relatively high. Particularly, SM and general MIMO schemes require the estimation of all channels between each Tx-Rx antenna. Thus, their receiver complexity and overhead of the channel estimation are quite high. Unlike these traditional techniques, in this letter we focus on novel IM-based low complexity transceiver designs with single antennas for use in applications such as IoT, M2M with high SE. Hermite-Gaussian pulses are widely used in ultra wideband communication [11], [12]. In [11], the authors proposed a spectrum efficient communication system that uses a summation of binary phase shift-keying (BPSK) modulated Hermite-Gaussian pulses with different orders. Since these pulses are orthogonal to each other, the transmission of a linear combination of these pulses that carries different symbols provides higher SE.
In this letter, we propose two novel IM schemes that activate certain Hermite-Gaussian pulse shapes for transmission instead of antenna indices according to the incoming information bits. Unlike SM and OFDM-IM systems, proposed pulse index The main contributions of the letter are summarized as follows:
• We introduce two novel PIM schemes for SISO systems.
Similar to SM, index bits are used as an extra dimension to convey data bits besides conventional constellation mapping.
• We propose a low complexity detector that requires only one Tx-Rx channel state information (CSI) estimation.
Since the overhead of channel estimation is low, more data can be transmitted during the coherence time.
• To increase the SE of the PIM technique, more than one pulse can be sent together owing to orthogonality property of Hermite-Gaussian pulses. Therefore, we introduce the GPIM scheme for SISO systems.
• We also obtain the average bit error probability (ABEP) for the maximum likelihood (ML) detector. ABEP results match well with the simulation results. The remainder of this letter is organized as follows. In Section II, we introduce the system model of PIM and GPIM schemes. In Section III, performance analysis of the proposed two schemes are presented. Simulation results and performance comparisons are given in Section IV. Finally, the letter is concluded in Section V.
Notation: Throughout the letter, scalar values are italicized, vectors/matrices are presented by bold lower/upper case symbols. The transpose and the conjugate transpose are denoted by (·) T and (·) H . ⌊.⌋, ||.||, and C(., .) represent the floor operation, Euclidean norm, and Binomial coefficient, respectively. CN (0, σ 2 ) represents the complex Gaussian distribution with zero mean and variance σ 2 and I n is the n× n identity matrix. Last, ∂ ∂t represents partial derivative.
II. SYSTEM MODEL OF PULSE INDEX MODULATION
We constitute a set of Hermite-Gaussian functions ψ v (t) that span the Hilbert space. These functions are known for their ability to be highly localized in time and frequency domains. They are defined by a Hermite polynomial modulated with a Gaussian function as
ψ v (t) = 2 1/4 √ 2 v v! H v ( √ 2πt)e −πt 2 ,(1)
where t represents time index, v is the order of Hermite-Gaussian function, and H v (t) is the Hermite polynomial series that is expressed as
H v (t) = (−1) v e t 2 ∂ v ∂t v e −t 2 .
A number of Hermite polynomials can be given for v = 0, 1, 2, 3 as follows:
H 0 (t) = 1, H 1 (t) = 2t, H 2 (t) = 4t 2 −2, H 3 (t) = 8t 3 −12t. (2)
One of the important properties of Hermite-Gaussian functions is orthogonality among them, which can be expressed as
∞ 0 ψ m (t)ψ n (t)dt = 0 for m = n [13].
Representation of Hermite-Gaussian pulses ψ 0 (t), ψ 1 (t), ψ 2 (t), and ψ 3 (t) in the time-domain are shown in Fig. 1(a) for v = 0, 1, 2, 3. As seen from Fig. 1(a), as the order of Hermite-Gaussian pulse increases, the oscillation of the pulse also increases. The frequency domain representation of the these Hermite-Gaussian pulses and square root raised cosine (SRRC) pulse with different roll-off factors (β) are given in Fig. 1(b). We can state two important issues from Fig. 1(b). First, the bandwidth of the Hermite-Gaussian pulses increases with the increase of order, as the first-null bandwidth is considered. Second, the bandwidth of the zeroth and the first order Hermite-Gaussian pulses are narrower than that of SRRC, while the bandwidth of the second and the third order Hermite-Gaussian pulses are wider. In other words, as can be seen from Fig. 1(b), the bandwidth usage of the proposed scheme is relatively higher than the SRRC with β = 1 .
In the following sections, to simplify presentation, only discrete signal samples will be used. Discrete representation of the Hermite-Gaussian pulses are obtained from the continuous Hermite-Gaussian pulses by using Nyquist sampling theorem (ψ j (t) = ψ j [lT s ], where l is an integer (l = 0, 1, . . . L, where L is number of samples of the j th Hermite-Gaussian pulse) and T s denotes the sampling interval). Thus, each pulse is represented by a vector with L samples, such as
ψ j = [ψ j,1 ψ j,2 . . . ψ j,L ] T . A. PIM Transmission Model (k = 1)
The transceiver block diagram of the proposed PIM scheme for k = 1 is represented in Fig. 2 (at the top of the next page), where k is the number of selected pulses. Firstly, incoming bit sequence b with the size of 1 × p PIM is splitted into pulse index selector and M -ary signal constellation blocks. While the first p 1 = log 2 n k bits determine the active pulse shape, where n denotes to total number of pulses, and the remaining p 2 = log 2 (M ) bits determine the modulated symbol according to modulation scheme, where M denotes the modulation order. In the PIM model, only one pulse is active for transmission. Thus, the baseband PIM-based pulse signal to be transmitted can be expressed as
x = s i ψ j ,(3)
where s i and ψ j represent modulated i th symbol and j th selected Hermite-Gaussian pulse vector, respectively, and i ∈
(t) ±2 1/4 e −πt 2 ±ψ 1 (t) ±2 1/4 (2 √ π)te −πt 2 ±ψ 2 (t) ± 2 1/4 2 √ 2 (8πt 2 − 2)e −πt 2 ±ψ 3 (t) ±2 1/4 4π √ 2πt 3 −3 √ 2πt √ 3 e −πt 2
Index Bits
Selected Pulse for PIM scheme (k = 1) Table I with s i = ±1. The number of transmitted bits per channel use (bpcu) can be calculated as p PIM = p 1 + p 2 . An example of the pulse index mapping rule for p 1 = 2 bits is given by Table II. In this case, the PIM scheme for BPSK (n = 4, k = 1, and p 1 = 2) transmits p PIM = p 1 + p 2 = 3 bits.
Selected Pulse for Generalized PIM scheme (k = 2) {0, 0} ψ 0 {ψ 0 , ψ 1 } {0, 1} ψ 1 {ψ 0 , ψ 2 } {1, 0} ψ 2 {ψ 1 , ψ 2 } {1, 1}
B. GPIM Transmission Model (k ≥ 2)
To increase the transmitted bpcu of the aforementioned method, we generalized it for k ≥ 2. In this letter, to simplify the analysis, we assume k = 2. The transmitter block diagram of this scheme and a look-up table, which maps the index bits to transmitted pulses, are given in Fig. 3 and Table II, respectively. Similar to the first method, incoming bit sequence b with the size of 1 × p GPIM is splitted into pulse index selector and M -ary signal constellation blocks. While the first p 1 = log 2 n k bits determine the active pulse shapes as given in last column of Table II (k = 2), the remaining p 2 = k log 2 (M ) bits determine the modulated symbol according to modulation scheme so that p GPIM = p 1 +p 2 . In Table II, we assume that n = 4, k = 2 for the GPIM scheme. However, there are C(4, 2) possible pulse shape pairs and we only use four out of them.
Possible Pulses
Analytical Expressions ±ψ 0 (t) ± ψ 1 (t)
2 1/4 (1 ± 2 √ πt)e −πt 2 ±ψ 0 (t) ± ψ 2 (t) 2 1/4 (1 ± 4πt 2 −1 √ 2 )e −πt 2 ±ψ 1 (t) ± ψ 2 (t) 2 1/4 (2 √ πt ± 4πt 2 −1 √ 2 )e −πt 2 ±ψ 1 (t) ± ψ 3 (t) 2 1/4 (2 √ πt ± 4π √ 2πt 3 −3 √ (2π)t √ 3 )e −πt 2
Then, the baseband GPIM based pulse signal to be transmitted is expressed as
x = 1 √ k (s i ψ j + s q ψ ℓ ),(4)
where 1 √ k is a normalization coefficient used to make the total symbol energy E s = 1. s i and s q denote i th and q th modulated symbols respectively, ψ j and ψ ℓ represent the selected Hermite-Gaussian pulses according to index bits, respectively, where i, q ∈ {1, 2, . . . , M } and j, ℓ ∈ {1, 2, . . . , 2 p1 }. For example, if the information bit block is given as b = [0 0 1 0], the selected Hermite-Gaussian pulses are zeroth order and the first order ones according to the first two bits, and selected BPSK symbols for the third and the fourth bits are s i = 1 and s q = −1. Thus the transmitted signal can be expressed as x = 1 √ 2 ψ 0 − ψ 1 , if BPSK modulation is applied. As Hermite-Gaussian pulses are orthogonal to each other, the modulation type can be thought as quadrature phase shiftkeying (QPSK).
Analytical expression of the possible transmitted pulses for BPSK modulation in the time domain is given in Table III for GPIM. SE of GPIM scheme can be calculated as p GPIM = log 2 n k + k log 2 (M ) bpcu. Without losing generality, we use four different Hermite-Gaussian functions of orders v = 0, 1, 2, and 3 for practical considerations and for simplicity (n = 4, M = 2 ). Thus, the SE of GPIM scheme equals to p GPIM = 4 bpcu while the classical SISO system with M = 2 obtains 1 bpcu.
1) ML Detection of PIM and GPIM Schemes: The vector representation of the received baseband signal for PIM and GPIM schemes can be expressed, respectively as follows:
r PIM = xh + n = s i ψ j h + n,(5)r GPIM = xh + n = s i ψ j + s q ψ ℓ h + n,(6)
where n ∈ C L×1 is the noise vector with elements following CN (0, N0 2 I L ) and h represents the complex Rayleigh fading coefficient.
The ML detectors for PIM and GPIM schemes can be expressed respectively as follows:
ŝ i ,ĵ = arg max i,j Pr r PIM |s i , ψ j = arg min i,j r PIM − hs i ψ j 2 ,(7)ŝ i ,ŝ q ,ĵ,l = arg max i,j,q,ℓ Pr r GPIM |s i , s q ψ j , ψ ℓ = arg min i,j,q,ℓ r GPIM − s i ψ j + s q ψ ℓ h 2(8)
where, i, q ∈ {1, 2, . . . , M } and j, ℓ ∈ {1, 2, . . . , 2 p1 }. Finally, using the detected (ŝ i ,ŝ q ,ĵ,l) values, the originally transmitted bit sequenceb is reconstructed at the receiver with the help of the index to bits mapping technique as shown at the receiver block of the PIM and GPIM systems.
III. PERFORMANCE ANALYSIS
In this section, we analyze the ABEP performance of the PIM and GPIM schemes. Accordingly, using the well-known union bounding technique as in [14], the expression of ABEP P for proposed two schemes can be given as follows:
P ≤ 1 2p 2p d=1 2p z=1 P e ξ d →ξ z N (d, z) p ,(9)
wherep = p 1 + p 2 is the number of bits transmitted in active pulse indices and modulated symbols, N (d, z) is expressed as the number of bits in errors between the vectors ξ d andξ z . P e ξ d →ξ z is the APEP of decidingξ z giving that ξ d is transmitted and it can be expressed as
P e ξ d →ξ z = 1 2 1 − σ 2 k,α 1 + σ 2 k,α ,(10)
where, k ∈ {1, 2}. Note that, for k = 1 and for k = 2, the ABEPs of PIM and GPIM schemes are obtained, respectively. For the PIM and GPIM schemes σ 2 k,α is given by: whereŝ i andŝ q are estimates of s i and s q , respectively. σ 2 h is the variance of Rayleigh fading channel coefficient and σ 2 h = 1.
σ 2 1,α = Es 2N0 σ 2 h |s i | 2 + |ŝ i | 2 if ψ j = ψĵ Es 2N0 σ 2 h |s i −ŝ i | 2 if ψ j = ψĵ(11)σ 2 2,α = Es 2N0 σ 2 h |s i | 2 +|ŝ i | 2 +|s q | 2 +|ŝ q | 2 if ψ = ψĵ, ψ ℓ = ψl Es 2N0 σ 2 h |s i −ŝ i | 2 +|s q | 2 +|ŝ q | 2 if ψ j = ψĵ, ψ ℓ = ψl Es 2N0 σ 2 h |s i | 2 +|ŝ i | 2 +|s q −ŝ q | 2 if ψ j = ψĵ, ψ ℓ = ψl Es 2N0 σ 2 h |s i −ŝ i | 2 +|s q −ŝ q | 2 if ψ j = ψĵ, ψ ℓ = ψl(12)
Consequently, by substituting (11) and (10) into (9), we obtain the ABEP for PIM system, and similarly, by substituting (12) and (10) into (9), we obtain the ABEP for GPIM system.
IV. SIMULATION RESULTS
To demonstrate the improved performance of the proposed techniques, the bit error rate (BER) of PIM systems is evaluated with different system setups. SM, quadrature SM (QSM), traditional M PSK/QAM schemes are selected as benchmarks. The SNR used in computer simulations herein is defined as E s /N 0 where E s is energy per symbol and N 0 is the noise power. At the receiver, ML detector is used for all systems. Each Hermite-Gaussian pulse consists of 127 samples. Since the head and tail of the blows contain a large number of zero-value samples, we truncate their edges. Thus, each pulse includes 61 samples. All simulations are performed over frequency-flat Rayleigh fading channels. We assume that the channel is constant during one symbol duration, and the CSI is perfectly known at the receiver.
The theoretical and simulation average BER performance curves of the PIM scheme with M -PSK, M = 4, 8, 16, 32, n = 4, and k = 1 are presented for p PIM = 4, 5, 6, and 7 bits in Fig. 4. Here, the PIM technique transmits 4, 5, 6, and 7 bits by 2 bits with active pulse indices and 2, 3, 4, and 5 bits with the transmitted symbols, respectively. As can be seen from Fig. 4, analytical results match simulation results well particularly at high modulation order.
The average BER performance curves of the PIM, GPIM and benchmarks schemes are shown in Fig. 5 for M -PSK/QAM at 6 bpcu. GPIM technique carries 2 bits with active pulse indices and 4 bits with the transmitted symbol; PIM scheme transmits 2 bits with with active pulse indices and 4 bits with the transmitted symbol; the QSM technique Fig. 5, the GPIM scheme provides better performance with approximately 1 dB SNR gain compared to PIM system when QAM is used. Also, the analytical and the simulation results match well. The proposed GPIM and PIM schemes have also better BER performance compared to SM, QSM, and traditional QAM schemes. Fig. 7 (b) are 2 and 8 bits for GPIM; 6 and 4 bits for QSM; 3 and 7 bits for SM schemes, respectively. In M -QAM, all 8 and 10 bits are carried on a modulated symbol with M = 256 and M = 1024, respectively. We can see from Fig. 6 that GPIM scheme has a considerable SNR gain compared to SM and QSM schemes for the same bpcu. At BER = 10 −2 , the proposed scheme requires almost 18 dB less power compared to SM and QSM schemes for M = 16 case.
V. CONCLUSIONS
We have proposed two new IM schemes, namely PIM and GPIM, which exploit the indices of Hermite-Gaussian pulses for SISO systems. These methods are suitable for systems that need low complexity owing to their SISO structure. For this reason, we think that our schemes can be utilized especially in M2M and IoT applications. Analytical expressions for average BER of the PIM and GPIM systems have been derived and their superiority have been shown.
Fig. 1 .
1Different Hermite-Gaussian pulses (a) with L = 127 in time domain (b) bandwidth comparison with SRRC in frequency domain. modulation (PIM) and generalized PIM (GPIM) do not require multiple antennas or multiple subcarriers, and the transmitters of PIM and GPIM have a relatively low complexity. Thus they are suitable for M2M or IoT applications.
Fig. 2 .Fig. 3 .
23The transceiver block diagram of the PIM scheme for k = 1. The transceiver block diagram of the GPIM scheme for k = 2.
ψ 3 {ψ 1 , ψ 3 }
13{1, 2, . . . , M } and j ∈ {1, 2, . . . , 2 p1 }. For example, analytical expressions of the possible transmitted symbols for BPSK modulation in the time domain are given in
Fig. 4 .
4BER performance curves of the PIM scheme with PSK modulation for various M values (n = 4, k = 1).
Fig. 5 .
5Performance comparisons of GPIM, PIM, QSM, SM, and traditional PSK/QAM systems for 6 bpcu. carries either 4 bits with antenna indices and 2 bits with the transmitted symbol or 2 bits with antenna indices and 4 bits with the transmitted symbol. In the SM technique, 2 bits are transmitted in antenna indices and 4 bits are transmitted with symbols. In PSK/QAM, all 6 bits are carried on a modulated symbol with M = 64. As seen from
Fig. 6
6presents average BER performance curves of GPIM, QSM, SM, and traditional QAM schemes for (a) 8 bpcu and (b) 10 bpcu. For Fig. 6 (a), GPIM carries 2 bits with active pulse indices and 6 bits with the transmitted symbol; the QSM technique carries 4 bits with antenna indices and 4 bits with the transmitted symbol; the SM scheme carries 3 bits with antenna indices and 5 bits with the transmitted symbol. The corresponding values in
Fig. 6 .
6Performance comparisons of GPIM, QSM, SM, and traditional QAM systems for (a) 8 bpcu (b) 10 bpcu.
et al. proposed a novel transmission scheme, namely spatial modulation (SM) to overcome the aforementioned S. Aldırmaz-Ç olak is with the Department of Electronics and Communication Engineering, Kocaeli University, Kocaeli 41380, Turkey (e-mail: [email protected]) E. Aydın is with the Department of Electrical and Electronics Engineering, Istanbul Medeniyet University, Istanbul 34857, Turkey (e-mail: [email protected]) Y. Celik is with the Department of Electrical and Electronics Engineering, Aksaray University, Aksaray 68100, Turkey (e-mail: [email protected]) Y. Acar is with the Group of Electronic Warfare Systems, STM Defense Technologies Engineering, Inc., 06510, Ankara, Turkey (e-mail: [email protected]) E. Basar is with the CoreLab, Department of Electrical and Electronics Engineering, Koç University, Istanbul 34450, Turkey (e-mail: [email protected]) IM) has emerged. For instance, Basar et al. proposed orthogonal frequency division multiplexing index modulation (OFDM-IM) that provides not only higher SE but also improved performance compared to classical OFDM [6]. In OFDM-IM, subcarriers are divided into groups and in each group only a few subcarriers are activated according to index bits to convey modulated symbols. To further improve SE compared to OFDM-IM, Mao et al. proposed dual-mode OFDM-IMManuscript received April 19, 2021; revised August 26, 2021.
problems of MIMO systems [3]. SM has attracted quite a
lot of attention from researchers. According to this technique,
only one antenna is activated among all transmit antennas
for transmission at one symbol duration. Incoming bits are
separated into index bits, which determine the active antenna
indices, and modulated bits, which constitute symbols. Hence,
incoming bits are conveyed not only by symbols but also by
active antenna indices. Since only one antenna is activated
at one symbol time, IAI and IAS requirement are avoided.
However, it is stated in [4], [5] that antenna switching in
each symbol duration results in decreasing spectral efficiency
(SE) in practice. Moreover, only one RF chain is sufficient
in the SM, still multiple antennas are needed. To further
exploit indexing mechanisms, the concept of SM has been
generalized to other resources of communication systems and
index modulation (
TABLE I POSSIBLE
IPULSES AND THEIR ANALYTICAL EXPRESSIONS FOR PIM.Possible Pulses for BPSK
Analytical Expressions
±ψ 0
TABLE II A
IIREFERENCE LOOK-UP TABLE FOR n = 4; k ∈ {1, 2}, AND p 1 = 2.
TABLE III POSSIBLE
IIIPULSES AND THEIR ANALYTICAL EXPRESSIONS FOR GPIM.
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"Andrew W Mann \nDepartment of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA\n\nDepartment of Astronomy\nColumbia University\n550 West 120th Street10027New YorkNYUSA\n",
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"Samuel N Quinn \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n"
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"Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Department of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Department of Astronomy\nThe University of Texas at Austin\n78712AustinTXUSA",
"Department of Astronomy\nColumbia University\n550 West 120th Street10027New YorkNYUSA",
"Department of Physics\nMassachusetts Institute of Technology\nCambridgeMAUSA",
"NASA Exoplanet Science Institute\nCalifornia Institute of Technology\nPasadenaCAUSA",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA"
]
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| We report the discovery of three small transiting planets orbiting GJ 9827, a bright (K = 7.2) nearby late Ktype dwarf star. GJ 9827 hosts a 1.62 ± 0.11 R ⊕ super Earth on a 1.2 day period, a 1.269 +0.087 −0.089 R ⊕ super Earth on a 3.6 day period, and a 2.07 ± 0.14 R ⊕ super Earth on a 6.2 day period. The radii of the planets transiting GJ 9827 span the transition between predominantly rocky and gaseous planets, and GJ 9827 b and c fall in or close to the known gap in the radius distribution of small planets between these populations. At a distance of 30 parsecs, GJ 9827 is the closest exoplanet host discovered by K2 to date, making these planets well-suited for atmospheric studies with the upcoming James Webb Space Telescope. The GJ 9827 system provides a valuable opportunity to characterize interior structure and atmospheric properties of coeval planets spanning the rocky to gaseous transition. | 10.3847/1538-3881/aaa292 | [
"https://arxiv.org/pdf/1709.01957v2.pdf"
]
| 55,459,523 | 1709.01957 | e1ce835484a7894d65d09f8f5e2b2d151273be5f |
A SYSTEM OF THREE SUPER EARTHS TRANSITING THE LATE K-DWARF GJ 9827 AT THIRTY PARSECS
DECEMBER 18, 2017 December 18, 2017
Joseph E Rodriguez
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Andrew Vanderburg
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Department of Astronomy
The University of Texas at Austin
78712AustinTXUSA
Jason D Eastman
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Andrew W Mann
Department of Astronomy
The University of Texas at Austin
78712AustinTXUSA
Department of Astronomy
Columbia University
550 West 120th Street10027New YorkNYUSA
Ian J M Crossfield
Department of Physics
Massachusetts Institute of Technology
CambridgeMAUSA
David R Ciardi
NASA Exoplanet Science Institute
California Institute of Technology
PasadenaCAUSA
David W Latham
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Samuel N Quinn
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
A SYSTEM OF THREE SUPER EARTHS TRANSITING THE LATE K-DWARF GJ 9827 AT THIRTY PARSECS
DECEMBER 18, 2017 December 18, 2017DRAFT VERSION Preprint typeset using L A T E X style emulateapj v. 01/23/15 NASA Sagan Fellow and † NASA Hubble Fellow Draft versionSubject headings: planetary systemsplanets and satellites: detectionstars: individual (GJ 9827)
We report the discovery of three small transiting planets orbiting GJ 9827, a bright (K = 7.2) nearby late Ktype dwarf star. GJ 9827 hosts a 1.62 ± 0.11 R ⊕ super Earth on a 1.2 day period, a 1.269 +0.087 −0.089 R ⊕ super Earth on a 3.6 day period, and a 2.07 ± 0.14 R ⊕ super Earth on a 6.2 day period. The radii of the planets transiting GJ 9827 span the transition between predominantly rocky and gaseous planets, and GJ 9827 b and c fall in or close to the known gap in the radius distribution of small planets between these populations. At a distance of 30 parsecs, GJ 9827 is the closest exoplanet host discovered by K2 to date, making these planets well-suited for atmospheric studies with the upcoming James Webb Space Telescope. The GJ 9827 system provides a valuable opportunity to characterize interior structure and atmospheric properties of coeval planets spanning the rocky to gaseous transition.
INTRODUCTION
With the confirmation of over 3500 planets to date and an additional ∼4500 candidates from Kepler (Thompson et al. 2017), the focus of studying exoplanets has largely shifted from pure discovery to understanding planetary demographics, system architectures, interior structures, and atmospheres. In particular, planets which transit their host stars are valuable for understanding the properties of small planets in detail. Like an eclipsing binary star, combining the transit light curve with radial velocity observations yields a measurement of the mass and radius of a planet relative to its star, which constrain the planet's interior structure. Planetary atmospheres can also be studied if the planet transits. The opacity of a planet's atmosphere depends on its chemical composition and the wavelength of the observation. This causes the apparent size of the planet to change as a function of wavelength. Therefore, by measuring the depth of the transit as a function of wavelength, it is possible to gain insight into the composition and temperature of the planet's atmosphere (this technique is known as transit transmission spectroscopy, Seager & Sasselov 2000;Brown 2001;Fortney et al. 2003).
Our ability to study the interior structures and atmospheres of planets, especially small planets (<3 R ⊕ ) with small radial velocity and atmospheric signals, is highly dependent on the brightness of its host star. The brighter the host star, the easier it is to attain high enough signal-to-noise ratios to search for the small signals produced by small planets. The relative size of the planet to its host star is also highly important for transit transmission spectroscopy. It is easier to detect the small, wavelength-dependent changes in transit depth when planets are larger compared to their host stars, so small stars are more favorable targets than large stars for transit spectroscopy measurements. Therefore, nearby bright small stars with planets are excellent targets for atmospheric characteri-zation (Burrows 2014).
Multi-planet systems provide the opportunity to compare the atmospheres and interior structures of different planets while accounting for many confounding variables, like formation history and composition. In some cases, like the recently discovered seven-planet system transiting the nearby late M-dwarf TRAPPIST-1 (Gillon et al. 2016(Gillon et al. , 2017, it is possible to study similarly sized planets across orders of magnitude in incident flux. In terms of the stellar irradiation of the seven planets, TRAPPIST-1 c resembles Venus, TRAPPIST-1 d resembles the Earth, and TRAPPIST-1 f is similar to Mars (Gillon et al. 2017).
However, it would also be desirable to find a multi-planet system suitable for characterization which has planets with different sizes in order to understand the compositions of small planets ranging in size from similar to Earth to about four times the size of Earth. The Kepler mission has found a nearly ubiquitous population of planets with radii larger than the Earth but smaller than Neptune (Batalha et al. 2013;Howard et al. 2012;Petigura et al. 2013;Morton & Swift 2014;Christiansen et al. 2015;Dressing & Charbonneau 2015), for which we have no analogue in our own solar system. Recently, the California Kepler Survey (CKS) measured precise radii for over 2000 Kepler planets and found a bimodal distribution in the radii of small planets, with a deficit of planets with radii between 1.5 and 2.0 R ⊕ , and two peaks in the radius distribution at about 1.3 R ⊕ and 2.5 R ⊕ (Fulton et al. 2017). The deficit in radii around 1.5-2 R ⊕ is coincident with the transition (Weiss & Marcy 2014;Rogers 2015) between predominantly rocky planets (typically smaller than 1.6 R ⊕ ) and planets with substantial gaseous envelopes (typically larger than 1.6 R ⊕ ) as determined from mass measurements of a large number of sub-Neptune-sized planets discovered by Kepler (Wu & Lithwick 2013;Marcy et al. 2014;Hadden & Lithwick 2014, 2017. Since most of these planets with mass measurements orbit very close to their host stars (P<100 days), they receive a large amount of high-energy irradiation that can evaporate gaseous envelopes made of H/He (Yelle 2004;Tian et al. 2005;Murray-Clay et al. 2009;Owen & Jackson 2012). The observed lack of planets with radii of 1.5-2.0 R ⊕ could be due to these gaseous envelopes being evaporated away and leaving the smaller denser cores (Owen & Wu 2017;Jin & Mordasini 2017).
In this paper, we present the discovery of three transiting planets orbiting the nearby (d=30.3 ± 1.6 pc) star GJ 9827 using data from the K2 mission. The planets transiting GJ 9827 are the closest planets discovered by K2 (surpassing K2-18, at 34±4 pc Montet et al. 2015;Crossfield et al. 2016;Benneke et al. 2017). GJ 9827 b, c, and d are all super-Earth sized with radii R b = 1.62 ± 0.11 R ⊕ , R c = 1.269 +0.087 −0.089 R ⊕ , R d = 2.07 ± 0.14 R ⊕ . Planets b (P b = 1.209d) and c (P c = 3.648d) orbit about half a percent outside of a 1:3 mean motion resonance, while planet d (P d = 6.201) orbits far from integer period ratios with the other two planets. The host is a bright (J≈ 8, H ≈7.4, K ≈ 7.2) nearby late K star, making it an excellent target for atmospheric characterization with the upcoming James Webb Space Telescope (Gardner et al. 2006). The planets span the transition from rocky to gaseous planets, so the characteristics of their atmospheres and interior structures may illuminate how the structure and composition of small planets change with radius.
2. OBSERVATIONS AND ARCHIVAL DATA 2.1. K2 Photometry In May 2013 the Kepler spacecraft experienced a failure of the second of its four reaction wheels, ending its primary mission. However, the Kepler spacecraft has been re-purposed to obtain high precision photometry for ∼80 days at a time on a set of fields near the ecliptic in its extended K2 mission (Howell et al. 2014). During K2 Campaign 12, GJ 9827 was observed from UT 2016 December 16 until UT 2017 March 04. We identified GJ 9827 as a candidate planet host after downloading all of the Kepler-pipeline calibrated target pixel files from the Mikulski Archive for Space Telescopes, producing light curves, and correcting for K2 spacecraft systematics following Vanderburg & Johnson (2014) and Vanderburg et al. (2016b). We then searched the resulting light curves for transiting planet candidates using the pipeline described by Vanderburg et al. (2016b). Among the objects uncovered in our search were three super-Earth-sized planet candidates with periods of 1.2, 3.6, and 6.2 days around the nearby star GJ 9827. After we identified the signals, we re-processed the K2 light curve to simultaneously fit the transits, stellar variability, and K2 systematics. We flattened the light curve by dividing away the best-fit stellar variability (which we modeled as a basis spline with breakpoints every 0.75 days) from our simultaneous fit to the light curve. The final lightcurve has a noise level of 39 ppm per 30 minute cadence exposure, and a 6 hour photometric precision of 9 ppm. See Figure 1 for the final light curve.
The K2 light curve shows rotational stellar variability on GJ 9827 with a typical amplitude of about 0.2% peak to peak (See Figure 1). We calculated the autocorrelation function of the K2 light curve, and find a rotation period of 31 ± 1 days, although it is possible the true rotation period is at about 16 days, or half our best estimate. The autocorrelation function preferred a 31 day period most likely because of the flatness at BJD TDB -2454833 = 2945 instead of another peak. 9 Reid et al. (1995) 2.2. Archival Spectroscopy As part of a survey of nearby Solar-type stars, GJ 9827 was observed on UT 2000 Aug 31 using the Center for Astrophysics (CfA) Digital Speedometer on the 1.5 m Wyeth Reflector at the Oak Ridge Observatory in the town of Harvard, Massachusetts. The Digital Speedometer measured an absolute RV of 31.2 km s −1 with an approximate accuracy of ∼0.3 km s −1 (Latham, private communication). GJ 9827 was also observed on UT 2010 Oct 08 and UT 2011 Aug 06 using a CORAVEL-type spectrometer at Vilnius University Observatory, which measured absolute RVs of GJ 9827 on these dates of 32.6 km s −1 and 31.1 km s −1 , respectively (Sperauskas et al. 2016). Using the equations given in Johnson & Soderblom (1987), the UVW space velocities of GJ 9827 were estimated to be (U,V,W) = (−59.2, 20.9, 30.6) km s −1 (Sperauskas et al. 2016). Using the probability distributions of Reddy et al. (2006), GJ 9827 is predicted to be a member of the Galactic thin disk. From these observations, we see no evidence of any large RV variation over the span of over 10 years.
GJ 9827 was also observed twice in 2004 with the High Accuracy Radial Velocity Planet Searcher (HARPS) spectrograph as part of the guaranteed time collaboration's planet search, but not enough observations were taken to identify the small planet candidates we find. Later, Houdebine et al. companions that may dilute our observed transit depths. GJ 9827 has a high proper motion (µ α = 374.4 mas and µ δ = 215.7 mas), and has moved nearly 30 from its original position when the POSS image was taken in 1953 (See Figure 2). In 1953, GJ 9827 was outside of the region of sky enclosed within the photometric aperture we use to produce its modern K2 light curve. No background stars are present inside our K2 photometric aperture down to the POSS limiting magnitude of about R = 20, a full 10 magnitudes fainter than GJ 9827. Since all three transit signals around GJ 9827 have depths greater than 100 ppm, the maximum depth of a transit caused by a background star 10 magnitudes fainter than GJ 9827, we can use "patient imaging" to confidently rule out background stars as the sources of these transit signals.
2.4. Keck/NIRC2 AO Imaging Using the Near Infrared Camera 2 (NIRC2) behind the natural guide star adaptive optics system at the W. M. Keck Observatory, we obtained high resolution images of GJ 9827 using the Br-γ filter on UT 2017 August 19. NIRC2 has a 1024×1024 pixel array with a 9.942 mas pix −1 pixel scale. The lower left quadrant of the NIRC2 array suffers from a higher noise level and a 3-point dither pattern was adopted excluding this regime of the detector. After flat-fielding and sky subtraction, each observation was shifted and co-added, resulting in the final image shown in Figure 2. No other star was detected in the 10 field-of-view. To determine our sensitivity to companions, we inject simulated sources into the final image that have a signal to noise of 5. Figure 2 shows the 5σ sensitivity as a function of spatial separation from GJ 9827, and the inset shows the image itself.
SYSTEM MODELING
Making use of the flattened K2 lightcurves, the Hipparcos parallax, and stellar parameters, we perform a global fit of FIG. 3.-A diagram of the GJ 9827 system geometry shown with all planets at their respective transit centers. From top to bottom, the planets are d, b, and c. The color of the star matches its effective temperature, the planets are to scale with respect to each other and the host star, and the limb darkening matches our best-fit model in the Kepler band. The grey dots trace the orbital path of the planet, with a dot every three minutes. The curvature of planet b's orbit is plainly visible. Ω for each planet (a rotation of the path about the center of the star) is assumed to be zero. Note the mutual inclinations may be much larger than implied here due to the ambiguity between the inclination and 180 degrees minus the inclination. Also note that, while this is the most likely model, the uncertainty in the impact parameters for planets b and c allow them to be non-overlapping (see Figure 4). the GJ 9827 system using EXOFASTv2 Eastman 2017, Eastman et. al., in prep). EXOFASTv2 is based heavily on EXOFAST, but a large fraction of the code has been rewritten to be more flexible. EXOFASTv2 can now, among other things, simultaneously fit multiple planets, incorporate characterization observations (like Doppler Tomography), and simultaneously perform an SED within the global fit. EXOFASTv2 has a few major conceptual changes. First, an the error scaling term for the transit photometry is now fit within the Markov chain Monte Carlo (MCMC). Also, the fit uses the stepping parameters log(M * ) and age instead of a/R * and log g. EXOFASTv2 has previously been used to determine parameters for the HD 106315 system (Rodriguez et al. 2017).
Because GJ 9827 is relatively low-mass with marginal applicability to both the Torres relations (Torres et al. 2010) and YY isochrones (Yi et al. 2001), we disable those constraints within the global model. To determine the mass and radius of GJ 9827, we interpolated the absolute K S -band magnitude onto a grid of stellar evolutionary models and the semi-empirical M K -M * and M K -R * relations from Mann et al. (2015a). We assumed a main-sequence but unknown age (0.5-10 Gyr), a metallicity of -0.15±0.2, and a solar [α/Fe]. This metallicity is based on the star's color-magnitude position (Neves et al. 2012) and JHK colors (Mann et al. 2013;Newton et al. 2014). However, the K-band is selected specifically because it shows a weak dependence with M * and R * , so adopting a lower metallicity as found by our spectral fitting does not significantly change the result. We tried both the Mesa Isochrones and Stellar Tracks (MIST, Dotter 2016; Choi et al. 2016) and Dartmouth Stellar Evolution Program (DSEP, Dotter et al. 2008) models, yielding radii of 0.60±0.02R * and 0.59±0.03R * and masses of 0.63±0.03M * and 0.61 ± 0.04M * respectively. The relations from Mann et al. (2015b), which are anchored in radii from long-baseline optical interferometry (Boyajian et al. 2012), produced a radius estimate of 0.64 ± 0.03R * and mass of 0.66 ± 0.02M * . Errors account for uncertainties in the parallax and K S -band magnitude.
GJ 9827 lands in a region of parameter space where weak molecular bands can form, where models are known to systematically underestimate the radii. However it also lands at the bright limit of the Mann et al. (2015a) relations, around which the calibration stars are preferentially metal-rich when compared to GJ9827 (which would lead to an overestimated radius). Instead, we adopt more conservative parameters of 0.63±0.03 M * and 0.61±0.03R R * for GJ 9827, which encompasses all values above with comparable uncertainties. These values for R * and M * were used as priors for the global fit.
We performed a separate global fit using the broad band photometry summarized in Table 1, the Hipparcos parallax, an upper limit on extinction from Schlegel et al. (1998), to derive the radius of the star. This fit recovered a consistent stellar radius and uncertainty to Houdebine et al. (2016), but the stellar metalicity was driven too high, perhaps biased unfairly by the lack of SED models for such metal-poor stars.
From the Houdebine et al. (2016) analysis of HARPS South spectra combined with an SED analysis of GJ 9827, we set a prior on T eff of 4270±100 K. Additionally, we imposed a prior on the parallax from Hipparcos (van Leeuwen 2007) Such a metal poor, low-mass star may suffer from systematic biases in the limb darkening and gaps in the parameter tables. While a small error in the limb darkening is well within the uncertainty of the K2 lightcurve, allowing it to be fit within the the global fit may work backward to bias the log g, T eff , and [Fe/H] from which they are derived. Therefore, while the limb darkening values can be derived within EXOFASTv2 using the Claret & Bloemen (2011), we place a uniform prior of µ1 = 0.44 ± 0.1 and µ2 = 0.26 ± 0.1. The starting values were determined using the EXOFAST online tool 1 Eastman et al. (2016).
The system parameters determined from our global fit are shown in Table 2 and a diagram of the system geometry is shown in Figure 3.
STATISTICAL VALIDATION
To validate the planetary nature of the candidates identified to be transiting GJ 9827, we use the statistical techniques of Morton (2012) implemented in the vespa software package (Morton 2015). Using the location of the system in the sky and observational constraints, vespa calculates the astrophysical false positive probability (FPP) of the transiting planet candidates. This takes into account the possibility of hierarchical companions or background objects that could lead to a false identification of a transiting planet. Since GJ 9827 hosts multiple planets it is very unlikely that all three planet candidates are false positives 2 . Previous works have calculated a "multiplicity boost" that reduces the false positive probability for multi-planet systems transiting a star in the original Kepler and K2 fields (Lissauer et al. 2012;Sinukoff et al. 2016;Vanderburg et al. 2016c). After applying the multiplicity boost to the vespa determined FPP for the planets transiting GJ 9827, we estimate a FPP of 2 × 10 −6 , 6 × 10 −7 , and 6 × 10 −10 for b, c, and d. Therefore, GJ 9827 b, c, and d are validated exoplanets.
DISCUSSION
The proximity of GJ 9827 and its planetary architecture make it a compelling system worth further characterization. At ≈30 parsecs, this is the closest exoplanet system discovered by K2 to date and one of the few stars to have multiple transiting terrestrial sized exoplanets that are well-suited for both mass measurements and atmospheric characterization. The host star is quite bright (V=10.3, J=8) and the measured planet radii of GJ 9827 b, c, and d are 1.62 ± 0.11 R ⊕ , 1.269 +0.087 −0.089 R ⊕ , and 2.07 ± 0.14 R ⊕ . As mentioned before, there is a known dichotomy in the sizes of short period (<100 days) small planets where planets are more commonly found to be less than 1.5 R ⊕ or larger than 2.0 R ⊕ (Fulton et al. 2017). Based on the mass measurements of planets in these two regimes, the larger planets are less dense and consistent with having a H/He envelope. It is thought that planets smaller than ∼1.6 R ⊕ have lost this outer H/He envelope leaving the rocky core, explaining their higher densities and a lack of planets with radii of 1.5 to 2.0 R ⊕ (Weiss & Marcy 2014;Rogers 2015). The three known planets orbiting GJ 9827 provide a rare opportunity to perform a comparative study since GJ 9827 c is <1.5 R ⊕ , GJ 9827 d is > 2.0 R ⊕ , and GJ 9827 b lands right in this deficiency gap. This system may shed light on the evolution of planets within this radius regime.
Using the Weiss & Marcy (2014) (2016) measured the rotational velocity of GJ 9827 to be <2 km s −1 , making the planets around GJ 9827 well-suited for precise RV observations with current spectroscopic facilities to measure their masses. The rotation period of GJ 9827 is either 31d or 16d, well separated from the orbital periods of the planets, so it should be possible to filter away signals from stellar activity using techniques like Gaussian process regression (Haywood et al. 2014).
To better understand the feasibility of characterizing the atmospheres of the three planets orbiting GJ 9827, we calculate the atmospheric scale height and an expected signal-tonoise per transit following the description given in Vanderburg et al. (2016c). We repeat this calculation for all known planets where R p < 3 R ⊕ using NASA's Exoplanet archive (Akeson et al. 2013). It is expected that both GJ 9827 b and d might have thick gaseous atmospheres (Weiss & Marcy 2014), while GJ 9827 d likely does not have a thick envelope. We find that GJ 9827 b and d are two of the best small (R p < 3 R ⊕ ) exoplanets for detailed atmospheric characterization (See Table 3) 3 . By studying their atmospheric compositions, we may better understand the observed dichotomy in planetary composition observed at ∼1.6 R ⊕ . All calculations are done using the H-band magnitude of the stars to test the feasibility of characterizing the planet's atmosphere with the Hubble Space Telescope's Wide Field Camera 3 instrument and the upcoming suite of instruments that will be available on the James Webb Space Telescope. At a J-band magnitude of 8, GJ 9827 is near the expected saturation limit of the JWST instruments but should be accessible to all four instrument suites allowing for a high S/N with a relatively short exposure time: Near Infared Camera (NIRCam), Near Infrared Imager and Slitless Spectrograph (NIRISS), Near-Infrared Spectrograph (NIRSpec), and the Mid-Infrared Instrument (MIRI) (Beichman et al. 2014). The brightness of the GJ 9827 system makes it a great target for NIRCam's Dispersed Hartmann Sensor 3 We note that signal to noise is not everything. This calculation makes no assumptions about clouds or the presence of high-mean molecular weight atmospheres. The potential pitfalls of making these assumptions are illustrated by GJ 1214, which according to our calculation is the most amenable small planet to atmospheric characterization, but which shows no atmospheric features, likely due to the presence of clouds, hazes, or aerosols (Kreidberg et al. 2014). (Schlawin et al. 2017). The short orbital periods of the three GJ 9827 planets and the near 1 to 3 period commensurability between GJ 9827 b and c provides opportunities to observe overlapping transits of the three planets, as shown in Figure 4. The simultaneous transit on UT 2017 Feb 11 of GJ 9827 b and c shows one discrepant datapoint which misses the EXOFASTv2 model. This kind of discrepancy might be explained by a mutual transit, where GJ 9827 c actually transits both GJ 9827 and planet b simultaneously, which is not modeled by EXOFASTv2. However, at the observed time of this observation, the transit of planet b likely would have already completed (unless there was a significant transit timing variation). We do not find any convincing evidence of mutual transits in our analysis but based on the probability of each planet's impact parameters (See Figure 4), we are not able to rule out this possibility.
CONCLUSION
We present the discovery of three transiting planets orbiting the nearby late K-type star, GJ 9827. Two of the three planets are in near resonance orbits with periods of 1.2d and 3.6d, while the outer planet has a period of 6.2d. All three planets are super-Earth in size with radii of 1.62 ± 0.11 R ⊕ , 1.269 +0.087 −0.089 R ⊕ , and 2.07 ± 0.14 R ⊕ , for GJ 9827 b, c, and d, respectively. At only 30 pc from the Sun, this is the closest exoplanet system discovered by the K2 mission. The proximity and brightness of the host star combined with the similarity in the size of the three transiting planets make GJ 9827 NOTES: a The predicted signal-to-noise ratios relative to GJ 1214 b. All values used in determining the signal-to-noise were obtained from the NASA Exoplanet Archive (Akeson et al. 2013). If a system did not have a reported mass on NASA Exoplanet Archive or it was not a 2σ result, we used the Weiss & Marcy (2014) Mass-Radius relationship to estimate the planet's mass. b Our calculation for the S/N of 55 Cnc e assumes a H/He envelope since it falls just above the pure rock line determined by Zeng et al. (2016). However, 55 Cnc e is in a ultra short period orbit, making it unlikely that it would hold onto a thick H/He envelope.
an excellent target for comparative atmospheric characterization. The expected radial velocity semiamplitudes of the three planets are small but detectable with current instrumentation, especially given the star's fairly bright optical magnitude of V = 10.25. Radial velocity observations should be undertaken to measure the mass of each planet, to determine their interior structures for comparative studies. Mass measurements will also be critical for properly interpreting any atmospheric characterization through transit spectroscopy. Note added in review: During the referee process of this paper, our team became aware of another paper reporting the discovery of a planetary system orbiting GJ 9827 (Niraula et al. 2017).
We thank Laura Kriedberg and Caroline Morley for their valuable conversations. Work performed by J.E.R. was supported by the Harvard Future Faculty Leaders Postdoctoral fellowship. This work was performed in part under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. This paper includes data collected by the Kepler/K2 mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts. Some of the data presented herein were obtained at the WM Keck Observatory (which is operated as a scientific partnership among Caltech, UC, and NASA). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
K5V 9 NOTE
9. -References are: 1 van Leeuwen (2007), 2 Høg et al. (2000) 3 Henden et al. (2015), 4 Cutri et al. (2003), 5 Skrutskie et al. (2006), 6 Cutri & et al. (2014), 7 Zacharias et al. (2004), 8 Houdebine et al. (2016),
FIG. 1 .
1-(Top) The full K2 light curve of GJ 9827 from Campaign 12, corrected for systematics using the technique described inVanderburg & Johnson (2014) andVanderburg et al. (2016b).(Middle) The corrected K2 lightcurve with best-fit low frequency variability removed. (Bottom) Phase folded K2 light curves of GJ 9827 b, c. and d. The observations are plotted in open black circles, and the best fit models are plotted in red. (2016) used a principal component analysis based method to analyze the HARPS spectra and estimate stellar parameters. They found: T eff = 4270±100, [Fe/H]= -0.5±0.1 dex, log g= 4.9±0.2, and v sin i = 1.3 +1.5 −1.3 km s −1 . From the Hipparcos parallax and an analysis of the spectral energy distribution (SED), Houdebine et al. (2016) estimated the radius of GJ 9827 to be R * of 0.623±0.082 R . In this paper, we adopt the spectroscopic parameters from Houdebine et al. (2016) but derive our own stellar mass and radius for our global modeling (described in Section 3). 2.3. Archival Seeing-Limited Imaging Using archival observations from the National Geographic Society Palomar Observatory Sky Survey (NGS POSS) from 1953 and 1991 (ESO/SERC), we looked for nearby bright FIG. 2.-(Top Left) Archival imaging from the National Geographic Society Palomar Observatory Sky Survey (NGS POSS) of GJ 9827 taken with a red emulsion in 1953. (Top Middle) Archival imaging from the ESO/SERC survey of GJ 9827 taken with a red emulsion in 1991. (Top Right) Summed image of GJ 9827 from K2 observations. The aperture selection is described inVanderburg et al. (2016a). (Bottom) The Keck Br-γ contrast curve and image (inset) of GJ 9827. We find no evidence of any additional components in the system.
FIG. 4 .
4-(Left) The corrected K2 lightcurve for GJ 9827 showing a simultaneous transit of b and c with the EXOFASTv2 model shown in red. (Right) The probability distribution of the impact parameter for GJ 9827 b (black), c (red), and d (blue). We cannot rule out the possibility of mutual transits of GJ 9827 b and c.
TABLE 1
1S . . . . . . . . . . . . 2MASS K S mag. . . . . . . . . WISE2 . . . . . . . . WISE2 mag. . . . . . . . . . . . . WISE3 . . . . . . . . WISE3 mag. . . . . . . . . . . . . WISE4 . . . . . . . . WISE4 mag. . . . . . . . . . . . . v sin i . . . . . . . . . Rotational velocity . . . 1.3±1.5 km s −1 8 [m/H] . . . . . . . . Metallicity . . . . . . . . . . -0.5±0.1 8 T eff . . . . . . . . . . . g) . . . . . . . . . Surface Gravity . . . . . . 4.9±0.2 (cgs) 8 π . . . . . . . . . . . . . Hipparcos Parallax (mas) 32.98 ± 1.76 1 d . . . . . . . . . . . . . Distance (pc) . . . . . . . . . . . 30.32±1.62 1 Spec. Type . . . . Spectral Type . . . . . . . . . . .GJ 9827 MAGNITUDES AND KINEMATICS
Other identifiers HIP 115752
2MASS J23270480-0117108
EPIC 246389858
Parameter
Description
Value
Source
α J2000 . . . . . . . .
Right Ascension (RA) . . . 23:27:04.83647
1
δ J2000 . . . . . . . . . Declination (Dec) . . . . . . .
-01:17:10.5816
1
B T . . . . . . . . . . . . Tycho B T mag. . . . . . . . . .
12.10 ± 0.178
2
V T . . . . . . . . . . .
Tycho V T mag. . . . . . . . . . 10.648 ± 0.069
2
B . . . . . . . . . . . . . APASS Johnson B mag. .
11.569 ± 0.034
3
V . . . . . . . . . . . . . APASS Johnson V mag. .
10.250 ± 0.138
3
g . . . . . . . . . . . .
APASS Sloan g mag. . . .
10.995 ± 0.021
3
r . . . . . . . . . . . . . APASS Sloan r mag. . . .
9.845
3
i . . . . . . . . . . . . . APASS Sloan i mag.. . . .
9.394
3
J . . . . . . . . . . . . .
2MASS J mag. . . . . . . . . .
7.984 ± 0.02
4, 5
H . . . . . . . . . . . . . 2MASS H mag. . . . . . . . .
7.379 ± 0.04
4, 5
K 7.193 ± 0.020
4, 5
WISE1 . . . . . . . . WISE1 mag. . . . . . . . . . . . .
6.990 ± 0.041
6
7.155 ± 0.02
6
7.114 ± 0.017
6
6.957 ±0.107
6
µα . . . . . . . . . . .
NOMAD proper motion .
374.4 ± 2.2
7
in RA (mas yr −1 )
µ δ . . . . . . . . . . . . NOMAD proper motion .
215.7 ± 1.9
7
in DEC (mas yr −1 )
Effective Temperature
4270±100 K
8
log(
TABLE 2 MEDIAN
2VALUES AND 68% CONFIDENCE INTERVAL FOR GJ 9827. T A . . . . . . Time of Ascending Node (BJD TDB ). Mass ratio . . . . . . . . . . . . . . . . . . . . . . . P T ,G . . . . A priori transit prob . . . . . . . . . . . . . . T FW HM . FWHM duration (days) . . . . . . . . . . . . τ . . . . . . . Ingress/egress duration (days) . . . . . . T 14 . . . . . Total duration (days) . . . . . . . . . . . . . . Wavelength Parameters: Kepler u 1,Kepler . linear limb-darkening coeff . . . . . . . . Transit Parameters: Kepler σ 2 . . . . . . Added Variance . . . . . . . . . . . . . . . . . . −0.000000000002 +0.000000000037 −0.000000000036 F 0 . . . . . . Baseline flux . . . . . . . . . . . . . . . . . . . . .Parameter
Units
Values
Stellar Parameters
M * . . . . . Mass ( M ) . . . . . . . . . . . . . . . . . . . . . .
0.614 +0.030
−0.029
R * . . . . .
Radius ( R ) . . . . . . . . . . . . . . . . . . . . .
0.613 +0.033
−0.034
ρ * . . . . . . Density (cgs) . . . . . . . . . . . . . . . . . . . . .
3.76 +0.75
−0.57
log g . . . . Surface gravity (cgs) . . . . . . . . . . . . . .
4.651 +0.055
−0.050
T e f f . . . . Effective Temperature (K) . . . . . . . . .
4269 +98
−99
Planetary Parameters:
b
c
d
a . . . . . . . Semi-major axis (AU) . . . . . . . . . . . . .
0.01888 +0.00030
−0.00031
0.03942 +0.00062
−0.00064
0.05615 +0.00089
−0.00091
P . . . . . . . Period (days) . . . . . . . . . . . . . . . . . . . . .
1.2089802 +0.0000084
−0.0000081
3.648083 +0.000060
−0.000058
6.201467 +0.000062
−0.000061
M P . . . . . Mass ( M ⊕ ) . . . . . . . . . . . . . . . . . . . . . .
3.42 +1.2
−0.76
2.42 +0.75
−0.49
5.2 +1.8
−1.2
R P . . . . . . Radius ( R ⊕ ) . . . . . . . . . . . . . . . . . . . . .
1.62 ± 0.11
1.269 +0.087
−0.089
2.07 ± 0.14
i . . . . . . .
Inclination (Degrees) . . . . . . . . . . . . . .
85.73 +1.2
−0.96
88.05 +0.64
−0.48
87.39 +0.20
−0.18
ρ P . . . . . . Density (cgs) . . . . . . . . . . . . . . . . . . . . .
4.50 +1.5
−0.98
6.4 +2.0
−1.1
3.23 +1.1
−0.72
logg P . . . Surface gravity . . . . . . . . . . . . . . . . . . .
3.110 +0.12
−0.098
3.163 +0.11
−0.082
3.07 +0.12
−0.10
Teq . . . . .
Equilibrium temperature (K) . . . . . . .
1172 ± 43
811 ± 30
680 ± 25
Θ . . . . . .
Safronov Number . . . . . . . . . . . . . . . .
0.00460 +0.0015
−0.00096
0.0086 +0.0025
−0.0016
0.0162 +0.0052
−0.0035
F . . . . . Incident Flux (10 9 erg s −1 cm −2 ) . . . .
0.429 +0.066
−0.060
0.098 +0.015
−0.014
0.0485 +0.0074
−0.0068
T C . . . . . . Time of Transit (BJD TDB ) . . . . . . . . .
2457738.82588 +0.00030
−0.00031
2457742.19944 +0.00063
−0.00068
2457740.96111 ± 0.00044
T P . . . . . . Time of Periastron (BJD TDB ). . . . . . .
2457738.82588 +0.00030
−0.00031
2457742.19944 +0.00063
−0.00068
2457740.96111 ± 0.00044
T S . . . . . . Time of eclipse (BJD TDB ) . . . . . . . . .
2457739.43037 ± 0.00030
2457744.02348 +0.00061
−0.00066
2457744.06185 ± 0.00041
2457738.52363 +0.00030
−0.00031
2457741.28742 +0.00065
−0.00069
2457739.41074 ± 0.00045
T D . . . . . . Time of Descending Node (BJD TDB )
2457739.12812 +0.00030
−0.00031
2457743.11146 +0.00062
−0.00067
2457742.51148 +0.00042
−0.00043
K . . . . . .
RV semi-amplitude (m/s) . . . . . . . . . .
2.84 +0.97
−0.64
1.39 +0.44
−0.29
2.50 +0.86
−0.58
logK . . .
Log of RV semi-amplitude . . . . . . . .
0.45 +0.13
−0.11
0.14 +0.12
−0.10
0.40 +0.13
−0.12
M P sin i .
Minimum mass ( M ⊕ ) . . . . . . . . . . . . .
3.41 +1.2
−0.76
2.42 +0.75
−0.49
5.2 +1.8
−1.2
M P /M *
0.0000168 +0.0000058
−0.0000038
0.0000119 +0.0000037
−0.0000025
0.0000254 +0.0000089
−0.0000060
R P /R * .
Radius of planet in stellar radii . . . .
0.02420 +0.00040
−0.00047
0.01899 +0.00034
−0.00037
0.03093 +0.00065
−0.00059
a/R * . . . Semi-major axis in stellar radii . . . .
6.62 +0.41
−0.35
13.83 +0.86
−0.74
19.7 +1.2
−1.0
d/R * . . . Separation at mid transit . . . . . . . . . .
6.62 +0.41
−0.35
13.83 +0.86
−0.74
19.7 +1.2
−1.0
b . . . . . . . Impact parameter . . . . . . . . . . . . . . . . .
0.493 +0.080
−0.12
0.469 +0.085
−0.13
0.896 +0.012
−0.016
δ . . . . . . . Transit depth . . . . . . . . . . . . . . . . . . . . .
0.000586 +0.000019
−0.000023
0.000361 +0.000013
−0.000014
0.000957 +0.000041
−0.000036
Depth . .
Flux decrement at mid transit . . . . . .
0.000586 +0.000019
−0.000023
0.000361 +0.000013
−0.000014
0.000957 +0.000041
−0.000036
P T . . . . . . A priori non-grazing transit prob . . .
0.1474 +0.0082
−0.0086
0.0710 +0.0040
−0.0042
0.0492 +0.0027
−0.0029
0.1547 +0.0088
−0.0092
0.0737 +0.0042
−0.0044
0.0524 +0.0030
−0.0031
0.05083 +0.00080
−0.00072
0.0743 +0.0011
−0.0012
0.04398 +0.0010
−0.00094
0.00163 +0.00023
−0.00022
0.00181 +0.00025
−0.00024
0.00708 +0.00098
−0.00094
0.05249 +0.00074
−0.00071
0.0761 ± 0.0011
0.0511 +0.0011
−0.0010
0.417 +0.069
−0.053
u 2,Kepler . quadratic limb-darkening coeff . . . .
0.240 +0.075
−0.059
0.99999999 +0.00000069
−0.00000070
(DHS) mode
TABLE 3 THE
3BEST CONFIRMED PLANETS FOR TRANSMISSION SPECTROSCOPY WITH R P < 3 R ⊕Planet
R P ( R ⊕ )
S/N a
Reference
GJ 1214 b
2.85±0.20
1.00
Charbonneau et al. (2009)
55 Cnc e b
1.91±0.08
0.41
Dawson & Fabrycky (2010)
HD 97658 b
2.34 +0.17
−0.15
0.36
Dragomir et al. (2013)
TRAPPIST-1f
1.045±0.038
0.24
(Gillon et al. 2017)
GJ 9827 b
1.62 ± 0.11 R ⊕
0.14
this work
HD 3167 c
2.85 +0.24
−0.15
0.14 Vanderburg et al. (2016c); Christiansen et al. (2017)
HIP 41378 b
2.90±0.44
0.14
Vanderburg et al. (2016a)
GJ 9827 d
2.07 ± 0.14 R ⊕
0.13
this work
K2-28 b
2.32±0.24
0.12
Hirano et al. (2016)
HD 106315 b
2.5±0.1
0.10
(Crossfield et al. 2017; Rodriguez et al. 2017)
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[
"ON REALIZATION OF SOME TWISTED TOROIDAL LIE ALGEBRAS",
"ON REALIZATION OF SOME TWISTED TOROIDAL LIE ALGEBRAS"
]
| [
"Naihuan Jing ",
"Chad R Mangum ",
"Kailash C Misra "
]
| []
| []
| Toroidal Lie algebras are generalizations of affine Lie algebras. In 1990, Moody, Rao and Yokonuma gave a presentation for untwisted toroidal Lie algebras. In this paper we give a presentation for the twisted toroidal Lie algebras of type A and D constructed by Fu and Jiang. | 10.1090/conm/695/14000 | [
"https://arxiv.org/pdf/1605.05300v1.pdf"
]
| 119,150,970 | 1605.05300 | 21057f03451fb7ecaf70241396db418e9617fe26 |
ON REALIZATION OF SOME TWISTED TOROIDAL LIE ALGEBRAS
17 May 2016
Naihuan Jing
Chad R Mangum
Kailash C Misra
ON REALIZATION OF SOME TWISTED TOROIDAL LIE ALGEBRAS
17 May 2016
Toroidal Lie algebras are generalizations of affine Lie algebras. In 1990, Moody, Rao and Yokonuma gave a presentation for untwisted toroidal Lie algebras. In this paper we give a presentation for the twisted toroidal Lie algebras of type A and D constructed by Fu and Jiang.
Introduction
Since the seventies, infinite dimensional Kac-Moody Lie algebras [K] have had numerous applications in mathematics and physics, mostly thanks to their subclass of affine Lie algebrasĝ. Affine Lie algebras consist of two types: the untwisted and twisted ones. Every automorphism σ of the finite dimensional simple Lie algebra g gives rise to a "twisted" affine Lie algebraĝ σ , which is a genuinely twisted affine Lie algebra if and only if σ is a Dynkin diagram automorphism of g.
In connection with resonance models in physics, Frenkel [F] has constructed loop Kac-Moody Lie algebras using vertex operators [FLM]. Double affine Lie algebras are special examples of such extended new algebras beyond affine Lie algebras. The n-toroidal Lie algebras are the universal central extensions of the n-loop algebras [BK] based on a finite dimensional simple Lie algebra, which have since found new applications in geometry and physics [S, NSW].
As a generalization of the untwisted affine Lie algebrasĝ, Moody, Rao and Yokonuma [MRY] have given a loop algebra realization of the 2-toroidal Lie algebras T(g) with generators and relations. The MRY realization has led to new constructions of the toroidal Lie algebras using vertex operators [FM,T1,T2], other similar techniques used in the affine Lie algebras [L, JMT], and restricted modules and simple modules for extended affine Lie algebras [BB, G]. Moreover, new realizations of the 2-toroidal Lie algebras of ADE types in a new form of the McKay correspondence [FJW] were also based upon the MRY realization.
Similar to twisted affine Lie algebras, Fu and Jiang [FJ] have considered twisted n-toroidal Lie algebras in an abstract setting and studied their integrable modules. In [EM] vertex representations of general toroidal Lie algebras and Virasoro-toroidal Lie algebras have been constructed.
In this paper, we give an MRY-like presentation to the Fu-Jiang twisted 2-toroidal Lie algebras. In recognition of the similarity between the MRY presentation and Drinfeld realization [D] for quantum affine algebras, we have determined fixed-point subalgebras under a Dynkin diagram automorphism for the toroidal Lie algebras in the A 2n+1 , D n+1 , D 4 types and therefore the fixed-point subalgebras are also realized as central extensions of certain twisted 2-toroidal Lie algebras using field-like generators. This shows that these twisted toroidal Lie algebras also enjoy the similar property as twisted affine Lie algebras.
Twisted Toroidal Lie Algebras
Let g be the finite dimensional simple Lie algebra A 2n−1 , (n ≥ 3), D n+1 , (n ≥ 2), or D 4 over the field of complex numbers C. We denote the Chevalley generators of g by
{e ′ i , f ′ i , h ′ i | 1 ≤ i ≤ N} where N = 2n − 1, n + 1, 4, respectively. Then h ′ = span{h ′ i | 1 ≤ i ≤ N} is the Cartan subalgebra of g. Let {α ′ i | 1 ≤ i ≤ N} ⊂ h ′ * denote
the simple roots and ∆ be the set of roots for g.
Note that α ′ j (h ′ i ) = a ′ ij where A ′ = (a ′ ij ) N i, j=1
is the Cartan matrix associated with g. Let ( | ) be the nondegenerate symmetric invariant bilinear form on g defined by (x|y) = tr(xy), 1 2 tr(xy), 1 2 tr(xy) for all x, y ∈ g.
Then (h ′ i |h ′ i ) = 2, 1 ≤ i ≤ N.
Since the Lie algebra g is simply-laced, we can identify the invariant form on h ′ to that on the dual space h ′ * and normalize the inner product by (α|α) = 2, α ∈ ∆.
Let Γ denote the Dynkin diagram for g and σ be the following Dynkin diagram automorphism of order r = 2, 2, 3 respectively:
σ(h ′ i ) = h ′ N−i+1 , i = 1, · · · , N, for type A 2n−1 σ(h ′ i ) = h ′ i , i = 1, · · · , n − 1 = N − 2; σ(h ′ n ) = h ′ n+1 , for type D n+1 σ(h ′ 1 , h ′ 2 , h ′ 3 , h ′ 4 ) = (h ′ 3 , h ′ 2 , h ′ 4 , h ′ 1 ) for type D 4 .
Then the Lie algebra g is decomposed as a Z/rZ-graded Lie algebra:
g = g 0 ⊕ · · · ⊕ g r−1 ,
where g i = {x ∈ g|σ(x) = ω i x} and ω = e 2π √ −1/r . It is well-known that the subalgebra g 0 is the simple Lie algebra of types C n , B n and G 2 respectively. Let I = {1, 2, · · · , n} for g = A 2n−1 , D n+1 and I = {1, 2} for g = D 4 . The Chevalley generators {e i , f i , h i | i ∈ I} of g 0 are given by:
e i = e ′ i , f i = f ′ i , h i = h ′ i , if σ(i) = i; e i = r−1 j=0 e ′ σ j (i) , f i = r−1 j=0 f ′ σ j (i) , h i = r−1 j=0 h ′ σ j (i) , if σ(i) i.
The Cartan subalgebra of g 0 is h 0 = span{h i | i ∈ I} and the simple roots {α i | i ∈ I} ⊂ h * 0 are given by:
α i = 1 r r−1 s=0 α ′ σ s (i) .
Then we have
(2.1) (α i |α j ) = d i a ij , for all i, j ∈ I
where A = (a ij ) i, j∈I is the Cartan matrix for g 0 and (d 1 , · · · , d n ) = (1/2, · · · , 1/2, 1), (1, · · · , 1, 1/2), or (1/3, 1), for g = A 2n−1 , D n+1 or D 4 respectively. Then correspondingly r = 2, 2 and 3. Note that A = (a ij ) i, j∈I is given as follows:
a ii = 2, i ∈ I a i,i+1 = a i+1,i = −1, g 0 G 2
a 12 = −3, a 21 = −1, g 0 = G 2 a n−1,n = −2, a n,n−1 = −1, g 0 = C n a n−1,n = −1, a n,n−1 = −2, g 0 = B n a ij = 0, otherwise.
Recall that
θ = α ′ 1 + · · · + α ′ 2n−2 + α ′ 2n−1 , for A 2n−1 , α ′ 1 + 2α ′ 2 + · · · + 2α ′ n−1 + α ′ n + α ′ n+1 , for D n+1 , α ′ 1 + 2α ′ 2 + α ′ 3 + α ′ 4 , for D 4 , is the highest root in g. Let f ′ 0 denote the θ-root vector, e ′ 0 denote the (−θ)-root vector such that [h ′ 0 , e ′ 0 ] = 2e ′ 0 , [h ′ 0 , f ′ 0 ] = −2 f ′ 0 where h ′ 0 = [e ′ 0 , f ′ 0 ]. Let A = C[s, s −1 , t, t −1 ]
be the ring of Laurent polynomials in the commuting variables s, t and L(g) = g ⊗ C A be the multi-loop algebra with the Lie bracket given by:
[x ⊗ s j t m , y ⊗ s k t l ] = [x, y] ⊗ s j+k t m+l ,
for all x, y ∈ g, j, k, m, l ∈ Z. For j ∈ Z we define 0 ≤j < r such that j ≡j mod r. For all j ∈ Z we define g j = gj. We extend the automorphism σ of g to an automorphismσ of L(g) by defining:
σ(x ⊗ s j t m ) = ω −j σ(x) ⊗ s j t m where x ∈ g, j, m ∈ Z.
We denote theσ fixed points of L(g) by L(g, σ). Note that the subalgebra L(g, σ) has the Z-gradation:
L(g, σ) = ⊕ j∈Z L(g, σ) j , where L(g, σ) j = g j ⊗ A j , A j = span C {s j t m | m ∈ Z} = s j C[t, t −1 ]. Set F = A ⊗ A. Then F is a two sided A-module via the action a(b 1 ⊗ b 2 ) = ab 1 ⊗ b 2 = (b 1 ⊗ b 2 )a for all a, b 1 , b 2 ∈ A. Let G be the A-submodule of F generated by {1 ⊗ ab − a ⊗ b − b ⊗ a | a, b ∈ A}. The A-quotient module Ω A = F/G is called the A -module of Kähler differentials. The canonical quotient map d : A −→ Ω A given by da = (1 ⊗ a) + G, a ∈ A is the differential map. Let − : Ω A −→ Ω A /dA = K ′ be the canonical linear map. Since d(ab) = 0, we have a(db) = −(da)b = −b(da) for all a, b ∈ A. Then K ′ = span C {bda | a, b ∈ A}. Set K = span C {bda | a ∈ A k , b ∈ A l , k + l ≡ 0(modr)} which is a subalgebra of K ′ .
We note that {s j−1 t m ds, s j t −1 dt, s −1 ds | j ∈ Z, m ∈ Z 0 } is a basis for K and the following relations are easy to check.
(2.2) s ℓ ds k = δ k,−ℓ ks −1 ds, s ℓ t −1 d(s k t) = δ k,−ℓ ks −1 ds + s k+ℓ t −1 dt. Let T(g) = L(g, σ) ⊕ K, with the Lie bracket given by [x ⊗ a, y ⊗ b] = [x, y] ⊗ ab + (x|y)bda, [T(g), K] = 0, where x ∈ g i , y ∈ g j , a ∈ A i , b ∈ A j for i, j ∈ Z. Using [BK, Proposi- tion 2.2], it is shown [FJ, Theorem 2.1] that T(g), with the canonical projection map η : T(g) → L(g, σ)
, is the universal central extension of L(g, σ). T(g) is called the twisted toroidal Lie algebra of type g. The algebra T ′ (g) = L(g) ⊕ K ′ is called the untwisted toroidal Lie algebra of type g. Letà ′ = (a ′ ij ) N i, j=0 be the affine Cartan matrix of the untwisted affine Lie algebra g (1) . In [MRY], Moody, Rao and Yokonuma (MRY) gave a presentation of untwisted toroidal Lie algebra T ′ (g) analogous to the Drinfeld realization [D] of affine Lie algebras as follows.
Let t ′ (g) be the algebra generated by
{ c, a ′ i (k), X(±a ′ i , k) | 0 ≤ i ≤ N, k ∈ Z} satisfying the relations: (2.3) [ c, a ′ i (k)] = [ c, X(±a ′ i , k)] = 0 [a ′ i (k), a ′ j (m)] = (α ′ i |α ′ j )δ k,−m k c [a ′ i (k), X(±a ′ j , m)] = ±(α ′ i |α ′ j )X(±a ′ j , m + k) [X(a ′ i , m), X(−a ′ j , k)] = −δ ij a ′ i (m + k) + 2mδ m,−k (α ′ i |α ′ j ) c [X(±a ′ i , m), X(±a ′ i , k)] = 0 adX(±a ′ i , m) 1−a ′ i j X(±a ′ j , k) = 0, for i j, where i, j ∈ {0, 1, · · · , N} and k, m ∈ Z.
Let z, w, z 1 , z 2 , ... be formal variables. We define formal power series with coefficients from the toroidal Lie algebra t ′ (g):
a ′ i (z) = n∈Z a ′ i (n)z −n−1 , X(±a ′ i , z) = n∈Z X(±a ′ i , n)z −n−1 ,
for i = 0, 1, · · · , N. We will use the delta function
δ(z − w) = n∈Z w n z −n−1 Using 1 z − w = ∞ n=0
z −n−1 w n , |z| > |w|, we have the following useful expansions:
δ(z − w) = ι z,w ((z − w) −1 ) + ι w,z ((w − z) −1 ), ∂ w δ(z − w) = ι z,w ((z − w) −2 ) − ι w,z ((w − z) −2 ),
where ι z,w means the expansion at the region |z| > |w|. For simplicity in the following we will drop ι z,w if it is clear from the context. Now we can write the relations in (2.3) in terms of power series as follows:
(2.4) [ c, a ′ i (z)] = [ c, X(±a ′ i , z)] = 0 [a ′ i (z), a ′ j (w)] = (α ′ i |α ′ j )∂ w δ(z − w) c [a ′ i (z), X(±a ′ j , w)] = ±(α ′ i |α ′ j )X(±a ′ j , w)δ(z − w) [X(a ′ i , z), X(−a ′ j , w)] = −δ ij 2 (α i |α j ) a ′ i (z)δ(z − w) + ∂ w δ(z − w) c [X(±a ′ i , z), X(±a ′ i , w)] = 0 adX(±a ′ i , z 2 )X(a ′ j , z 1 ) = 0, if a ′ ij = 0, i j adX(±a ′ i , z 3 )adX(±a ′ i , z 2 )X(a ′ j , z 1 ) = 0, if a ′ ij = −1, i j adX(±a ′ i , z 4 )adX(±a ′ i , z 3 )adX(±a ′ i , z 2 )X(a ′ j , z 1 ) = 0, if a ′ ij = −2, i j, where i, j ∈ {0, 1, · · · , N}. Define the map π : t ′ (g) −→ L(g) by: c → 0, a ′ j (k) → h ′ j ⊗ s k , X(a ′ 0 , k) → e ′ 0 ⊗ s k t, X(−a ′ 0 , k) → − f ′ 0 ⊗ s k t −1 , X(a ′ i , k) → e ′ i ⊗ s k , X(−a ′ i , k) → − f ′ i ⊗ s k , for 0 ≤ j ≤ N, 1 ≤ i ≤ N.
The following theorem shows that t ′ (g) gives a realization of the untwisted toroidal Lie algebra T ′ (g).
Theorem 2.1. ([MRY, Proposition 3.5]):
The map π is a surjective homomorphism, the kernel of π is contained in the center Z(t ′ (g)) and (t ′ (g), π) is the universal central extension of L(g).
MRY presentation of T(g)
In this section we give an MRY type presentation for the twisted toroidal Lie algebra T(g) which is the main result in this paper. For g = A 2n−1 , r = 2, 1 ≤ i ≤ n − 1 we define:
(3.1) a 0 (k) = a ′ 0 (k) for k ∈ 2Z, a i (k) = a ′ i (k) + (−1) k σ(a ′ i )(k) for k ∈ Z, a n (k) = 2a ′ n (k) for k ∈ 2Z, X(±a 0 , k) = X(±a ′ 0 , k) for k ∈ 2Z, X(±a i , k) = X(±a ′ i , k) + (−1) k X(±σ(a ′ i ), k) for k ∈ Z, and X(±a n , k) = 2X(±a ′ n , k) for k ∈ 2Z. For g = D n+1 , r = 2, 1 ≤ i ≤ n − 1 we define: (3.2) a 0 (k) = a ′ 0 (k) for k ∈ 2Z, a i (k) = 2a ′ i (k) for k ∈ 2Z, a n (k) = a ′ n (k) + (−1) k a ′ n+1 (k) for k ∈ Z, X(±a 0 , k) = X(±a ′ 0 , k) for k ∈ 2Z, X(±a i , k) = 2X(±a ′ i , k) for k ∈ 2Z, X(±a n , k) = X(±a ′ n , k) + (−1) k X(±a ′ n+1 , k) for k ∈ Z. For g = D 4 , r = 3 we define: (3.3) a 0 (k) = a ′ 0 (k) for k ∈ 3Z a 1 (k) = a ′ 1 (k) + ω −k a ′ 3 (k) + ω −2k a ′ 4 (k) for k ∈ Z a 2 (k) = 3a ′ 2 (k) for k ∈ 3Z X(±a 0 , k) = X(±a ′ 0 , k) for k ∈ 3Z X(±a 1 , k) = X(±a ′ 1 , k) + ω −k X(±a ′ 3 , k) +ω −2k X(±a ′ 4 , k), for k ∈ Z X(±a 2 , k) = 3X(±a ′ 2 , k) for k ∈ 3Z. DenoteĨ = I ∪ {0}
and extend the Cartan matrix A = (a ij ) i, j∈I tõ A = (a ij ) i, j∈Ĩ by defining a 00 = 2, a 01 = −1 = a 10 for g = A 2n−1 , a 02 = −1 = a 20 for g = D n+1 or D 4 , and a 0 j = 0 = a j0 otherwise.
Let t(g) be the subalgebra of t ′ (g) generated by { c, a 0 (2k), a i (k), a n (2k), a 1 (k), a 2 (3k), X(±a 0 , 3k), X(±a 1 , k), X(±a 2 , 3k) | k ∈ Z} for g = A 2n−1 , D n+1 and D 4 respectively. Then using the untwisted relations (2.3) we obtain the following relations among the generators of t(g) (with c being central). Below we assume k, k i , l ∈ Z, 2Z or 3Z depending on the generators of t(g) as above.
X(±a 0 , 2k), X(±a i , k), X(±a n , 2k) | 1 ≤ i ≤ n − 1, k ∈ Z}, { c, a 0 (2k), a i (2k), a n (k), X(±a 0 , 2k), X(±a i , 2k), X(±a n , k) | 1 ≤ i ≤ n − 1, k ∈ Z}, and { c, a 0 (3k),
(1) [a 0 (k), a 0 (l)] = a 00 kδ k,−l c
(2) [a 0 (k), a j (l)] = ra 0 j kδ k,−l c where 1 ≤ j ≤ n.
( (D n+1 ) ra ij δ k,0modr kδ k,−ℓ c (D 4 ) where 1 ≤ i ≤ j ≤ n and (i, j) (n − 1, n), (n, n).
3) [a i (k), a j (l)] = ra ij kδ k,−ℓ c (A 2n−1 ) r 2 a ij kδ k,−ℓ c
(4) [a n−1 (k), a n (l)] = ra n−1,n kδ k,−ℓ c (A 2n−1 , D 4 ) r 2 a n−1,n kδ k,−ℓ c (D n+1 ) (5) [a n (k), a n (l)] = r 2 a nn kδ k,−ℓ c (A 2n−1 , D 4 )
ra nn kδ k,−ℓ c (D n+1 ) (6) [a 0 (k), X(±a j , l)] = ±a 0 j X(±a j , k + l)
where 0 ≤ j ≤ n.
(7) [a i (k), X(±a 0 , l)] = ±rδ k,0modr a i0 X(±a 0 , k + l) (A 2n−1 , D n+1 ) ±ra i0 X(±a 0 , k + l) (D 4 ) where 1 ≤ i ≤ n. (8) [a i (k), X(±a j , l)] = ±a ij X(±a j , k + l) (A 2n−1 , D 4 ) ±ra ij X(±a j , k + l) (D n+1 ) where 1 ≤ i, j ≤ n and (i, j) (n − 1, n), (n, n − 1), (n, n).
(9) [a n−1 (k), X(±a n , l)] = ±δ k,0modr a n−1,n X(±a n , k + l) (A 2n−1 , D 4 ) ±ra n−1,n X(±a n , k + l) (D n+1 ) (10) [a n (k), X(±a n−1 , l)] = ±ra n,n−1 X(±a n−1 , k + l) (A 2n−1 , D 4 ) ±δ k,0modr a n,n−1 X(±a n−1 , k + l) (D n+1 ) (11) [a n (k), X(±a n , l)] = ±ra nn X(±a n , k + l) (A 2n−1 , D 4 ) ±a nn X(±a n , k + l) (D n+1
) (12) [X(±a i , k), X(±a i , l)] = 0 where 0 ≤ i ≤ n. (13) [X(a i , k), X(−a j , l)] = −δ i, j a i (k + l)(1 + δ i,n ) +kδ k,−ℓ (1 + (1 − δ i,0 ) + 2δ i,n ) c (A 2n−1 ) −δ i, j a i (k + l) 1 + (1 − δ i,0 − δ i,n ) +kδ k,−ℓ 4 − 3δ i,0 − 2δ i,n c (D n+1 ) −δ i, j a i (k + l)(1 + 2δ i,2 ) +kδ k,−ℓ (1 + 2δ i,1 + 8δ i,2 ) c (D 4 ) where 0 ≤ i, j ≤ n(
14) adX(±a p , k 2 )X(±a m , k 1 ) = 0 for 0 ≤ p m ≤ n and a pm = 0 (15) adX(±a p , k 3 )adX(±a p , k 2 )X(±a m , k 1 ) = 0 for 0 ≤ p m ≤ n and a pm = −1 (16) adX(±a p , k 4 )adX(±a p , k 3 )adX(±a p , k 2 )X(±a m , k 1 ) = 0 for 0 ≤ p m ≤ n and a pm = −2 (17) adX(±a p , k 5 )adX(±a p , k 4 )adX(±a p , k 3 )adX(±a p , k 2 )X(±a m , k 1 ) = 0 for 0 ≤ p m ≤ n and a pm = −3
The above defining relations of t(g) can be written in terms of power series as follows.
(1) [a 0 (z), a 0 (w)] = 1 r a 00
r−1 j=0 ∂ w δ(z − ω −j w) c (2) [a 0 (z), a j (w)] = a 0 j r−1 j=0 ∂ w δ(z − ω −j w) c where 1 ≤ j ≤ n. (3) [a i (z), a j (w)] = ra ij ∂ w δ(z − w) c (A 2n−1 ) ra ij r−1 j=0 ∂ w δ(z − ω −j w) c (D n+1 ) a ij r−1 j=0 ∂ w δ(z − ω −j w) c (D 4 ) where 1 ≤ i ≤ j ≤ n and (i, j) (n − 1, n), (n, n).
(4) [a n−1 (z), a n (w)] = a n−1,n
r−1 j=0 ∂ w δ(z − ω −j w) c (A 2n−1 , D 4 ) ra n−1,n r−1 j=0 ∂ w δ(z − ω −j w) c (D n+1 )
(5) [a n (z), a n (w)] = ra nn
r−1 j=0 ∂ w δ(z − ω −j w) c (A 2n−1 , D 4 ) ra nn ∂ w δ(z − w) c (D n+1 ) (6) [a 0 (z), X(±a j , w)] = ± 1 r a 0 j X(±a j , w) r−1 j=0 δ(z − ω −j w) where 0 ≤ j ≤ n. (7) [a i (z), X(±a 0 , w)] = ±a i0 X(±a 0 , w) r−1 j=0 δ(z − ω −j w) where 1 ≤ i ≤ n. (8) [a i (z), X(±a j , w)] = ±a ij X(±a j , w)δ(z − w) (A 2n−1 , D 4 ) ±a ij X(±a j , w) r−1 j=0 δ(z − ω −j w) (D n+1 ) where 1 ≤ i, j ≤ n and (i, j) (n − 1, n), (n, n − 1), (n, n).
(9) [a n−1 (z), X(±a n , w)] = ± 1 r a n−1,n X(±a n , w) r−1
j=0 δ(z − ω −j w) (A 2n−1 , D 4 ) ±a n−1,n X(±a n , w) r−1 j=0 δ(z − ω −j w) (D n+1 ) (10) [a n (z), X(±a n−1 , w)] = ±a n,n−1 X(±a n−1 , w) r−1 j=0 δ(z − ω −j w) (A 2n−1 , D 4 ) ± 1 r a n,n−1 X(±a n−1 , w) r−1 j=0 δ(z − ω −j w) (D n+1 ) (11) [a n (z), X(±a n , w)] = ±a nn X(±a n , w) r−1 j=0 δ(z − ω −j w) (A 2n−1 , D 4 ) ±a nn X(±a n , w)δ(z − w) (D n+1 ) (12) [X(±a i , z), X(±a i , w)] = 0 where 0 ≤ i ≤ n. (13) [X(a i , z), X(−a j , w)] = −δ i, j (1 − 1 2 δ i,0 )a i (w) δ(z − w) + (δ i,0 + δ i,n )δ(z + w) + (2 − 3 2 δ i,0 ) ∂ w δ(z − w) + (δ i,0 + δ i,n )∂ w δ(z + w) c (A 2n−1 ) −δ i, j (1 − 1 2 δ i,0 )a i (w) δ(z − w) + (1 − δ i,n )δ(z + w) + (2 − 3 2 δ i,0 ) ∂ w δ(z − w) + (1 − δ i,n )∂ w δ(z + w) c (D n+1 ) −δ i, j (1 − 2 3 δ i,0 )a i (w) δ(z − w) + δ(z − ω −1 w) + δ(z − ω −2 w) − δ i,1 (δ(z − ω −1 w) + δ(z − ω −2 w)) + (3 − 8 3 δ i,0 ) ∂ w δ(z − w) + ∂ w δ(z − ω −1 w) + ∂ w δ(z − ω −2 w) − δ i,1 (∂ w δ(z − ω −1 w) + ∂ w δ(z − ω −2 w)) c (D 4 ) where 0 ≤ i, j ≤ n.
(14) adX(±a p , z 2 )X(±a m , z 1 ) = 0 for 0 ≤ p m ≤ n and a pm = 0 (15) adX(±a p , z 3 )adX(±a p , z 2 )X(±a m , z 1 ) = 0 for 0 ≤ p m ≤ n and a pm = −1 (16) adX(±a p , z 4 )adX(±a p , z 3 )adX(±a p , z 2 )X(±a m , z 1 ) = 0 for 0 ≤ p m ≤ n and a pm = −2 (17) adX(±a p , z 5 )adX(±a p , z 4 )adX(±a p , z 3 )adX(±a p , z 2 )X(±a m , z 1 ) = 0 for 0 ≤ p m ≤ n and a pm = −3 By definition of L(g, σ) and t(g) it is clear that π(t(g)) ⊆ L(g, σ). Henceπ = π| t(g) : t(g) −→ L(g, σ). Explicitly the mapπ is given by:
c → 0, a 0 (k) → h ′ 0 ⊗ s k a i (k) → r−1 j=0 σ j (h ′ i ) ⊗ (ω −j s) k X(a 0 , k) → e ′ 0 ⊗ s k t X(−a 0 , k) → − f ′ 0 ⊗ s k t −1 X(a i , k) → r−1 j=0 σ j (e ′ i ) ⊗ (ω −j s) k X(−a i , k) → r−1 j=0 σ j ( f ′ i ) ⊗ (ω −j s) k for 1 ≤ i ≤ n.
The following theorem which is the main result of this paper shows that t(g) gives a realization of the twisted toroidal Lie algebra T(g).
Theorem 3.1. The mapπ is a surjective homomorphism, the kernel of π is contained in the center Z(t(g)) and (t(g),π) is the universal central extension of L(g, σ).
Proof. Using [K,Proposition 8.3], it follows that the mapπ is surjective. By Theorem 2.1, we have ker(π) ⊆ Z(t ′ (g)). Hence ker(π) = ker(π) ∩ t(g) ⊆ Z(t ′ (g)) ∩ t(g) = Z(t(g)). Therefore, (t(g),π) is a central extension of L(g, σ). It is left to show that (t(g),π) is the universal central extension of L(g, σ). Suppose (V, γ) is a central extension of L(g, σ). Since (T(g), η) is the universal central extension of L(g, σ), we have a unique map λ : T(g) → V such that γλ = η. Define the map ψ : t(g) → T(g) by:
(3.4) c → s −1 ds a 0 (k) → h ′ 0 ⊗ s k + s k t −1 dt a i (k) → r−1 j=0 σ j (h ′ i ) ⊗ (ω −j s) k X(a 0 , k) → e ′ 0 ⊗ s k t X(−a 0 , k) → − f ′ 0 ⊗ s k t −1 X(a i , k) → r−1 j=0 σ j (e ′ i ) ⊗ (ω −j s) k X(−a i , k) → − r−1 j=0 σ j ( f ′ i ) ⊗ (ω −j s) k
for 1 ≤ i ≤ n. Note that the map ψ andπ differ only on c and a 0 (k) by elements of K. Hence ηψ =π. Sinceπ is a homomorphism, to show that ψ is a homomorphism it suffices to show that ψ preserves the defining relations involving a 0 (k) by direct calculations. For example, using (2.2) when g = A 2n−1 or D n+1 we have:
ψ a 0 (k) , ψ a j (l) = h ′ 0 ⊗ s k + s k t −1 dt, h ′ j ⊗ s l + σ(h ′ j ) ⊗ (−s) l = (h ′ 0 |h ′ j )s l ds k + (h ′ 0 |σ(h ′ j ))s l ds k (−1) l = ra 0 j kδ k,−l ψ( c) since for g = A 2n−1 , (h ′ 0 |h ′ j ) = (h ′ 0 |σ(h ′ j )) = −δ j,1 = a 0 j and for g = D n+1 ,
(h ′ 0 |h ′ j ) = (h ′ 0 |σ(h ′ j )) = −δ j,2 = a 0 j .
Similarly, when g = D 4 using (2.2) we have: ψ a 0 (k) , ψ a j (l)
= h ′ 0 ⊗ s k + s k t −1 dt, h ′ j ⊗ s l + σ(h ′ j ) ⊗ (ω −1 s) l + σ 2 (h ′ j ) ⊗ (ω −2 s) l = (h ′ 0 |h ′ j ) + (h ′ 0 |σ(h ′ j ))(ω −1 ) k+l + (h ′ 0 |σ 2 (h ′j
))(ω −2 ) k+l s l ds k = a 0 j 1 + (ω −1 ) k+l + (ω −2 ) k+l s l ds k = ra 0 j kδ k,−l ψ( c), since k ∈ 3Z and (h ′ 0 |h ′ j ) = (h ′ 0 |σ(h ′ j )) = (h ′ 0 |σ 2 (h ′ j )) = −δ j,2 = a 0 j . Thus we have a homomorphism λψ : t(g) −→ V and γλψ = ηψ =π giving the following commuting diagram:
T(g) t(g) V L(g, σ) λ η γ ψπ
Since T(g) is the universal central extension of L(g, σ), the lower triangle commutes which implies that the map ψ is unique.
Irreducible representations for toroidal Lie algebras. S Berman, Y Billig, J. Algebra. 221S. Berman and Y. Billig, Irreducible representations for toroidal Lie algebras, J. Algebra 221 (1999), 188-231.
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A new realization of Yangians and quantum affine algebras. V G Drinfeld, 36Soviet DokladyV. G. Drinfeld, A new realization of Yangians and quantum affine algebras, Soviet Doklady 36 (1987), 212-216.
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I B Frenkel, Applications of group theory in physics and mathematical physics. Chicago; ProvidenceAmer. Math. Soc21Representations of Kac-Moody algebras and dual resonance modelsI. B. Frenkel, Representations of Kac-Moody algebras and dual resonance models, in: Applications of group theory in physics and mathematical physics (Chicago, 1982), Lect. Appl. Math. 21, Amer. Math. Soc., Providence, 1985. pp. 325-353.
Integrable representations for the twisted full toroidal Lie algebras. J Fu, C Jiang, J. Algebra. 307J. Fu and C. Jiang. Integrable representations for the twisted full toroidal Lie algebras, J. Algebra 307 (2007), 769-794.
Vertex representations via finite groups and the McKay correspondence. I Frenkel, N Jing, W Wang, Int. Math. Res. Not. 4I. Frenkel, N. Jing and W. Wang, Vertex representations via finite groups and the McKay correspondence, Int. Math. Res. Not. 4 (2000), 195-222.
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Irreducible representations of Virasoro-toroidal Lie algebras. M Fabbri, R V Moody, Comm. Math. Phys. 159M. Fabbri and R. V. Moody, Irreducible representations of Virasoro-toroidal Lie algebras, Comm. Math. Phys. 159 (1994), 1-13.
Fermionic and bosonic representations of the extended affine Lie algebra gl N (C q ). Y Gao, Canad. Math. Bull. 45Y. Gao, Fermionic and bosonic representations of the extended affine Lie algebra gl N (C q ). Canad. Math. Bull. 45 (2002), 623-633.
Bosonic realizations of higher level toroidal Lie algebras. N Jing, K C Misra, S Tan, Pacific J. Math. 219N. Jing, K. C. Misra and S. Tan, Bosonic realizations of higher level toroidal Lie algebras, Pacific J. Math. 219 (2005), 285-302.
Infinite dimensional Lie algebras, 3rd. V G Kac, Cambridge University PressCambridgeV. G. Kac, Infinite dimensional Lie algebras, 3rd. Ed., Cambridge University Press, Cambridge, 1990.
Bosonic and fermionic representations of Lie algebra central extensions. M Lau, Adv. Math. 194M. Lau, Bosonic and fermionic representations of Lie algebra central extensions. Adv. Math. 194 (2005), 225-245.
R V Moody, S E Rao, T Yokonuma, Toroidal Lie algebras and vertex representations. 35R. V. Moody, S. E. Rao and T. Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990), 283-307.
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| []
|
[
"arXiv:0807.4373v2 [astro-ph] Dark matter from cosmic defects on galactic scales?",
"arXiv:0807.4373v2 [astro-ph] Dark matter from cosmic defects on galactic scales?"
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"N Guerreiro \nCentro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nDepartamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n",
"P P Avelino \nCentro de Física do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n\nDepartamento de Física da Faculdade de Ciências da Universidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n",
"J P M De Carvalho \nCentro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nDepartamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal\n",
"C ",
"J A P Martins \nCentro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nDAMTP\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUnited Kingdom\n"
]
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"Centro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal",
"Departamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal",
"Centro de Física do Porto\nRua do Campo Alegre 6874169-007PortoPortugal",
"Departamento de Física da Faculdade de Ciências da Universidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal",
"Centro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal",
"Departamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto\nRua do Campo Alegre 6874169-007PortoPortugal",
"Centro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal",
"DAMTP\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUnited Kingdom"
]
| []
| We discuss the possible dynamical role of extended cosmic defects on galactic scales, specifically focusing on the possibility that they may provide the dark matter suggested by the classical problem of galactic rotation curves. We emphasize that the more standard defects (such as Goto-Nambu strings) are unsuitable for this task, but show that more general models (such as transonic wiggly strings) could in principle have a better chance. In any case, we show that observational data severely restricts any such scenarios. | 10.1103/physrevd.78.067302 | [
"https://arxiv.org/pdf/0807.4373v2.pdf"
]
| 118,342,459 | 0807.4373 | d8116b4e4acddd515f6a390842a08f292e62238a |
arXiv:0807.4373v2 [astro-ph] Dark matter from cosmic defects on galactic scales?
12 Sep 2008 (Dated: 26 July 2008)
N Guerreiro
Centro de Astrofísica da Universidade do Porto
Rua das Estrelas4150-762PortoPortugal
Departamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre 6874169-007PortoPortugal
P P Avelino
Centro de Física do Porto
Rua do Campo Alegre 6874169-007PortoPortugal
Departamento de Física da Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre 6874169-007PortoPortugal
J P M De Carvalho
Centro de Astrofísica da Universidade do Porto
Rua das Estrelas4150-762PortoPortugal
Departamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre 6874169-007PortoPortugal
C
J A P Martins
Centro de Astrofísica da Universidade do Porto
Rua das Estrelas4150-762PortoPortugal
DAMTP
University of Cambridge
Wilberforce RoadCB3 0WACambridgeUnited Kingdom
arXiv:0807.4373v2 [astro-ph] Dark matter from cosmic defects on galactic scales?
12 Sep 2008 (Dated: 26 July 2008)
We discuss the possible dynamical role of extended cosmic defects on galactic scales, specifically focusing on the possibility that they may provide the dark matter suggested by the classical problem of galactic rotation curves. We emphasize that the more standard defects (such as Goto-Nambu strings) are unsuitable for this task, but show that more general models (such as transonic wiggly strings) could in principle have a better chance. In any case, we show that observational data severely restricts any such scenarios.
I. INTRODUCTION
Symmetry breaking phase transitions in the early universe are expected to have produced networks of topological defects [1]. The possible roles of these defect networks in key cosmological scenarios will depend both on the type of defect considered and on the corresponding dynamics. For example, if defects are to significantly contribute to the dark energy, they should have a negative equation of state. If so the best possible situation is that of a frustrated network, in which case
w ≡ p ρ = − N 3(1)
where N is the defect's spatial dimension (N = 1, 2 respectively for cosmic strings and domain walls, while N = 3 corresponds to a cosmological constant). It is clear that only the cases with N = 2 and N = 3 can lead to the recent acceleration of the Universe. A crucial property shared by these two cases is that the ratio between the dark energy density and the background density grows rapidly with time, with dark energy being dynamically dominant only around today. The defect network should therefore have a present density close to critical (Ω 0 de ∼ 1), but compatibility with other cosmological observables requires it to have a characteristic scale several orders of magnitude below the horizon ξ ≪ H −1 .
Defects can also act as seeds for structure formation [2]. This may be the case if the ratio between the average defect energy density and the background density is approximately a constant and the characteristic scale * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] of the network is roughly proportional to the Hubble radius (ξ ∝ H −1 ). Moreover, the defect fluctuations on the Hubble scale must be small (δ ∼ < 10 −5 ), and if the characteristic scale of the defects is of the order of the Hubble scale then it follows that their average energy density must also be very small (Ω def ∼ < 10 −5 ) for consistency with cosmic microwave background anisotropies. It is well known that a scaling cosmic string network has all these properties [1]. Notice that the required network properties are mutually incompatible: although there are unified dark energy scenarios in which dark matter and dark energy are described by a single entity, it is not possible for a given defect network to simultaneously be the dark energy and act as a seed for structure formation.
The possible role of domain walls as dark energy candidates has been investigated in detail in [3,4] where it was shown that the dynamics of realistic domain wall networks appears to be incompatible with a dark energy role. Here we take a closer look at the role of defects on smaller scales. Specifically we will be mostly interested in kiloparsec (that is, galactic) scales. In particular, in doing this we will also explore another possibility that has been recently put forward [5,6], viz. that cosmic strings could be the dark matter and explain the observed flat rotation curves of spiral galaxies.
II. DARK MATTER
The oldest evidence for the existence of dark matter comes from the pioneering work of Zwicky [7] in the 1930s, on the velocity dispersion of galaxies in clusters. On the other hand the evidence for the presence of dark matter on galactic scales comes from the behavior of circular velocities of stars and gas as a function of their distance from the galactic center [8,9,10], and has been known since the 1970s.
Measurements are normally carried out by combining observations of the 21 cm hydrogen line with optical surface photometry. The observed rotation curves usually exhibit a characteristic flat behavior at large distances, around and well beyond the edge of the visible disks. If one assumes the validity of Newtonian gravity, the circular velocity is given by v(r) 2 = GM (r) r .
(2)
In this framework, the observation of a constant largescale velocity is usually explained by the existence of a dark mater halo, with a mass M ∝ r. There is also evidence for dark matter in elliptical galaxies, in particular coming from strong gravitational lensing measurements [11]. However, the dark matter scenario presents some puzzles. There is no particle, predicted by the standard model of particle physics, whether elementary or composite, that can account for the amount of dark matter required by modern cosmological and astrophysical observations. Big bang nucleosynthesis determines the total amount of baryonic matter present in the Universe to be 0.017 ≤ Ω B h 2 ≤ 0.024 [12] , where h ≡ H 0 /100Kms −1 M pc −1 ∼ 0.7, while the 5-year WMAP data [13,14] implies Ω B h 2 = 0.02273 ± 0.00062, with the density of luminous matter being Ω lum ≃ 0.0024h −1 [15]. We may therefore conclude that some of the baryons are dark, although the WMAP data and nucleosynthesis constraints clearly show that they are not sufficient to account for most of the dark matter needed. At the same time, these results rule out the so-called dark bodies (such as black holes, planets or brown dwarfs) as primary dark matter candidates. Therefore, the conclusion that one may draw is that most of dark matter has a nonbaryonic nature.
Given the success of the particle physics standard model, this could therefore point to the existence of new physics beyond it. Indeed, this is one of the key motivations for the forthcoming LHC experiments at CERN. Whatever this new physics may be it should come with new particles-one or more of which could be responsible for the dark matter-in fact, there is no shortage of candidates [16].
Alternatively, the new physics could be in the gravitational sector. For example, the modified Newtonian dynamics (MOND) [17] is an empirically-based modification of Newtonian gravity in the limit of low accelerations which describes reasonably well the dynamics in most galaxies and other larger astrophysical systems. A tensor-vector-scalar gravity (TeVeS) theory has also recently been proposed by Bekenstein [18] which reduces to MOND in the appropriate limit, but it can also be used to make cosmological predictions or to describe gravitational lensing, which could not be addressed by MOND. Although these and other theories [19] present some successful results on different scales, they face several challenges, such as those posed by the latest cosmic microwave background results [20] or the ones implied by the inferred dynamics of the bullet cluster [21].
In what follows we will explore a new mechanism where the flatness of observed rotation curves in spiral galaxies is due to the presence of cosmic strings. Doing so in fact leads us to a broader analysis of the possible roles of defects on galactic scales.
III. COSMIC STRINGS AND SPIRAL GALAXIES
In order to explain the galaxy rotation curves one would need a cosmic string to go through the center of each galaxy. The string network would need to have a characteristic scale of the order of the average intergalaxy distance ∼ 1Mpc and the string velocities would have to be very small (basically equal to the velocity of the galaxy). Such scenarios can in principle exist, although they will in general lead to string-dominated universes.
The first such example, using simple Goto-Nambu strings, was proposed by Kibble [22]. Such a stringdominated universe would have ξ ∝ a 3/2 ∝ t (neglecting energy losses to loops), and the present-day ratio (ξ/t) 0 = √ 30Gµ then corresponds to ξ ∼ 1 − 30 kpc. Kibble himself points out that such a scenario relies on a particular set of (somewhat unphysical) phenomenological parameters. Notice that this is no solution to the dark matter problem: long strings could hardly be bound in galaxies-indeed, the analysis also neglects the dynamical effect of the string velocities.
Much more recently, Alexander [6] suggests that dark matter stems from a frozen (rigid) string network, which would provide an accretion mechanism at recombination and thread galaxies, leading to flat rotation curves. Note that a viable mechanism requires that strings explain the dark matter on both large (cosmological) and small (galactic/cluster) scales. Clearly this scenario requires a rigid network (v ∼ 0) whose strings have wiggles (for an attractive radial force leading to accretion), and obviously dark matter should have an equation of state w ∼ 0. These are mutually incompatible for Goto-Nambu strings, since v 2 ∼ 1/2 in the matter limit (w = 0). However, things are different when we consider transonic wiggly models [23,24]. For Goto-Nambu strings, T = µ = µ 0 = const, and µ and T are the effective mass per unit length and tension of the string. Wiggly models obey T µ = µ 2 0 , and have an equation of state
3w = 1 + T µ v 2 − T µ .(3)
Notice that this does allow for the desired behavior: in the frozen limit we have
w = − T 3µ(4)
while matterlike behavior (w = 0) requires v 2 = T /µ 1 + T /µ .
The two regimes coincide in the tensionless limit T /µ → 0: a frozen tensionless network has a matterlike equation of state. Moreover, for such a frozen network
ρ = µ ξ 2 = µ a 2 ∝ a −3 ,(6)
which implies that µ ∝ a −1 . This behavior is therefore quite different from that of an ordinary Goto-Nambu string network. In the context of 'standard' cosmological scenarios, one would need to invoke rather fine-tuned models to achieve it. On the other hand, it is certainly conceivable that such a behavior can ensue in cosmic (super)string models where some of the network's energy momentum gradually leaks into extra dimensions. This issue is beyond the scope of the present work, and its study is left for future work. For our present purposes the relevant point is that, unlike the simplest Goto-Nambu strings, wiggly cosmic strings have a nonvanishing gravitational potential, Φ, which leads to the following gravitational force on a test particle [
1] ∇Φ = 2G (µ − T ) r ,(7)
where r is the distance from the center of the galaxy. Of course in the relevant tensionless limit T ≪ µ, so we will henceforth neglect the string tension. If a wiggly string threads the center of a spiral galaxy, the above contribution will add to the usual Newtonian one, so in the weak field limit the circular velocity is
v 2 = GM r + 2Gµ.(8)
This immediately shows that asymptotically the squared velocity is dominated by a constant term due to the string, so at least naively √ 2Gµ can be identified with the observational asymptotic velocities of spiral galaxies.
We have analyzed several normal spiral galaxies with a range of total masses and different asymptotic velocities (in what follows we will show only four examples). We divided the galaxies in two groups: one group with the galaxies whose structure can be modeled assuming only a baryonic disk component (here represented by NGC3198 and NGC7331), and the other one with those where a bulge can not be ignored (here the galaxies NGC2841 and NGC2903).
Considering a range for the parameter √ 2Gµ of between 40 and 100 km s −1 , which in natural units corresponds to
10 −9 ∼ < Gµ ∼ < 10 −8 ,(9)
one can easily show that it is possible to adjust the asymptotic velocities for most of normal spiral galaxies. For the galaxies shown in Figs. 1 and 2, our best fits were obtained for significantly different values of √ 2Gµ, although they are all in the chosen range. However, the Only a baryonic disc and a cosmic string threading each galactic center are assumed. The dotted (blue) line represents the observational data, and the dot-dashed one is the disk contribution. The solid (red) line is the total (disk + string) rotation curve. The horizontal (yellow) bands and the bars represent, respectively, the ranges for √ 2Gµ and for the total rotation curve discussed in the main text. For NGC3198 the best fit is obtained using √ 2Gµ = 90 km s −1 , while for NGC7331 that is achieved for √ 2Gµ = 77 km s −1 . Fig. 1, except that for these two galaxies one has to take into account a baryonic central bulge whose contribution to the total rotation curve (bulge + disk + string) is represented by the dashed line. Best fits were obtained with √ 2Gµ = 90 km s −1 for NGC2841, and √ 2Gµ = 40 km s −1 for NGC2903. constant term, due to the string, in Eq. (8), also increases the velocities in the central region of the galaxies, which is in complete contradiction with observational data. As can be seen in both figures, the central region of the galaxies is never fitted. The discrepancy is less dramatic for NGC7331, but for this galaxy there is an intermediate region where no fit is possible.
IV. LARGE SCALE STRUCTURE
Dark matter should be approximately homogeneous and isotropic on cosmological scales, for otherwise that would yield strong (unobserved) signatures on the cosmic microwave background. If we take into account that the amplitude of the cosmic microwave background temperature perturbations generated by dark matter on scales of order ∼ H −1 0 /100 cannot be much larger than 10 −5 and assume that the power spectrum of density perturbations associated with the defects is white noise on scales larger than ξ, with the dispersion of order unity at the characteristic scale ξ itself, we require (100H 0 ξ) 3/2 ∼ < 10 −5 implying that ξ ∼ < 10 kpc. Of course, an important aspect one should also consider is the compensation of defect perturbations on large scales. In standard defect models where the characteristic scale of the network is of the order of the Hubble radius, its main effect is to provide a cut-off to the spectrum of density fluctuations seeded by the defects at a scale of the order of the horizon (see for example [25,26,27]).
In this case, because compensation only acts on very large scales, the above calculation holds. However, if the defect network becomes frozen then compensation may become effective at a much smaller scale (still always larger than ξ). Nevertheless, at the characteristic scale, ξ, the defect energy density perturbation is of order unity and on scales larger than ξ the power spectrum of the defect energy density is white noise. Consequently, even if we assume that compensation becomes effective on a scale ξ c with ξ < ξ c ≪ H −1 , the fact that during the matter era the growth factor is approximately equal to z eq ∼ 3.2 × 10 3 would lead to dramatic implications for the growth of small scale structures from very early on if ξ ∼ 1 Mpc. This would change the amplitude of small scale density fluctuations from an early stage strongly affecting the reionization history of the Universe [28].
In fact we expect that ξ c > H −1 eq . On one hand, if the network freezes during the radiation era then in that epoch the compensation scale is comparable to the horizon if the decay products of the defect network move at relativistic speeds. On the other hand, even if this is not the case, perturbations in the radiation component propagate close to the speed of light and consequently one might expect that ξ c ∼ H −1 during the radiation era. Deep in the matter era the fluctuations in the radiation component become less and less important and consequently if the source does not move then the comoving compensation scale should also not change much.
V. CONCLUSIONS
In this report we have considered the possible dynamical role of cosmic defects on scales significantly smaller than the horizon, In particular, we have discussed the possibility that a network of cosmic strings could provide the dark matter whose presence on galactic scales is suggested by the galactic rotation curves.
Our results show that even in principle only a nonstandard string network would have the key properties that could make the scenario viable. While no such scenarios are currently known (at least in quantitative detail) it is certainly conceivable that they may arise in cosmic (super)string models where some of the network's energy momentum gradually leaks into extra dimensions-such an energy leakage seems to be the only mechanism that could slow down the network to the required very low (effectively zero) velocities.
In any case, from the observational point of view the situation is quite clear. By adding a string contribution it is indeed quite easy to reproduce the asymptotic velocities for most of normal spiral galaxies, and the energy scale of the appropriate strings would in fact be the cosmologically interesting Gµ ∼ 10 −8 . However, any such string contribution will also increase the velocities in the central region of the galaxies, which is in complete contradiction with observational data. We thus conclude that although defects can still play a number of interesting astrophysical roles on galactic scales, they are unable to shed light on the dark matter problem.
FIG. 1 :
1Rotation curves for two spiral galaxies with no bulge.
FIG. 2 :
2Same as for
AcknowledgmentsWe thank Catarina Lobo and Y. Sofue for valuable advice and enlightening comments. The work of C.M. is funded by a Ciencia2007 Research Contract.
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| []
|
[
"Hidden fermion as milli-charged dark matter in Stueckelberg Z ′ model",
"Hidden fermion as milli-charged dark matter in Stueckelberg Z ′ model"
]
| [
"Kingman Cheung [email protected]†emailaddress:[email protected] \nDepartment of Physics\nPhysics Division\nNational Center for Theoretical Sciences\nNational Tsing Hua University\nHsinchu, HsinchuTaiwan, Taiwan\n",
"Tzu-Chiang Yuan \nDepartment of Physics\nPhysics Division\nNational Center for Theoretical Sciences\nNational Tsing Hua University\nHsinchu, HsinchuTaiwan, Taiwan\n"
]
| [
"Department of Physics\nPhysics Division\nNational Center for Theoretical Sciences\nNational Tsing Hua University\nHsinchu, HsinchuTaiwan, Taiwan",
"Department of Physics\nPhysics Division\nNational Center for Theoretical Sciences\nNational Tsing Hua University\nHsinchu, HsinchuTaiwan, Taiwan"
]
| []
| We augment the hidden Stueckelberg Z ′ model by a pair of Dirac fermions in the hidden sector, in which the Z ′ has a coupling strength comparable to weak scale coupling. We show that this hidden fermion-antifermion pair could be a milli-charged dark matter candidate with a viable relic density. Existing terrestrial and astrophysical searches on milli-charged particles do not place severe constraints on this hidden fermion. We calculate the flux of monochromatic photons coming from the Galactic center due to pair annihilation of these milli-charged particles and show that it is within reach of the next generation γ-ray experiments. The characteristic signature of this theoretical endeavor is that the Stueckelberg Z ′ boson has a large invisible width decaying into the hidden fermion-antifermion pair. We show that existing Drell-Yan data do not constrain this model yet. Various channels of singly production of this Z ′ boson at the LHC and ILC are explored. * | 10.1088/1126-6708/2007/03/120 | [
"https://arxiv.org/pdf/hep-ph/0701107v2.pdf"
]
| 53,756,183 | hep-ph/0701107 | b0dd8b0681fd032a46cf6c0176ca1e1fe73a5b06 |
Hidden fermion as milli-charged dark matter in Stueckelberg Z ′ model
May 2007
Kingman Cheung [email protected]†emailaddress:[email protected]
Department of Physics
Physics Division
National Center for Theoretical Sciences
National Tsing Hua University
Hsinchu, HsinchuTaiwan, Taiwan
Tzu-Chiang Yuan
Department of Physics
Physics Division
National Center for Theoretical Sciences
National Tsing Hua University
Hsinchu, HsinchuTaiwan, Taiwan
Hidden fermion as milli-charged dark matter in Stueckelberg Z ′ model
May 2007(Dated: October 2, 2018)arXiv:hep-ph/0701107v2 28
We augment the hidden Stueckelberg Z ′ model by a pair of Dirac fermions in the hidden sector, in which the Z ′ has a coupling strength comparable to weak scale coupling. We show that this hidden fermion-antifermion pair could be a milli-charged dark matter candidate with a viable relic density. Existing terrestrial and astrophysical searches on milli-charged particles do not place severe constraints on this hidden fermion. We calculate the flux of monochromatic photons coming from the Galactic center due to pair annihilation of these milli-charged particles and show that it is within reach of the next generation γ-ray experiments. The characteristic signature of this theoretical endeavor is that the Stueckelberg Z ′ boson has a large invisible width decaying into the hidden fermion-antifermion pair. We show that existing Drell-Yan data do not constrain this model yet. Various channels of singly production of this Z ′ boson at the LHC and ILC are explored. *
I. INTRODUCTION
The standard model (SM) of particle physics has been blessed with her elegant way of giving masses to the weak gauge bosons by the Higgs mechanism. However, the crucial ingredient of this mechanism, the Higgs boson, is still missing. In addition, a scalar Higgs boson mass is not stable under perturbative calculation. It will receive an enormous amount of radiative corrections to its mass such that a delicate cancellation between its bare mass and radiative corrections is needed so as to obtain a mass in the electroweak scale -this is the famous hierarchy problem. An alternative way to give mass to an abelian U(1) gauge boson is known as the Stueckelberg mechanism. Although it is very difficult to give masses to nonabelian gauge bosons without sacrificing renormalizability within the Stueckelberg approach, it is worthwhile to study the consequence of this mechanism as an extension to the SM with extra abelian U(1) factors.
Recently, Kors and Nath [1] showed that the SM extended by a hidden sector described by a Stueckelberg U(1) X and the gauge field C µ associated with it can pass all the existing constraints from electroweak data as well as direct search limits from the Tevatron. Through the combined Stueckelberg and Higgs mechanisms, the SM SU L (2) × U(1) Y gauge fields B µ and W 3 µ mix with the hidden sector gauge field C µ . After rotation from the interaction basis (C µ , B µ , W 3 µ ) to the mass eigenbasis (Z ′ µ , Z µ , A µ ), one obtains a massless state identified to be the photon γ and two massive eigenstates which are the SM-like Z boson and an additional Z ′ boson. As long as the mixing is small, the Z ′ boson only couples very weakly to the SM fermions, and so it can evade all the existing constraints on conventional Z ′ models. The allowed mass range for the Stueckelberg Z ′ can be anywhere from 200 GeV to a few TeV [2]. Typically, the mass of the Stueckelberg Z ′ is above the SM Z boson mass.
In this work, a pair of hidden Dirac fermions is introduced in the Stueckelberg Z ′ model. Such a possibility has been mentioned in Ref. [1], but its phenomenology was not explored.
There could be various types or generations of fermions in the hidden sector, just like our visible world. Since the abelian U(1) X is assumed to be the only gauge group in the hidden sector and there is no connector sector between our visible world and the hidden one in this class of models, all hidden fermions in this sector are stable. 1 Thus the hidden fermion-antifermion pair that we add in the model can be viewed as the lightest ones in the hidden sector, should there be more than one type of them. The SM fermions are neutral under this hidden U(1) X . Since this hidden fermion pair is stable, it can be the dark matter candidate of our Universe.
In the next Section, we will present some details of the Stueckelberg Z ′ extension of the SM with an additional pair of Dirac fermion-antifermion in the hidden sector. In Section III, we discuss milli-charged dark matter. Treating the hidden fermion as our candidate of dark matter, we calculate its relic density and explore the parameter space allowed by the WMAP measurement. We also calculate the monochromatic photon flux coming from the Galactic center due to pair annihilation of these hidden fermions. In Section IV, we explore some novel collider phenomenology of the Stueckelberg Z ′ with the presence of the hidden fermion. Since the width of the Stueckelberg Z ′ is no longer narrow, compared to the scenario studied in [2], its phenomenology is rather different. Comments and conclusions are given in Section V.
II. THE MODEL
The Stueckelberg extension [1] of the SM (StSM) is obtained by adding a hidden sector associated with an extra U(1) X interaction, under which the SM particles are neutral. 2 We explicitly specify the content of the hidden sector: a gauge boson C µ and a pair of fermion and antifermion χ andχ.
The Lagrangian describing the system is L StSM = L SM + L St , where
L SM = − 1 4 W a µν W aµν − 1 4 B µν B µν + D µ Φ † D µ Φ − V (Φ † Φ) + iψ f γ µ D µ ψ f ,(1)L St = − 1 4 C µν C µν + iχγ µ D X µ χ + 1 2 (∂ µ σ + M 1 C µ + M 2 B µ ) 2 ,(2)D µ = ∂ µ + ig 2 τ a 2 W a µ + ig Y Y 2 B µ ,(3)D X µ = ∂ µ + ig X Q X C µ ,(4)
where W a µν (a = 1, 2, 3), B µν , and C µν are the field strength tensors of the gauge fields W a µ , B µ , and C µ , respectively. The SM fermions f were explicitly forbidden from carrying the U(1) X charges, as implied by Eq. (3), while the hidden fermion pair only carries the U(1) X charge, as implied by Eq. (4). One can show that the scalar field σ decouples from the theory after gauge fixing terms are added upon quantization.
The mass term for V ≡ (C µ , B µ , W 3 µ ) T , after electroweak symmetry breaking Φ = v/ √ 2, is given by
[1] − 1 2 V T MV ≡ − 1 2 C µ , B µ , W 3 µ M 2 1 M 1 M 2 0 M 1 M 2 M 2 2 + 1 4 g 2 Y v 2 − 1 4 g 2 g Y v 2 0 − 1 4 g 2 g Y v 2 1 4 g 2 2 v 2 C µ B µ W 3 µ .(5)
A similarity transformation can bring the mass matrix M into a diagonal form
C µ B µ W 3 µ = O Z ′ µ Z µ A µ , O T M O = diag(m 2 Z ′ , m 2 Z , 0) .(6)
The m 2 Z ′ and m 2 Z are given by
m 2 Z ′ , Z = 1 2 M 2 1 + M 2 2 + 1 4 (g 2 Y + g 2 2 )v 2 ± (M 2 1 + M 2 2 + 1 4 g 2 Y v 2 + 1 4 g 2 2 v 2 ) 2 − (M 2 1 (g 2 Y + g 2 2 )v 2 + g 2 2 M 2 2 v 2 ) .(7)
The orthogonal matrix O is parameterized as 3
O =
where θ w is the Weinberg angle.
The interactions of fermions with the neutral gauge bosons before rotating to the mass eigenbasis are given by
− L N C int = g 2 W 3 µψ f γ µ τ 3 2 ψ f + g Y B µψf γ µ Y 2 ψ f + g X C µχ γ µ Q X χ ,(10)
where f denotes the SM fermions. The neutral gauge fields are rotated into the mass eigenbasis using Eq. (8), and the above neutral current interaction becomes
− L N C int =ψ f γ µ ǫ f L Z ′ P L + ǫ f R Z ′ P R Z ′ µ + ǫ f L Z P L + ǫ f R Z P R Z µ + eQ em A µ ψ f +χγ µ ǫ χ γ A µ + ǫ χ Z Z µ + ǫ χ Z ′ Z ′ µ χ ,(11)
where
ǫ χ γ = g X Q χ X (−c θ s φ ) , ǫ χ Z = g X Q χ X (s ψ c φ + s θ s φ c ψ ) , ǫ χ Z ′ = g X Q χ X (c ψ c φ − s θ s φ s ψ ) , ǫ f L,R Z = c ψ g 2 2 + g 2 Y c 2 φ −c 2 φ g 2 Y Y 2 + g 2 2 τ 2 + s ψ s φ g Y Y 2 , ǫ f L,R Z ′ = s ψ g 2 2 + g 2 Y c 2 φ c 2 φ g 2 Y Y 2 − g 2 2 τ 2 + c ψ s φ g Y Y 2 .(12)
In the above, we have used the relations
e = g 2 s θ = g Y c φ c θ and Q em = τ 3 2 + Y 2 ,
where Q em is the electric charge operator. From Eqs.(11)- (12), it is clear that in this class of model, the SM fermions interact with the hidden world through Z ′ and the hidden fermion interacts with our visible world through γ and Z. In our computation, we assume the following input parameters at the electroweak scale [3] α em (m Z ) = 1 128.91
, G F = 1.16637 × 10 −5 GeV −2 , m Z = 91.1876 GeV , sin 2 θ w = 0.231 , and the following three inputs related to the hidden sector δ ≡ tan φ, g X , and M Z ′ .
Since Q χ X always enters in the product form g X Q χ X , one can set Q χ X to be unity without loss of generality. We derive from α em , G F , m Z , and sin 2 θ w the values of
e = √ 4πα em , v = √ 2G F −1/2 , m W = m Z 1 − sin 2 θ w , and g 2 = 2m W /v .
We then fix the value of g Y by the following equation
e = g 2 g Y c φ g 2 2 + g 2 Y c 2 φ .
The other two angles θ and ψ are determined from the last two formulas in Eq. (9).
It is clear from Eqs. (11)-(12) that the chiral couplings of the SM Z boson are affected by the mixing. In fact, even the mass of the Z boson is modified in this model, as shown by
Eq. (7). It has been shown in Ref. [2] that in order to keep the Z boson mass within the experimental uncertainty, the mixing angle must satisfies
δ < ∼ 0.061 1 − (m Z /M 1 ) 2 .(13)
When δ is small and m Z ′ is large,
M 1 ≈ m Z ′ + O(g 2 v)
. The constraint coming from the electroweak precision data is more or less the same as in Eq. (13) [2].
The limits obtained in Ref. [2] also included the analysis from direct Z ′ production at the Tevatron. They showed that with the current Drell-Yan data,
m Z ′ > 250 GeV for δ ≈ 0.035 , m Z ′ > 375 GeV for δ ≈ 0.06 .(14)
If including the presence of a hidden fermion that the Stueckelberg Z ′ can couple to, the limit from direct Z ′ direction would be relaxed because of the smaller production rate into visible lepton pairs [2]. In Sec. IV, we will show that with a hidden fermion χ fulfilling the dark matter constraint, the Z ′ would dominantly decay into the hidden sector fermion pair provided that m Z ′ > 2m χ . It would therefore entirely remove the constraint in Eq. (14) from direct production. On the other hand, if m Z ′ < 2m χ the Z ′ boson cannot decay into the hidden sector fermions, and so the constraint in Eq. (14) stands.
In the following numerical works, we will apply the constraints on δ and m Z ′ given by Eqs. (13) and (14) respectively, but when m Z ′ > 2m χ the latter constraint will be dropped.
III. DARK MATTER PHENOMENOLOGY
A. Milli-charged dark matter
Milli-charged dark matter was first discussed by Goldberg and Hall [4], motivated by the work of Holdom [5] in which milli-charged particles in the hidden sector can interact with the visible photon due to kinetic mixing between the visible photon and the hidden or shadow photon. Numerous constraints for the milli-charged particles, including accelerator experiments, invisible decay in ortho-positronium, SLAC milli-charged particle search, Lamb shift, big-bang nucleosynthesis, dark matter search, search of fractional charged particles in cosmic rays, and other astrophysical reactions, were summarized in [6] (see Fig. 1 of the first reference in [6]). Study of the constraints on milli-charged particles from neutron stars and CMB measurements were discussed in Refs. [7] and [8] respectively. In summary, millicharged particles of mass from MeV to TeV with a fractional electric charge (10 −6 − 10 −1 ) of a unit charge are still allowed. We note that integral charged dark matter was contemplated in [9] and composite dark matter was studied in [10]. More recently, PVLAS [11] reported a positive signal of vacuum magnetic dichroism. It has been suggested [12] that photoninitiated pair production of milli-charged fermions with a mass range between 0.1 to a few eV and a milli-charge O(10 −6 ) of a unit charge can explain the signal. However, this signal has not been confirmed by other experiments like the Q & A experiment [13]. For detailed analysis of various experiments of vacuum magnetic dichroism, we refer our readers to Ref. [14].
B. Relic density and WMAP measurement
The first set of processes we consider in our relic density calculation are
χχ → Z ′ , Z, γ → ff
where f is any SM fermion. The amplitude for the annihilation
χ(p 1 )χ(p 2 ) → f (q 1 )f (q 2 )
can be written as
M =v(p 2 ) γ µ u(p 1 ) ×ū(q 1 ) γ µ (ξ L P L + ξ R P R ) v(q 2 )(15)
where P L,R = (1 ∓ γ 5 )/2, and
ξ L,R = ǫ χ γ eQ f em s + ǫ χ Z ǫ f L,R Z s − m 2 Z + ǫ χ Z ′ ǫ f L,R Z ′ s − m 2 Z ′ .(16)
The differential cross section is given by
dσ dz = N f 32π β f sβ χ (ξ 2 L + ξ 2 R )(u 2 m + t 2 m + 2m 2 χ (s − 2m 2 f )) + 4 m 2 f ξ L ξ R (s + 2m 2 χ )(17)
where
β f,χ = (1 − 4m 2 f,χ /s) 1/2 , N f = 3 (1) for f being a quark (lepton), t m = t − m 2 χ − m 2 f = −s(1−β f β χ z)/2, u m = u−m 2 χ −m 2 f = −s(1+β f β χ z)/2,
s is the square of the center-of-mass energy, and z ≡ cos Θ with Θ the scattering angle.
We also consider pair annihilation of χχ into two neutral gauge bosons,
χχ → V 1 V 2 with V 1,2 = γ, Z, Z ′(18)
in our relic density calculation 4 . The differential cross section is given by
dσ(χχ → V 1 V 2 ) dΩ = S(ǫ χ V 1 ) 2 (ǫ χ V 2 ) 2 β V 1 V 2 64π 2 sβ χ − 2 2 m 2 χ + m 2 V 1 2 m 2 χ + m 2 V 2 1 u 2 χ + 1 t 2 χ + 2 t χ u χ + u χ t χ − 4 1 u χ + 1 t χ 2 m 2 χ + m 2 V 1 + m 2 V 2 − 4 u χ t χ 2 m 2 χ + m 2 V 1 + m 2 V 2 2 m 2 χ − m 2 V 1 − m 2 V 2 θ(2m χ − m V 1 − m V 2 ) (19) where β V 1 V 2 = λ 1/2 1, m 2 V 1 /s, m 2 V 2 /s with λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca)
is the Mandelstam function, t χ and u χ are given by t χ = t − m 2 χ and u χ = u − m 2 χ respectively, and S is the statistical factor. We note that processes χχ → γγ, ZZ are doubly suppressed by the small mixing angles and χχ → Z ′ Z ′ are either suppressed or forbidden by phase space, and therefore their contributions are negligible in the annihilation rates.
The quantity that is relevant in the relic density calculation is the thermal averaged cross section σv , where v is the relative velocity of two annihilating particles. In the nonrelativistic approximation, v ≃ 2 β χ . To estimate the relic density of a weakly-interacting massive particle, we use the following formula [15]
Ω χ h 2 ≃ 0.1 pb σv .
With the most recent WMAP [16] result on dark matter density
Ω CDM h 2 = 0.105 ± 0.009 ,
where we have used the WMAP-data-only fit and taken Ω CDM = Ω matter − Ω baryon , one can translate this WMAP data into σv ≃ 0.95 ± 0.08 pb . In estimating the annihilation rate during the freeze-out, we assume that the species has a velocity-squared v 2 ≃ 0.1. To get a crude estimation, we ignore the thermal average and evaluate σv directly.
In Fig.1(a) and (b), we show the contours of σv = 0.95 ± 0.16 pb (2 σ range) in the plane of (g X , δ) for various input values of m Z ′ and m χ . We have included χχ → γZ ′ , ZZ ′ , and
ff , with f = ν e , ν µ , ν τ , e − , µ − , τ − , u, d, s, c, b, and t that are kinematically allowed. From GeV and a larger m Z ′ = 600 GeV, δ ≃ 0.03 and a slightly larger g X ≃ 0.7 can also do the job. For comparison, we note that e ≃ 0.31 and g 2 ≃ 0.65 in the SM. Thus, the value of the hidden U X (1) coupling g X that we deduced from the WMAP measurement has the same order of the weak coupling g 2 of the visible sector. In Fig.1(c), we show the annihilation rate σv versus m χ for δ = 0.03, g X = g 2 , and a fixed m Z ′ = 500 GeV. Clear resonance structures of Z and Z ′ are seen. In Fig.1(d), and m Z ′ in order to give the correct dark matter density. In the latter branch, the Tevatron bound of m Z ′ > 250 GeV for δ ≈ 0.03 has to be imposed. Therefore, the case of m χ < m Z ′ /2
is more preferred theoretically.
The hidden fermion χ couples to the photon via the mixing angles c θ s φ , the value of which is about 0.9 × 0.03 ≈ 0.03. Therefore, effectively the fermion χ "acquires" an electric charge of g X Q χ X c θ s φ /e ≈ 0.06, when its coupling to the photon is considered. Therefore, the range of m χ ∼ O(100) GeV and the size of effective electric charge ≃ 0.06 implied by dark matter density requirement in our calculation are consistent with the constraints on milli-charged particles [6].
C. Indirect detection
If milli-charged hidden fermions like χ andχ are the dark matter, their pair annihilation into γγ, γZ, and γZ ′ in regions of high dark matter density, e.g. the Galactic center, can give rise to monochromatic γ-ray line that can reach our Earth for their indirect detection.
The cross sections for these processes can be obtained from Eq. (19) readily. The observed γ-ray flux along the line-of-sight between the Earth and the Galactic center is given by [15]
Φ γ (ψ, E) = σv dN γ dE γ 1 4πm 2 χ line of sight dsρ 2 (r(s, ψ)) ,(21)
where the coordinate s runs along the line of sight in a direction making an angle ψ from the direction of the Galactic center, dN γ /dE γ is the energy spectrum of the γ-rays, and v ≈ 2β χ is the relative velocity of the dark matter χ andχ, and the present value of v ≈ 10 −3 . The flux from a solid angle ∆Ω is often expressed as
Φ γ (∆Ω, E) ≈ 5.6 × 10 −12 dN γ dE γ σv pb 1 TeV m χ 2 J(∆Ω)∆Ω cm −2 s −1 ,(22)χ -χ -> γ γ χ -χ -> γ Z χ -χ -> γ Z' m Z' = 300 GeV
For the process of χχ → γγ, we would have a mono-energetic γ-ray line with dN γ /dE γ ≈ 2δ(E γ − m χ ). Such a line, if observed, would be a distinctive signal for dark matter annihilation. Similarly, processes χχ → γZ and χχ → γZ ′ will have a photon energy spectrum
as dN γ /dE γ ≈ δ(E γ − m χ (1 − m 2 Z,Z ′ /4m 2 χ )
). The contributions from these processes to the photon flux are shown in Fig. 2 with J = 100 and ∆Ω = 10 −3 . In this plot, we have taken a moderate value for J = 100 (averaged over 10 −3 sr at the Galactic center). From Table 7 of Ref. [15], we know the value of J varies from 2.166 × 10 (Kra profile) to 1.352 × 10 3 (NFW profile) and to 1.544 × 10 5 (Moore profile). There are also cold dark matter profiles with dense spikes [17] near the Galactic center due to the accretion by central black holes that can give rise significant enhancement to the quantity J . With a rather conservative choice of J = 100, the flux of the γ-rays from the process χχ → γγ is quite small due to double suppression of (ǫ χ γ ) 2 . The process χχ → γZ contributes at a somewhat lower flux level. The process χχ → γZ ′ can also contribute to the monochromatic photon flux, provided that 2m χ > m Z ′ . Since this process is only suppressed by one power of the mixing angle, it could be more substantial than the doubly-suppressed process χχ → γγ. Note that since 2m χ > m Z ′ , the Tevatron bound Eq. (14) of m Z ′ > 250 GeV for δ ≈ 0.035 must be enforced. When kinematics allowed, the photon flux from this process is three orders of magnitude higher than that from χχ → γγ. For comparison, we note that the photon flux from the neutralino pair annihilationχ 0χ0 → γγ [18] has been estimated to be about 1.5 × 10 −14 (2 × 10 −13 ) cm −2 s −1 if the neutralino is a Higgsino-LSP (Wino-LSP), using the same moderate value of J = 100 [19,20].
The expected sensitivities for the new Atmospheric Cerenkov Telescope (ACT) experiments such as HESS [21] and VERITAS [22] are at the level of (10 −14 − 10 −13 ) cm −2 s −1 with an angular coverage of about 10 −3 . They are sensitive to the γ-rays from a few hundred GeV to TeV. On the other hand, the Gamma-ray Large Area Space Telescope (GLAST) experiment [23] due for launch in Fall this year, can probe γ-rays from 20 MeV to 300 GeV, but at a lower sensitivity level about 2 × 10 −9 − 10 −10 cm −2 s −1 . From Fig. 2, there is a small range of m χ ( m χ < 100 GeV) such that χχ → γγ contributes at a level larger than 10 −14 cm −2 s −1 . The process χχ → γZ contributes at a level below the sensitivities of all these experiments for all ranges of m χ , whereas the process χχ → γZ ′ can contribute at a much larger flux and it is indeed above the sensitivity level of ACT experiments for m χ < 600 GeV. Since GLAST can only be sensitive to γ-rays of 300 GeV or less with lower sensitivity, it is hard to detect the γ-rays from the lighter milli-charged dark matter. For heavier milli-charged dark matter, the γ-rays can be above a few hundred GeV and thus above the sensitivity reaches of HESS and VERITAS.
The continuum γ-ray background from astrophysical sources near Galactic Center is largely an unsettled issue due to astrophysical uncertainties. There have been data showing excess of γ-rays in different energy regimes near the Galactic center. The EGRET experiment [24] has reported an excess of γ-rays in the region of the Galactic center, including the galactic longitude and latitude position at l = 0 • and b = 0 • degrees. The level of excess is above the expectation of primary cosmic rays interacting with interstellar medium. The EGRET excess region is around 1 GeV. However, there may be some other unknown sources of γ-rays around the Galactic center. It is hard to establish the fact that the excess is due to dark matter annihilation, because the excess does not have specific features. This is in contrast to the monochromatic γ-ray flux, which is a clean signature of the dark matter annihilation.
In the Galactic center region, excesses of γ-rays were also reported by VERITAS [25] in the range above 2.8 TeV and by CANGAROO collaborations [26] in the range of 250 GeV to 1 TeV. Such excesses are also hard to be explained by conventional dark matter candidates.
The HESS Collaboration also had a measurement of TeV gamma rays from the Galactic Center [27], which is, to some extent, in disagreement with the CANGAROO results. It was pointed out [28] that this TeV γ−ray excess is likely to be of astrophysical origin and thus it constitutes a background for detecting dark matter annihilation. The origin of these backgrounds may be due to violent acceleration of cosmic protons and other particles by the chaotic magnetic fields near the Galactic center black hole [29]. After escaping the black hole environment and fly off into the interstellar medium, these extremely high energy protons collide with low energy protons (hydrogen gas) to form pions, which subsequently decay into high energy γ-rays that can radiate in all directions.
Due to its unknown astrophysical origin, it is hard to establish accurately the true continuum γ-ray background which could be used for comparison with dark matter annihilation.
Thus, using the continuum γ-ray signal is difficult to provide strong evidence for dark matter, unless the dark matter annihilation rate is very large. On the other hand, provided that the annihilation cross section is sufficiently large, the monochromatic photon line would be a "smoking gun" signal for dark matter annihilation, since the energy of the γ-ray is uniquely determined by the mass of the milli-charged dark matter (and the Z ′ mass as well in the χχ → γZ ′ channel). Nevertheless, the EGRET and HESS continuum backgrounds still pose a serious challenge to detecting the monochromatic photon line due to dark matter annihilation in the Galactic center region [28]. It was shown in Ref. [28] that in order for a photon line to be detected above the continuum background, the quantity (σv/10 −29 cm 3 s −1 ) J ∆Ω must be larger than 10 − 100. This implies that the photon flux to be larger than 1.9 × (TeV/m χ ) 2 × (10 −14 − 10 −13 ) cm −2 s −1 . From Fig. 2, it is easy to see that for m χ between 150 and 300 GeV, the photon flux in the χχ → γZ ′ channel is within the detectability level.
One may also give a rough estimate for continuum photon flux arises from the millicharged dark matter annihilation into light quark pairs. The continuum photon spectrum mainly comes from the light quark fragmentation into neutral pions, which subsequently decay into secondary photons. The differential spectrum dN γ /dE γ can be obtained by Monte Carlo event generators, e.g. PYTHIA, and then parameterized as a quark fragmentation function. We can use Eq. (21) with dN γ /dE γ given by a fragmentation-like function [30]:
dN γ dx = η x a exp(b + cx + dx 2 + ex 3 ) ,(24)
where x = E γ /m χ and for a light quark, e.g. u or d quark at an energy of 500 GeV, η = 1, a = −1.5, b = 0.047, c = −8.7, d = 9.14, and e = −10.3. These coefficients depend only mildly on the energy of the light quarks [30]. Putting all these factors together, we estimate the integrated photon flux with E γ > 1 GeV to be of the order of 10 −10 (10 −11 ) cm −2 s −1 for m χ = 100 (500) GeV. It is at most around or slightly below the sensitivity level of GLAST.
Since the VERITAS and HESS experiments are sensitive to higher energy and the above spectrum Eq.(24) falls off rapidly as x increases, their integrated photon fluxes are at least an order of magnitude smaller than that of GLAST. Despite challenging by the uncertain astrophysical backgrounds, this continuous secondary photon spectrum together with the monochromatic photon line from milli-charged dark matter annihilation can be probed by the next generation of γ-ray experiments.
IV. COLLIDER PHENOMENOLOGY
Phenomenology of the Stueckelberg Z ′ with the presence of the hidden fermionantifermion χ andχ that the Z ′ can decay into is quite different from the one studied before in Refs. [1,2].
The partial width of Z ′ into a SM fermion pair ff is given by
Γ(Z ′ → ff ) = N f β f 24π m Z ′ ǫ f L Z ′ 2 + ǫ f R Z ′ 2 1 − m 2 f m 2 Z ′ + 6 ǫ f L Z ′ ǫ f R Z ′ m 2 f m 2 Z ′(25)
and into hidden fermion pair χχ is simply
Γ(Z ′ → χχ) = β χ 12π m Z ′ ǫ χ Z ′ 2 1 + 2 m 2 χ m 2 Z ′ .(26)
Here, β f,χ = (1 − 4m 2 f,χ /m 2 Z ′ ) 1/2 . The total width of Z ′ is evaluated by summing over all partial widths, including both the SM modes and the hidden mode. We show in Fig. 3 the various branching ratios for Z ′ as a function of its mass with the following inputs g X = g 2 , δ = 0.03, and m χ = 60 GeV. Since the mixing angle is so small (δ = 0.03), the Z ′ is mainly composed of the C µ boson of the hidden sector. Hence, the Z ′ dominantly decays into the hidden sector fermion pair while it has only a small fraction of 10 −3 into visible fermions. The strategy for the search of this Z ′ would be very different from all the previous conventional Z ′ models including the hidden Stueckelberg Z ′ studied in [1,2]. Before we explore for the possible collider phenomenology of the Stueckelberg Z ′ boson and the hidden sector fermion χ, we have to make sure that the new particles and the hidden sector interactions will not upset the existing data.
A. Constraints from invisible decays of Z and quarkonia
Firstly, the SM Z boson that is observed at LEP would decay into a pair of hidden fermions χχ, giving rise to additional invisible width other than the neutrinos. Because of the mixings among the three neutral gauge bosons, the Z boson can couple to the χχ pair via the mixing angle s φ . We have calculated the partial width of Z → χχ for g X = g 2 , δ = 0.03 (consistent with the dark matter requirement), and m χ = 0 − 45 GeV. The partial width is about 0.24 MeV, which is much smaller than the uncertainty 1.5 MeV of the invisible width of the Z boson [3]. Even if we allow a larger mixing angle δ = 0.061 (its maximum value allowed by Eq. (13)), the invisible width of Z would be at most 1 MeV, which is still within the 1σ uncertainty of the data. If the mass of χ is beyond half of the Z boson mass, the invisible width of the Z boson would not constrain the model.
The hidden fermion χ can also couple to the photon via the mixing angles c θ s φ , the maximum of which is about 0.9×0.03 ≈ 0.03. Therefore, effectively the fermion χ "acquires" an electric charge of g X Q χ X c θ s φ /e ≈ 0.06 when its coupling to the photon is considered. If χ is very light, of the order of MeV, it could be produced in J/ψ and Υ decays as invisible particles. Constraints on invisible decays of J/ψ and Υ exist (for a comprehensive review on constraints on light dark matter: see Ref. [31]). A very recent update on the Υ(1S) invisible width is given in Ref. [32]. The invisible widths of J/ψ and Υ are respectively
B(J/ψ → invisible) < 7 × 10 −3 and B(Υ(1S) → invisible) < 2.5 × 10 −3 .
However, the partial width of J/ψ into χχ is suppressed by the milli-charged factor of (0.06) 2 relative to the partial width into e − e + . Thus B(J/ψ → χχ) ≈ (0.06) 2 × B(J/ψ → e − e + ) ≈ 10 −4 , which is well below the above limit. The situation for Υ invisible decay is very similar: B(Υ(1S) → χχ) ≈ (0.06) 2 × B(Υ(1S) → e − e + ) ≈ 10 −4 , which is also safe. Indeed, a recent study [33] using 400 fb −1 luminosity collected at the Υ(4S), the B-
factory can limit B(Υ(1S) → invisible) < ∼ 10 −3 .
There are also other decays modes, such as J/ψ or Υ → γ + invisible, but it is straightforward to check that with an effective charge of 0.06 the experimental limits of these radiative invisible modes do not constrain the model at all. If the mass m χ is above 5 GeV, it has no constraint at all from these invisible decays of the quarkonia.
B. Constraint from singly production of Z ′ at LEPII Singly production of the Z ′ at LEPII is possible via e − e + → γZ ′ followed by the invisible decay of the Z ′ . This process is very similar to the SM process e − e + → γZ → γνν. The differential cross section for e − e + → γZ ′ is given by
dσ(e − e + → γZ ′ ) d cos Θ = β Z ′ e 2 Q 2 e 32πs ǫ e L Z ′ 2 + ǫ e R Z ′ 2 1 ut t 2 + u 2 + 2sm 2 Z ′ ,(27)
where Θ is the scattering angle of the photon, t = −sβ Z ′ (1 − cos Θ)/2, u = −sβ Z ′ (1 + cos Θ)/2, and β Z ′ = (1 − m 2 Z ′ /s). We show the production cross section at LEPII energy √ s = 205 GeV in Fig. 4 as a function of m Z ′ . Since the Z ′ would decay into invisible χχ, the signal of which would be a mono-photon. The recoil mass spectrum would then indicate the mass of the Z ′ . In the figure, we also show the 95% C.L. upper limits on mono-photon production as a function of the missing mass obtained by DELPHI [34]. A small mass range of Z ′ , 180 GeV < ∼ m Z ′ < ∼ 200 GeV, is disfavored by the data. However, one has to be cautious The production cross section of Z ′ followed by the leptonic decay at the Tevatron is given by
σ(pp → Z ′ → ℓ − ℓ + ) = 1 144 1 s m Z ′ Γ Z ′ ǫ ℓ L Z ′ 2 + ǫ ℓ R Z ′ 2 q=u,d,s,c ǫ q L Z ′ 2 + ǫ q R Z ′ 2 1 r dx x f q (x) fq r x(28)
where √ s = 1960 GeV, r = m 2 Z ′ /s, Γ Z ′ is the total width of Z ′ given in Eqs. (25) and (26), and ǫ f L,R Z ′ can be found in Eq. (12). This Drell-Yan cross section for the Z ′ boson is plotted in Fig. 5, where the 95% C.L. upper limits on σ(Z ′ ) · B(Z ′ → e − e + ) from the CDF preliminary results [35] are also shown. It is clear that the present CDF limits do not constrain the model at all. This is in sharp contrast to the results studied in Ref. [2] because the Z ′ boson that we consider here has a very small branching fraction into charged lepton pair. The Z ′ boson would decay preferably into the hidden sector fermions instead Perhaps one has to rely on the invisible decay mode of the Stueckelberg Z ′ of this model to identify its presence. Here we calculate the predictions of singly Z ′ production at the LHC and ILC. Other than the Drell-Yan process that we have considered, the next relevant process to probe this invisible Z ′ is via qq → gZ ′ followed by Z ′ → χχ, which gives rise to monojet events. The subprocess cross section can be easily adapted from Eq. (27): p Tj > 20 GeV, |η j | < 2.5 g X = g 2 , δ = 0.03 FIG. 6: Production cross section for the process pp → j + Z ′ followed by invisible decay of the Z ′ with g X = g 2 and δ = 0.03 at the LHC. We imposed p T j > 20 GeV and |η j | < 2.5 on the jet.
dσ(qq → gZ ′ ) d cos θ * = β Z ′ g 2 s 72πŝ ǫ q L Z ′ 2 + ǫ q R Z ′ 2 1 ut t 2 +û 2 + 2ŝ m 2 Z ′ .(29)
Other cross channels, e.g., qg → qZ ′ , can be obtained from Eq. (29) using the crossing symmetry. The branching ratio B(Z ′ → χχ) is very close to unity. We show in Fig. 6 the production rate of monojet events versus m Z ′ with g X = g 2 and δ = 0.03 at the LHC under the jet cuts of p T j > 20 GeV and |η j | < 2.5. The qqZ ′ coupling is suppressed by the small mixing angle, the same as in the Drell-Yan process, but unlike the Drell-Yan process, this monojet amplitude is suppressed by only one power of the mixing angle instead of two.
Therefore, the rate is not negligible. Also, the true SM background for monojet is rather rare. Thus, the monojet event actually signals the presence of such an invisible Z ′ .
Another place to detect such an invisible Z ′ is at the ILC with the process e − e + → γZ ′ → γχχ, which we have considered above for the mono-photon limits from LEP. We extend the energy to 0.5 − 1.5 TeV and calculate the event rates for the mono-photon final state. We show in Fig. 7 the production rates at √ s = 0.5, 1, 1.5 TeV with g X = g 2 and δ = 0.03.
V. CONCLUSIONS
We have proposed an extension of the Stueckelberg Z ′ standard model by adding a pair of fermion and antifermion in the hidden sector, which has only a U(1) X symmetry. The stability of the hidden fermion pair with its weak sized interaction makes it a suitable dark matter candidate with a correct amount of dark matter density. We have calculated the photon flux from the Galactic center due to the annihilation of this milli-charged dark matter. If 2m χ < m Z ′ , there is only a small range of m χ that the photon flux is above the sensitivity level of the future γ-ray experiments. However, if 2m χ > m Z ′ there is a wide range of m χ that the photon flux is above the sensitivity level. The collider phenomenology may be different from those studied in Ref. [2], because the dominant decay of the Z ′ is into the invisible χχ if kinematically allowed. In this case, the present Drell-Yan data cannot constrain the model at all. We have proposed the monojet signal at the LHC and the mono-photon signal at the future ILC to probe this invisibly decaying Stueckelberg Z ′ boson.
FIG. 1 :
1(a)-(b) are contours of σv = 0.95 ± 0.16 pb (2 σ range) in the plane of (g X , δ) for various m Z ′ and m χ . Part (c) shows the annihilation rate σv versus m χ with m Z ′ = 500 GeV, g X = g 2 , and δ = 0.03. Part (d) shows the contour of σv = 0.95 ± 0.16 pb (2 σ range) in the (m χ , m Z ′ ) plane.
Fig. 1 (
1a) for m χ = 60 GeV and m Z ′ = 300 GeV, we can see that δ ≃ 0.03 and g X ≃ 0.6 can give the correct amount of dark matter. Similarly, fromFig.1(b)with the same m χ = 60
we show the contours of σv in the (m χ , m Z ′ ) plane. There are two branches: (i) the upper branch where m χ < m Z ′ /2 and the band relating m χ and m Z ′ is relatively wide; (ii) the lower branch where 2m χ > m Z ′ and the band relating m χ and m Z ′ is quite narrow. A narrow band implies the need of a fine-tuned relation between m χ
FIG. 2 :
2The resulting photon flux from annihilation processes χχ → γγ, γZ, and γZ ′ . We have used typical values of J = 100, ∆Ω = 10 −3 , and the present value of v ≈ 10 −3 .with the quantity J(ψ) defined by
FIG. 3 :
3Branching ratios for Z ′ with g X = g 2 , δ = 0.03, and m χ = 60 GeV.
FIG. 4 :
4Comparison with the DELPHI data on the mono-photon production. The theory prediction is for g X = g 2 and δ = 0.03.in this relatively soft photon region where large theoretical uncertainties are expected to be important.C. Drell-Yan production of Z ′ at the Tevatron
FIG. 5 :
5Drell-Yan cross sections pp → Z ′ → e − e + versus m Z ′ for g X = g 2 and δ = 0.03. We also show the 95% C.L. upper limits on σ(Z ′ ) · B(Z ′ → e − e + ) from the CDF preliminary results [35]. of visible particles. On the other hand, the Stueckelberg Z ′ in Ref. [2] only decays into the SM particles. In our case the Z ′ only has a branching ratio of ∼ 10 −4 − 10 −3 into leptonic pairs, and it would not be easily detected in the Drell-Yan channel. Neither the hadronic decay modes of Z ′ can afford it to be detected. Even in the future runs of the Tevatron with a sensitivity reaching the level of 10 −3 − 10 −2 pb, it is still not possible to detect this kind of Z ′ boson through the Drell-Yan channel.D. Singly production of Z ′ at LHC and ILC
FIG. 7 :
7Production cross section for the process e − e + → γ + Z ′ followed by invisible decay of the Z ′ with g X = g 2 and δ = 0.03 at ILC ( √ s = 0.5, 1, 1.5 TeV). We imposed E γ > 10 GeV and | cos γ | < 0.95 on the photon.
This is in analogous to the pure QED case, muon does not decay into an electron plus a photon.
It was shown in Ref.[1] that the SM fermions are neutral under the extra U (1) X has the advantage of maintaining the neutron charge to be zero.
We note that the middle column is chosen to be different from that of Ref.[1] by an overall minus sign.
We have ignored the channel χχ → γ, Z, Z ′ → W + W − which may contribute to certain extent.
AcknowledgmentWe would like to thank Pran Nath for encouragement and Holger Gies for useful comments on the manuscript. K. C. also thanks K. S. Cheng and the Centre of Theoretical and Computational Physics at the University of Hong Kong for hospitality. This research was supported in part by the National Science Council of Taiwan R. O. C. under Grant No. NSC
Since only a U X (1) symmetry is assumed in the hidden sector, each hidden fermion is stable against decay. Therefore, if we assume more fermion pairs in the hidden sector, their relic densities are additive. Thus, a larger coupling constant is needed to ensure larger annihilation cross sections. One can also consider multiple hidden Stueckelberg U(1) extension of the SM. We refer to Ref.[1] for the discussion for this possibilitySince only a U X (1) symmetry is assumed in the hidden sector, each hidden fermion is stable against decay. Therefore, if we assume more fermion pairs in the hidden sector, their relic densities are additive. Thus, a larger coupling constant is needed to ensure larger annihilation cross sections. One can also consider multiple hidden Stueckelberg U(1) extension of the SM. We refer to Ref.[1] for the discussion for this possibility.
It can be easily detected in the Drell-Yan channel. The existing data constrains the model, as given by Eq. (14) originally obtained by the authors in Ref. When m Z ′ < 2m χ , the Z ′ decays dominantly into visible particles. Photon flux from pair annihilation of χχ → γZ ′ at the Galactic center is also within reach at the next generation of γ-ray experimentsWhen m Z ′ < 2m χ , the Z ′ decays dominantly into visible particles. It can be easily detected in the Drell-Yan channel. The existing data constrains the model, as given by Eq. (14) originally obtained by the authors in Ref.[2]. Photon flux from pair annihilation of χχ → γZ ′ at the Galactic center is also within reach at the next generation of γ-ray experiments.
When m Z ′ > 2m χ , the Z ′ decays dominantly into invisible χχ. The present Drell-Yan data cannot constrain the model, neither can the invisible decays of J/ψ and Υ for a very light χ. However, the mono-photon production limits obtained by DELPHI. When m Z ′ > 2m χ , the Z ′ decays dominantly into invisible χχ. The present Drell-Yan data cannot constrain the model, neither can the invisible decays of J/ψ and Υ for a very light χ. However, the mono-photon production limits obtained by DELPHI
With a mass of O(100) GeV and an effective charge 0.06 of a unit charge, the hidden fermions are consistent with the existing constraints on millicharged particles [6]. As milli-charged particle is of very recent interests [12], an update on the terrestrial and astrophysical constraints on this hidden milli-charged particle is desirable. The hidden fermion appears to have a milli-charge as it acquires a small effective coupling to the photon through the mixing induced by the combined Higgs and Stueckelberg mechanisms. 95-2112-M-007-001 and by the National Center for Theoretical Sciences. Note added. Stueckelberg Z ′ extension with kinetic mixing has been studied recently inThe hidden fermion appears to have a milli-charge as it acquires a small effective cou- pling to the photon through the mixing induced by the combined Higgs and Stueck- elberg mechanisms. With a mass of O(100) GeV and an effective charge 0.06 of a unit charge, the hidden fermions are consistent with the existing constraints on milli- charged particles [6]. As milli-charged particle is of very recent interests [12], an update on the terrestrial and astrophysical constraints on this hidden milli-charged particle is desirable. 95-2112-M-007-001 and by the National Center for Theoretical Sciences. Note added. Stueckelberg Z ′ extension with kinetic mixing has been studied recently in
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[
"Time discretisation and L 2 stability of polynomial summation-by-parts schemes with Runge-Kutta methods",
"Time discretisation and L 2 stability of polynomial summation-by-parts schemes with Runge-Kutta methods"
]
| [
"Hendrik Ranocha \nInstitute Computational Mathematics\n\n",
"PhilippÖffnerJan Glaubitz \nInstitute Computational Mathematics\n\n",
"Thomas Sonar \nInstitute Computational Mathematics\n\n"
]
| [
"Institute Computational Mathematics\n",
"Institute Computational Mathematics\n",
"Institute Computational Mathematics\n"
]
| []
| Splitting the discretisation of a partial differential equation by the method of lines in a semidiscretisation in space and a following integration in time, the coupling of these schemes has to be considered for fully discrete stability. This article focuses at first on the extension of adaptive techniques developed by the authors for semidiscretisations in the correction procedure via reconstruction / flux reconstruction framework. Additionally, general results about stability of some Runge-Kutta methods for linear problems are established, followed by numerical examples. | null | [
"https://arxiv.org/pdf/1609.02393v1.pdf"
]
| 119,747,163 | 1609.02393 | 4bd5506d2a0fc5e399eaa68f82a98245f77ea601 |
Time discretisation and L 2 stability of polynomial summation-by-parts schemes with Runge-Kutta methods
September 8, 2016
Hendrik Ranocha
Institute Computational Mathematics
PhilippÖffnerJan Glaubitz
Institute Computational Mathematics
Thomas Sonar
Institute Computational Mathematics
Time discretisation and L 2 stability of polynomial summation-by-parts schemes with Runge-Kutta methods
September 8, 2016
Splitting the discretisation of a partial differential equation by the method of lines in a semidiscretisation in space and a following integration in time, the coupling of these schemes has to be considered for fully discrete stability. This article focuses at first on the extension of adaptive techniques developed by the authors for semidiscretisations in the correction procedure via reconstruction / flux reconstruction framework. Additionally, general results about stability of some Runge-Kutta methods for linear problems are established, followed by numerical examples.
Introduction
A common idea in the numerical treatment of time-dependent partial differential equations, such as conservation laws
∂ t u + ∂ x f (u) = 0,(1)
equipped with adequate initial and boundary conditions, is to split the discretisations of space and time by the method of lines. There, a space discretisation is developed first, resulting in a system of ordinary differential equations that have to be solved by an appropriate scheme.
Here, Runge-Kutta methods will be considered. For conservation laws with periodic boundary conditions, L 2 stability can often be proven, and is an adequate concept, especially for linear equations. For nonlinear scalar conservation laws with convex flux f (u), this concept of entropy stability is similarly successful.
L 2 stability of the semidiscretisation means, that an appropriate scalar product of the time derivative with the solution is non-positive
∂ t 1 2 u 2 = u, ∂ t u ≤ 0,(2)
resulting in strong stability of the form u(t) 2 ≤ u(0) 2 . In the continuous setting, such estimates can often be obtained by integration by parts, at least for linear problems. Using summation-by-parts (SBP) as a discrete analogue, these results can be transferred to the semidiscrete setting. As references, the review articles of Fernández, Hicken, and Zingg (2014); Nordström and Eliasson (2015); Svärd and Nordström (2014) and references cited therein can be used. Originating in finite-difference methods, these concepts have been transferred to nodal discontinuous Galerkin methods as described by Gassner (2013). Based on a reformulation of the correction procedure via reconstruction / flux reconstruction introduced by Huynh (2007), a framework of high-order semidiscretisations for conservation laws, in this setting of SBP operators by Ranocha,Öffner, andSonar (2015, 2016), Glaubitz, Ranocha,Öffner, and Sonar (2016); Ranocha, Glaubitz,Öffner, and Sonar (2016) investigated fully discrete (strong) stability using the explicit Euler method as time discretisation. Looking at strong stability, several results for linear problems have been obtained by Levy and Tadmor (1998); Tadmor (2002). Recently, Sun and Shu (2016) conducted further investigations about the classical fourth order Runge-Kutta method.
There are other stability concepts, adapted especially to linear (or linearised) problems, as described inter alia in the monograph by Gustafsson, Kreiss, and Oliger (2013) and references cited therein. However, they are not equivalent or can be adapted to nonlinear problems similarly. On the other hand, corresponding results, inter alia of Kreiss and Wu (1993); Wu (1995), about the extension of semidiscrete stability to the fully discrete scheme are far more developed.
This article is organised as follows. In section 2, general Runge-Kutta time discretisations will be investigated and the adaptive modal filtering of the authors will be extended to these schemes. Section 3 focuses on linear problems, providing new results about the strong stability of some Runge-Kutta methods. Afterwards, some examples will be presented in section 4. Finally, the results are summed up in section 5. Some calculations of section 3 have been moved to the appendix A.
Runge-Kutta time discretisations and adaptive modal filtering
In this section, a stable semidiscretisation of a scalar conservation law
∂ t u + ∂ x f (u) = 0 (3)
in one spatial dimension with adequate initial and periodic boundary conditions will be used for a fully discrete scheme. Common model equations are linear advection with flux f (u) = u and Burgers' equation with nonlinear flux f (u) = u 2 2 . Semidiscretisations using SBP operators can be derived that are both conservative across elements and stable in the discrete norm induced by M , the mass matrix associated to the given SBP operator, as described inter alia by Ranocha,Öffner, andSonar (2015, 2016). These are obtained by dividing the domain into non-overlapping elements and representing the numerical solution as a polynomial in each element. If a nodal basis is used, they can be seen as nodal discontinuous Galerkin (DG) methods.
Explicit Euler and strong stability preserving methods
The semidiscrete schemes have to be rendered fully discrete by an adequate time integration method. Ranocha,Öffner, and Sonar (2016) observed that a simple explicit Euler method
u → u + := u + ∆t ∂ t u(4)
is conservative (across elements), but not stable, since
u + 2 M = u T + M u + = u T + M u + + 2∆t u T M ∂ t u + (∆t) 2 ∂ t u T + M ∂ t u + = u + 2 M + 2∆t u, ∂ t u M + (∆t) 2 ∂ t u 2 M ,(5)
where the second term can be estimated via the stable semidiscretisation, whereas the last term of order (∆t) 2 is non-negative and may render instabilities. At least, the fully discrete scheme does not obey the desired estimate of the semidiscrete one. As a remedy, Ranocha, Glaubitz,Öffner, and Sonar (2016) and Glaubitz, Ranocha,Öffner, and Sonar (2016) introduced adaptive artificial viscosity and modal filtering to remove the undesired increase of the L 2 approximating norm induced by M .
The enforced stability carries over to strong stability preserving (SSP) methods, since these can be written as a convex combination of explicit Euler steps, see inter alia the monograph of Gottlieb, Ketcheson, and Shu (2011) and references cited therein. However, since the triangle inequality is invoked for the resulting estimates, an undesired additional decrease of the norm may result. Therefore, the adaptive modal filtering should be applied only after a full time step and not for every stage. Additionally, this renders the computation more efficient.
Runge-Kutta methods
In this subsection, an explicit Runge-Kutta method with s stages, given by its Butcher tableau
c A b(6)
is considered. Here, A ∈ R s×s and b, c ∈ R s . Since there is no explicit dependence on the time in the semidiscretisation, one step from u 0 to u + is given by
u i := u 0 + ∆t s j=1 a ij ∂ t u j , u + := u 0 + ∆t s i=1 b i ∂ t u i .(7)
Here, the u i are the stage values of the Runge-Kutta method. It is also possible to express the method via the slopes k i = ∂ t u i , as done inter alia by Hairer, Lubich, and Wanner (2006, Definition II.1.1). Using the stage values u i as in (7),
u + 2 M − u 0 2 M (8) =2∆t u 0 , s i=1 b i ∂ t u i M + (∆t) 2 s i=1 b i ∂ t u i 2 M (7) = 2∆t s i=1 b i u i − ∆t s j=1 a ij ∂ t u j , ∂ t u i M + (∆t) 2 s i=1 b i ∂ t u i 2 M =2∆t s i=1 b i u i , ∂ t u i M + (∆t) 2 s i=1 b i ∂ t u i 2 M − 2 s i,j=1 b i a ij ∂ t u i , ∂ t u j M =2∆t s i=1 b i u i , ∂ t u i M + (∆t) 2 s i,j=1 b i b j − b i a ij − b j a ji ∂ t u i , ∂ t u j M ,
where the symmetry of the scalar product has been used in the last step. Here, the first term on the right hand side is consistent with
t 0 +∆t t 0 2 u, ∂ t u , if the Runge-Kutta method is consistent, i.e. s i=1 b i = 1.
Additionally, it can be estimated via the semidiscretisation. The second term of order (∆t) 2 is undesired. Depending on the method (and the stages, of course), it may be positive or negative. However, if it is positive, then a stability error may be introduced. As a special case, if the method fulfils b i b j = b i a ij +b j a ji , i, j ∈ {1, . . . , s}, this term vanishes. These methods can conserve quadratic invariants of ordinary differential equations, a topic of geometric numerical integration, see Theorem IV.2.2 of Hairer, Lubich, and Wanner (2006), originally proved by Cooper (1987). A special kind of these methods are the implicit Gauß methods, see section II.1.3 of Hairer, Lubich, and Wanner (2006).
For an explicit method (a ij = 0 for j ≥ i), the undesired term of order (∆t) 2 in (8) can be rewritten as
s i=1 b i ∂ t u i 2 M − 2 s i=1 i−1 j=1 b i a ij ∂ t u i , ∂ t u j M = s i=1 b 2 i ∂ t u i 2 M + 2 s i=1 i−1 j=1 b i (b j − a ij ) ∂ t u i , ∂ t u j M .(9)
This undesired increase of the norm may be remedied by the application of an adaptive modal filter F . Similar to Glaubitz, Ranocha,Öffner, and Sonar (2016), the adaptive filter strength ε may be estimated via
F u + 2 M ! ≤ RHS := u 0 2 M + 2∆t s i=1 b i u i , ∂ t u i M ≤ RHS + (∆t) 2 s i,j=1 b i b j − b i a ij − b j a ji ∂ t u i , ∂ t u j M ,(10)
if the term of order (∆t) 2 is non-negative. In a modal Legendre basis {ϕ i }, the (exact) modal filter F may be written as
F = diag exp [−ε λ s n ] p n=0 ,(11)
where λ n = n(n + 1) ≥ 0. Thus,
p n=0 exp[−2ε λ s n ] u 2 +,n ϕ n 2 ! ≤ RHS.(12)
Here, u +,n are the coefficients of the polynomial u + , expressed in the Legendre basis of polynomials of degree ≤ p. Since
exp[x] ≥ 1 + x, x ∈ R,(13)
the filter strength ε can be estimated by
p n=0 (1 − 2ε λ s n ) u 2 +,n ϕ n 2 ≤ RHS ⇔ p n=0 u 2 +,n ϕ n 2 − RHS p n=0 2λ s n u 2 +,n ϕ n 2 −1 ≤ ε,(14)
for p n=0 2λ s n u 2 +,n ϕ n 2 > 0. Using p n=0 u 2 +,n ϕ n 2 ≈ u + 2 M (since · M approximates the exact L 2 norm on the left hand side), this results in
ε ≥ u + 2 M − u 0 2 M − 2∆t s i=1 b i u i , ∂ t u i M p n=0 2λ s nũ 2 +,n ϕ n 2 −1 .(15)
This kind of adaptive filtering can be seen as a special case of projection, enforcing the constraint on the squared norm (a quadratic form) and not violating conservation, i.e. a constraint on the integral of the solution (a linear form). This is visualised in Figure 1. However, there are various possibilities to conduct this projection. As noted in section IV.4 of Hairer, Lubich, and Wanner (2006), projection methods can be useful, but can also destroy good properties. Therefore, they have to be investigated thoroughly.
As an example, the linear advection equation with constant velocity as in section 4.1 is simulated in the time interval [0, 4] using 20 000 time steps of the explicit Euler method, the explicit Euler method with adaptive modal filtering, and the explicit Euler method with a simple projection, given by a scaling of all the non-constant Legendre modes by the same factor, resulting in the desired norm inequality and conservation.
If this projection is not really necessary, the results are very similar to the ones of the filtered Euler method, as can be seen in Figure 2a, where the smooth initial condition
u 0 (x) = exp(−20x 2 )(16)
has been evolved and all solutions are visually nearly indistinguishable. However, the nonsmooth initial datum results in Gibbs oscillations and the projection / filter has to be applied a lot more. In this case, the results of the Euler method using a simple projection obeying the constraints described above are totally useless, as can be seen in Figure 2b. It may be conjectured that the boundary values between cells are influenced in such a way that the numerical upwind flux adds further dissipation.
u 0 (x) = 1, − 1 4 ≤ x ≤ 1 4 , 0, otherwise,(17)u =const ∧ u 2 =const u 2 =const u =const
More numerical results will be presented in section 4.
Possible optimisation
The (∆t) 2 term in (8) can not be estimated via the semidiscrete analysis in general. Therefore, one may want to optimise an explicit Runge-Kutta method with regard to this error term (9)
err = err(∂ t u i , A, b) := s i=1 b 2 i ∂ t u i 2 M + 2 s i=1 i−1 j=1 b i (b j − a ij ) ∂ t u i , ∂ t u j M .(18)
One possibility is to minimise the error term in the mean, i.e.
min A,b B 1 (0) s err(∂ t u i , A, b).(19)
Here, the time derivatives ∂ t u i can be seen as iid random variables, distributed uniformly in the unit ball B 1 (0). Another possibility is to minimise the maximal error (for ∂ t u i ∈ R)
min A,b max ∂tu i ∈[−1,1] err(∂ t u i , A, b).(20)
A general Runge-Kutta method of second order using two stages is given by the Butcher tableau 0
a 21 a 21 b 1 b 2(21)
with b 1 + b 2 = 1 and b 2 a 21 = 1 2 . Using b 2 as a free parameter, b 1 = 1 − b 2 and a 21 = 1 2b 2 . The mean value of the error term is given by B 1 (0) 2 b 2 2 and therefore, the optimal value regarding (19) can be computed via
min b 2 (1 − b 2 ) 2 + b 2 2 .(22)
This minimum is attained at b 2 = 1 2 and yields the SSPRK(2,2) method given by Gottlieb and Shu (1998), which is described in section 4.2.
Abbreviating
∂ t u 1 = x, ∂ t u 2 = y, the maximal error term is max x,y b 2 1 x 2 + b 2 2 y 2 + 2b 2 (b 1 − a 21 )xy = max x,y (1 − b 2 ) 2 x 2 + b 2 2 y 2 + 2 b 2 1 − b 2 − 1 2b 2 =−(b 2 − 1 2 ) 2 − 1 4 xy .(23)
Lengthy calculations yield surprisingly the same optimal value b 2 = 1 2 as in the previous case. This analysis may be a starting point for further optimisation of Runge-Kutta methods. Note however, that the minimisation of the mean value is equivalent to the minimisation of the 2 norm of b, which has been used as a design criterion for Runge-Kutta methods.
Linear problems
Using a stable semidiscretisation, a linear problem yields an ordinary differential equatioṅ
u = Lu,(24)
where L is a linear operator in a finite-dimensional real Hilbert space. If this operator is coercive,
i.e. ∃η > 0 ∀u : u, Lu ≤ −η Lu 2 ,(25)
then the explicit Euler method is clearly stable under the time step restriction ∆t ≤ 2η, as described inter alia by Levy and Tadmor (1998); Tadmor (2002), since
u + ∆t Lu 2 − u 2 = 2∆ u, Lu + (∆t) 2 Lu 2 ≤ (∆t − 2η)∆t Lu 2 .(26)
The adaptive dissipation of Glaubitz, Ranocha,Öffner, and Sonar (2016); Ranocha, Glaubitz, Offner, and Sonar (2016) can be seen as a means to add just enough dissipation to get the coercivity needed for a stable scheme. However, since this coercivity is tied to dissipative operators and not fulfilled by stable semidiscretisations of transport equations in general, another condition is interesting: semiboundedness. If the semidiscretisation is stable, L is semibounded, i.e.
∀u : u, Lu ≤ 0.
This property is considered in more detail in section 3.
Linear ordinary differential equations and explicit Runge-Kutta methods
In this section, the ordinary linear differential equation
d dt u = Lu (28)
in a real Hilbert space is considered. The operator L is assumed to be bounded (i.e. continuous) and semibounded in the sense u, Lu ≤ 0, ∀u.
In a complex Hilbert space, the corresponding property would be
Lu, u + u, Lu ≤ 0, ∀u.(30)
However, only the real case will be considered in the following to simplify the arguments. Applying a Runge-Kutta method to (28), the solution after one step of size ∆t is given by the stability polynomial P of the method:
u + = P (L) u, L := ∆t L.(31)
For an explicit method of order p using s stages, P is a polynomial of degree s and approximates the exponential function up to order p.
Since the linear operator L is semibounded, the norm of the solution does not increase in time, since d dt
u 2 = d dt u, u + u, d dt u = Lu, u + u, Lu ≤ 0.(32)
In the following, this stability estimate will be considered in the discrete case.
First order method using one stage
The only first order explicit Runge-Kutta method using one stage is the explicit Euler method with stability polynomial P (L) = I +L.
Thus, the energy u 2 after one step is given by
u + 2 − u 2 = P (L) u 2 − u 2 = I u 2 + 2 P (L) − I u, I u + P (L) − I u 2 − u 2 = P (L) − I u, 2u + P (L) − I u = P (L) − I u, P (L) + I u = Lu, (2 I +L) u = 2 u, Lu ≤0 + Lu 2 ≥0 .(34)
The first term is non-positive, since L is semibounded for ∆t > 0. However, the second term can not be controlled by the first one for a general semibounded operator L. For a skew-symmetric L, the first term vanishes and the second one introduces an erroneous growth of the energy u 2 . For a coercive operator L fulfilling
u, Lu ≤ − c 2 Lu 2 , ∀u,(35)
with c > 0, the second term is controlled by the first one if ∆t ≤ c, since L = ∆t L. This idea has been pursued inter alia by Levy and Tadmor (1998) and extends to strong-stability preserving methods that can be rewritten as explicit Euler steps, as described inter alia in the monograph by Gottlieb, Ketcheson, and Shu (2011).
Second order methods using two stages
The stability polynomial of a second order Runge-Kutta method using two stages is
P (L) = I +L + 1 2 L 2 .(36)
An example of this class is the strong-stability preserving method SSPRK(2,2) given by Gottlieb and Shu (1998). Analogously to the explicit Euler method (34), the energy after one step obeys
u + 2 − u 2 = P (L) − I u, P (L) + I u = L + 1 2 L 2 u, 2 + L + 1 2 L 2 u .(37)
In order to use the semiboundedness (29) of L, the following procedure will be applied:
Idea 1. In order to estimate the energy after one step, proceed as follows.
1. If there is only one term of lowest order in L on either the left or the right hand side of the scalar product estimating u + 2 − u 2 , remove this lowest order term using the semiboundedness of L (29).
2. If the lowest order terms on the left and right hand side are of equal order, estimate the remaining terms with respect to L = ∆t L .
Applying these ideas to (37) yields
u + 2 − u 2 = L + 1 2 L 2 u, 2 + L + 1 2 L 2 u = L + 1 2 L 2 u, (2 + L) u + 1 2 L + 1 2 L 2 u, L 2 u ≤ 1 2 L + 1 2 L 2 u, L 2 u ≤ 1 4 L 2 u 2 .(38)
Thus, as in the case of a first order method using one stage, Idea 1 does not lead to a viable stability estimate. This is due to the non-inclusion of an imaginary interval in the stability region of these methods.
Third order methods using three stages
The stability polynomial of a third order Runge-Kutta method using three stages is
P (L) = I +L + 1 2 L 2 + 1 6 L 3 .(39)
An example of this class is the strong-stability preserving method SSPRK(3,3) given by Gottlieb and Shu (1998). Analogously to the explicit Euler method (34), the energy after one step obeys
u + 2 − u 2 = P (L) − I u, P (L) + I u = L + 1 2 L 2 + 1 6 L 3 u, 2 + L + 1 2 L 2 + 1 6 L 3 u .(40)
Applying Idea 1 to (40) yields
u + 2 − u 2 = L + 1 2 L 2 + 1 6 L 3 u, 2 + L + 1 2 L 2 + 1 6 L 3 u(41)= L + 1 2 L 2 + 1 6 L 3 u, 2 + L + 1 3 L 2 u + L + 1 2 L 2 + 1 6 L 3 u, 1 6 L 2 + 1 6 L 3 u ≤ 1 6 L + 1 2 L 2 + 1 6 L 3 u, L 2 + L 3 u = 1 6 L + L 2 u, L 2 + L 3 u + − 1 2 L 2 + 1 6 L 3 u, L 2 + L 3 u ≤ 1 12 −L 2 + 1 3 L 3 u, L 2 + L 3 u .
Thus, the first stage of Idea 1 is completed, since the lowest order terms L 2 u appear in both arguments of the scalar product. The remaining terms are estimated in the second stage by
u + 2 − u 2 ≤ 1 12 1 3 L 3 u 2 − 2 3 L 3 u, L 2 u − L 2 u 2 ≤ 1 12 1 3 L 3 u 2 + 2 3 L 3 u L 2 u − L 2 u 2 ≤ 1 12 1 3 L 2 + 2 3 L − 1 =:p( L ) L 2 u 2 .(42)
The term in brackets is a polynomial in L with negative constant term. Thus, for sufficiently small L , it is non-positive. Specifically, this term will be non-positive for ∆t L = L ≤ 1, since 1 is the smallest positive root of the polynomial p(x) = (x − 1) 1 3 x + 1 . If L is skew-symmetric, this estimate can be sharpened, since
u + 2 − u 2 = 1 12 1 3 L 3 u 2 − 2 3 L 3 u, L 2 u =0 − L 2 u 2 ≤ 1 12 1 3 L 2 − 1 L 2 u 2 ,(43)
and this is non-positive for ∆t L = L ≤ √ 3. These results are summarised in the following Theorem 1 (Levermore; see Theorem 2 of Tadmor (2002)). The third-order, three stage explicit Runge-Kutta method applied to the linear ordinary differential equationu = Lu (28) with semibounded L (29) is strongly stable under the time step restriction ∆t L ≤ 1.
If L is skew-symmetric, the method is strongly stable under the relaxed time step restriction
∆t L ≤ √ 3.(45)
SSPRK(10,4)
The stability polynomial of the strong-stability preserving fourth order Runge-Kutta method using ten stages SSPRK(10,4) given by Ketcheson (2008) is P (L) = L 10 251942400 + L 9 4199040 + L 8 155520 + L 7 9720 + 7L 6 6480 + 17L 5 2160 + L 4 24 + L 3 6 + L 2 2 + L + I . (46) Analogously to the proceeding case, the energy after one step obeys This proves the following Theorem 2. The fourth-order, ten stage, strong-stability preserving explicit Runge-Kutta method SSPRK(10,4) of Ketcheson (2008) applied to the linear ordinary differential equationu = Lu (28) with semibounded L (29) is strongly stable under the time step restriction ∆t L ≤ 0.67493.
63474972917760000 u + 2 − u 2(
Of course, this estimate is not optimal. Various terms in equation (70) can be bounded not only by multiples of L and L 3 u , but also by L 4 u , L 5 u , or L 6 u . Thus, to get an optimal estimate using Idea 1, the terms in equation (70) have to be distributed optimally to these four negative terms L k u , k ∈ {3, 4, 5, 6}.
However, a simple numerical example shows that the estimate in Theorem 2 is less than a factor a ten smaller than an optimal estimate. The initial value problem d dt
u 1 (t) u 2 (t) u 3 (t) = 0 1 0 −1 0 0 0 0 0 =:L u 1 (t) u 2 (t) u 3 (t) =:u(t) , u(0) = 1 1 1 =:u 0 ,(49)
is described by a skew-symmetric operator L with norm L = 1 (with respect to the identity matrix I). Solving this initial value problem (49) using the SSPRK(10,4) method with 203 and 204 time steps in the interval [0, 1000] yields the energies plotted in Figure 4. As can be seen there, the energy is bounded by the initial energy u 0 2 = 3 for 204 steps, corresponding to ∆t L ≈ 4.902. However, the energy grows if only 203 steps are used, corresponding to ∆t L ≈ 4.926. Thus, an optimal bound in Theorem 2 has to be smaller than ≈ 4.926.
Other fourth order methods
A fourth order method with up to six stages has a stability polynomial
P (L) = I +L + 1 2 L 2 + 1 6 L 3 + 1 24 L 4 + a 5 L 5 + a 6 L 6 .(50)
If the method uses only five or four stages, then a 6 = 0 or a 6 = a 5 = 0, respectively. Using Idea 1, the energy after one time step can be estimated by 24 2 u + 2 − u 2 = 576 P (L) − I u, P (L) + I u (51) = 24a 6 L 5 + 24a 5 L 4 + L 3 + 4L 2 + 12L + 24 Lu, 24a 6 L 6 + 24a 5 L 5 + L 4 + 4L 3 + 12L 2 + 24L + 48 u = 24a 6 L 5 + 24a 5 L 4 + L 3 + 4L 2 + 12L + 24 Lu, 48a 6 L 5 + 48a 5 L 4 + 2L 3 + 8L 2 + 24L + 48 u + 24a 6 L 5 + 24a 5 L 4 + L 3 + 4L 2 + 12L + 24 Lu, 24a 6 L 4 + (24a 5 − 48a 6 )L 3 + (1 − 48a 5 )L 2 + 2L + 4 L 2 u ≤ 24a 6 L 5 + 24a 5 L 4 + L 3 + 4L 2 + 12L + 24 Lu, 24a 6 L 4 + (24a 5 − 48a 6 )L 3 + (1 − 48a 5 )L 2 + 2L + 4 L 2 u = 144a 6 L 4 + (144a 5 − 288a 6 )L 3 + (6 − 288a 5 )L 2 + 12L + 24 Lu, 24a 6 L 4 + (24a 5 − 48a 6 )L 3 + (1 − 48a 5 )L 2 + 2L + 4 L 2 u + 24a 6 L 3 + (24a 5 − 144a 6 )L 2 + (1 − 144a 5 + 288a 6 )L + 288a 5 − 2 L 3 u,
24a 6 L 4 + (24a 5 − 48a 6 )L 3 + (1 − 48a 5 )L 2 + 2L + 4 L 2 u ≤ 24a 6 L 3 + (24a 5 − 144a 6 )L 2 + (1 − 144a 5 + 288a 6 )L + 288a 5 − 2 L 3 u, 24a 6 L 4 + (24a 5 − 48a 6 )L 3 + (1 − 48a 5 )L 2 + 2L + 4 L 2 u .
If 288a 5 − 2 = 0, Idea 1 can not be used further in this estimate, since the lowest order terms on both sides have different orders in L. Thus, Idea 1 can not be used to prove stability for fourth order methods using only four stages. If 288a 5 − 2 < 0, the semiboundedness (29) of L can not be used as in the previous steps, since it would yield an estimate in the wrong direction. However, if a 5 > 1 144 , the energy can be further estimated by
576 u + 2 − u 2 (52) ≤ 24a 6 L 3 + (24a 5 − 144a 6 )L 2 + (1 − 144a 5 + 288a 6 )L + 288a 5 − 2 L 3 u,
24a 6 L 4 + (24a 5 − 48a 6 )L 3 + (1 − 48a 5 )L 2 + 2L + 4 L 2 u = 24a 6 L 3 + (24a 5 − 144a 6 )L 2 + (1 − 144a 5 + 288a 6 )L + 288a 5 − 2 L 3 u, 48a 6 L 3 + (48a 5 − 288a 6 )L 2 + (2 − 288a 5 + 576a 6 )L + 576a 5 − 4 1 144a 5 − 1 L 2 u + 24a 6 L 3 + (24a 5 − 144a 6 )L 2 + (1 − 144a 5 + 288a 6 )L + 288a 5 − 2 L 3 u, (3456a 5 a 6 − 24a 6 )L 3 + (3456a 2 5 − 6912a 5 a 6 − 24a 5 )L 2
+ (144a 5 + 288a 6 − 1 − 6912a 2 5 )L + 576a 5 − 576a 6 − 4 1 144a 5 − 1 L 3 u ≤ 24a 6 L 3 + (24a 5 − 144a 6 )L 2 + (1 − 144a 5 + 288a 6 )L + 288a 5 − 2 L 3 u, (3456a 5 a 6 − 24a 6 )L 3 + (3456a 2 5 − 6912a 5 a 6 − 24a 5 )L 2 + (144a 5 + 288a 6 − 1 − 6912a 2 5 )L + 576(a 5 − a 6 ) − 4 1 144a 5 − 1 L 3 u .
Since a 5 > 1 144 , this can be estimated similarly to the previous cases, if 576(a 5 − a 6 ) − 4 < 0. In that case, the term
(288a 5 − 2) >0 576(a 5 − a 6 ) − 4 144a 5 − 1 <0 L 3 u(53)
can be used to bound the other terms. Thus, the proof using Idea 1 does not work for fourth order methods using only four or five stages. This may be the reason for Tadmor's anticipatory conjecture that more than four stages are necessary (Tadmor, 2002, Remark 3 after Theorem 2). Indeed, Sun and Shu (2016) proved recently that the fourth order Runge-Kutta method is not strongly stable in the sense considered here. However, applying two steps of it yields a stable method of order four using eight stages. This stability can be proven using Idea 1 as before.
Examples
In this section, some Runge-Kutta methods will be treated as examples.
Explicit and implicit Euler
In order to demonstrate the impact of the dissipation introduced by an implicit time integration method and the ability to estimate these, the explicit and implicit Euler method will be compared.
The explicit Euler method
u + := u 0 + ∆t ∂ t u 0(54)
introduces an erroneous growth of energy of size (∆t) 2 ∂ t u 0 2 , whereas the implicit Euler
method u + := u 0 + ∆t ∂ t u +(55)
yields artificial dissipation of size (∆t) 2 ∂ t u + 2 per time step.
Similarly to Glaubitz, Ranocha,Öffner, and Sonar (2016); Ranocha, Glaubitz,Öffner, and Sonar (2016), the estimate of the semidiscretisation can be mimicked by a filtering of strength
afterwards yields a filtered explicit Euler method mimicking the dissipation introduced by an implicit Euler method.
These estimates are applied to the linear advection equation
∂ t u + ∂ x u = 0(58)
in [−1, 1] with periodic boundary conditions. The initial condition
u 0 (x) = 1, − 1 4 ≤ x ≤ 1 4 , 0, otherwise,(59)
is evolved during the time interval [0, 4] on a grid of N = 8 elements using polynomials of degree ≤ p = 9 and an upwind numerical flux. The results using 20 000 time steps are plotted in Figure 5. The initial condition u 0 (dashed, grey) is also the exact solution of the PDE at t = 4, i.e. after two periods. However, ignoring errors of the spatial discretisation, the semidiscrete exact solution (green) shows a slightly oscillatory behaviour around the discontinuities at x = ± 1 4 . Contrary, the explicit Euler method (blue) introduces strong oscillations. Applying the adaptive modal filtering once after each time step yields a solution (dashed, magenta) that is visually indistinguishable from the semidiscrete exact solution (green) computed using the matrix exponential. The implicit Euler method (red) introduces artificial dissipation and yields a non-oscillatory solution. Therefore, it does not mimic the semidiscrete exact solution. However, the estimate of the dissipation yields a solution of the explicit Euler method with modal filtering applied twice (dashed, cyan) that is nearly indistinguishable from the implicit one.
Although the estimate of the filter strength is conservative (i.e. only necessary), the energy of the twice filtered explicit solution is slightly less than the energy of the implicitly computed solution. The reason is probably the appearance of some changes of boundary values due to the filtering that triggers additional dissipation by the upwind flux. The results of the implicit and filtered explicit Euler method using only 1000 time steps are plotted in Figure 6. Similarly to the case before, the filtered solutions approximate their targets very well, although more deviations can be seen.
SSPRK(10,4)
The optimal ten-stage, fourth order method given by Ketcheson (2008) and can be implemented in a low-storage format. The error term (9) also has no fixed sign in this case.
Summary and conclusions
The target of this article is twofold. At first, the adaptive techniques developed by the authors for the extension of semidiscrete stability obtained in a summation-by-parts framework using the explicit Euler method to fully discrete stability have been broadened to general Runge-Kutta methods in section 2. Additionally, a new interpretation of this adaptive modal filtering has been described, together with an example demonstrating the importance of a careful choice of the type of filtering / dissipation. Moreover, some ideas about possibilities for further optimisation of Runge-Kutta methods have been presented.
Additionally, general stability results of some Runge-Kutta methods for general linear and semibounded operators have been developed in section 3. There, a conjecture of Tadmor (2002) has been proven, related to the recent investigation of the stability of fourth order Runge-Kutta methods using four stages by Sun and Shu (2016). In addition, some examples have been presented.
Further research includes the deeper investigation of projections as described in section 2 such as modal filtering. Moreover, the possibilities for optimisation may be further pursued. Last but not least, the enhancement of the semidiscretisations itself will be an important subject of further investigations.
A. Appendix
The stability polynomial of the strong-stability preserving fourth order Runge-Kutta method using ten stages SSPRK(10,4) given by Ketcheson (2008) is
P (L) = L 10 251942400 + L 9 4199040 + L 8 155520 + L 7 9720 + 7L 6 6480 + 17L 5 2160 + L 4 24 + L 3 6 + L 2 2 + L + I .(65)
Analogously to the other cases, the energy after one step obeys 251942400 2 u + 2 − u 2 = 63474972917760000 P (L) − I u, P (L) + I u
= L 9 + 60L 8 + 1620L 7 + 25920L 6 + 272160L 5 + 1982880L 4 + 10497600L 3 + 41990400L 2 + 125971200L + 251942400 Lu, L 10 + 60L 9 + 1620L 8 + 25920L 7 + 272160L 6 + 1982880L 5 + 10497600L 4 + 41990400L 3 + 125971200L 2 + 251942400L + 503884800 u = L 9 + 60L 8 + 1620L 7 + 25920L 6 + 272160L 5 + 1982880L 4 + 10497600L 3 + 41990400L 2 + 125971200L + 251942400 Lu, 2L 9 + 120L 8 + 3240L 7 + 51840L 6 + 544320L 5 + 3965760L 4 + 20995200L 3 + 83980800L 2 + 251942400L + 503884800 u + L 9 + 60L 8 + 1620L 7 + 25920L 6 + 272160L 5 + 1982880L 4 + 10497600L 3 + 41990400L 2 + 125971200L + 251942400 Lu, L 10 + 58L 9 + 1500L 8 + 22680L 7 + 220320L 6 + 1438560L 5 + 6531840L 4 + 20995200L 3 + 41990400L 2 u ≤ L 9 + 60L 8 + 1620L 7 + 25920L 6 + 272160L 5 + 1982880L 4 + 10497600L 3 + 41990400L 2 + 125971200L + 251942400 Lu, L 8 + 58L 7 + 1500L 6 + 22680L 5 + 220320L 4 + 1438560L 3 + 6531840L 2 + 20995200L + 41990400 L 2 u = 6L 8 + 348L 7 + 9000L 6 + 136080L 5 + 1321920L 4 + 8631360L 3 + 39191040L 2 + 125971200L + 251942400 Lu, L 8 + 58L 7 + 1500L 6 + 22680L 5 + 220320L 4 + 1438560L 3 + 6531840L 2 + 20995200L + 41990400 L 2 u + L 9 + 54L 8 + 1272L 7 + 16920L 6 + 136080L 5 + 660960L 4 + 1866240L 3 + 2799360L 2 Lu, L 8 + 58L 7 + 1500L 6 + 22680L 5 + 220320L 4 + 1438560L 3 + 6531840L 2 + 20995200L + 41990400 L 2 u .
Thus,
63474972917760000 u + 2 − u 2(67)
≤ L 7 + 54L 6 + 1272L 5 + 16920L 4 + 136080L 3 + 660960L 2 + 1866240L + 2799360 L 3 u, L 8 + 58L 7 + 1500L 6 + 22680L 5 + 220320L 4 + 1438560L 3 + 6531840L 2 + 20995200L + 41990400 L 2 u = L 7 + 54L 6 + 1272L 5 + 16920L 4 + 136080L 3 + 660960L 2 + 1866240L + 2799360 L 3 u, 15L 7 + 810L 6 + 19080L 5 + 253800L 4 + 2041200L 3 + 9914400L 2 + 27993600L + 41990400 L 2 u + L 7 + 54L 6 + 1272L 5 + 16920L 4 + 136080L 3 + 660960L 2 + 1866240L + 2799360 L 3 u,
L 8 + 43L 7 + 690L 6 + 3600L 5 − 33480L 4 − 602640L 3 − 3382560L 2 − 6998400L L 2 u ≤ L 7 + 54L 6 + 1272L 5 + 16920L 4 + 136080L 3 + 660960L 2 + 1866240L + 2799360 L 3 u, L 7 + 43L 6 + 690L 5 + 3600L 4 − 33480L 3 − 602640L 2 − 3382560L − 6998400 L 3 u .
Again, the first stage of Idea 1 is completed, since the lowest order terms L 3 u appear in both arguments of the scalar product. In the second stage, the remaining terms have to be estimated. This will be possible, since the squared norm of the lowest order terms appear with a negative coefficient as in section 3.3. Expanding the scalar product results in
63474972917760000 u + 2 − u 2 (68) ≤ − 19591041024000 L 3 u 2 − 6312668774400 L 4 u 2 − 398320934400 L 5 u 2
− 4555958400 L 6 u 2 − 22529697177600 L 3 u, L 4 u − 6312668774400 L 3 u, L 5 u − 1046064844800 L 3 u, L 6 u − 3360407731200 L 4 u, L 5 u − 108335232000 L 3 u, L 7 u − 522780480000 L 4 u, L 6 u − 6970406400 L 3 u, L 8 u − 50514451200 L 4 u, L 7 u − 104136192000 L 5 u, L 6 u − 257541120 L 3 u, L 9 u − 3014910720 L 4 u, L 8 u − 7817212800 L 5 u, L 7 u − 4199040 L 3 u, L 10 u − 102409920 L 4 u, L 9 u − 310495680 L 5 u, L 8 u − 76593600 L 6 u, L 7 u − 1516320 L 4 u, L 10 u − 4121280 L 5 u, L 9 u + 51308640 L 6 u, L 8 u + 60912000 L 7 u 2 + 58320 L 5 u, L 10 u + 4043520 L 6 u, L 9 u + 16254000 L 7 u, L 8 u + 102600 L 6 u, L 10 u + 921960 L 7 u, L 9 u + 877680 L 8 u 2 + 20520 L 7 u, L 10 u + 91956 L 8 u, L 9 u + 1962 L 8 u, L 10 u + 2322 L 9 u 2 + 97 L 9 u, L 10 u + L 10 u 2 .
Using again the semiboundedness (29) of L, the terms L 7 u, L 8 u , L 8 u, L 9 u , L 9 u, L 10 u can be ignored, since they have positive coefficients, yielding 63474972917760000 u + 2 − u 2
≤ − 19591041024000 L 3 u 2 − 6312668774400 L 4 u 2 − 398320934400 L 5 u 2 − 4555958400 L 6 u 2 − 22529697177600 L 3 u, L 4 u − 6312668774400 L 3 u, L 5 u − 1046064844800 L 3 u, L 6 u − 3360407731200 L 4 u, L 5 u − 108335232000 L 3 u, L 7 u − 522780480000 L 4 u, L 6 u − 6970406400 L 3 u, L 8 u − 50514451200 L 4 u, L 7 u − 104136192000 L 5 u, L 6 u − 257541120 L 3 u, L 9 u − 3014910720 L 4 u, L 8 u − 7817212800 L 5 u, L 7 u − 4199040 L 3 u, L 10 u − 102409920 L 4 u, L 9 u − 310495680 L 5 u, L 8 u − 76593600 L 6 u, L 7 u − 1516320 L 4 u, L 10 u − 4121280 L 5 u, L 9 u + 51308640 L 6 u, L 8 u + 60912000 L 7 u 2 + 58320 L 5 u, L 10 u + 4043520 L 6 u, L 9 u + 102600 L 6 u, L 10 u + 921960 L 7 u, L 9 u + 877680 L 8 u 2 + 20520 L 7 u, L 10 u + 1962 L 8 u, L 10 u + 2322 L 9 u 2 + L 10 u 2 .
The most simple estimate for this expression can be obtained by bounding the scalar products using the Cauchy-Schwartz inequality against products of the lowest order term L 3 u and multiples of L . This results in 63474972917760000 u + 2 − u 2 (70) ≤ − 19591041024000 L 3 u 2 − 6312668774400 L 4 u 2 − 398320934400 L 5 u 2 − 4555958400 L 6 u 2 + 22529697177600 L 3 u L 4 u + 6312668774400 L 3 u L 5 u + 1046064844800 L 3 u L 6 u + 3360407731200 L 4 u L 5 u + 108335232000 L 3 u L 7 u + 522780480000 L 4 u L 6 u + 6970406400 L 3 u L 8 u + 50514451200 L 4 u L 7 u + 104136192000 L 5 u L 6 u + 257541120 L 3 u L 9 u + 3014910720 L 4 u L 8 u + 7817212800 L 5 u L 7 u + 4199040 L 3 u L 10 u + 102409920 L 4 u L 9 u + 310495680 L 5 u L 8 u + 76593600 L 6 u L 7 u + 1516320 L 4 u L 10 u + 4121280 L 5 u L 9 u + 51308640 L 6 u L 8 u + 60912000 L 7 u 2 + 58320 L 5 u L 10 u + 4043520 L 6 u L 9 u + 102600 L 6 u L 10 u + 921960 L 7 u L 9 u + 877680 L 8 u 2 + 20520 L 7 u L 10 u + 1962 L 8 u L 10 u + 2322 L 9 u 2 + L 10 u
Figure 1 :
1Visualisation of the requirements for projections such as filtering.
Figure 2 :
2Solutions at t = 4 computed using 20 000 time steps of the unmodified, filtered, and projected explicit Euler method.
Figure 3 Figure 3 :
33in appendix A. The term in brackets is again a polynomial p L in the norm of L with negative constant term. Thus, it is negative for L = 0 and small values of L . Its smallest positive root is approximately 0.67493 as visualised in Polynomial p L appearing in the simple estimate (47) for SSPRK(10,4). The smallest positive root is approximately 0.67493.
Figure 4 :
4Energies (M = I) of the solutions computed with SSPRK(10,4) in [0, 1000] using 203 (blue, dashed) and 204 (green line) steps.
time step. Similarly, application of this filter and an additional filter of strength ε = (∆t) 2 ∂ t u
Figure 5 :
5Energies and solutions computed using 20 000 time steps of the implicit and explicit Euler methods and modal filtering.
(b )
)Energies for t ∈ [0, 4].
Figure 6 :
6Energies and solutions computed using 1000 time steps of the implicit and filtered explicit Euler methods.
+
− 19591041024000 + 22529697177600 L + 6312668774400 L 2 + 4406472576000 L 3 + 631115712000 L 4 + 161621049600 L 5 + 11089664640 L 6 + 493698240 L 7 + 117858240 L 8 + 4101840 L 9 + 1902240 L 10 + 20520 L 11 + 4284 L 12 + L 14
SSPRK(2,2)The optimal two-stage, second order method given byGottlieb and Shu (1998)Thus, u 1 = u 0 andHere, the additional term is non-negative and has to be balanced by modal filtering.SSPRK(3,3)The optimal three-stage, third order method given byGottlieb and Shu (1998)In this case,and the additional term has no fixed sign [consider e.g. ∂ t u 1 = ∂ t u 2 = 4 3 ∂ t u 3 and ∂ t u 1 = ∂ t u 2 = ∂ t u 3 ]. Thus, the time discretisation can introduce artificial dissipation by itself. However, if the (∆t) 2 term is positive, modal filtering should be applied in the general case.
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| []
|
[
"SOME PROPERTIES OF TRANSFORMS IN CULTURAL THEORY",
"SOME PROPERTIES OF TRANSFORMS IN CULTURAL THEORY"
]
| [
"Culture ",
"Transforms "
]
| []
| [
"UNDER REVIEW AT INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS"
]
| It is shown that, in certain circumstances, systems of cultural rules may be represented by doubly stochastic matrices denoted Π, called "possibility transforms," and by certain real valued "possibility densities" π=(π 1 , π 2 , ..., π n ) with inner product <π,π>=Σ i π i 2 =1. Using such objects we may characterize a certain problem of ethnographic and ethological description as a problem of prediction, in which observations are predicted by properties of fixed points of transforms of "pure systems", or by properties of convex combinations of such "pure systems". That is, ethnographic description is an application of the Birkhoff theorem regarding doubly stochastic matrices on a space whose vertices are permutations. | 10.1007/s10773-010-0438-7 | [
"https://arxiv.org/pdf/0903.1366v1.pdf"
]
| 115,169,955 | 0903.1366 | c1ee554fd29aa882c74aa764d33f66a1919a813b |
SOME PROPERTIES OF TRANSFORMS IN CULTURAL THEORY
POLANDCopyright POLAND2008
Culture
Transforms
SOME PROPERTIES OF TRANSFORMS IN CULTURAL THEORY
UNDER REVIEW AT INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
POLAND2008
It is shown that, in certain circumstances, systems of cultural rules may be represented by doubly stochastic matrices denoted Π, called "possibility transforms," and by certain real valued "possibility densities" π=(π 1 , π 2 , ..., π n ) with inner product <π,π>=Σ i π i 2 =1. Using such objects we may characterize a certain problem of ethnographic and ethological description as a problem of prediction, in which observations are predicted by properties of fixed points of transforms of "pure systems", or by properties of convex combinations of such "pure systems". That is, ethnographic description is an application of the Birkhoff theorem regarding doubly stochastic matrices on a space whose vertices are permutations.
This paper follows from and adopts the definitions used in previous papers [4,5] to which the reader is referred for details. In brief we assume a finite non-empty set P whose members are called individuals. Such P is organized into an evolutionary structure S as a quintuple (P, R, D, B, M) where R is a non-empty set of "rules", and D, B, and M are binary relations on P. We assume that each evolutionary structure satisfies the following four axioms: (1) D is totally non-symmetric and transitive; (2) M is symmetric;
(3) if bDc and there exists no d∈P, d ≠ b,c for which bDd and dDc, then we write cPb, and then require B = { (b,c) | there exists d∈P with both dPb and dPc }; and (4) #bM ≤ 2, where bM:={c∈P | (b,c) ∈M}. A rule R∈R is a statement concerning the relationships between the D, B, and/or M, which does not violate those four axioms. Given a family of subsets G t ∈P, indexed by a set t∈T of consecutive non-negative integers starting with 0, each G t is thought of as the generation at time t. Then G={G t | t∈T} is called a descent sequence of S in case, for all G t ∈G, each cell B occurs in only one generation, each subset M occurs in only one generation, and when G t ∈G, b∈G t , and cPb, then c∈G t-1 (that is, the set G t+1 contains all of, and only, the immediate descendants of individuals in G t ). Thus the generations of each descent sequence G partitions its total population. We assume a "Darwinian Sequences axiom" which says that all descent sequences of a given evolutionary structure can be traced back through a chain of descent in an unbroken series of non-empty generations, to the same date of initial origin.
We define by enumeration a set of closed (cyclic) objects which we call regular structures. We illustrate using a dot to represent an individual, a circle around two dots (say, b and c) shows bMc, a line between two dots (say, d,e) shows dBe. We give the simple closed cycles names Mn, where n = the number of M-sets in each figure. For example a diagram of relations and resulting M2 regular structure is: are the regular structures with 2, 3, 4 and 5 M-sets. All individuals in a particular regular structure are necessarily of the same generation. Each generation is composed of a set of regular structures. An ordered list counting the numbers of regular structures present in a particular G t is a configuration C:=(m 1 , …, m j , …) where the coefficient m j is the number of regular structures of type Mj in C. Thus a configuration consisting only of 2 of the M2 structures would be written (0,0,2,0, …). We shall denote that C = { C i | i = 1,2, ..., n} is a finite non-empty set of n distinct configurations C i . We here consider only finite sets C. In general, if C i and C j are configurations then C i +C j is also a configuration, though C i +C j ∈C is not required (since C is finite). If C is a configuration, then μ(C)=Σ(jm j ), or simply "μ" when C is understood, is the number of reproducing marriages in C (the "marriage number of C i "); β is the number of cells induced by B (sibships) in C; and γ=2μ is the total population of the generation G t on which C i is formed. If G is a descent sequence and, for t∈T, G t ∈G is a generation in G of size γ t , μ t is the number of marriages on G t , and β t the number of sibships on G t , then (when defined) μ t-1 =β t and μ t+1 ≤μ t , since a generation can not have more sibships (which arise by descent) than are reproducing marriages in its predecessor generation.
We define P(C):={ξ | ξ ⊆ C}. Let ξ∈P(C). We let each ξ∈P(C) correspond to a unique "row vector" (ξ 1 , ξ 2 , …, ξ n ) where ξ j =1 if C i ∈ξ, and ξ i =0 otherwise, called the content list of ξ. We denote Ξ:={(ξ 1 , ξ 2 , …, ξ n ) | ξ∈P(C), ξ=(ξ 1 , ξ 2 , …, ξ n ) is the content list of ξ}. For simplicity by abuse of notation we use the notation ξ∈P(C) and also ξ∈Ξ for the corresponding content list (ξ 1 , ξ 2 , …, ξ n ). We denote the empty configuration as C 0 but list only non-empty configurations. For example, if n=3 so that C={ξ 1 , ξ 2 , ξ 3 }, and if ξ={ξ 1 , ξ 2 } then the content list ξ= (1,1,0). The transpose of a vector ξ=(ξ 1 , ξ 2 , …, ξ n ) is denoted ξ Τ =(ξ 1 , ξ 2 , … , ξ n ) Τ . The objects ξ and ξ Τ represent the same set of configurations. Let C i ∈C and let μ C =s, s an integer >0, be the marriage number of C i . Then: C s :={ C i | C i ∈C and μ C =s} is a set of configurations of order s, and P s :=P(C s ) is a set of subsets of C s .
A rule can be represented by defining a transform R=[r ij ] where r ij =1 if the rule allows a transition from C j to C j otherwise r ij =0. We arrange the r ij on the ordered pairs (C i , C j ), 1 ≤ i,j ≤ n, in rows r i =(r i1 , r i2 ,…,r in ), 0 ≤ i,j ≤ n in the standard order; there are n such rows. If R=[r ij ] is a transform and r i =(r i1 , r i2 , …, r in ) is a row of R, then r i shows which transitions, from each ξ j to ξ i are allowed by R.
Definition 1: Let S be an evolutionary structure with a finite non-empty set C of configurations on generations of S, and let Ř be a finite non-empty set of transforms on C. Then 1. Let Ĥ be the free semigroup generated by Ř, called the set of transforms generated by Ř; 2. If α∈Ĥ then α is called a history generated by Ř .
A history is thus a string of one or more transforms applied in succession on a descent sequence. For example, if Ř is a set of transforms and Q,R,S∈Ř then α=QRS is a string of transforms of Ř and thus α is a history. Histories are associative under composition by concatenation. We do not allow that an empty chain is a history. Histories occur on an evolutionary structure S, so existence of a history (that is, of application of certain rules in a stated sequence) implies existence of some non-empty descent sequence G of S on whose generations the rules are applied in that order. If Ĥ is the set of histories generated by a set Ř of transforms then Ř ⊆ Ĥ and if R∈Ř, R∈Ĥ.
Notational Conventions: Hereafter we adopt the conventions that C is a finite non-empty set of n ≥ 1non-empty configurations, P(C) is a set of all subsets of C, Ξ the set of contents lists for ξ∈P(C), Ř a non-empty set of transforms, Ĥ a set of transforms generated by Ř, and we let α∈Ĥ, α=[α ij ] 0 ≤ i ≤ n, 0 ≤ j ≤ n and for ξ,φ∈Ξ, αξ Τ =φ Τ .
Definition 2: Let C be a non-empty set of configurations, let Ř be a non-empty set of transforms on C, let R,S∈Ř, and let R=[r ij ] and S=[s ij ]. Let α=SR. Let r i =(r i1 , …, r in ) be the i th row of R and s j =(s 1j , …, s nj ) be the j th column of S, where 0 ≤ i,j ≤ n. Then α ij =r i s j =(r i1 s 1j r i2 s 2j … r iν s υj ) where xy = min{x,y}, and x y = max{x,y}.
Then α=[α ij ] is the transform of α, which we call the logical product (by matrix multiplication) of the transforms S and R. Definition 2 simply incorporates the result of Theorem 3 of [1]. We apply the same logical product multiplication to the product of transforms with vectors ξ∈Ξ. Thus if C is a set of configurations, ξ∈Ξ then αξ Τ =φ Τ , where also φ∈Ξ.
For example, let C be a set of configurations, α=RS be a history acting on configurations of C, and α=[α ij ]. Then α ij =0 if S allows no transition to C i from any C j ∈C; while α ij =1 if S allows a transition from any C j ∈C to C i ∈C, and R allows a transition to C j from at least one of the C k allowed under S. This operation is associative, so given a history (a string of transforms) α, using Definition 2 we can construct a single transform that represents the effect of α. viable if there exists ξ∈Ξ, ξ>0, Σ i ξ i ≥1 such that αξ Τ =ξ Τ , and for ϕ∈Ξ , ϕ>0 and ϕ≤ξ then αϕ Τ =ϕ Τ . Then we also say that α is viable on ξ. (b) A minimal structure of α is a non-empty configuration C such that ξ C =1 in ξ, α is viable on ξ, and if there exists a nonempty configuration D such that when ϕ∈Ξ, ϕ>0, α is viable on ϕ and ϕ D =1 then Σ(jc j ) ≤ Σ(jd j ).
If α is viable on ξ then α is viable on any ϕ∈Ξ, ϕ ≤ ξ, provided ϕ>0. If α is viable, then α has at least one minimal structure. If C is a minimal structure of α, then Σ(jc j ) = s is the structural number of α. Clearly, every minimal structure of a history α has the same structural number, and every such minimal structure is a "fixed point" of α (see also [4]).
A history α acting only on a minimal structure of α is viable.
Let C be a non-empty set of configurations, and let T be the set of all transforms on C which are allowed by the rules of construction of configurations. Then we say that T is a full set of transforms on C. Let α=[α ij ] be a transform and let α T =[ β ji ] where β ji = α ij be a matrix which is the transpose of α. Such α T is also seemingly a transform, provided there is a rule generating α T , which "reverses" the action of α, since β ji =α ij ; the object α T would thus appear to allow j to transform to i, if α allows i to transform to j. However, if we again let μ C =Σ(jc j ), μ D =Σ(jd j ) and μ D >μ C , then α DC =0 even if α CD =1, because under the rules of construction of the M and B relations, there can not be more cells defined by the relation B created in one generation, than is the marriage number of the preceding generation. Thus, there can be no rule creating such α T and thus not every matrix which is a transpose of a transform, is itself a transform allowed by the rules of construction of configurations. For similar reasons, the inverse of a transform in a full set is not a necessary member of that set. Thus a full set of transforms in general is also not a group.
Note that each ξ i (respectively φ i ) is a number either 0 or 1, so if αξ Τ =φ Τ and ξ,φ ≠ 0, then Σξ i ≥ 1 and Σφ i ≥ 1. If Σφ i =1 then α has exactly one possible outcome, and if Σξ i =1 then α acted on a set consisting of only one initial configuration. We wish to describe the "relative possibilities" when more than one possible outcome might occur. This problem includes the following: how to describe the "relative possibilities" of outcomes when more than one immediately previous configuration might have existed, and we do not know which intermediate configuration actually occurred.
Definition 4
1. If α∈Ĥ is a history, a possibility transform of α is Π=[p ij ] in which 0 ≤ p ij ≤ 1, Σ j p ij ≤ 1, p ij >0 iff α ij =1. Then Π:={ Π | Π is a possibility transform} is a set of possibility transforms. 2. Let Π be a non-empty set of possibility transforms of dimension n>0, let Π,Θ∈Π, and let ξ,ω∈ Ξ, where Σ i ξ i =w ≤ n (respectively Σ i ω i =x ≤ n). Let the products ΠΘ be computed using ordinary arithmetic product of two matrices, and the products ξΠ (respectively Πξ Τ ) as the arithmetic product of a vector and a matrix.
Noticing that Σ i ξ i =w, w ≤ n, then the possibility density π of ξΠ (respectively Πω Τ ) is π:=(π 1 , π 2 , ...π n ) (respectively ω Τ :=(ω 1 , ω 2 , ...ω n ) Τ ) where π i =Σ j (p ij ξ j /w) (respectively ω i =Σ j (p ji ω j /w).
3. If π,ω are possibility densities then <π,ω>=Σ i (π i ω i ) is the inner product of π and ω.
Comment 1: Since each ξ j =0 or ξ j =1, and each p ij ξ j >0 iff ξ j =1, then if ξ j =1, p ij ξ j =p ij , otherwise p ij =0. So while Definition 4.1 allows Σ j (p ij ξ j ) ≤ 1, Σ j (p ij ξ j )=1 iff Σ j p ij =1, which implies Σξ j >0.
A possibility density π of ξΠ answers "if we know the transition ends in one of the non-empty configurations of ξ and the descent sequence applies the possibility transform Π, what if any are the non-empty configurations from which it may have started, and the relative possibility of each?" A possibility density ω Τ of Πξ Τ answers "if we know the transition starts in one of the non-empty configurations of ξ and the descent sequence applied the possibility transform Π, what are the non-empty configurations from which the transition might end and the relative possibility of each?" The inner product answers both questions at once. Given these interpretations, it is reasonable to impose this axiom:
Axiom 1: If π=(π 1 , π 2 , ...π n ) is a possibility density then Σ i π i ≤1.
Since a viable history must have a non-empty descent sequence, we would thus like to know when <π,ω>=1.
Theorem 1: Under conditions of Definitions 3 and 4 and the notational conventions, if α,β∈Ĥ, Π,Θ∈Π are possibility transforms of α, β respectively, ξ,φ∈ Ξ, π the possibility density of ξΠ, ω Τ the possibility density of Θφ Τ , Σ i ξ i =w, all these lists and matrices have dimension n, then <π,ω>=1 iff all of: (i) φ ≠ 0; (ii) ξ ≠ 0; (iii) for each i, Σ j p ij =1 and Σ j q ij =1; (iv) φ=ξ; (v) Σξ j =Σφ j =w.
Proof: The products ξΠ and Θφ Τ are defined since the matrices and vectors all have the same dimension, and thus the respective possibility densities π, ω Τ are defined. Note that <π,ω Τ >=Σ i ((Σ j (p ij ξ j /w)(Σ j (q ij φ j /w)). Conditions (i) and (ii) are required since if either φ,ξ=0, <π,ω Τ >=0, since all of the products then involve at least one term = 0, thus all products = 0. Condition (iv) results since if there is any i such that φ i =1 but ξ i =0, or any φ i =0 but ξ i =1, then there will be a product p ij ξ i q ji φ i =0 in a numerator, but for which not both p ij =0 and q ij =0. But in that case even if Σ j p ij =1 and Σ j q ij =1, at least one of the p ij ≠0 or q ij ≠0 values will be disregarded in computing sums, so Σ i ((Σ j (p ij ξ j /w)(Σ j (q ij φ j /w))<1. Thus necessarily φ=ξ so (iv) is shown. To show conditions (iii) recall that 0 ≤ p ij ≤ 1 and 0 ≤ q ij ≤ 1, Σ j p ij ≤ 1, Σ j q ij ≤ 1, and thus every sum Σ j (p ij ξ i /w) ≤ 1, and every Σ j (q ij φ i /w) ≤ 1. And if not both: (1) Σ j p ij =1 for all i then there is at least one i such that Σ j (p ij ξ ij /w)<1, and (2) Σ j q ij =1 for all i then there is at least one i such that Σ j (q ij φ i /w)<1; so <π,ω Τ >=1 only if both Σ j p ij =1 and Σ j q ij =1. Thus (iv) is met, and since φ=ξ then If α is viable then conditions (i), (ii), (iv) and (v) of Theorem 1 are met, but these conditions are requisite for (iii), so a history α is viable if its transform Π meets Theorem 1. From Theorem 1 (iv) φ=ξ and Comment 1, if any row or column of a possibility transform Π or Θ is that of some φ i =ξ i =0 then the corresponding p ij =0 and q ij =0. Thus when discussing possibility transforms meeting Theorem 1, we can consider such possibility transform to be only its non-zero rows and columns, and the corresponding lists only the non-zero entries, and all thus of dimension w; we call such forms reduced.
CULTURE
Lemma 1: If π=(π 1 , π 2 , ..., π n ) and ω Τ =(ω 1 , ω 2 , ..., ω n ) Τ are possibility densities meeting Theorem 1, then Σ i π i =1 and Σ i ω i =1.
Proof: From Theorem 1(iii) each non zero row of Π is such that Σ j p ij =1, 0≤p ij ≤0. From Definition 4.2, using the multiplication defined in 4.3, and from Theorem 1 (v), each ω is a convex combination of w rows of Π, each of which row is weighted by 1/w. Thus, Σ i ω i =w(1/w)1=1. By a similar argument, each Σ i ξ i is a sum of exactly all non-zero entries p ij , with exactly the same weights since there are w non-zero columns, and the denominators are all w, and, though possibly taken in a different sequence, that sum is thus given by Σ i π i =Σ i ω i =1.// A doubly stochastic matrix is a matrix, each of whose rows and columns are nonnegative numbers that sum to 1.
Theorem 2:
A possibility transform Π meeting Lemma 1 and Theorem 1 is doubly stochastic on those rows and columns for which φ i =ξ i =1 Proof: Let Π be a possibility transform satisfying Theorem 1. We consider only rows and columns for which φ i =ξ i =1 (we need consider only the reduced form of Π). There are w such rows or columns. Theorem 1 establishes the result as to rows of Π. We extend the same reasoning as in Lemma 1. Given a non-empty set of configurations ξ, the possibility density ω is a sum of w columns, each weighted by 1/w, whose sum is w, so the sum of each such column is w(1/w)=1.// Theorem 3: Let α be a history, let ξ∈Ξ, let α be viable on ξ, let be Π the possibility transform of α, let and π be the possibility density of ξΠ then <π,π Τ >=Σ i π i 2 =1.
Proof: Obvious. We note Theorem 1, set Θ=Π, set φ=ξ, then find the possibility densities π of ξΠ and ω Τ of Πξ Τ . But since we assume Π is symmetric, then π Τ =ω Τ so substituting in Theorem 1 gives <π,π>=1. Since π=(π 1 , π 2 , ..., π n ), then also <π,π>=Σ i π i 2 =1.//
CULTURE THEORY TRANSFORMS, PAGE 3 OF 8 BALLONOFF IQSA SOPOT PRESENTED AT: INTERNATIONAL QUANTUM STRUCTURES ASSOCIATION MEETINGS, POLAND, 2008 UNDER REVIEW AT INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS , IQSA SPECIAL ISSUE
CULTURE THEORY TRANSFORMS, PAGE 4 OF 8 BALLONOFF IQSA SOPOT PRESENTED AT: INTERNATIONAL QUANTUM STRUCTURES ASSOCIATION MEETINGS, POLAND, 2008 UNDER REVIEW AT INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS , IQSA SPECIAL ISSUE Definition 3: Under the notational conventions: (a) A history α is
CULTURE THEORY TRANSFORMS, PAGE 5 OF 8 BALLONOFF IQSA SOPOT PRESENTED AT: INTERNATIONAL QUANTUM STRUCTURES ASSOCIATION MEETINGS, POLAND, 2008 UNDER REVIEW AT INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS , IQSA SPECIAL ISSUE
THEORY TRANSFORMS, PAGE 6 OF 8 BALLONOFF IQSA SOPOT PRESENTED AT: INTERNATIONAL QUANTUM STRUCTURES ASSOCIATION MEETINGS, POLAND, 2008 UNDER REVIEW AT INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS , IQSA SPECIAL ISSUE Σξ j =Σφ j =w, so (v) is met. Thus only if (i) through (v) then <π,ω>=1. But also by similar arguments, if <π,ω>=1 then conditions (i) through (v) are met.//
I thank Dick Greechie for his generous comments on earlier versions of this text. All errors are my own.
An important example of a set ξ of configurations allowing to meet Theorems 1 through 3 is a set of the minimal structures of a transform (history, rule) α.Definition 5: Let C s be a non-empty set of configurations of order s, and let P s (C s )={ξ | ξ ⊆ C s }. Let Ĥ s be the full set of transforms on C s . Let Π s :={Π α | α∈Ĥ s , Π α is the possibility transform for α}.1. Then S s :=(C s ,P s (C s ), Ĥ s , Π s ) is a cultural structure of order s. 2. If S s is a cultural structure of order s, α∈Ĥ s a history with minimal structure ξ m ∈C s and structural number s, α=[α ij ], α ij =1 iff i=j=m, and Ĥ s ={α}, then S s is a pure system of α.A pure system of α is viable. Let α be a history with structural number s>0, and S s a pure system of α, then Ĥ s ={α}, and since α acts only on a particular minimal structure of α, then in a pure system α=α -1 =α 2 . In a pure system, the possibility transform Π of α has but one entry p ii =1, where i indexes the minimal structure configuration ξ m and all other p jk =0, jk≠ii. Then for such Π, trΠ=1, indeed trΠΠ=1, and such Π is obviously symmetric.Definition 6: Under conditions of Definition 4 let Ĥ be a set of transforms, let α∈Ĥ, let Π α be the possibility transform of α, and let Π ={ Π α | α∈Ĥ, and Π α is the possibility transform of α }.(1)and(2). The equalities occur in (2) only if α is viable. If we assume the set Π of Definition 6 includes only possibility transforms of pure systems, then clearly also, trΨ=1, and also tr(Σ α v α Π α )=trΨ=1. Then for each such pure system, if Π α =[p ij ]∈θ there exists an α such that exactly one α ij =1. And therefore if y ij >0 there also exists at least one α with at least one α ij =1. This shows that when Ψ= Σ α v α Π α , α∈Ĥ, Π α ∈Π, then also Ψ∈Π. We may then need to construct a corresponding history α (which is what anthropologists typically do in creating a minimally structured genealogy to describe the action of a "cultural rule"), but we know that (at least) one exists.Consider an ethnologist (or field biologist) observing a species of individuals in a descent sequence of some evolutionary structure. At the outset of work, the ethnologist hypothesizes that trΨ=1 meaning, that the descent sequence has some marriage rule(s), which can be characterized by some pure system(s). After observing, the ethnologist claims a subset of rules θ ⊆ Ĥ s such that trΨ=tr(Σ α v α Π α )=1 for α∈θ. Of course θ may consist of a single rule α (for which therefore v α =1). It is known[1,2,3], that that if a descent sequence follows a rule α with structural number s, that certain "demographic" measures can be computed from the convex combination of structural numbers of the rules thus observed. Thus the ethnologist can perform several simultaneous empirical observations on the descent sequence at date t, including: ask if they observed at least one rule α with structural number s>0 (that is, observed a non-empty set θ ⊆ Ĥ such that trΨ=tr(Σ α v α Π α )=1, for α∈θ); and ask if the statistics are those predicted from the structural number or convex combination of structural numbers of the claimed rule(s).Where other features such as "kinship terminologies" are also studied, the ethnologist can also ask if the kinship terminology of each α∈θ ⊆ Ĥ maps "properly" to a descent sequence composed of a pure system using the minimal structure of α. Ethnographic description thus amounts to asking which if any transform(s) α have a fixed point that describes the empirically observable properties of a descent sequence following a rule.The fact that the possibility densities of viable histories are doubly stochastic matrices opens to culture theory the application of the "Birkhoff theorem"[6], that the set of doubly stochastic matrices of order n is the convex closure of the set of permutation matrices of the same order, and the vertices (extreme points) of that set are those permutation matrices. At least many marriage and kinship systems are known to be described as permutations[7,8]. Since pure systems are a very simple example of permutations, we have used only a very basic form of that result. The breadth of application this suggests for culture theory is large, and we shall exploit it in later papers.
Statistical Theory of Marriage Structures. P Ballonoff, Mathematical Models of Social and Cognitive Structures. P. BallonoffUrbanaUniversity of Illinois PressBallonoff, P. Statistical Theory of Marriage Structures, in P. Ballonoff (ed) Mathematical Models of Social and Cognitive Structures, pp. 11 -27 University of Illinois Press, Urbana, 1974, at http://www.ballonoffconsulting.com/pdf/STMS-1974.pdf
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MV-algebra for Cultural Rules. P Ballonoff, International Journal for Theoretical Physics. 471Ballonoff, P. MV-algebra for Cultural Rules, International Journal for Theoretical Physics, Volume 47 No. 1, 223-235, 2008, at http://www.springerlink.com/content/ep3579q5l0643038/fulltext.pdf
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Tres observaciones sobre el algebra lineal. G Birkhoff, Univ. Nac. Tucuman Revista. Ser. A. 5Birkhoff, G. Tres observaciones sobre el algebra lineal. Univ. Nac. Tucuman Revista. Ser. A 5 (1946) 147-151.
On some classes of kinship systems, I: Abelian systems. A Gottscheiner, Anthropology and Cultural Theory. 24to appearGottscheiner, A. On some classes of kinship systems, I: Abelian systems. Anthropology and Cultural Theory, Vol. 2 No.4, 2008, to appear.
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"Experimental determination of circuit equations",
"Experimental determination of circuit equations"
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| [
"Jason Shulman ",
"Frank Malatino ",
"Matthew Widjaja ",
"Gemunu H Gunaratne ",
"\nDepartment of Physics\nDepartment of Physics\nRichard Stockton College of New Jersey\n08205GallowayNJ\n",
"\nUniversity of Houston\n77204HoustonTX\n"
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| Kirchhoff's laws offer a general, straightforward approach to circuit analysis. Unfortunately, use of the laws becomes impractical for all but the simplest of circuits. This work presents a novel method of analyzing direct current resistor circuits. It is based on an approach developed to model complex networks, making it appropriate for use on large, complicated circuits. It is unique in that it is not an analytic method. It is based on experiment, yet the approach produces the same circuit equations obtained by more traditional means. | 10.1119/1.4895005 | [
"https://arxiv.org/pdf/1309.4848v1.pdf"
]
| 118,311,041 | 1309.4848 | fe8ed583dc8d48c613048069fb7328512dd0fb75 |
Experimental determination of circuit equations
19 Sep 2013
Jason Shulman
Frank Malatino
Matthew Widjaja
Gemunu H Gunaratne
Department of Physics
Department of Physics
Richard Stockton College of New Jersey
08205GallowayNJ
University of Houston
77204HoustonTX
Experimental determination of circuit equations
19 Sep 2013(Dated: May 22, 2014)arXiv:1309.4848v1 [physics.class-ph]
Kirchhoff's laws offer a general, straightforward approach to circuit analysis. Unfortunately, use of the laws becomes impractical for all but the simplest of circuits. This work presents a novel method of analyzing direct current resistor circuits. It is based on an approach developed to model complex networks, making it appropriate for use on large, complicated circuits. It is unique in that it is not an analytic method. It is based on experiment, yet the approach produces the same circuit equations obtained by more traditional means.
I. INTRODUCTION
In introductory physics, students are universally taught to analyze circuits using Kirchhoff's laws. The standard approach employs the loop and current laws in order to derive a set of equations that can be solved for the branch currents. This technique is, of course, very powerful in that it provides a systematic procedure for studying a variety of circuits. Unfortunately, it becomes unfeasible for any circuit of modest complexity. We are forced to search for an alternative method and/or engage a computer. There have been several interesting and clever alternatives presented in the literature that aim to teach or supplement the standard methodology presented in textbooks. [1][2][3][4] Here, we take a more bottom-line approach and present, not an analytical method, but an experimental one to obtain the equations that govern circuit behavior. These are the same equations found through an implementation of Kirchhoff's laws. The experimental method is based on a modeling approach developed to aid in controlling large, complex networks. 5 Thus, it is well suited for quite complicated circuits. It, however, is not intended to replace the learning and use of standard techniques;
it simply provides an efficient means of determining the equations when other methods of analysis are not practical or even feasible. This practicality is also accompanied by several didactic aspects, allowing students to gain insight into the nature of circuits and the method.
The network modeling methodology is presented in section II. Section III reviews the node method, a more traditional approach to circuit analysis. The results of both the circuit and network methods are compared in section IV. Finally, section V demonstrates the implementation of the network approach on a simple circuit.
II. A NETWORK MODEL
A. General considerations
The network modeling approach, upon which the experimental method is based, was developed specifically for genetic networks, [5][6][7] where accurate models are difficult or impossible to construct from experimental data. 8 We have since shown that the approach is appropriate for a host of large, nonlinear networks. 9 Circuits can certainly fall into this category; however, in the present work, we will restrict our attention to those appropriate for undergraduate students beyond the introductory courses; that is, we will focus on networks of linear (ohmic) resistors in dc circuits. First, we will briefly outline the network modeling methodology.
The details of the network modeling approach are provided elsewhere. [5][6][7] Here, we present only that which is relevant to our purpose. Consider the three node network in Fig. 1a. A parameter V is associated with each node. Later, when discussing circuits specifically, V will represent the potential at the node, and each arrow will symbolize an electrical component connecting two nodes. The vector V = [V 1 , V 2 , V 3 ] defines the state of the system. Each arrow represents the interaction between two nodes, corresponding to a possibly unknown nonlinear mathematical function. Changing V 1 , for example, will induce a change in the values of V 2 and V 3 , as determined by these functions. While the set of interactions and the associated mathematical functions are commonly unknown to us, we can often perform a set of experiments on the system in which some of the node parameters V are perturbed, and the corresponding effect on the remaining parameters is measured. An example of this would be fixing the potential at one point in a circuit and measuring the potential at the others in response to this change. When choosing nodes to perturb, we often select highly connected hub-like nodes which are able to influence many others. Imagine such an experiment on the three node network. Hub nodes cannot exist in such a small system; however, we choose to perturb nodes one and two (the circles) and observe the effect on node three (the square).
Upon sweeping the values V 1 and V 2 , a two dimensional surface can be formed (Fig. 1b).
When presented in this way, V 1 and V 2 are the independent variables, and V 3 is the dependent variable. In other words, the value V 3 is determined by V 1 and V 2 . The independent nodes are referred to as the master nodes (circles), which determine the behavior of the remaining slave nodes (square). In our network model, we make the approximation that the values of the slave nodes are exclusively determined by those of the master nodes. This approximation corresponds to ignoring the dashed lines in Fig. 1a, which run from the slave nodes to the masters. In keeping with the approximation, any slave/slave interactions will be ignored as well. The categorization of nodes into masters and slaves is not limited to small networks such as the one described here. Networks with an arbitrary number of master/slaves can be considered. In such cases, the surfaces corresponding to Fig. 1b will be higher dimensional.
By determining surfaces in this way, one can construct a model of the (directed) interactions between the highly coupled master nodes and the sparsely coupled slaves.
Unfortunately, it is often not feasible to perform the experiment imagined above for large systems. Such experiments on gene networks, for example, would be prohibitively expensive. 8,10 However, these surfaces exist in principle, and one can generally find an approximation to the surface by conducting only a few experiments. Consider again the network of Fig. 1a. Let the state of the system be V
(o) = V (o) 1 , V (o) 2 , V (o) 3
when the unperturbed network is measured. This state corresponds to one point (P o ) on the surface in Fig. 1b; the point at which the values of nodes one and two equal their unperturbed values. It is often relatively easy to perturb a system by fixing the parameter of one node to zero, e.g., grounding a node in a circuit. By forcing V 1 = 0, the remaining parameters will be modified from their natural values, and the state of the network will be V (1)
, where the superscript indicates the modification of node 1. This corresponds to point P 1 on the surface. Similarly, node two can be fixed to zero to obtain the state V (2) and point P 2 .
These three points, lying in a three dimensional space, uniquely define a plane. Furthermore, the points lie on both the plane and the surface, indicating close proximity between the two in the regions surrounding the points. The plane represents a good approximation to the surface if the surface is sufficiently smooth. While one may not be able to determine the surface, the plane can be found conducting relatively few experiments, in this case by measuring the state of the system three times, when it is unperturbed and upon perturbation of each master node. A series of linear equations can be written which describe the plane and relating the values of the slave node parameters to those of the master nodes. For the present case, in which there is only one slave node, a single equation approximating the dependence of V 3 on V 1 and V 2 is produced,
V 3 − V (o) 3 = B 11 V 1 − V (o) 1 + B 12 V 2 − V (o) 2 .
(1)
Here, the coefficients B describe the plane, and we have subtracted the unperturbed values
V (o)
, essentially stating that the deviation of V 3 from its unperturbed value is dependent on the deviations of V 1 and V 2 from theirs. Eq. (1) relates the slave parameters to those of the master nodes. For the model to be complete, one needs to account for interactions between the master nodes. In keeping with the linear approximation, we assume relationships between the master nodes take the form,
V 1 = A 11 V 1 − V (o) 1 + A 12 V 2 − V (o) 2 (2a) V 2 = A 21 V 1 − V (o) 1 + A 22 V 2 − V (o) 2 ,(2b)
where the coefficients A represent effective interactions between the master nodes.
This procedure generalizes nicely to larger systems. By measuring the states of the unperturbed network and situations in which each of the n master nodes is singly modified, we obtain n+1 points which lie on the surface and form an n-dimensional plane or, equivalently, the elements of the A and B matrices. This plane is the geometric representation of the network model, which describes the response of the system to modification of the master nodes. The plane is not the surface; however, it represents a reasonable approximation if the surface is relatively smooth. Of course, this approximation can be improved, by including quadratic (or higher) terms at the cost of performing more experiments to determine the new parameters.
B. Exact models
The approach described above does not generally produce an exact model of the network, rather an approximation to the system is generated, the quality of which is quantified by the proximity of the plane and the surface. The methodology has been shown to accurately represent several types of networks. 5-7 Furthermore, the models cease to be approximate under two conditions: 1) the surface itself is a plane and 2) all nodes are considered to be master nodes. The first condition is equivalent to having all interactions between nodes be linear in nature. In such a situation, the procedure described above fits a plane on to a planar surface. The second condition removes one of the key assumptions employed when constructing the model, namely that some set of nodes, the masters, exclusively control those considered to be slaves. The removal of this approximation, combined with a linear system, results in an exact mathematical representation of the network. 11 As there are no slave nodes, equations similar to Eq. (1) are not present in the model, and the dynamics of the nodes is described by the master node equations (Eq. (2)),
V i = j A ij V j − V (o) j ,(3)
where the sum is over all nodes. Under steady state conditions, allV i = 0, and Eqs. (3) simplify to, This represents an exact mathematical description of a linear system in the steady state.
0 = j A ij V j − V (o) j .(4)
Thus, by performing the experiments described in section II A, the matrix A can be calculated (see Supplementary Materials 12 ). The equations describing the behavior of the network will, therefore, have been determined.
The equations can be put in a more convenient form by solving for the node parameters themselves. For example, in the case of a three node network,
V 1 = − A 12 A 11 V 2 − A 13 A 11 V 3 + C 1 (5a) V 2 = − A 21 A 22 V 1 − A 23 A 22 V 3 + C 2 (5b) V 3 = − A 31 A 33 V 1 − A 32 A 33 V 2 + C 3 ,(5c)
where the values C i are constants containing the elements of A and V (o) .
Here, we will demonstrate that this procedure can be used to determine the equations for a circuit of resistors, even when implementation of Kirchhoff's laws is not practical. For a circuit containing N nodes, N + 1 experiments in which the potential V of each node is measured, need to be performed. It is not necessary to fix the node values to zero, as in the example described above. Any value can be used to construct the model; however, in the context of circuits, setting the node potential to zero (by grounding) is quite convenient and will be used throughout the work. The experiments described here can be performed efficiently with a computer and an analog/digital card. If such resources are unavailable, two people using a handheld multimeter, e.g., lab partners, can complete the experiments quickly as well. As a bonus, we find that applying these ideas to circuits provides some physical insight into the parameters of the model (A), something which has been elusive previously.
III. CIRCUITS AND THE NODE METHOD
Before we begin applying these ideas to circuits, it will be instructive to examine some properties of electrical networks and their equations. Consider the four node network shown in Fig. 2. Direct currents I 1 , I 2 , and I 3 flow into the network as shown and exit via node zero, which is held at zero potential. The potentials of the remaining nodes are V 1 , V 2 , and V 3 . For simplicity, we will utilize the node method 4,13 , a technique not typically taught in undergraduate courses, to analyze this network. The presentation will follow that of Ref. 4.
Kirchhoff's current law ensures that the current flowing into node one, I 1 , equals the sum currents flowing through R 10 , R 12 , and R 13 . This observation leads one to conclude Similar considerations lead to equations for nodes two and three,
I 1 = V 1 − 0 R 10 + V 1 − V 2 R 12 + V 1 − V 3 R 13 .(6)I 2 = V 2 − 0 R 20 + V 2 − V 1 R 12 + V 2 − V 3 R 23 (7a) I 3 = V 3 − 0 R 30 + V 3 − V 1 R 13 + V 3 − V 2 R 23 .(7b)
Upon rearrangement, the incoming currents can be written,
I 1 = 1 R 10 + 1 R 12 + 1 R 13 V 1 − 1 R 12 V 2 − 1 R 13 V 3 (8a) I 2 = − 1 R 12 V 1 + 1 R 20 + 1 R 12 + 1 R 23 V 2 − 1 R 23 V 3 (8b) I 3 = − 1 R 13 V 1 − 1 R 23 V 2 + 1 R 30 + 1 R 13 + 1 R 23 V 3 .(8c)
By replacing the resistances, R ij with the corresponding conductances, G ij = 1/R ij , the equations can be placed in a simplified form,
I 1 = G 11 V 1 − G 12 V 2 − G 13 V 3(9a)I 2 = −G 12 V 1 + G 22 V 2 − G 23 V 3(9b)I 3 = −G 13 V 1 − G 23 V 2 + G 3 V 3 ,(9c)
where G 11 = G 10 + G 12 + G 13 (10) and is called the coefficient of self-conductance for node one. 14 The self-conductances of the other nodes are similar and can be obtained from the terms within the parentheses of their respective equations. It is worth noting that the coefficient of self-conductance for a node is the sum of conductances from all resistive channels connecting the node in question to the others in the network. 15 This fact will become important when examining the physical significance of the parameters in the network equations.
The situation represented in Fig. 2 is not typically realized in the laboratory or classroom.
More commonly, the network is connected to a single voltage source, and current enters through one or more nodes. It is then distributed throughout network and returns to the power supply. If, for example, only node one is connected to the power supply, held at potential V s , via a resistor R 1s , the current I 1 is not fixed but depends on the potential difference across R 1s , i.e., (V s − V 1 ) /R 1s . The remaining currents are zero since no current is being injected into the other nodes. In this case, the updated equation for node one reads, G 1s V s = G 11 V 1 − G 12 V 2 − G 13 V 3 , and, with the additional connection, the coefficient of self-conductance has another term, G 11 = G 10 + G 12 + G 13 + G 1s .
The equations (9), updated to incorporate a power supply, can be solved for the node potentials to obtain,
V 1 = G 12 G 11 V 2 + G 13 G 11 V 3 + C ′ 1 (11a) V 2 = G 12 G 22 V 1 + G 23 G 22 V 3 + C ′ 2 (11b) V 3 = G 13 G 33 V 1 + G 23 G 33 V 3 + C ′ 3 ,(11c)
where the C ′ i are constants (possibly zero). In this form, the equations relate the potential at one node to the potentials of the others and a constant. They can be used in a variety of ways. For example, they can be solved simultaneously to obtain the node potentials of the unperturbed circuit. Further, if the circuit is perturbed by externally fixing the potentials of nodes two and three, Eq. (11a) will determine the new potential V 1 .
IV. NETWORK AND CIRCUIT EQUATIONS
If one needs to analyze a complex circuit of resistors, it might not be convenient to employ
Kirchhoff's laws. One can be assured, however, that equations in the form of Eqs. (11) will describe the node potentials. Alternatively, the network strategy described in sections II A and II B could be implemented to obtain network equations which would also describe the circuit. Specifically, for the circuit in Fig. 2, one would obtain Eqs. (5). Note the similarity in the form of Eqs. (5) and (11). Both systems represent an exact description of the circuit and demonstrate the node potential is a linear combination of the other potentials.
The coefficients preceding the node potentials as well as the constants in each system of equations must be equal, i.e.,
− A ij Aii = G ij Gii (12a) C i = C ′ i .(12b)
These equations represent a major conclusion of this work. By performing the N + 1 experiments described above and calculating the matrix A, one can obtain the equations describing the circuits without Kirchhoff's laws. We are assured, through Eqs. (12), that these equations will be the same as those obtained through more conventional means.
It is important to recognize that Eqs. (12) represent the most fundamental relationship between the circuit and network equations. For example, one cannot equate A ij and G ij .
First, the A ii presented here are equal to one by convention, 12 which is not generally true for G ii . Second, a glance at their respective equations indicates that their units are different,
i.e., s −1 for A ij and Ω −1 for G ij .
We will see below that this equivalence between the circuit and network equations can be exploited to easily obtain the equations governing circuit behavior. While this is the central focus of the work, it is also important to extract any physical insight, an important lesson for students. It has already been noted that G ii is the sum of all conducting channels connecting to node i. It is a measure quantifying the connectivity of node i to the rest of the circuit. It says nothing, however, about how those connections are distributed throughout the circuit. For example, a node with a single connection could possess a coefficient of self-conductance with the same value as a node with five connections. The distribution information is contained in G ij /G ii , the coefficients of Eqs. (11). Consider the circuit described by these equations. Written explicitly, the first coefficient of (11a) is, G 12 /G 11 = G 12 / (G 10 + G 12 + . . . + G 1s ). In other words, G 12 is some portion of G 11 , and, therefore, G 12 /G 11 describes the strength of the connection between nodes one and two relative to all the other connections nodes one makes with the rest of the circuit. It is a measure of how much influence node two has on one compared to the other nodes. Note also, j G ij /G ii = 100%. This has all been gained by examining the constants in the circuit equations. In light of Eq. (12a), A ij /A ii , the coefficients in the network equations, describe the proportion the ij channel contributes to the total connectivity of node i. More generally, such a situation arises when the interaction between two nodes in a network is proportional to the difference in the node parameters, as is the case with circuits. This physical insight into the constants of the network equations has been unavailable in prior studies. Equation (12b) indicates the constants in the network equations (C i ) represent the contributions from the power supply and ground, i.e., C i = (G js /G ii ) V s . For the example circuit described above, in which the power supply only connects to node one, only C 1 = 0 since the other nodes make no connection to the supply (G 2s = G 3s = 0).
V. RESULTS AND DISCUSSION
A. A simple example
We will illustrate the ideas discussed above with the simple four node circuit shown in Fig. 3. Such a circuit is simple enough to be studied with the standard techniques, and the results of such an analysis will be compared to that of the network method. In what follows, the results of simulations, 16 rather than experiments, will be presented so complicating factors such as experimental error and resistor tolerance will not hinder the exposition, hopefully allowing the reader to easily verify the calculations. For every simulation described, the corresponding experiment has been performed (by undergraduate students) and results verified (except for section V B).
A student in an introductory course would use Kirchhoff's laws to calculate the unknown branch currents and the potential differences across the resistors. A more advanced analysis could utilize the node method to obtain equations which illustrate the relationships, or interactions, between the nodes, e.g., Eqs. (11). For the circuit of Fig. 3, the equations are,
V 1 = 0.43V 2 + 0.14V 3 + 4.29 (13a) V 2 = 0.47V 1 + 0.47V 3 + 0.06V 4 (13b) V 3 = 0.18V 1 + 0.55V 2 + 0.27V 4 (13c) V 4 = 0.05V 2 + 0.19V 3 .(13d)
Alternatively, the network method could be implemented to determine the equations, which would take the form of Eqs. (4). The primary task is to determine the elements of A and V (o) . This requires experimental input, which consists of the node potentials of the unperturbed circuit V (o) and circuits in which each node potential has been individually fixed. Here, each node will be grounded, i.e., V = 0 V. The results of these five experiments are contained in Table I. These data can be used to calculate the elements of A. We
.(14)
Note the similarity of the elements to coefficients of Eqs. (13). This is a consequence of the relationship between the circuit and network equations, Eq. (12a). The network equations can be expanded and solved for the potentials to obtain the same equations found using the node method (Eqs. 13). Once these equations have been determined, by either method, all parameters of the circuit can be calculated. They can be solved for the node potentials of the unperturbed circuit V (o) or any perturbed circuit in which one or more potentials are fixed. Branch currents can be calculated with the node potentials and the resistor values.
This particular example is simple, and neither the network technique nor the node method is required to analyze the circuit. For more complicated circuits, however, the network method is quite straightforward and can be implemented quickly. Table I. The network method can be implemented again, now for the malfunctioning circuit.
It is non-invasive, only requiring the measurement of the new potentials for the normal and perturbed circuits. Previously, the perturbations were made by grounding each node individually, which was simply chosen for convenience. In this scenario, it might not be prudent to implement such a drastic change in the circuit while it is still in operation. A slight deviation from the normal node potentials can be used instead; the results will be unaffected. After completing the experiments and calculations for the burned circuit, the new A matrix is,
A b = 1 −0.43 −0.14 0 −0.47 1 −0.47 −0.06 −0.25 −0.75 1 0 0 −0.06 0 1 .(15)
Equations (14) and (15) is affected similarly. The coefficients of self-conductance for nodes three and four are also changed. They decrease due to the reduction of the conductivity. This has the effect of increasing all other elements in rows three and four, even if the associated resistors were unaffected by the burning. Finally, any elements not associated with the nodes three and four should remain unchanged. All of this can be observed by examining the difference between A and A b , i.e.,
∆A = |A| − |A b | = 0 0 0 0 0 0 0 0 −0.07 −0.20 0 0.27 0 −0.01 0.19 0 .(16)
The The node method for a circuit of resistors produces a linear system of equations. The network approach, approximate for many networks, is exact for such systems due to its linear nature. Here, we have exploited this mathematical relationship to study circuits.
Interestingly, before the advent of the handheld calculator, the relationship was used in reverse; circuits were used for algebraic calculations. 17 If nonlinear elements are present in the circuit, the network equations approximate the actual circuit behavior. In spite of the nonlinearity, the network model can accurately describe node potentials and can even be used to control the system, an important topic in network research. 9
VI. CONCLUSION
We have demonstrated that the network modeling methodology can be used to obtain equations describing complex resistor circuits. These equations are the same as those obtained through more traditional means. The approach is unique in that it is experimental, rather than analytical, and well suited for complex arrays of resistors. The ease of implementation, to the extent that it resembles a recipe, is convenient; however, we do not suggest it to be a substitute for traditional instruction. It is simply an efficient means of obtaining the circuit equations when other techniques are not practical.
The connection between the circuit and network equations has, for the first time, provided physical interpretation of the elements of the A matrix. Each A ij , (i = j), represents the amount of influence the ij channel has relative to all of the other connections to node i.
This feature is not unique to circuits; it arises in systems where in the interactions between nodes is proportional to the difference in node parameter values.
We have found that this material is suitable for undergraduate students beyond the introductory courses, perhaps as an exercise in intermediate or advanced laboratories. It can be used as a springboard into discussions of networks, systems of equations, and the agreement between theory and experiment. Furthermore, it demonstrates the connection between research and the classroom.
FIG. 1 :
1(Color online) (a) A simple three node network. In the network model, node three (the slave) is exclusively controlled by nodes one and two (the master nodes). Solid arrows represent interactions approximated by the network model. Dashed arrows, from the slave, are ignored. (b) The value V 3 of the slave node is determined by those of the master nodes. A surface, representing the values of the slave parameter, is formed upon sweeping the master node values. A plane, approximating the surface, can be determined by a few experiments. The plane is a good approximation to the surface if the surface is sufficiently smooth.
have included an algorithm and an explanation of the calculation in the SupplementaryMaterials.12 For the circuit inFig.
FIG. 3 :
3The circuit used to generate data in
∆A 34 and ∆A 43 elements are positive since their associated conductances decreased with the burning. The remaining elements in those rows are negative due to the decrease of the coefficients of self-conductance. All other elements have been unaffected by the burned component. Thus, the damaged component can be identified by searching for rows with non-zero elements.The concepts presented in this section have been verified on a variety of circuits, ranging from the simple to the quite complex. A circuit containing 18 nodes and 30 components cannot practically be analyzed by traditional methods, yet the network approach can be efficiently employed by students with common laboratory equipment. Unlike the straightforward implementation of Kirchhoff's laws, the network approach extracts the interactions between the nodes (conductances). This fact can be used to determine the resistors connecting nodes, and the equations describing the node potentials, for black box circuits in which the components are hidden. Furthermore, the approach automatically combines resistors in parallel and series since the elements of A are determined by the effective conductances between the nodes.
FIG. 2: Current is injected into nodes one, two, and three and exits via node zero, which is grounded. Potentials V 1 , V 2 , and V 3 are formed at the nodes.V 1
R 13
R 12
R 10
I 1
R 30
R 20
R 23
I 3
I 2
V 3
0
V 2
TABLE I :
INode potentials (in Volts) from experiments. V (i) denotes data from the circuit in which node i is grounded or the unperturbed circuit (o). It is not immediately obvious that R 34 was the component that suffered the damage.Experiment Node 1 Node 2 Node 3 Node 4
V (1)
0
0
0
0
V (2)
4.41
0
0.85
0.16
V (3)
5.37
2.54
0
0.12
V (4)
7.2
5.39
4.25
0
V (o)
7.55
5.95
4.95
1.23
B. Burned resistor detection
We have found that the experimental network technique can be taught to students using
interesting, real-world applications. For example, students have been asked to use the ap-
proach to detect burned resistors. Imagine a circuit running an important electrical device,
e.g., life support equipment, which must remain running. This complicates any repairs if, for
example, a resistor burns. The node potentials can be measured with a voltmeter, making it
clear that the device is malfunctioning; however, it is not generally possible to pinpoint the
burned resistor by examining potentials since most, if not all, will be modified due to the
connectivity of the circuit. Further, an ohmmeter cannot be used to measure the resistance
of the components while current is flowing through them. Fortunately, the network method
can identify the burned resistor, which can then be replaced, without having to shut down
the device.
This idea will, again, be demonstrated on the circuit of Fig. 3. We have already deter-
mined A for the circuit and know the normal operating potentials, V (o) = [7.55 5.95 4.95 1.23],
Eq. 14 and Table I, respectively. If a component burns, the conductance will decrease (re-
sisitance increases), resulting in a change in the elements of A, due to Eq. (12a), and a
modification of the node potentials. For example, if R 34 burns, increasing its resistance to
50 MΩ, the node potentials will be V
(o)
b = [9.03 8.25 8.45 0.49].
can be compared. The elements A b,34 and A b,43 have now gone to zero, indicating that the conductance between nodes three and four has decreased significantly. It can be concluded that this component was damaged.Such a comparison is not an efficient means of locating the burned component, especially if the circuit is large. A simple algorithm has been developed to easily identify the damaged resistor. It is based on the fact that not all of the elements of the matrix are modified when a component burns. In the example above, A 34 clearly changes since A ij = −G ij /G ii (by convention, A ii = 1). The drastic modification of G 34 is reflected in this element. A 43
The coefficient between the i th and j th nodes, G ij , is often called the mutual conductance
This is rather obvious from Eq. (10) and its generalization; however, to the best of the authors'
ACKNOWLEDGMENTSThe authors would like to thank Stephen Tsui for review of the manuscript as well as Jeniffer Allen and Jared Bland for discussions. This work was supported by the School of Natural Sciences and Mathematics and the Grants Office of The Richard Stockton College
The random walk method for dc circuit analysis. Raymond A Sorensen, Am. J. Phys. 5811Raymond A. Sorensen, "The random walk method for dc circuit analysis," Am. J. Phys. 58(11), 1056-1059 (1990).
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. Henry G Booker, An Approach to Electrical Science. McGraw HillHenry G. Booker, An Approach to Electrical Science (McGraw Hill, New York 1959).
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It should be noted that condition 1 is sufficient to produce an exact model of the network. Condition 2 is necessary to produce an exact and complete model. Without this condition, the effect of node three on nodes one and two would not be included. See Eqs. (2) for an exampleIt should be noted that condition 1 is sufficient to produce an exact model of the network. Condition 2 is necessary to produce an exact and complete model. Without this condition, the effect of node three on nodes one and two would not be included. See Eqs. (2) for an example.
In the archive. Can contact for supplementary materials. In the archive. Can contact for supplementary materials.
Anant Agarwal, Jeffrey Lang, Foundations of Analog and Digital Electronic Circuits. San FranciscoMorgan KaufmannAnant Agarwal and Jeffrey Lang, Foundations of Analog and Digital Electronic Circuits (Mor- gan Kaufmann, San Francisco 2005).
the most recent mention of this fact is from a textbook published in 1966, and reiteration seems appropriate. The citation for the textbook is Charles Close, The analysis of Linear Circuits. Harcourt, Brace & World, IncNew Yorkknowledge, the most recent mention of this fact is from a textbook published in 1966, and reiteration seems appropriate. The citation for the textbook is Charles Close, The analysis of Linear Circuits (Harcourt, Brace & World, Inc., New York, 1966).
Simulations were performed with 5spice software which can be. Simulations were performed with 5spice software which can be found at http://www.5spice.com/.
. Donald Johnson, University of WisconsinMasters ThesisDonald Johnson, Masters Thesis, University of Wisconsin, 1953.
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[
"THE UNIVERSE INSIDE HALL ALGEBRAS OF COHERENT SHEAVES ON TORIC RESOLUTIONS",
"THE UNIVERSE INSIDE HALL ALGEBRAS OF COHERENT SHEAVES ON TORIC RESOLUTIONS"
]
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"Boris Tsvelikhovskiy [email protected] \nDepartment of Mathematics\nUniversity of Pittsburgh\n15260PittsburghUSA\n"
]
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"Department of Mathematics\nUniversity of Pittsburgh\n15260PittsburghUSA"
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| Let g = so 8 be a simple Lie algebra of type A, D, E with g the corresponding affine Kac-Moody algebra and n − ⊂ g a nilpotent subalgebra. Given n − as above, we provide an infinite collection of cyclic finite abelian subgroups of SL 3 (C) with the following properties. Let G be any group in the collection,) the derived equivalence of Bridgeland, King and Reid. We present an (explicitly described) subset of objects in Coh G (C 3 ), s.t. the Hall algebra generated by their images under Ψ is isomorphic to U(n − ). In case the field k (in place of C) is finite and char(k) is coprime with the order of G, we conjecture the isomorphisms of the corresponding 'counting' Ringel-Hall algebras and the specializations of quantized universal enveloping algebras U v (n − ) at v = |k|.THE UNIVERSE INSIDE HALL ALGEBRAS OF COHERENT SHEAVES ON TORIC RESOLUTIONS• the skyscraper sheaf V ! = V ⊗ C O 0 , whose fiber at 0 is V and all the other fibers vanish;is an exact equivalence of triangulated categories. A natural question emerges. | null | [
"https://arxiv.org/pdf/2201.07847v2.pdf"
]
| 246,063,539 | 2201.07847 | d5c5912d1e70d281c762cb992d7986df23c80b5e |
THE UNIVERSE INSIDE HALL ALGEBRAS OF COHERENT SHEAVES ON TORIC RESOLUTIONS
2020
Boris Tsvelikhovskiy [email protected]
Department of Mathematics
University of Pittsburgh
15260PittsburghUSA
THE UNIVERSE INSIDE HALL ALGEBRAS OF COHERENT SHEAVES ON TORIC RESOLUTIONS
Mathematics Subject Classification
14172020
Let g = so 8 be a simple Lie algebra of type A, D, E with g the corresponding affine Kac-Moody algebra and n − ⊂ g a nilpotent subalgebra. Given n − as above, we provide an infinite collection of cyclic finite abelian subgroups of SL 3 (C) with the following properties. Let G be any group in the collection,) the derived equivalence of Bridgeland, King and Reid. We present an (explicitly described) subset of objects in Coh G (C 3 ), s.t. the Hall algebra generated by their images under Ψ is isomorphic to U(n − ). In case the field k (in place of C) is finite and char(k) is coprime with the order of G, we conjecture the isomorphisms of the corresponding 'counting' Ringel-Hall algebras and the specializations of quantized universal enveloping algebras U v (n − ) at v = |k|.THE UNIVERSE INSIDE HALL ALGEBRAS OF COHERENT SHEAVES ON TORIC RESOLUTIONS• the skyscraper sheaf V ! = V ⊗ C O 0 , whose fiber at 0 is V and all the other fibers vanish;is an exact equivalence of triangulated categories. A natural question emerges.
INTRODUCTION
We begin with describing different versions of McKay correspondence. Let G ⊂ GL n (C) be a finite subgroup and consider the categorical quotient X = C n /G := Spec(C[x 1 , x 2 , . . . , x n ] G ). In addition assume the following properties:
• X has an isolated singularity at 0; • there exists a projective resolution ρ : Y → X. Let Coh G (C n ) be the category of G-equivariant coherent sheaves on C n and Coh(Y) be the category of coherent sheaves on Y. In modern language the McKay correspondence is usually understood as an equivalence of triangulated categories D b G (Coh(C n )) and D b (Coh(Y)). Such an equivalence is known to hold in the following cases:
• G ⊂ SL 2 (C), any G ( [KV00]); • G ⊂ SL 3 (C), any G, Y = G -Hilb(C 3 ) ( [BKR01]); • G ⊂ SL 3 (C), any abelian G and any crepant resolution (Y, ρ) ( [CI04]); • G ⊂ SP 2n (C), any G and crepant symplectic resolution (Y, ρ) ( [BK04]); • G ⊂ SL n (C), any abelian G, any crepant symplectic resolution (Y, ρ) ( [Kaw16]). Any finite-dimensional representation V of G gives rise to two equivariant sheaves on C n : 1 Suppose G ⊂ GL n (C) satisfies the following assumptions:
(1) the McKay quiver Q(G, C n ) contains a subquiver Q ′ (without oriented cycles) with I ∩ P Q ′ = 0; (2) there is a derived equivalence Ψ : D b G (Coh(C n )) → D b (Coh(Y)); (3) Ψ sends the skyscraper sheaves χ ! ∈ Coh G (C n ), corresponding to the simple representations in Rep(Q(G, C n ), R) supported at the vertices of Q ′ , to pure sheaves concentrated in the same degree. Let H {Ψ(χ ! i )} i∈Q ′ 0 be the Hall algebra generated by the images of sheaves corresponding to simple representations of Q ′ under Ψ and n − ⊂ g Q ′ stand for the corresponding nilpotent subalgebra of g Q ′ . It follows from the discussion above that one has an isomorphism of algebras (see also diagram 13):
Θ : U(n − ) → H {Ψ(χ ! i )} i∈Q ′ 0 .
In this paper we present an infinite collection of cyclic finite abelian subgroups of SL 3 (C) satisfying conditions (1)−(3) above with Q ′ any simply laced Dynkin diagram of affine type except D 4 or A 0 , hence, produce isomorphisms U(n − )
≃ → H {Ψ(χ ! i )} i∈Q ′ 0
for n − ⊂ g with g = so 8 a simple Lie algebra of type A, D, E and g the corresponding affine Kac-Moody algebra. Let ε = e 2πi/r be the primitive root of unity. The indicated family consists of cyclic abelian groups ϕ : Z/rZ ֒→ SL 3 (C) with ϕ(1) = diag(ε, ε k , ε s ), where r = 1 + k + s and • s ≡ 0 (mod k), • s ≡ 0 (mod k + 1). Interestingly, as s goes to infinity, the proportion of nontrivial characters {χ | H 0 (Ψ(χ ! )) = 0} tends to k − 1 k + 1 (see Corollary 5.4). In particular, as both s and k tend to infinity, for a uniformly randomly chosen character χ, one has that Ψ(χ ! ) is concentrated in degree 0 with probability 1. The exposition in the paper is organized as follows. In Sections 2−4 we recall the generalities on McKay correspondence, quiver representations and Hall algebras, respectively. Each section has references for a more detailed overview of the corresponding topic. While these sections contain essentially no new results, the example presented in Section 2 is important for a better understanding of the construction. Finally, Section 5 introduces the families of finite cyclic subgroups in SL 3 (C) and establishes the isomorphisms of algebras announced above (Theorems 5.3 and 5.6).
Acknowledgement: I am grateful to Roman Bezrukavnikov and Timothy Logvinenko for enlightening discussions. I would like to thank Timothy Logvinenko for drawing my attention to derived Reid's recipe and explaining how it works. I am indebted to Michael Finkelberg and Olivier Schiffmann for valuable suggestions.
MCKAY CORRESPONDENCE
We start with a quick chronological overview of the subject. In [McK80] John McKay has observed that for a finite subgroup G ⊂ SL 2 (C) there is a bijection nontrivial irreducible representations of G 1:1 ←→ irreducible components of the central fiber ρ −1 (0) , where ρ : Y → C 2 /G is the minimal resolution of singularities. Notice that C[G] (the ring of representations of G) is naturally isomorphic to K G (C 2 ), the Grothendieck group of Gequivariant coherent sheaves on C 2 . Following this observation, in [GSV83], the McKay correspondence was realized geometrically as an isomorphism of Grothendieck groups K G (C 2 ) → K(Y). Next in [KV00] this isomorphism was lifted to an equivalence of triangulated categories of coherent sheaves:
D G (C 2 ) → D(Y)
. In particular, under this equivalence, χ ! := χ ⊗ O 0 , the skyscraper sheaf at 0 associated to a nontrivial irreducible G-representation χ, is mapped to the structure sheaf of the corresponding exceptional divisor (twisted by O(−1)). Then Bridgeland, King and Reid constructed the equivalence D G (C 3 ) → D(Y) for any finite subgroup G ⊂ SL 3 (C) and Y = G -Hilb(C 3 ) (see [BKR01]). They showed that G -Hilb(C 3 ) is a crepant resolution of C 3 . It was established that the images of χ ! s are concentrated in a single degree in case G is abelian and C 3 /G has a single isolated singularity (see [CL09]). The result was then extended to any finite abelian subgroup G of SL 3 (C) in [CCL17]. We briefly recall the setup.
2.1. G -Hilb(C) 3 as a toric variety. We refer the reader to Section 2 of [Cra05] and [CR02] for a more comprehensive and detailed exposition.
Let G ⊂ SL 3 (C) be a finite abelian subgroup of order r = |G|, and ε = e 2πi/r a primitive root of unity. We diagonalize the action of G and denote the corresponding coordinates on C 3 by x, y and z. The lattice of exponents of Laurent monomials in x, y, z will be denoted by L = Z 3 and the dual lattice by L ∨ . Associate a vector v g = 1 r (γ 1 , γ 2 , γ 3 ) to each group element g = diag(ε γ 1 , ε γ 2 , ε γ 3 ), define the lattice N := L ∨ + Definition 2.1. The junior simplex △ ⊂ N R is the triangle with vertices e x = (1, 0, 0), e y = (0, 1, 0) and e z = (0, 0, 1). It contains the lattice points 1 r (γ 1 , γ 2 , γ 3 ) with γ 1 + γ 2 + γ 3 = 1, γ i ≥ 0.
A subdivision of the cone σ gives rise to a fan Σ and, hence, a toric variety X Σ together with a birational map X Σ → X. A triangle inside the junior simplex is called basic in case the pyramid with this triangle as the base and origin as the apex has volume 1. If a fan Σ gives rise to a partition of the junior simplex into basic triangles, then the corresponding map X Σ → X is a crepant resolution of singularities. Notice that such a fan Σ is uniquely determined by the associated triangulation of the junior simplex into basic triangles (slightly abusing notation we will refer to such a triangulation as Σ as well).
Definition 2.2. A G-cluster is a G-invariant zero-dimensional subscheme Z ⊂ C n for which H 0 (O Z ) is isomorphic to the regular representation of G as a C[G]-module. The G -Hilbert scheme is the variety Y = G -Hilb(C n ), which is the fine moduli space parameterizing G-clusters.
The G -Hilbert scheme is a toric variety and for G ⊂ SL 2 (C) or SL 3 (C) the map Y → X is a crepant resolution of singularities. The partition of the junior simplex into basic triangles for finite abelian G ⊂ SL 3 (C), giving rise to the fan of Y, can be computed according to the 3-step procedure below (see [Nak01] and [Cra05]). Prior to giving the algorithm a definition is due.
Definition 2.3. Let r and a be coprime positive integers with r > a. The Hirzebruch-Jung continued fraction of r a is the expression
r a = a 1 − 1 a 2 − 1 a 3 − . . . .
We will also refer to the collection of numbers [a 1 : a 2 : . . . : a k ] as the Hirzebruch-Jung sequence of r a and denote it by HJ(r, a).
Remark 2.4. Each number a i in a Hirzebruch-Jung sequence [a 1 : a 2 : . . . : a k ] is greater than or equal to 2.
Example 2.5. Let r ≥ 2 be an integer. The Hirzebruch-Jung continued fraction for (r, r − 1) is
r r − 1 = 2 − 1 2 − 1 2 − . . . − 1 2
with the corresponding sequence HJ(r, r − 1) = [2 : 2 : . . . : 2] consisting of an (r − 1)-tuple of twos.
Next we recall an algorithm for producing the partition of △ into basic triangles corresponding to Y.
Step 1. Draw line segments connecting the vertices of △ to lattice points on the boundary of the convex hull of △ \ {e x , e y , e z } in such a way that the line segments do not cross the interior of Conv(△ \ {e x , e y , e z }). Let the Hirzebruch-Jung sequence at a vertex ζ ∈ {e x , e y , e z } be HJ ζ = [k 1 : k 2 : . . . : k s ]. There will be s + 2 line segments L ζ 0 , L ζ 1 , . . . , L ζ s , L ζ s+1 emanating from ζ with L ζ 0 and L ζ s+1 being parts of edges between ζ and the remaining two vertices of △. Moreover, L ζ i+1 = k i L ζ i − L ζ i−1 and we label the i th segment with k i (the edges L ζ 0 and L ζ s+1 on the boundary of △ receive no label).
Step 2. Extend the lines until they are 'defeated' according to the following rule: when lines meet at a point, the line with greatest label extends and its label value is reduced by 1 for every 'rival' it defeats; lines meeting with equal labels all terminate at that point.
Step 3. Draw k−1 lines parallel to the sides of each regular triangle of side k (lattice triangle with k + 1 lattice points on each edge) to produce its regular tesselation into k 2 basic triangles.
Example 2.6. Let G = Z/15Z ⊂ SL 3 (C) with v 1 = 1 15
(1, 2, 12). The exceptional divisors are (the rays corresponding to) the vectors with endpoints v k = 1 15 (k 15 , 2k 15 , 12k 15 ) ∈ △, i.e. k 15 + 2k 15 + 12k 15 = 15 (we use the notation s 15 for s modulo 15):
E 1 = 1 15
(1, 2, 12), E 2 = 1 15 (2, 4, 9)
E 3 = 1 15 (3, 6, 6), E 4 = 1 15 (4, 8, 3) E 5 = 1 15 (5, 10, 0), E 6 = 1 15 (8, 1, 6) E 7 = 1 15 (9, 3, 3), E 8 = 1 15 (10, 5, 0).
Step 1. We compute the Hirzebruch-Jung sequences:
1 15 (2, 12) ∼ 1 15 (1, 6) ∼ 1 5 (1, 2), 5 2 = 3 − 1 2 , HJ x = HJ(15, 2) = [3 : 2]; 1 15 (12, 1) ∼ 1 5 (4, 1) ∼ 1 5 (1, 4), 5 4 = 2 − 1 2 − 1 2 − 1 2
, HJ y = HJ(5, 4) = [2 : 2 : 2 : 2];
15 2 = 8 − 1 2 , HJ z = HJ(15, 2) = [8 : 2] and draw • • • • • E 1 E 2 E 3 E 4 E 5 • • • E 6 E 7 E 8 8 2 3 2 2 2 2 2 e z e y e x FIGURE 1.
Step 1 of the triangulation algorithm for G = 1 15
(1, 2, 12)
Step 2. Extending the lines according to their labels. Step 2 of the triangulation algorithm for G = 1 15
• • • • • E 1 E 2 E 3 E 4 E 5 • • • E 6 E 7 E
(1, 2, 12)
Step 3. Subdividing regular triangles into basic. (1, 2, 12) 2.2. Reid's recipe. Reid's recipe (see [Rei02] and [Cra05]) is an algorithm to construct the cohomological version of the McKay correspondence for abelian subgroups of SL 3 (C). It starts with marking internal edges and vertices of the triangulation Σ corresponding to G -Hilb(C 3 ) with characters of G. For the purposes of this paper only the marking of edges will be required, which we recall below and refer the reader to Section 3 of [Cra05] for details. An edge (e, f) in Σ is labeled by a character of G according to the following rule.
• • • • • E 1 E 2 E 3 E 4 E 5 • • • E 6 E 7 E
The one-dimensional ray in M perpendicular to the hyperplane e, f in M R has a primitive generator given by exponents of m 1 m 2 , where m 1 and m 2 are coprime regular monomials. As
M is the lattice of G-invariant Laurent monomials, m 1 and m 2 have the same character χ with respect to G-action. The edge (e, f) is marked with character χ.
Example 2.7. We determine the character that marks the edge (e z , E 1 ):
c = 0 a + 2b + 12c = 0 ⇔ a = −2b, c = 0,
hence, m 1 = x 2 and m 2 = y with χ = χ 2 . The computation below shows that the edge (E 6 , E 7 ) is marked by χ = χ 1 :
8a + b + 6c = 0 9a + 3b + 3c = 0 ⇔ 3a + b + c = 0 a + c = 0 ⇔ a = −c, b = 2c,
thus, m 1 = x and m 2 = y 2 z.
• • • • • E 1 E 2 E 3 E 4 E 5 • • • E 6 E 7 E 8 χ 2 χ 2 χ 2 χ 2 χ 2 χ 1 χ 1 χ 1 χ 3 χ 3 χ 3 χ 12 χ 9 χ 9
χ 10 χ 12 (1, 2, 12)
χ 6 χ 4 χ 6 χ 12 e z e y e x
2.3. Results for abelian subgroups G ⊂ SL 3 (C). Let G ⊂ SL n (C) be a finite subgroup.
The following result appeared in the celebrated paper of Bridgeland, King and Reid (see [BKR01]).
Theorem 2.8. Let G ⊂ SL n C be a finite subgroup with n ≤ 3.
(1) The variety G -Hilb(C n ) is irreducible and the resolution Y → X is crepant.
(2) The map Ψ :
D b G (C n ) → D b (Y)
is an exact equivalence of triangulated categories. Let χ be an irreducible representation of G. There are two natural G-equivariant sheaves on C n associated to χ:
• χ := χ ⊗ O C n • χ ! := χ ⊗ O 0 , where O 0 = O C n /m 0 = C[x 1 , x 2 , . . . , x n ]/(x 1 , x 2 , . . . , x n ) is the structure sheaf of the origin in C n .
The image Ψ( χ⊗O C n ) admits a straightforward description. It is isomorphic to L ∨ χ , where L χ is the corresponding tautological vector bundle. The tautological vector bundles on Y are defined as direct summands in the decomposition
p * (O Z ) = L χ ⊗ χ, with respect to the trivial G-action on Y (here Z ⊂ Y × C n is the universal subscheme and p the projection on Y).
However, the images of skyscraper sheaves χ ! are more complicated to describe. In case of abelian G the main result of [CCL17] provides such a description.
Theorem 2.9. Let G ⊂ SL 3 (C) be a finite abelian subgroup and let χ be an irreducible representation of G. Then H i (χ ! ) = 0 unless i ∈ {0, −1, −2}. Moreover, one of the following holds:
Reid's recipe H −2 (Ψ(χ ! )) H −1 (Ψ(χ ! )) H 0 (Ψ(χ ! )) χ marks a single divisor E 0 0 L −1 χ ⊗ O E χ marks a single curve C 0 0 L −1 χ ⊗ O C χ marks a chain of divisors starting at E and terminating at F 0 L −1 χ (−E − F) ⊗ O Z 0
χ marks three chains of divisors, starting at E x , E y and E z and meeting at a divisor P 0 ) is an abstract complex in D b (Y). It follows from Theorem 2.9 that for every nontrivial character χ the object Ψ(χ ! ) is a pure sheaf (i.e. some shift of a coherent sheaf).
L −1 χ (−E x − E y − E z ) ⊗ O V Z 0 χ = χ 0 w ZF 2 w ZF 1 (ZF 2 ) 0
QUIVERS
The next ingredient that we need is McKay quivers. A good reference for basic concepts of quivers and representations thereof is the book [DW17]. We also invite the reader to look in the paper [IN00] for the exposition on realizations of G -Hilbert schemes as moduli spaces of McKay quiver representations.
Generalities.
Definition 3.1. A quiver Q = (Q 0 , Q 1 ) is a finite directed graph with finitely many vertices enumerated by the set Q 0 and finitely many edges indexed by Q 1 . Each edge is uniquely determined by the pair of vertices it connects, which we will denote by t(a) and h(a) standing for 'tail' and 'head', respectively.
Definition 3.2. A path p in a quiver Q = (Q 0 , Q 1 ) is a sequence a ℓ a ℓ−1 . . . a 1 of arrows in Q 1 such that t(a i+1 ) = h(a i ) for i = 1, 2, . . . , ℓ − 1. In addition, for every vertex x ∈ Q 0 we introduce a path e x .
The path algebra P Q is a C-algebra with a basis labeled by all paths in Q. The multiplication in P Q is given by
p · q := pq, if t(p) = h(q) 0, otherwise,
where pq stands for the concatenation of paths subject to the conventions that pe x = p if t(p) = x, and e x p = p if h(p) = x.
Remark 3.3. Notice that P Q is of finite dimension over C if and only if Q has no oriented cycles. The path algebra has a natural grading by path length with the subring of grade zero spanned by the trivial paths e x for x ∈ Q 0 . It is a semisimple ring, in which the elements e x are orthogonal idempotents.
Definition 3.4. A representation of a quiver Q consists of a collection of vector spaces {V i } i∈Q 0 and linear homomorphisms α a ∈ Hom C (V ta , V ha ) for each arrow a ∈ Q 1 . Such representations form a category with morphisms being collections of C-linear maps ψ i : V i → W i for all i ∈ Q 0 such that the diagrams
V ta V ha W ta W ha αa ψta ψ ha α ′ a
commute. This category will be denoted by Rep C (Q).
Theorem 3.5. The category Rep C (Q) is equivalent to the category of finitely-generated left P Q -modules. In particular, Rep C (Q) is an abelian category.
Often, the algebra of interest is not the path algebra of a quiver Q, its quotient by an ideal of relations. A relation in P Q is a C-linear combination of paths of length at least two, each with the same head and the same tail. A quiver with relations is a quiver Q together with a finite set of relations R. A representation of such a quiver is a representation of Q where any composition of maps indexed by subsequent edges in a relation vanishes (is a zero map). Any finite set of relations R in Q determines a two-sided ideal I R ⊂ P Q . As before, finite-dimensional representations of (Q, R) form a category Rep C (Q, R). Moreover, the analogue of Theorem 3.5 holds true, i.e. there is an equivalence of categories (1)
Rep C (Q, R) ≃ (P Q /I R )mod.
McKay Quivers. Let G ⊂ SL(V) be a finite subgroup.
Definition 3.6. The McKay quiver Q(G, C n ) is the quiver whose vertices are enumerated by irreducible representations of G with dim(Hom G (χ k ⊗ C n , χ ℓ )) arrows (possibly none) from vertex k to vertex ℓ. Henceforth we assume that G is abelian. Then all irreducible representations of G are one-dimensional and correspond to characters of G:
char(G) := {χ : G → C * }.
In particular, as a representation of G, we have C n = n i=1 Cχ i =:
n i=1
Ce i and let x 1 , x 2 , . . . , x n ∈ (C n ) * be the dual basis to {e 1 , e 2 , . . . , e n } with R = C[x 1 , x 2 , . . . , x n ] the coordinate ring of C n . The chain of isomorphisms Hom
G (χ k ⊗ C n , χ ℓ ) ≃ Hom G (χ k ⊗ n i=1 Ce i , χ ℓ ) ≃ n i=1
Hom G (χ k ⊗ Ce i , χ ℓ ) provides a natural identification of the maps assigned to the arrows in the McKay quiver Q(G, C n ) with multiplication by x i 's and, hence, impose the relations corresponding to the commutation of the latter:
I := a χ⊗χ j i a χ j − a χ⊗χ i j a χ i | χ ∈ char(G), 1 ≤ i, j ≤ n ⊂ CQ(G, C n )
(for every vertex χ ∈ Q 0 there are n arrows with head at χ denoted by a χ k : χ ⊗ χ k → χ and label the arrow a χ k by the monomial x k ).
Example 3.7. Let G = Z/nZ be embedded in SL(C 2 ) via mapping 1 to ε 0 0 ε −1 with ε = e 2πi/n the primitive n th root of unity. There are n irreducible representations of G, one-dimensional, to be denoted by χ 0 , χ 1 , . . . , χ n−1 and C 2 ≃ χ 1 ⊕ χ n−1 . We label x k := a χ k χ 1 and y k := a χ k χ n−1 , then the ideal of relations is I = x i y i − x i+1 y i+1 . The following Proposition will play an essential role for the construction in Section 5 (see Lemma 7.5 and its proof in [Cra08]).
Proposition 3.8. Let G ⊂ GL(C n ) be a finite subgroup. There exists a set of relations R in the McKay quiver Q(G, C n ) such that the categories Rep C (Q(G, C n ), R) and Coh G (C n ) are equivalent.
HALL ALGEBRAS
We will use two versions of the Hall algebra construction. The first variant appeared in [KV00] and was described in detail in [Joy07]. A very good overview of Ringel-Hall algebras over finite fields can be found in [Sch12].
Let C be a C-linear abelian finitary category (the latter means that all extension groups between any two objects in C are finite dimensional). 4.1. Euler characteristic Hall algebra. The set of isomorphism classes of objects in C will be denoted by C iso . The space of functions Fun(C iso ) can be made into an associative algebra H(C), called the Hall algebra of C. Let G C be the stack formed by pairs (A, B) of objects of C, where A is a subobject of B, and morphisms of such pairs. There are three morphisms p 1 , p 2 , p 3 : G C → C iso associating to the pair (A, B) the objects A, B and B/A, respectively. The fibers of p 2 are algebraic varieties. The multiplication on H(C) is given by
f ⋆ g := p 2 * (p * 1 (f)p * 3 (g)).
Let A ∈ C be an object and [C] ∈ H(C) the characteristic function of A. The multiplicity
of [C] in [A] ⋆ [B] is χ(G C AB ), where G C AB = {A ′ ⊆ C | A ′ ≃ A, C/A ′ ≃ B}
and χ stands for Euler characteristic with compact support.
The following proposition appearing in [KV00] (see Section 3.1) is a consequence of the fact that the heart of a triangulated category is stable under extensions. It will be an essential ingredient of the main construction in this paper. Corollary 4.2. Let C 1 , C 2 be two finitary abelian categories, and ϕ : D b (C 1 ) → D b (C 2 ) an equivalence of triangulated categories. If A 1 , A 2 , . . . A n ∈ C 1 are such that ϕ(A 1 ), ϕ(A 2 ), . . . , ϕ(A n ) ∈ D b (C 2 ) have cohomology concentrated in a single degree and that degree is the same for all A i , then ϕ induces an isomorphism of Hall algebras H A 1 , A 2 , . . . , A n ≃ H ϕ(A 1 ), ϕ(A 2 ), . . . , ϕ(A n ) .
Ringel-Hall algebras over finite fields.
Assume that C is a finitary abelian category, such that
• gldim(C) < ∞;
• |Ext i (A, B)| < ∞ for any two objects A, B ∈ Ob(C) and all i ≥ 0. Let Q be a quiver without oriented cycles, Rep k (Q) the category of representations of Q over a finite field k and U q (g) the quantized enveloping algebra. Here g is the Lie algebra associated to the Dynkin diagram formed by Q. We denote the simple roots of g by E i and simple representations of Q by {S i } i∈Q 0 .
The following result was obtained by Ringel and Green (see Theorem 3.15 in [Sch12]).
Theorem 4.5. Let v = |k|. There is an embedding of algebras ϕ :
U v (n − ) ֒→ H fin (Rep k (Q)) with ϕ(E i ) = [S i ].
MAIN RESULT
In this section we will combine the results reviewed in Sections 2 − 4 to formulate and establish the main theorem of this paper. A few examples prior to giving the general statement will be helpful. (1, 1, 4) and G = 1 7
(1, 1, 5)
The data on the images of skyscraper sheaves under Ψ appearing in Table 2 is obtained via a direct application of Theorem 2.9. Notice that the McKay quiver Q(G, C 3 ) contains a Q ′ = A 1 subquiver supported on the vertices enumerated by characters χ 1 and χ 2 : (1, 1, 4) and G = 1 7
1 2. χ H −1 (Ψ(χ ! )) χ 1 L −1 χ 1 (−E z3 ) ⊗ O E 12 χ 2 L −1 χ 2 (−E xy ) ⊗ O E 2 χ 4 L −1 χ 4 (−E xy ) ⊗ O E 1 χ H −1 (Ψ(χ ! )) χ 1 L −1 χ 1 (−E xyz ) ⊗ V E 3 χ 3 L −1 χ 3 (−E xy ) ⊗ O E 2 χ 5 L −1 χ 5 (−E xy ) ⊗ O E 1
(1, 1, 5)
Example 5.2. We continue with Example 2.6. Recall that G = Z/15Z with ν 1 = 1 15
(1, 2, 12). See Figure 4 for the partition of junior simplex into basic triangles corresponding to G -Hilb(C 3 ) and marking of edges with characters. This time the McKay quiver Q(G, C 3 ) contains a Q ′ = A 2 subquiver supported on the vertices enumerated by characters χ 1 , χ 2 and χ 3 : (1, 2, 12)
χ H −1 (Ψ(χ ! )) χ 1 L −1 χ 1 (−E z5 ) ⊗ O E 1234 χ 2 L −1 χ 2 (−E z8 ) ⊗ O E 67 χ 3 L −1 χ 3 (−E xy ) ⊗ O E 47 1 2 3.
Theorem 5.3. Let r = k + s + 1 with s ≡ 0 (mod k) and s ≡ 0 (mod k + 1). Set t := r k + 1 and consider the sequence γ 0,t = 1 γ n,t := nt − n(n − 1).
(1) If a character χ c appears marking an edge, then c satisfies at least one of the following conditions:
• 1 ≤ c ≤ k + 1 • c is divisible by k • c is divisible by k + 1 (2)
The images of the k+1 skyscraper sheaves corresponding to the first k+1 nontrivial characters of G are as presented below. Proof. We follow Reid's recipe (see Section 2). Let ℓ = s k and compute the Hirzebruch-Jung fractions and sequences at vertices of △ (see Example 2.5): (we have used that t (t − 1)/k = k + 1 − (t − 1)/k − 1 (t − 1)/k to evaluate HJ X ).
χ H −1 (Ψ(χ ! )) χ 1 L −1 χ 1 (−E zγ k,t ) ⊗ O E γ k−1,t +2γ k−1,t +3...γ k,t −1 χ 2 L −1 χ 2 (−E zγ k−1,t ) ⊗ O E γ k−2,t +2γ k−2,t +3...γ k−2,t −1 . . . . . . χ k L −1 χ k (−E zγ 1,t ) ⊗ O E 123...γ 1,t −1 χ k+1 L −1 χ k+1 (−E xy ) ⊗ O E γ 1,t −1γ 2,t −1...γ k,t −1r k = ℓ + 2 − k − 1 k , HJ z = [ℓ + 2 : 2 : 2 : . . . : 2 k−1 ]; 1 r (s, 1) = 1 r/(k + 1) s k + 1 , 1 = 1 t (t − 1, 1) = 1 t (1, t − 1
As HJ x = [k + 1 : 2 : 2 : . . . : 2
(t−1)/k−1 ]
, the line segment connecting e x and E γ k−1,t−1 +1 (or E 13 on Figure 7) continues until E 1 . As ℓ + 2 = s/k + 2 > 2 = k + 1 − (k − 1) and s/k + 2 − t/k − (t − 1) = 3 > 2, it follows that each of the line segments emerging from e y and each of the line segments emanating from e z except L ez 1 does not continue beyond the first lattice point on it (see Step 2 of Algorithm 2.1). On the other hand L ez 1 defeats all the line segments on its way to E t+1 , which belongs to the edge [e x , e y ] of △. The remaining edges of triangulation of △ (see Step 3 of the algorithm) come in three families of parallel lines with slopes equal to those of
• [e z , E 1 ] • [e x , e y ]
• [e x , E (t+1)/2 ]. Next we verify that the edges of triangulation of △ are marked as in the statement of the theorem. Let 1 ≤ i ≤ k and consider the edge (e z , E γ k−i ,t+1 ). Denote α i := 1 + (r − 1)(i − 1) k , then c = 0 α i a + (k − i + 1)b = 0, hence, m 1 = x k−i and m 2 = y α i with χ = χ k−i+1 (the solution is (a, b, c) = (k − i + 1, −α i , 0)).
For each 1 ≤ i ≤ k the vertices E γ k−i ,t+1 , E γ k−i ,t+2 , . . . , E γ k−i+1 ,t lie on a line ℓ i , moreover, all these lines are parallel and have direction vector v = (1, k, r − k − 1) (see Figure 7). As the solution (a, b, c) = (k − i + 1, −α i , 0) satisfies the equation v · (a, b, c) ≡ 0 (mod r), it follows that all intervals on the line ℓ i are labeled by the character χ k−i+1 . Now we check that the edge (e y , E t−1 ) is marked with χ k+1 . As E t−1 = 1 r (t − 1, r − k − t, k + 1), we get the system of equations b = 0 (t − 1)a + (k + 1)c = 0, hence, m 1 = x k+1 and m 2 = z t−1 with χ = χ k+1 (the solution is (a, b, c) = (k + 1, 0, 1 − t)). The vertices E t−1 , E γ 2,t −1 , . . . , E γ k,t −1 lie on the same line, with the vector connecting any two subsequent vertices being (t, −t, 0). As (k + 1)t = (k + 1) r k + 1 = r, we (inductively) get that the solution (a, b, c) = (k + 1, 0, 1 − t) satisfies the equations imposed by E γg,t−1 and E γ g+1,t −1 , so each edge (E γg,t−1 , E γ g+1,t −1 ) is marked by χ k+1 . A straightforward computation shows that the edge (e x , E γ k,t ) is marked with χ k+1 as well. The remaining 'labeling assertions' can be checked similarly using the parallelism of corresponding lines. In particular, notice that
s→∞ |H 0 | r − 1 ≥ k − 1 k + 1 . In particular, lim s→∞ k→∞ |H 0 | r − 1 = 1.
Proof. It follows from (1) in Theorem 5.3 combined with Theorem 2.9 that Hence, lim
|H 0 | ≥ r − (k + 1) − r − 1 k − r − 1 k + 1 + r − 1 k(k + 1) = = sk(k + 1) − (k + 1)(s + k) − k(s + k) + s + k k(k + 1) = (k 2 − k)s − 2k 2 k(k + 1) = (k − 1)s − 2k k + 1 .s→∞ |H 0 | r − 1 ≥ lim s→∞ (k − 1)s − 2k (k + 1)(k + s) = k − 1 k + 1 .
Remark 5.5. For s ≫ k ≫ 0 the cohomology of Ψ(χ ! ) tend to concentrate in degree 0.
Theorem 5.6. Let r = k + s + 1 with k and s satisfying the assumptions of Theorem 5.3 and s ≫ 0.
(1) Let n − ⊂ sl k+1 (C) be a nilpotent subalgebra and Q ′ ⊂ Q(G, C 3 ) the subquiver supported on vertices Q ′ 0 = {1, 2, . . . , k + 1}. There is an isomorphism of algebras
U(n − ) Θ → H {Ψ(χ ! i )} i∈Q ′ 0 . 1 2 3 4 ... k+1 FIGURE 8. Q ′ = A k ⊂ Q(G, C 3 )
(2) Assume, in addition, that k = 2q + 1 ≥ 5 is odd, 5 ≤ n ≤ k + 1 and α satisfies the properties:
• α ≡ q (mod k), α ≡ −2 (mod k + 1);
• α + k(n − 3) + k + 1 ≤ r − 1 ⇔ α ≤ s − k 2 + 3k − 1. Let n − ⊂ so 2n (C) with 5 ≤ n ≤ k+1 be a nilpotent subalgebra and Q ′ ⊂ Q(G, C 3 ) the subquiver supported on vertices Q ′ 0 = {α−k−1, α, α+1, α+k, α+2k, . . . , α+ k(n − 3), α + k(n − 3) − 1, α + k(n − 3) + k + 1}. There is an isomorphism of algebras U(n − ) Θ → H {Ψ(χ ! i )} i∈Q ′ 0 . α+1 α+k(n−3)−1 α α+k ... α+k(n−3) α−k−1 α+kn−2k+1 FIGURE 9. Q ′ = D n+1 ⊂ Q(G, C 3 )
(3) Assume that k > 8 and let g be the Lie algebra of type E 6 with n − ⊂ g a nilpotent subalgebra and Q ′ ⊂ Q(G, C 3 ) the subquiver supported on vertices Q ′ 0 = {k + 6, 2k + 6, 3k + 4, 3k + 5, 3k + 6, 4k + 7, 5k + 8}. There is an isomorphism of algebras
U(n − ) Θ → H {Ψ(χ ! i )} i∈Q ′ 0 . 3k+4 3k+5 3k+6 4k+7 5k+8 2k+6 k+6 FIGURE 10. Q ′ = E 6 ⊂ Q(G, C 3 )
(4) Assume that k > 9 and let g be the Lie algebra of type E 7 with n − ⊂ g a nilpotent subalgebra and Q ′ ⊂ Q(G, C 3 ) the subquiver supported on vertices Q ′ 0 = {k + 6, 2k + 3, 2k + 4, 2k + 5, 2k + 6, 3k + 7, 4k + 8, 5k + 9}. There is an isomorphism of algebras U(n − )
Θ → H {Ψ(χ ! i )} i∈Q ′ 0 .
2k+3 2k+4 2k+5 2k+6 3k+7 4k+8 5k+9 k+6 FIGURE 11. Q ′ = E 7 ⊂ Q(G, C 3 )
(5) Assume that k > 10 and let g be the Lie algebra of type E 8 with n − ⊂ g a nilpotent subalgebra and Q ′ ⊂ Q(G, C 3 ) the subquiver supported on vertices Q ′ 0 = {k + 5, 2k + 3, 2k + 4, 2k + 5, 3k + 6, 4k + 7, 5k + 8, 6k + 9, 7k + 10}. There is an isomorphism of algebras
U(n − ) Θ → H {Ψ(χ ! i )} i∈Q ′ 0 . 2k+3 2k+4 2k+5 3k+6 4k+7 5k+8 6k+9 7k+10 k+5 FIGURE 12. Q ′ = E 8 ⊂ Q(G, C 3 )
Proof. We present an argument for (1) with the verification of (2) − (5) being completely analogous. The McKay quiver Q(G, C 3 ) contains an A k subquiver Q ′ supported on the vertices 1, 2, . . . , k + 1. Notice that Q ′ has no oriented cycles, hence, U(n − ) is isomorphic to the composition subalgebra of H(Rep(Q ′ )) (subalgebra generated by characteristic functions of simple representations), see Example 4.25 in [Joy07]. Then subsequent application of (2), Theorem 2.8 and Proposition 3.8 together with Theorem 5.3 and Corollary 4.2 gives rise to the proposed isomorphism, see diagram on Figure 13 with embeddings, isomorphisms and correspondences below for a schematic summary.
Remark 5.7. Let G = Z/rZ ֒→ SL 3 (C) be a finite abelian subgroup with ν 1 = 1 r (1, k, s) and 1 + k + s = r. The McKay quiver Q(G, C 3 ) can not contain a D 4 subquiver. This is due to the fact that D 4 has a vertex of valency 4, which implies the existence of an oriented 3-cycle, supported on these vertex and 2 of the vertices connected to it. Indeed, let the vertex of valency 4 correspond to the character (irreducible representation) χ i . Then, inevitably, there are two vertices indexed by χ i−a and χ i+b with a = b ∈ {1, k, s} that are included in the subgraph as well. Therefore, the subgraph contains an oriented 3-cycle supported on the vertices enumerated by χ i , χ i−a and χ i+b .
D b (Coh G (C n )) D b (Coh(Y)) Rep C (Q(G, C n ), R) Coh G (C n ) Rep C (Q ′ ) Coh G,Q ′ 0 (C n ) U(n − ) H(Rep C (Q ′ )) H(Coh G,Q ′ 0 (C n )) H {Ψ(χ ! i )} i∈Q ′ 0 ≃ ≃ ≃ ≃ Ψ Θ FIGURE 13
. Diagrammatic overview of the construction Conjecture 5.8. Let k be a finite field and char(k) coprime with the order of G. Under the assumptions (1) − (5) of Theorem 5.6 one has the corresponding isomorphisms
U v (n − ) Θ fin → H fin {Ψ(χ ! i )} i∈Q ′ 0 with v = √ k.
Remark 5.9. We provide some facts in support of the conjecture. The Bridgeland-King-Reid equivalence ((2) in Theorem 2.8) is known to hold in the above setup (see the comment after Conjecture 2.24 in [Rou06]). Another required modification to the proof of Theorem 5.6 is to use Theorem 4.5 in place of the isomorphism from Example 4.25 in [Joy07]. The analogue of Corollary 4.2 holds true as well.
g∈G Z · v g (with N R = N ⊗ Z R) and use M := Hom(N, Z) for the dual lattice of G-invariant Laurent monomials. The categorical quotient X = Spec C[x, y, z] G is the toric variety Spec C[σ ∨ ∩ M] with the cone σ being the positive octant σ = R ≥0 e i ⊂ N R .
FIGURE 2. Step 2 of the triangulation algorithm for G = 1 15 (1, 2, 12)
FIGURE 3 .
3Step 3 of the triangulation algorithm for G = 1 15
FIGURE 4 .
4Σ fan and character marking for G = 1 15
FIGURE 5 .
5McKay quiver Q(Z/nZ, C 2 ).
Proposition 4 . 1 .
41Let C 1 , C 2 be two finitary abelian categories, and ϕ : D b (C 1 ) → D b (C 2 ) an equivalence of triangulated categories. If A, B, C ∈ C 1 are such that ϕ(A), ϕ(B) ∈ C 2 with G C AB = ∅, then ϕ(C) ∈ C 2 and ϕ is an isomorphism of complex varieties G
Definition 4. 3 .
3The multiplicative Euler form ·, · : K(C × C) → C is the form given byA, B := ( ∞ i=0 |Ext i (A, B)| (−1) i ) 1/2 . Let P C A,B be the number of short exact sequences 0 → B → C → A → 0 and P C A,B := P C A,B | End(A)|| End(B)| . Consider the vector space H fin (Remark 4.4. The unit i : C → H fin (C) is given by i(λ) = λ[0], where 0 is the initial object of C.
Example 5 . 1 .Figure 6 FIGURE 6 .
5166Consider the groups G = Z/6Z and G = Z/7Z with ν The partitions of junior simplices into basic triangles corresponding to G -Hilb(C 3 ) and G -Hilb(C 3 ) are presented on Σ fans and character labels for G = 1 6
), HJ y = [2 : 2 : . . . x = [k + 1 : 2 : 2 : . . .
• a move by one unit to the right in the family of lines with slope equal to the one of[e z , E 1 ] results in decrease of the corresponding label by k; • a move by one unit to the right in the family of lines with slope equal to the one of [e x , e y ] results in decrease of the corresponding label by k + 1; • a move by one unit to the right in the family of lines with slope equal to the one of [e x , E (t+1)/2 ] results in decrease of the corresponding label by k to the left and by k + 1 to the right of the line segment [e x , E (t+1)/2 ]. Corollary 5.4. Let k and s satisfy the assumptions of Theorem 5.3 with k fixed. Denote the set of characters χ with H 0 (Ψ(χ ! )) = 0 by H 0 . Then lim
FIGURE 7 .
7Σ fans and character marking for G = 1 28(1, 3, 24)
TABLE 1 .
1Observed valuesRemark 2.10. Apriori, each object Ψ(χ !
TABLE 2 .
2Images of χ !
i 's under Ψ for G =
1
6
TABLE 3 .
3Images of χ !
1,2,3 under Ψ for G =
1
15
TABLE 4 .
4Images of χ !
1,2,...,k+1 under Ψ for G =
1
r
(1, k, s)
McKay equivalence for symplectic resolutions of quotient singularities. R Bezrukavnikov, D Kaledin, Algebr. Geom. Metody, Svyazi i Prilozh. 246Tr. Mat. Inst. Steklova. Russian, with Russian summaryR. Bezrukavnikov and D. Kaledin, McKay equivalence for symplectic resolutions of quotient singu- larities, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 20-42 (Russian, with Russian summary);
. English Transl, Proc. Steklov Inst. Math. 3246English transl., Proc. Steklov Inst. Math. 3(246) (2004), 13-33. ↑1
The McKay correspondence as an equivalence of derived categories. T Bridgeland, A King, M Reid, J. Amer. Math. Soc. 143T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535-554. ↑1, 4, 8
A derived approach to geometric McKay correspondence in dimension three. S Cautis, T Logvinenko, J. Reine Angew. Math. 636S. Cautis and T. Logvinenko, A derived approach to geometric McKay correspondence in dimension three, J. Reine Angew. Math. 636 (2009), 193-236. ↑4
Derived Reid's recipe for abelian subgroups of SL 3 (C). S Cautis, A Craw, T Logvinenko, J. Reine Angew. Math. 7279S. Cautis, A. Craw, and T. Logvinenko, Derived Reid's recipe for abelian subgroups of SL 3 (C), J. Reine Angew. Math. 727 (2017), 1-48. ↑2, 4, 9
An explicit construction of the McKay correspondence for A-Hilb C 3. A Craw, J. Algebra. 2852A. Craw, An explicit construction of the McKay correspondence for A-Hilb C 3 , J. Algebra 285 (2005), no. 2, 682-705. ↑4, 5, 8
arXiv:0807.2191v1Quiver representations in toric geometry. 11, Quiver representations in toric geometry, arXiv:0807.2191v1 (2008). ↑11
Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient. A Craw, A Ishii, Duke Math. J. 12421A. Craw and A. Ishii, Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), no. 2, 259-307. ↑1
How to calculate A-Hilb C 3 , Geometry of toric varieties. A Craw, M Reid, Sémin. Congr. 6Soc. Math. FranceA Craw and M. Reid, How to calculate A-Hilb C 3 , Geometry of toric varieties, Sémin. Congr., vol. 6, Soc. Math. France, Paris, 2002, pp. 129-154. ↑4
An introduction to quiver representations. H Derksen, J Weyman, Graduate Studies in Mathematics. Providence, RIAmerican Mathematical Society1849H. Derksen and J. Weyman, An introduction to quiver representations, Graduate Studies in Mathe- matics, vol. 184, American Mathematical Society, Providence, RI, 2017. ↑9
McKay correspondence and Hilbert schemes in dimension three. Y Ito, H Nakajima, Topology. 3969Y. Ito and H. Nakajima, McKay correspondence and Hilbert schemes in dimension three, Topology 39 (2000), no. 6, 1155-1191. ↑9
Construction géométrique de la correspondance de McKay. G Gonzalez-Sprinberg, J.-L Verdier, Ann. Sci. École Norm. Sup. 44FrenchG. Gonzalez-Sprinberg and J.-L. Verdier, Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 409-449 (1984) (French). ↑4
Ringel-Hall algebras. D Joyce, Configurations in abelian categories. II. 21020D. Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635-706. ↑2, 12, 19, 20
Kleinian singularities, derived categories and Hall algebras. M Kapranov, E Vasserot, Math. Ann. 316312M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), no. 3, 565-576. ↑1, 2, 4, 12
Derived categories of toric varieties III. Y Kawamata, Eur. J. Math. 21Y. Kawamata, Derived categories of toric varieties III, Eur. J. Math. 2 (2016), no. 1, 196-207. ↑1
The Santa Cruz Conference on Finite Groups. J Mckay, Graphs, singularities, and finite groups. Santa CruzAmer. Math. Soc37Univ. CaliforniaPure Math.J. McKay, Graphs, singularities, and finite groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Prov- idence, R.I., 1980, pp. 183-186. ↑3
Hilbert schemes of abelian group orbits. I Nakamura, J. Algebraic Geom. 104I. Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001), no. 4, 757-779. ↑5
. M Reid, La Mckay, Séminaire Bourbaki. 2768AstérisqueM. Reid, La correspondance de McKay, Astérisque 276 (2002), 53-72. Séminaire Bourbaki, Vol. 1999/2000. ↑8
Hall algebras and quantum groups. C M Ringel, Invent. Math. 1013C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583-591. ↑
Derived equivalences and finite dimensional algebras. Raphael Rouquier, International Congress of Mathematicians. IIEur. Math. Soc.Raphael Rouquier, Derived equivalences and finite dimensional algebras, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 191-221. ↑20
Geometric methods in representation theory. O Schiffmann, Lectures on Hall algebras. ParisEnglish, with English and French summariesII13O. Schiffmann, Lectures on Hall algebras, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 1-141 (English, with English and French summaries). ↑12, 13
| []
|
[
"CAN HIGH ENERGY COSMIC RAYS BE VORTONS ?",
"CAN HIGH ENERGY COSMIC RAYS BE VORTONS ?"
]
| [
"Silvano Bonazzola ",
"Patrick Peter ",
"\nDépartement d'Astrophysique Relativiste et de Cosmologie\nObservatoire de Paris-Meudon\nUPR 176\n",
"\nCNRS\n92195MeudonFrance\n"
]
| [
"Département d'Astrophysique Relativiste et de Cosmologie\nObservatoire de Paris-Meudon\nUPR 176",
"CNRS\n92195MeudonFrance"
]
| []
| A simple model is exhibited in which the remnant density of charged vortons is used to provide candidates for explaining the observed ultra high energy cosmic rays (above 10 20 eV). These vortons would be accelerated in active galaxies and propagated through intergalactic medium with negligible losses of energy. The expected number density of observable events is shown to be consistent with extrapolation of the observations. The spectrum is predicted to be spatially isotropic while its shape is that of an atomic excitation-ionisation, i.e. with a few peaks followed by a continuum; there is also an energy threshold below which no vorton is visible. | 10.1016/s0927-6505(97)00015-7 | [
"https://arxiv.org/pdf/hep-ph/9701246v1.pdf"
]
| 118,977,557 | hep-ph/9701246 | 13275c29506ffe507219a8f251ab584994bce111 |
CAN HIGH ENERGY COSMIC RAYS BE VORTONS ?
arXiv:hep-ph/9701246v1 8 Jan 1997
Silvano Bonazzola
Patrick Peter
Département d'Astrophysique Relativiste et de Cosmologie
Observatoire de Paris-Meudon
UPR 176
CNRS
92195MeudonFrance
CAN HIGH ENERGY COSMIC RAYS BE VORTONS ?
arXiv:hep-ph/9701246v1 8 Jan 1997(March 26, 2021)numbers: 1110Lm1127+d9640-z9870Sa9870-f9880Cq
A simple model is exhibited in which the remnant density of charged vortons is used to provide candidates for explaining the observed ultra high energy cosmic rays (above 10 20 eV). These vortons would be accelerated in active galaxies and propagated through intergalactic medium with negligible losses of energy. The expected number density of observable events is shown to be consistent with extrapolation of the observations. The spectrum is predicted to be spatially isotropic while its shape is that of an atomic excitation-ionisation, i.e. with a few peaks followed by a continuum; there is also an energy threshold below which no vorton is visible.
INTRODUCTION
The problem of very high energy cosmic rays [1] is still an open one : most models based on reasonable astrophysical assumptions seem to indicate a likely maximum value for the energy of any kind of emitted particle at most of ∼ 10 19 eV [2] and the existence of the microwave background makes it impossible for a proton say to propagate with such energy on scales much larger than a few tens of Mpc ; this is the celebrated Greisen-Zatsepin-Kuz'min (GZK) cutoff [3]. An observation dated October 1991 by the Fly's Eye detector showed evidence for a cosmic ray at an energy of (3.2 ± 0.6)10 20 eV [1] and a few others reported showers above 10 20 eV [4] (and well above the GZK cutoff). There does not seem to date to be any completely satisfactory suggestion for explaining these observations (see however Ref. [5]), and it is the purpose of this paper to propose such a possible explanation that has been overlooked, namely that cosmic rays may consist of (as opposed to originate in) topological defects (as was first proposed in Ref. [6]).
A fairly common (neither confirmed nor excluded) prediction of particle physics models at energies beyond that of the electroweak scale is the existence of topological defects [7,8]. Among these, only cosmic strings seem to be consistent with cosmological constraints (with the possible exception of textures which, as non-localized objects, are not considered here). In particular, strings have been proposed as seeds for galaxy formation as well as sources for the microwave background fluctuations observed by COBE. This requires that the strings appear at the Grand Unified (GUT) scale, a scenario that may breakdown in the case suggested by Witten [9] in which they would be endowed with superconducting properties. In the latter case, string loops can form equilibrium configurations called vortons [10,11], whose stability is currently under investigation [12,13], that would easily overfill the Universe [10,11], thus provoking an unobserved cosmological catastrophe. It has been suggested, as a possible way out of this problem, that cosmic strings might have appeared at a much lower symmetry breaking scale, estimated in the range ∼ 10 4 − 10 9 GeV [11,14] (or at least that the formation of superconducting currents may have been postponed until this stage).
A second reason to consider such comparatively low energy cosmic strings instead of the more popular GUT strings is that the latter have been shown [15] to yield a negligible flux of cosmic ray types events as compared to what is currently observed, whatever the mechanism involved. This is mostly based on the idea that, a string being a topologically stable object, emission of very high energy particles must proceed via some removal of the topological stability, which for a GUT string implies either extremely highly energetic phenomena (and we're back to the previous problem of reaching the required high energies) or relatively rare situations (cusp evaporation, intercommutation and loop collapse). Adding up both cases, the expected flux turns out to be very tiny (roughly 10 −10 times that observed). Our aim here is to show that it is possible in principle to consider high energy rays as cosmic string loops themselves if the scale is not that of GUT but rather much below that scale. This, as we shall see, gives a flux that is comparable with observations.
The paper is organized as follows: in Section I, we recall the basic facts about vortons and how one may expect them to interact with ordinary particles. Then, Section II is devoted to the interaction of a vorton with a proton at rest in order to estimate the order of magnitude of the various phenomena involved in the detection of a 10 20 eV event, including the probability that a vorton interacts with the atmosphere. It is predicted that the typical expected spectrum form should resemble that of an excitation-ionisation spectrum, namely it should consist of lines followed by a continuum. Besides, since the interaction probability is found to be quite low (the cross-section is typical of neutrino-hadron interactions at the same energies [16]), we predict important horizontal showers. In Section III, we present a plausible acceleration mechanism for vortons, which turns out to be much simpler than most of the standard acceleration mechanisms for protons (Fermi-like mechanisms), without sharing their drawbacks. We then discuss propagation in Section IV where it is shown that no GZK cutoff is to be expected for vortons. It is then argued that this model implies that the most energetic events should be distributed isotropically. Finally we summarize our findings in the conclusion where we also compare the expected properties of these rays with the anticipated detection capabilities of the future Auger observatory [17].
I. THE PHYSICS OF VORTONS
The objects we shall now consider are vortons [10,11,14], namely loops of cosmic strings endowed with superconducting properties and stabilized by a current. They arise in symmetry breaking theories above the electroweak scale for which the vacuum manifold is not simply connected. In other words, if the high energy vacuum is invariant under the transformation of a symmetry group G and the low energy vacuum under that of the group H, the necessary and sufficient condition for the existence of cosmic strings is that the first homotopy group π 1 of the quotient G/H be nontrivial, π 1 (G/H) ∼ {0}. A typical example is the scheme U (1) → {0}, which is also the scheme at work in the Landau-Ginzburg model for superconductivity [18], and the strings are the corresponding vortices, much studied experimentally [19]. The vortices appear as infinite or in the form of closed loops which decay via emission of gravitational radiation. The symmetry breaking is achieved thanks to a Higgs field Φ which in turn can be coupled to other fields, particularly charged ones, which we shall generically write as Σ. Here we consider only the case where this charge carrier is a scalar particle, noting anyway that because of the formal equivalence between bosons and fermions in the two-dimensional worldsheet generated by the string, one expects similar behaviors for any kind of particle. Besides, as was shown to be sufficient in previous work [20], we shall only consider the coupling of Σ with the electromagnetic field as a perturbative effect not modifying much the overall dynamics of the vorton.
Let us here summarize the basic properties of vortons [14] as we expect to observe them. They can be characterized by essentially two integer numbers, namely a topological one, N say, which specifies the winding of the current carrier phase around the loop, and a dynamically conserved number, Z, related in the charge-coupled case (to which the present analysis is restricted) with the total amount of electric charge Q that it holds through Q = Ze, with e the electron electric charge. Note that both N and Z can be positive or negative, although for N it is just a matter of convention, while in the case of Z, the sign is important when there is an external interaction (as in the case of acceleration for instance).
Both these numbers are conserved at a classical level and one expects them to be of the same order of magnitude Z ≃ N ∼ 100 [14]. From these, on constructs the loop angular momentum as J = ZN ∼ 10 4 : although a bit hypothetical and thus often seen as "exotic", vortons can actually be considered as classical objects.
Since it can be shown that electromagnetic selfcoupling is mostly negligible [21], one can evaluate the total mass-energy of the vorton simply in terms of its characteristic radius r V , its energy per unit length U ∼ m 2 and tension T ∼ m 2 as [22]
M V = 2πr V (U + T ) ∼ m 2 r V ,(1)
with m the energy scale at which the strings are formed (essentially the string forming Higgs mass). Moreover, we shall assume in what follows that the current carrier mass scale m σ is also of order m so that we can keep only one energy scale. Note however that the following analysis can be easily generalized for two different mass scales since in practice it is the current scale that is relevant in most calculations. Then, knowing the angular momentum to be given by [22] J 2 = U T r V /2π permits to calculate the characteristic vorton circumference as
2πr V = (2π) 1/2 |N Z| 1/2 (U T ) −1/4 ∼ Z/m(2)
with a corresponding mass
M V ∼ Zm.(3)
Moreover, the mass scale is constrained: depending on various assumptions about the string network evolution and the rate of loop formation as well as the probability that arbitrarily shaped loops end up in vorton states, it can be shown that, if the vortons are stable [12,13], then in order to avoid a cosmological mass excess (Ω V < 1), the condition m < 10 9 GeV,
must be satisfied, Having discussed the basic properties of vortons, let us now turn to a rough evaluation of the typical mass scale expected for m if those vortons were to be seen as cosmic rays, developing air showers. As we shall see in the following section, interaction between a vorton and whatever other particle occurs mainly via inelastic scattering resulting in the extraction of a trapped Σ particle. In other words, the current flowing along the string loop can be seen as a bunch of bound states which can be excited provided the interaction energy is large enough. We therefore conclude at the existence of an energy threshold above which the typical expected spectrum should change qualitatively. Besides, and we now come to the second firm prediction of the model, since those particles form bound states, we expect them to show up in the form of a line spectrum, bound states being always quantized; this will be shown on a specific vorton model in section II.
For the time being, what really matters is the existence of bound state energies the order of magnitude of which we shall now attempt to evaluate.
Let ∆E be the variation of energy between two energy levels in the vorton, calculated in its rest frame, and γ its Lorentz factor in the rest frame of the particle it interacts with (recall we are at the end interested in air showers occurring in the atmosphere where the particles interacting with the vorton, namely essentially quarks composing protons and neutrons, are supposed to be at rest). The energy ǫ at which the interaction is then seen is obtained by transforming back to the particle's rest frame,
ǫ ∼ γ∆E.(5)
Denoting by m the particle's mass and requiring the interaction to actually take place gives
γ m ∼ ∆E,(6)
so that altogether, Eqs. (5) and (6) imply
γ 2 ∼ ǫ m ,(7)
from which the characteristic cosmic string scale can be deduced in the following way. From the fact that the angular momentum of the vorton is J ∼ Z 2 , it is seen that the density of states scales like Z −2 . Therefore, ∆E ∼ m/Z 2 , so that using (6) and (7) yields
m ∼ Z 2 √ ǫ m.(8)
We shall now use these relations together with the observations that have been realized on cosmic rays to normalize the energy levels.. Let us apply this evaluation to ultra high energy cosmic rays [1] so that the characteristic observed energy is normalized to ǫ ∼ 10 20 eV. The particles vortons interact with are essentially quarks composing hadrons, so the mass m can be taken to be that of the proton (at this level of approximation, the mass difference between a quark and a proton is negligible). With these numbers in mind, Eq. (7) transforms to γ 2 ∼ 10 11 and the cosmic string scale (8) m ∼ 10 9 GeV which, surprisingly enough, turns out to be right at the closure limit given by Eq. (4). It now remains to calculate the effective crosssection between a vorton and a quark in order to know the expected flux and check that it is compatible with the current observational limits.
II. CROSS-SECTION VORTON-HADRON
There is no privileged model for cosmic strings, and therefore neither is there for describing vortons. Thus, we can at best evaluate rough orders of magnitude for the cross section we are looking for. There are however various levels of approximation at which a vorton can be described. As mentioned earlier, the characteristic length scale associated with a vorton configuration is expected to be a hundred times larger than its thickness, so the most obvious description of a vorton is that of a classical circle. This is unfortunately of absolutely no use when one is looking for the trapped Σ particles along the loop. Hence, as a second level of approximation, we shall consider a vorton to be a torus like configuration in which the field Σ feels a confining potential. This is beyond the scope of the present calculation and is left for further investigation [23]. So let us describe a vorton as a sphere of radius R where in fact the mass of Σ is a radially dependent function. In order to try and recover the circular geometry, the confined field, expanded as it should in the eigenvectors of angular momentum (spherical harmonics), will be described only by those high values of the total angular momentum (to take into account the fact that J ≫ 1) as well as its projection along a fixed axis.
The basis for the possible bound states of Σ will therefore be given as the set of quantized solutions of the Klein-Gordon equation
(✷ + M 2 )Σ = 0,(9)
where the mass M = M (r). The effective radius R is adjusted in such a way that the geometrical section of the corresponding sphere πR 2 is that of the equivalent ring if it hits a particle face-on, i.e.
πR 2 = π[(r V + r X ) 2 − (r V − r X ) 2 ], or R 2 = 4r V r X .
In these formula, r V stands as before for the vorton radius, while r X ∼ m −1 is its thickness, i.e. R ∼ Z 1/2 m −1 .
In the particular example which was used to calculate numerically the curve on Fig. 1, the mass function in Eq. (9) is taken as M (r) = mΘ(R − r), and the trapped field is assumed the separated form Σ(x α ) = u nℓ (r)Y ℓM (θ, φ)e iω nℓM t , with Y ℓM (θ, φ) a spherical harmonic and the radial function provides the quantized energy levels for each value of the angular momentum by imposing equality of the logarithmic derivative of the normalizable solutions at r = R (in the simplified model we investigate, those are spherical Bessel and Hankel functions of the first kind). As discussed above, we shall restrict our attention to M ∼ ℓ ∼ Z 2 ∼ 10 4 in order to take into account, at least partially, the effectively circular (as opposed to spherical) symmetry. For the actual calculations of effective cross-sections, we shall adopt the same normalization convention for all the states, i.e. the bound and free states (including hadron states) are covariantly normalized at 2E particles in the volume V for a state of energy E (ω nℓM in the case of a bound state).
We now calculate the cross section σ V for an incident electromagnetically charged particle, a hadron say for definiteness, in the frame of the loop, to interact inelastically with a vorton. We consider the situation in the rest frame of the loop in which the incident hadron with momentum p in = (E 1 , p 1 ) hits the Σ particle bound state characterized by its angular quantum number ℓ i and bound energy ω 1 to yield an outgoing hadron with momentum p out = (E 2 , p 2 ) and a final state for the Σ particle which could in principle be either another bound state with quantum number ℓ f or a free particle state with momentum k f = (ω 2 , k); in the notation of the previous section, one has ∆E = ω 2 − ω 1 . In both cases, we assume the decay rates dΓ I of the resulting products, namely either the excited state or the unstable particle itself, to be much smaller than the energies of the particles so that the cross-section for producing those unstable states effectively factorizes as dσ = dσ I dΓ I /Γ I . Integrating over the various decay possibilities ( dΓ I = Γ I ) shows that it suffices to calculate only the intermediate cross sections dσ I and sum them up. Note also that because our vorton model is very crude, we cannot expect more than a rough order of magnitude for the cross section, a detailed quantum examination of the same process being held for another work [23]. This is the reason why we now restrict our attention to the ionisation process, including the possibility of final bound state by considering various energy eigenvalues.
The interaction is assumed to be only electromagnetic. Hence, if ψ in and ψ out are the wave functions of the hadron under consideration before and after the collision, the hadronic current J h we are interested in reads
J h µ ≡ ie(ψ * out ∂ µ ψ in − (∂ µ ψ * out )ψ in ),(10)
a similar definition holding for the Σ current J Σ where the "in" state is a bound state, while the "out" state is a free Σ particle. With the electromagnetic field A µ , the total Hamiltonian for describing the collision is then
H = J h α A α J Σ β A β + H Σ + H h ,(11)
where the self interaction Hamiltonians H Σ and H h contain in principle all the information about the internal structure on the interacting particles. In practice, this means that we can work with eigenstates of both these self-interacting Hamiltonians to take into account exactly the vorton structure (this is supposedly done by calculating the bound states) as well as the strong interaction effects binding the quarks in the hadron together; thus, both protons and neutrons can interact with a vorton, a point which is understandable by noting that the vorton being much smaller than the hadron, it interacts essentially with the quarks. Neglecting these corrections of order unity, and working in the Lorentz gauge ∂ µ A µ = 0, the amplitude for this process is easily calculated semiclassically as (note we do not consider the fermionic degrees of freedom of the hadrons) (12) where q = p out −p in is the exchanged momentum. At this point, a completely quantum field analysis could be made by expanding the Σ operator in bound and free particle creation and annihilation operators and the hadron field in plane waves and imposing suitable commutation relations (anticommutation to describe properly the hadron states) [23]. We recall that our purpose here is simply to derive a rough order of magnitude estimate for the interaction cross section, which explains why we perform only a semiclassical analysis and neglect the fermionic behaviour of hadrons. Use of Eq. (12) shall be made later for deriving the details of our oversimplified spherical model, but for the time being, let us derive the expected order of magnitude of the interaction cross-section. The total cross-section, being a sum over all the confined particles of terms like Eq. (12) squared is seen to have a factor Z 2 e 4 , with e the electromagnetic coupling constant. Then, on dimensional grounds, it can be nothing but proportional to (∆E) −2 . Thus, one ends up with
p in , ℓ i |p out , k(ℓ f ) = i d 4 xg µν J h µ (x)q −2 J Σ ν (x)σ V = Z 2 e 4 (∆E) 2 F (∆E/m) ≃ 10 −32 10 9 GeV m 2 Z 6 100 cm 2 ,(13)
where Z 100 ≡ Z/100 and the function F is a dimensionless quantity thats need a specific model to be evaluated.
In the spherical symmetry approximation and for a few energy levels, it is exemplified on Fig. 1. The line and continuum spectrum characteristic of bound state interactions is clearly shown on the figure, as well as the existence of an energy threshold below which no interaction takes place. To conclude this section, we consider the probability that a vorton interacts in the way discussed above in the atmosphere. Using the values derived in the previous section for the mass and charge of the vorton shows that the characteristic ionisation cross section is typical of neutrino interaction at these energies [16]. The probability we are looking for is therefore, using ρ Atm ∼ 10 3 g·cm −2 for the mean atmospheric depth
α V = σ V ρ Atm /m P ≃ 10 −5 − 10 −4 Z 6 100 .(14)
This quantity we shall use later to compute the expected flux of this type of events on earth.
III. ACCELERATION MECHANISM FOR VORTONS
In this section and the following, we turn back to ordinary units in which c andh have their usual values.
Accelerating a particle of charge Ze like a vorton to energies larger than 10 20 eV by means of an electric mechanism requires a potential difference larger than 10 20 /Z Volts. There are basically two classes of astrophysical objects in which such potential differences may be found, namely pulsars and accreting black holes (BH). In fact, potential differences as high as 10 18 Volts are known to exist in the magnetosphere of young pulsars and in that of the accretion disks around massive (10 7 − 10 8 M ⊙ ) Kerr BH.
Pulsars as sources of high energy vortons can be immediately ruled out since vortons are not expected to be present at the surface of neutron stars. Their mass is so high, as we have just seen, that they would sink toward the center of the neutron star. We shall therefore for now on consider the case of accreting BH, supposing those to be the power engine for active galactic nuclei (AGN) radio galaxies and quasars (QSO). The underlying idea is the following: radio jets, X and γ rays emissions are due to the acceleration of electrons, positrons and protons by electrostatic fields generated by a Blandford type mechanism [24] near the horizon of the BH. The model is based on the assumption that a fraction α of the total matter is made of vortons [14], which behaves essentially as cold dark matter. Therefore, one can expect their spatial distribution to look like that of ordinary matter. The main idea is then that in an accelerating object, whatever it is, the same fraction α of vortons is accelerated by the same mechanism that works for the protons. However, in reasonable models, the total mass of the vorton is M V ∼ Zm where Z is the charge carried by the vorton and m the scale of symmetry breaking at which the strings were first formed. As shown in Sec. II, a Lorentz acceleration factor of 10 5 − 10 6 is actually sufficient to reproduce the data. This means that a very basic acceleration by means of an electric field is enough since at these energies, losses in radiation can be considered negligible. In what follows, we shall justify the above model to show that it encompasses no major difficulties.
It should first be clear, and we want to emphasize that point again, that no exotic mechanism is required for the acceleration (even though the accelerated particles themselves may be considered exotic). Indeed, the acceleration mechanism that produces ultra-relativistic jets in radio galaxies and X and γ rays in AGNs and QSOs is enough to accelerate vortons to energies up to 10 20 eV and higher. We want to point out that electrostatic fields are very likely to be responsible for the acceleration of particles in jets of radio galaxies and for the generation of high energy photons (with energies exceeding 1 TeV). Although the origin of such electrostatic fields is not completely understood yet, they may well originate through the quite appealing Blandford-Znajecs mechanism [24]. Let us first describe shortly such a mechanism. Consider a magnetized rotating neutron star. Its surface is supposed to be a perfect conductor, so that in the rest frame of the surface of the neutron star, the tangential component E θ of the electrostatic field must vanish, a condition which, when written in the inertial frame takes the form
E θ − Ω × B c R * θ = 0,(15)
where Ω, B and R * are respectively the angular velocity, surface magnetic field and radius of the neutron star. This gives very energetic electric field lines along which charged particles can be easily accelerated. An analogous mechanism for generating electric field lines works for Kerr BHs, with the unfortunate difference that a bare BH cannot have a magnetic field, the latter being supplied by the accretion disk via a dynamo mechanism. The surface of the BH behaves like a rotating conductor with angular velocity Ω =âc/M , witĥ a and M the Kerr solution parameters somehow identifiable with the total mass (M ) and the angular momentum (for 0 <â < 1) of the source. Now because of the boundary conditions on the surface of the BH, an electrostatic field is created which yields a potential difference given by [25] ∆V =â
B M D 2 ,(16)
where B is the poloidal component of the magnetic field generated by the accretion disk and D the typical length scale of the electrostatic field, D ≃ M , i.e., ∆V ≃âBM (see Fig. 2). The magnetic field is estimated to be 10 4 gauss for the most active galaxies. In fact, under the hypothesis that the magnetic energy density is in equipartition equilibrium with the thermal energy of the disk, one finds
B = L E c 1/2 c 2 GM ≃ 4 × 10 4 M −1/2 8 gauss,(17)
where L E is the Eddington luminosity Volts, (19) or, equivalently
L E = 4πGcM m p σ T ≃ 10 46 M 8 erg · s −1 ,(18)∆V = L câ ≃ 5 × 10 19â L 45 Volts,(20)
L 45 being the luminosity in units of 10 45 erg/sec [25]. Estimates of B [Eq. (17)] and ∆V [Eq. (16)] are very rough and depend crucially on the conversion efficiency of gravitational energy of the disk into electromagnetic energy. As already stated, the parameterâ is less than unity and is related with the angular momentum J of the BH through
J =â c GM 2 ,(21)
where M is now expressed in ordinary units. For old AGNs and QSOs,â can be larger than 0.9. So high a potential difference is very efficient to accelerate particles and can thus be the source for the electromagnetic energy of AGNs, QSOs and radio galaxies. Similar powerful sources of electromagnetic energy can be found in the magnetosphere of the accretion disks and by radiation accelerated winds from the disk. Interested readers will find details and useful references in the review article [27] Hot spots in radio galaxy jets are also good candidates as acceleration regions of very high energy particles. Shock waves in the jets of radio galaxies are the main mechanism allowing acceleration of protons at energies ≃ 10 20 eV, while protons in the near zone of the BH cannot be accelerated at so high energies because of the losses due to the unfavorable environment (synchrotron and Compton losses) [26]. Vortons on the other hand do not suffer from this drawback since because of their high mass, they have a Lorentz factor which is at most of the order of 10 6 as we have seen above.
The synchrotron energy loss per unit length is
dE dx = − 2 3 (Ze) 2 ρ 2 γ 4 ,(22)
with ρ the curvature radius of the vorton trajectory along the magnetic field lines. If we take ρ equal to the Schwarzchild radius r S = 2GM/c 2 of the BH, we obtain the total energy loss δE as
δE = (Ze) 2 3GM c 2 Aγ 4 = 5.2 × 10 −33 Z 2 AM −1 8 γ 4 erg (23)
where A is the acceleration path expressed in Schwarzchild units. For Z = 100 and γ = 10 6 , the synchrotron losses are ∼ 10 −4 erg (with A = 1), a very small quantity indeed with respect to the required final energy of 10 8 erg.
A similar calculation can be performed for the Compton losses; let us first compute the Thompson cross section σ T h V of a vorton:
σ T h V = 8π (Ze) 4 3M 2 V c 4 = 1.6 × 10 −45 Z 2 100 10 9 m P m 2 cm 2(24)
(recall M V = Zm). It is now easy to compute the mean free path of a vorton in the photon bath around a BH. For a given luminosity L, the photon density n ph reads
n ph ≃ L 4πR 2 chν(25)
where R is the radius of the source and ν is a typical frequency of the radiated spectrum. Again fixing R to be the Schwarzchild radius of the BH, we have
n ph = 2 × 10 15 L 45 M −2 8 1 keV E ph photons/cm 3 ,(26)
so that the vorton mean free path λ V = (n ph σ T h V ) −1 is
λ V = 2 × 10 29 Z −2 100 m 10 9 m p 2 M 2 8 E ph 1 keV L −1 45 cm(27)
which is much greater than the typical acceleration length ℓ s ≃ 3 × 10 13 M 8 cm.
In brief, the acceleration of vortons in the hot spots of radio galaxies is more favorable than the same mechanism applied to protons because synchrotron and Compton losses scale respectively like
Z 2 m p M V 4 ≃ 10 −22 Z −2 100 (synchrotron)
and
Z 4 m p M V 4 ≃ 10 −18 Z 0 100 (Compton).
In what follows, we postulate that vortons that are uniformly distributed in the accretion disk around a BH can reach the region of high electrostatic field. Taking into account Eq. (19), we see that they can gain energies of
E V = 10 20 ZâL 45 eV,(28)
so that even active galaxies with luminosities as low as 10 43 erg/s can accelerate vorton to 10 20 eV if the angular momentum of the BH is large enough thatâ ∼ 1.
As already stated, acceleration at these energies of vortons can happen also in the radio galaxy hot spots. The possibility of accelerating them near the central BH means that under the hypothesis that monopolar electrostatic and rotating BH is the power engine of all active galaxies [27], Seyfert I, II, QSO and AGN are potential vorton acceleration sites, thereby tremendously increasing the number of sites as compared to the proton case since there is no GZK cutoff in their case, as we shall now see.
IV. ENERGY CONSIDERATIONS AND
PROPAGATION.
The flux of cosmic rays with energy higher than 10 20 eV is estimated of the order of 10 −20 cm −2 ·s −1 ·ster −1 , i.e. a flux 10 −19 cm −2 ·s −1 , so the energy flux of particles having energies E = E 20 × 10 20 eV is
Φ = 10 −22 E 20 erg · cm −2 · s −1 .(29)
If P V is the probability that a vorton interacts with the atmosphere and if all the particles with energies greater than 10 20 eV are vortons, then the actual flux (as opposed to the observed one) is
Φ = 10 −6 E 20 P 5 erg · cm −2 · s −1 ,(30)
with P 5 = P V /10 −5 (see Sec. II). In order to achieve this flux, extragalactic sources up to a distance D must supply a power W W = 10 47 E 20 P 5 100 Mpc D 2 erg · s −1 .
Assuming 10 5 potential sources (Sy I, I, QSO) [28] means that each source must supply a power
W = 10 44 E 20 P 5 1 Gpc D 2 erg · s −1 ,(32)
power too uncomfortably close to the averaged power radiated by active galaxies. Note the neutrino hypothesis suffers even more of the same drawback. Until now, we have assumed the typical mean free path of a 10 20 eV vorton to be much larger than the Hubble radius, with the meaning that a vorton can reach us, even if emitted at large redshifts. We shall now prove that point by estimating the energy losses the vorton has to experience on his way.
We start with synchrotron radiation due to intergalactic magnetic field B since this is supposed to be the most efficient mechanism. The energy loss per unit time is
dE dt = 2 3 (Ze) 4 E 2 cγ 4 B 2 ,(33)
in a mean magnetic field B = B 2 1/2 if the Larmor radius of the vorton is less than the correlation length of the magnetic field B. The typical slow down timescale for the vorton is thus given by which is clearly sufficiently large that there is not even a cosmological cutoff for vortons.
t V = 3 2 M 4 V c 7 E B 2 (Ze) 4
A similar conclusion holds for the energy losses due to the microwave background. Note however the essential difference in this case between protons and vortons: if protons were pointlike particles with no internal quark structure, they would be able to propagate almost freely in the microwave background, even at these energies, and they decay dominantly because the binding energy between quarks and gluons is much less than the proton mass. Vortons, on the other hand, not only have energy levels which are much higher than protons, but also they propagate with a relatively low velocity (γ ∼ 10 6 ). Therefore, vortons, once accelerated can arrive on earth with undegraded energy: supposing they were accelerated at the time of formation of the galaxies (z ∼ 1) [29], they would by now have lost a mere factor of two because of cosmological redshift. The interesting point in that observation is that for z ∼ 1, the ratio between active and normal galaxies is about 0.1 [29,30]. Keeping that in mind, we can now calculate the high energy vorton density under the hypothesis that vortons were accelerated at the birth of galaxies.
If one assumes that all the unexplained cosmic rays with energy larger that 10 20 eV are made of vortons, their density will be
n V = 4π c Φ V P V = 4.2 × 10 −25 Φ 20 P −1 5 V · cm −1 ,(35)
where Φ V (and Φ 20 = Φ V /10 −20 cm −2 ·s −1 ·ster −1 ) is the high energy cosmic ray flux. The total number of vortons N V in a sphere of radius corresponding to z = 1 is
N V = 3 × 10 60 h −3 Φ 20 P 5 ,(36)
and their energy is 10 20 (1 + z)N V eV = 6 × 10 68 h −3 Φ 20 /P 5 ergs, where h is the Hubble constant H 0 in units of 75 km·s −1 ·Mpc −1 . Such an amount of energy can be released by a number of active galaxies N G
N G = 2 × 10 9 ε h −3 Φ 20 P 5 T −1 9 L 43 ,(37)
with T 9 the duty time of the galaxies in millions of years and ε the ratio between the power used to accelerate vortons and the electromagnetic luminosity; note this latter parameter is not constrained and can in fact exceed unity. The baryonic mass in the sphere with z = 1 is M B = 5.8 × 10 54 h −1 g, so that the total number of galaxies is
N G ≃ 3 × 10 10 h −1 M −1 10 ,(38)
where M 10 is the galaxy mass in 10 10 solar masses. From Refs. [29,30], we know the ratio between active and normal galaxies to be of order 10%, so the required efficiency is, expressed as a function of solar masses accreted by the BH per yearṀ ,
ε = 10 −4 h −2 Φ 20 P 5 T −1 9 M 10 Ṁ M ⊙ −1 .(39)
Finally, let us compute the required number of vortons present in the universe in the form of dark matter. Let α be the ratio between the total vorton and baryon masses in the universe. The ratio η between vorton and baryon number densities is
η = α m P M V ,(40)
and assuming the baryon density to be 10% of the critical density (n b = 6.6 × −7 h 2 baryons/cm 3 ) we finally obtain
α = 6.4 × 10 −4 Φ 20 P 5 Z 100 m 10 9 m P .(41)
This result is understandable in two ways. Either all vortons present in AGNs are indeed accelerated, in which case vortons are, according to this calculation, expected to represent roughly less than a thousandth of the matter in the universe, or, conversely, vortons do fill the universe so that the actual value of α should exceed unity. In the latter case, our calculation reveals that either the interaction probability is lower than what we evaluated above, or only a small fraction of vortons get accelerated.
In any case, it should be clear that our model provides a very high energy cosmic ray flux which can be made to agree with observations.
CONCLUSIONS
We have exhibited a model for explaining extremely high energy cosmic rays that have recently been observed. More of these events are expected to be observed in the near future by the Auger Observatory [17]. We propose that they are bound states of very massive particles in vortons, i.e. loops of superconducting cosmic strings stabilized by a current, that are are freed by inelastic collision with atmospheric hadrons. The model, in its simplest form, has only one free parameter, namely the energy scale at which the current sets up in cosmic strings. This could easily be extended to take into account a possible difference between this mass scale and the string energy scale itself [14]. In this particular framework however, the mass scale is constrained by requiring that vortons do not overfill the universe, a bound that is almost saturated by the demand that high energy cosmic rays are made of vortons. This interesting coincidence could imply, if verified, that a non negligible fraction of the dark matter in the universe consists of vortons.
Let us here summarize our findings concerning the model. First, as mentioned earlier, it essentially has only one free parameter, namely the mass scale at which the superconducting current sets up in the core of the cosmic strings. As it turns out, demanding these objects to be candidates for the few 10 20 eV events through interaction with atmospheric protons completely fixes this parameter to 10 9 GeV. There doesn't seem to be any way out of this prediction, which renders the model very falsifiable.
Another point worth mentioning is that the vortonproton (or neutron) cross section is quite weak, giving, at these energies, a total interaction probability between 10 −5 and 10 −4 with the earth atmosphere. Note again that this probability depends on nothing but the fixed energy scale, and that, in order to fit the observed data, it implies a high energy vorton flux whose numerical value is also therefore determined.
No high energy cosmic ray model is complete without specifying acceleration and propagation processes. In our case, acceleration is performed very simply by kicking the vortons (at least those which have been ionized somehow beforehand) with the high electrostatic fields that are expected to be present in AGNs. The required quantity of energy turns out to exist in these objects and because of the relatively low amount of acceleration required (γ < ∼ 10 6 ), losses are negligible and the vortons can escape the acceleration zone. This is, to the best of our knowledge, the most efficient mean of extracting 10 20 eV in a single particle out of any astrophysical object.
Similarly to the fact that energy losses are negligible in the region of acceleration, and for the same reason, the propagation in intergalactic medium is done almost without any collision, thus making this medium effectively transparent to very high energy vortons. As a result, there is no reason for a cosmological (GZK) cutoff and vortons can come from as far as a redshift of a few. This is satisfactory because the ratio of active to normal galaxies increases as one goes farther away, and the final vorton flux we end up with thanks to this fact is actually quite large and indeed requires only a very small fraction of the total mass density in the universe to consist in vortons (of the order of 10 −4 ) if all vortons are ionized (and therefore accelerated), or, if one believes the standard vorton predictions [14] that for a mass scale of 10 9 GeV they should be the dominant part of the dark matter, then one just needs the same tiny fraction of vorton to be accelerated. The absence of any cutoff also means we predict a spatial isotropy together with a correlation with active galaxies.
Interacting with the atmosphere with a very low probability (and then presenting a threshold followed by a line spectrum, presumably unobservable unfortunately), vortons are here predicted to give rise to mostly horizontal air showers, just like the neutrinos. Let us stress however that until now, it has not been possible to find any astrophysical mechanism that would give neutrinos such high energies. Hence, although our vortons are indeed hypothetical in the sense that we don't know what is the actual theory that describes physics beyond the electroweak scale at which we expect to find them, it should however be clear that they do not, apart in their existence itself, imply any new mechanism of any kind.
Finally it is worth pointing out that the acceleration mechanism we have proposed is just one amongst a few other acceleration mechanisms already proposed in the same context such as the Fermi mechanism. Vortons can be accelerated as well across the shocks of the jets of the radio galaxies. We prefer however the direct acceleration mechanism because of its high efficiency and therefore the energy budget constraints can be more easily fulfilled. Note that this is not the case for the neutrinos: in fact, as already stated, 10 20 eV neutrinos have interaction cross sections (with hadrons) that are roughly equivalent to vorton's. Consequently, it is highly questionable whether the energetic constraints required to accelerate the parent charged particle of the neutrinos in the lobes of radio galaxies can be satisfied.
To end up this conclusion, let us compare our expectations with those of the Auger Observatory [17], a project specifically designed to accumulate more statistics on cosmic rays with energies in excess of 10 19 eV. The Auger Observatory will have an energy resolution less than 20 % which is presumably going to be insufficient to actually allow definite conclusion about the line features. In spite of this difficulty, it is however not impossible that part of the line signal might be found using correlation function techniques after a power-law substraction will have been made in the data: no significant feature should be left by almost all the competing candidates. Another characteristic of the Auger Observatory concerns its ability to identify the primary of an air shower through accurate measurement of the shower maximum (∆X max ∼ 20g·cm −2 ). In a bound state model such as the one we propose here, the line part of the spectrum should be initiated by ra-diative decay of the excited vortons, hence the primary should be a photon. Then once the continuum is reached, some other particle (the Σ in our model) plays the part of the primary, initiating a shower whose maximum is expected somewhere else. It is not clear yet, because very model dependent, whether the Auger Observatory accuracy in this measurement will be sufficient, but it is also a firm prediction of any bound state model. Finally, according to our calculation, the interaction probability with the atmosphere is very low, with the result that many horizontal showers are expected. Contrary to most other cosmic ray detectors, the Auger Observatory will be very efficient to detect those, being also understandable as a neutrino detector with an acceptance of 10 km 3 ·sr water equivalent for horizontal air showers at 10 19 eV [31]. Finally, although the angular resolution of the two detectors (giant array and air fluorescence) in the Auger Observatory should be of the order of less than 2 o , some events will be observed in hybrid mode by both with a resolution of 0.3 o , a precision that is expected to be enough to conclude on the isotropy of the cosmic ray sources.
All these facts lead to the conclusion that our model has a very high potential for being either confirmed or ruled out shortly after the Auger Observatory will be started.
FIG. 1 .
1Expected qualitative form of the cosmic ray spectrum for vortons: it consists of a line spectrum followed by a continuum. This function F(∆E/m) [see Eq. (13)] modulates the cross-section and was calculated using 6 energy levels with ℓ = 10 4 , taking everything numerically into account.
FIG. 2 .
2The electromagnetic field configuration surrounding a Kerr BH in the poloidal Blandford Znajecs mechanism. The circle represents the event horizon of the BH (r = r+) and the shaded area stands for the accretion disk.
m p being the mass of the proton, σ T the Thompson crosssection [24] and M 8 = (M/10 8 M ⊙ ) being the BH mass in units of 10 8 solar masses. Taking into account Eq. (16), we then end up with a potential difference given by ∆V ≃ 1.7 × 10 20â M
ACKNOWLEDGMENTSWe wish to thank E. Audit, B. Carter, A.C. Davis, A. Gangui, T. Kibble, J. P. Lasota and H. Sol for many illuminating conversations, as well as many people from the Pierre Auger Project, in particular J. Cronin and M. Boratav who accepted to spend some time explaining the project.
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| []
|
[
"TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES",
"TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES"
]
| [
"Ken Richardson "
]
| []
| []
| In these lectures, we investigate generalizations of the ordinary Dirac operator to manifolds with additional structure. In particular, if the manifold comes equipped with a distribution and an associated Clifford algebra action on a bundle over the manifold, one may define a transversal Dirac operator associated to this structure. We investigate the geometric and analytic properties of these operators, and we apply the analysis to the settings of Riemannian foliations and of manifolds endowed with Lie group actions. Among other results, we show that although a bundle-like metric on the manifold is needed to define the basic Dirac operator on a Riemannian foliation, its spectrum depends only on the Riemannian foliation structure. Using these ideas, we produce a type of basic cohomology that satisfies Poincaré duality on transversally oriented Riemannian foliations. Also, we show that there is an Atiyah-Singer type theorem for the equivariant index of operators that are transversally elliptic with respect to a compact Lie group action. This formula relies heavily on the stratification of the manifold with group action and contains eta invariants and curvature forms. These notes contain exercises at the end of each subsection and are meant to be accessible to graduate students.2000 Mathematics Subject Classification. 53C12; 53C21; 58J50; 58J60. Key words and phrases. Riemannian foliation, Dirac operator, transverse geometry, index.for all sections s ∈ Γ (E) and vector fields V, W ∈ Γ (T M). We also require that Clifford multiplication of vectors is skew-adjoint with respect to the L 2 inner product, meaning thatfor all v ∈ Γ (T M), s 1 , s 2 ∈ Γ (E). It can be shown that the expression for D above is independent of the choice of orthonormal frame of T M. In the case where E has the minimum possible rank k = 2 ⌊n/2⌋ , we call E a complex spinor bundle and D a spin c Dirac operator. If such a bundle exists over a smooth manifold M, we say that M is spin c . There is a mild topological obstruction to the existence of such a structure; the third integral Stiefel-Whitney class of T M must vanish.Often the bundle E comes equipped with a grading E = E + ⊕E − such that D maps Γ (E + ) to Γ (E − ) and vice-versa. In these cases, we often restrict our attention to D :Examples of ordinary Dirac operators are as follows:• The de Rham operator is defined to be d + δ : Ω even (M) → Ω odd (M) from even forms to odd forms. In this case, the Clifford multiplication is given bywhere v ∈ T x M and i (v) denotes interior product, and ∇ is the ordinary Levi-Civita connection extended to forms. • If M is even-dimensional, the signature operator is defined to befrom self-dual to anti-self-dual forms. This grading is defined as follows. Let * denote the Hodge star operator on forms, defined as the unique endomorphism of the bundle of forms such that * : Ω r (M) → Ω n−r (M) and α ∧ * β = (α, β) dvol, α, β ∈ Ω r (M) .Then observe that the operator ⋆ = i r(r−1)+ n 2 * : Ω r (M) → Ω n−r (M) satisfies ⋆ 2 = 1. Then it can be shown that d + δ anticommutes with ⋆ and thus maps the +1 eigenspace of ⋆, denoted Ω + (M), to the −1 eigenspace of ⋆, denoted Ω − (M). Even though the bundles have changed from the previous example, the expression for Clifford multiplication is the same. | 10.1007/978-3-0348-0871-2_4 | [
"https://arxiv.org/pdf/1006.0185v1.pdf"
]
| 53,520,657 | 1006.0185 | c0bb23ec9608cb400ef046cae02c30694d1523a8 |
TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES
1 Jun 2010
Ken Richardson
TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES
1 Jun 2010
In these lectures, we investigate generalizations of the ordinary Dirac operator to manifolds with additional structure. In particular, if the manifold comes equipped with a distribution and an associated Clifford algebra action on a bundle over the manifold, one may define a transversal Dirac operator associated to this structure. We investigate the geometric and analytic properties of these operators, and we apply the analysis to the settings of Riemannian foliations and of manifolds endowed with Lie group actions. Among other results, we show that although a bundle-like metric on the manifold is needed to define the basic Dirac operator on a Riemannian foliation, its spectrum depends only on the Riemannian foliation structure. Using these ideas, we produce a type of basic cohomology that satisfies Poincaré duality on transversally oriented Riemannian foliations. Also, we show that there is an Atiyah-Singer type theorem for the equivariant index of operators that are transversally elliptic with respect to a compact Lie group action. This formula relies heavily on the stratification of the manifold with group action and contains eta invariants and curvature forms. These notes contain exercises at the end of each subsection and are meant to be accessible to graduate students.2000 Mathematics Subject Classification. 53C12; 53C21; 58J50; 58J60. Key words and phrases. Riemannian foliation, Dirac operator, transverse geometry, index.for all sections s ∈ Γ (E) and vector fields V, W ∈ Γ (T M). We also require that Clifford multiplication of vectors is skew-adjoint with respect to the L 2 inner product, meaning thatfor all v ∈ Γ (T M), s 1 , s 2 ∈ Γ (E). It can be shown that the expression for D above is independent of the choice of orthonormal frame of T M. In the case where E has the minimum possible rank k = 2 ⌊n/2⌋ , we call E a complex spinor bundle and D a spin c Dirac operator. If such a bundle exists over a smooth manifold M, we say that M is spin c . There is a mild topological obstruction to the existence of such a structure; the third integral Stiefel-Whitney class of T M must vanish.Often the bundle E comes equipped with a grading E = E + ⊕E − such that D maps Γ (E + ) to Γ (E − ) and vice-versa. In these cases, we often restrict our attention to D :Examples of ordinary Dirac operators are as follows:• The de Rham operator is defined to be d + δ : Ω even (M) → Ω odd (M) from even forms to odd forms. In this case, the Clifford multiplication is given bywhere v ∈ T x M and i (v) denotes interior product, and ∇ is the ordinary Levi-Civita connection extended to forms. • If M is even-dimensional, the signature operator is defined to befrom self-dual to anti-self-dual forms. This grading is defined as follows. Let * denote the Hodge star operator on forms, defined as the unique endomorphism of the bundle of forms such that * : Ω r (M) → Ω n−r (M) and α ∧ * β = (α, β) dvol, α, β ∈ Ω r (M) .Then observe that the operator ⋆ = i r(r−1)+ n 2 * : Ω r (M) → Ω n−r (M) satisfies ⋆ 2 = 1. Then it can be shown that d + δ anticommutes with ⋆ and thus maps the +1 eigenspace of ⋆, denoted Ω + (M), to the −1 eigenspace of ⋆, denoted Ω − (M). Even though the bundles have changed from the previous example, the expression for Clifford multiplication is the same.
Introduction to Ordinary Dirac Operators
∂u (t, x) ∂t + ∆ x u (t, x) = 0
Wave equation :
∂ 2 u (t, x) ∂t 2 + ∆ x u (t, x) = 0
The sign of the Laplacian is chosen so that it is a nonnegative operator. If u, v denotes the L 2 inner product on complex-valued functions on R n , by integrating by parts, we see that
∆u, u = R n (∆u) u = R n |∇u| 2
if u is compactly supported, where ∇u = ∂u ∂x 1 , ..., ∂u ∂xn is the gradient vector. The calculation verifies the nonnegativity of ∆.
The same result holds if instead the Laplace operator acts on the space of smooth functions on a closed Riemannian manifold (compact, without boundary); the differential operator is modified in a natural way to account for the metric. That is, if the manifold is isometrically embedded in Euclidean space, the Laplacian of a function on that manifold agrees with the Euclidean Laplacian above if that function is extended to be constant in the normal direction in a neighborhood of the embedded submanifold. One may also define the Laplacian on differential forms in precisely the same way; the Euclidean Laplacian on forms satisfies
∆ u (x) dx i 1 ∧ dx i 2 ∧ ... ∧ dx ip = (∆u) (x) dx i 1 ∧ dx i 2 ∧ ... ∧ dx ip .
These standard formulas for the Laplace operator suffice if the Riemannian manifold is flat (for example, flat tori), but it is convenient to give a coordinate-free description for this operator. If (M, g) is a smooth manifold with metric g = (•, •), the volume form on M satisfies dvol = √ det gdx. The metric induces an isomorphism v p → v ♭ p between vectors and one forms at p ∈ M, given by v ♭ p (w p ) := (v p , w p ) , w p ∈ T p M. Thus, given an orthonormal basis {e j : 1 ≤ j ≤ n}of the tangent space T p M, we declare the corresponding dual basis e ♭ j : 1 ≤ j ≤ n to be orthonormal, and in general we declare e ♭ α = e ♭ α 1 ∧ ... ∧ e ♭ αr |α|=k to be an orthonormal basis of r-forms at a point. Then the L 2 inner product of r-forms on M is defined by
γ, β = M (γ, β) dvol .
Next, if d : Ω r (M) → Ω r+1 (M) is the exterior derivative on smooth r-forms, we define δ : Ω r+1 (M) → Ω r (M) to be the formal adjoint of d with respect to the L 2 inner product. That is, if ω ∈ Ω r+1 (M), we define δω by requiring γ, δω = dγ, ω for all γ ∈ Ω r (M). Then the Laplacian on differential r-forms on M is defined to be ∆ = δd + dδ : Ω r (M) → Ω r (M) .
It can be shown that ∆ is an essentially self-adjoint operator. The word essentially means that the space of smooth forms needs to be closed with respect to a certain Hilbert space norm, called a Sobolev norm.
We mention that in many applications, vector-valued Laplacians and Laplacians on sections of vector bundles are used. Exercise 1. Explicitly compute the formal adjoint δ for d restricted to compactly supported forms in Euclidean space, and verify that the δd + dδ agrees with the Euclidean Laplacian on r-forms.
Exercise 2. Show that a smooth r-form α ∈ Ω r (M) is harmonic, meaning that ∆α = 0, if and only if dα and δα are both zero.
Exercise 3. Explicitly compute the set of harmonic r-forms on the 2-dimensional flat torus T 2 = R 2 Z 2 . Verify the Hodge Theorem in this specific case; that is, show that the space of harmonic r-forms is isomorphic to the r -dimensional de Rham cohomology group H r (M).
Exercise 4. Suppose that α is a representative of a cohomology class in H r (M). Show that α is a harmonic form if and only if α is the element of the cohomology class with minimum L 2 norm. Exercise 5. If (g ij ) is the local matrix for the metric with g ij = ∂ ∂x i , ∂ ∂x j , show that the matrix (g ij ) defined by g ij = (dx i , dx j ) is the inverse of the matrix (g ij ).
Exercise 6. If α = n j=1 α j (x) dx j is a one-form on a Riemannian manifold of dimension n, where g ij = (dx i , dx j ) is the local metric matrix for one-forms, verify that the formal adjoint δ satisfies
δ (α) = − 1 √ g i,j ∂ ∂x i g ij √ gα j .
Exercise 7. Show that if f ∈ C ∞ (M), then M ∆f dvol = 0.
1.2. The ordinary Dirac operator. The original motivation for constructing a Dirac operator was the need of a first-order differential operator whose square is the Laplacian. Dirac needed such an operator in order to make some version of quantum mechanics that is compatible with special relativity. Specifically, suppose that D = n j=1 c j ∂ ∂x j is a first-order, constant-coefficient differential operator on R n such that D 2 is the ordinary Laplacian on R n . Then one is quickly led to the equations
c 2 i = −1, c i c j + c j c i = 0, i = j
Clearly, this is impossible if we require each c j ∈ C. However, if we allow matrix coefficients, we are able to find such matrices; they are called Clifford matrices. In the particular case of R 3 , we may use the famous Pauli spin matrices
c 1 = 0 i i 0 , c 2 = 0 1 −1 0 , c 3 = i 0 0 −i .
The vector space C k on which the matrices and derivatives act is called the vector space of spinors. It can be shown that the minimum dimension k satisfies k = 2 ⌊n/2⌋ . The matrices can be used to form an associated Clifford multiplication of vectors, written c (v), defined by
c (v) = n j=1 v j c j ,
where v = (v 1 , ..., v n ). Note that c : R n → End C k such that
c (v) c (w) + c (w) c (v) = −2 (v, w) , v, w ∈ R n .
If M is a closed Riemannian manifold, we desire to find a Hermitian vector bundle E → M and a first-order differential operator D : Γ (E) → Γ (E) on sections of E such that its square is a Laplacian plus a lower-order differential operator. This implies that each E x is a Cl (T x M)-module, where Cl (T x M) is the subalgebra of End C (E x ) generated by a Clifford multiplication of tangent vectors. Then the Dirac operator associated to the Clifford module E is defined for a local orthonormal frame (e j ) n j=1 of T M to be
D = n j=1 c (e j ) ∇ e j ,
where c denotes Clifford multiplication and where ∇ is a metric connection on E satisfying the compatibility condition
• If M is complex, then the Dolbeault operator is defined to be ∂ + ∂ * : Ω 0,even (M) → Ω 0,odd (M) ,
where the differential forms involve wedge products of dz j and the differential ∂ differentiates only with respect to the z j variables. • The spin c Dirac operator has already been mentioned above. The key point is that the vector bundle S → M in this case has the minimum possible dimension. When M is even-dimensional, then the spinor bundle decomposes as S + ⊕ S − , and D : Γ (S + ) → Γ (S − ). The spinor bundle S is unique up to tensoring with a complex line bundle.
For more information on Dirac operators, spin manifolds, and Clifford algebras, we refer the reader to [38] and [53]. Often the operators described above are called Dirac-type operators, with the word "Dirac operator" reserved for the special examples of the spin or spin c Dirac operator. Elements of the kernel of a spin or spin c Dirac operator are called harmonic spinors.
Exercise 8. Let the Dirac operator D on the two-dimensional torus T 2 = R 2 Z 2 be defined using c 1 and c 2 of the Pauli spin matrices. Find a decomposition of the bundle as S + ⊕S − , and calculate ker (D| S + ) and ker ( D| S − ). Find all the eigenvalues and corresponding eigensections of D + = D| S + . Exercise 9. On an n-dimensional manifold M, show that * 2 = (−1) r(n−r) and ⋆ 2 = 1 when restricted to r-forms.
Exercise 10. On R 4 with metric ds 2 = dx 2 1 + 4dx 2 2 + dx 2 3 + (1 + exp (x 1 )) 2 dx 2 4 , let ω = x 2 1 x 2 dx 2 ∧ dx 4 . Find * ω and ⋆ω. Exercise 11. Calculate the signature operator on T 2 , and identify the subspaces Ω + (T 2 ) and Ω − (T 2 ).
Exercise 12. Show that −i ∂ ∂θ is a Dirac operator on S 1 = e iθ : θ ∈ R . Find all the eigenvalues and eigenfunctions of this operator.
Exercise 13. Show that if S and T are two anticommuting linear transformations from a vector space V to itself, then if E λ is the eigenspace of S corresponding to an eigenvalue λ, then T E λ is the eigenspace of S corresponding to the eigenvalue −λ.
Exercise 14. Show that if δ r is the adjoint of d : Ω r−1 (M) → Ω r (M), then δ r = (−1) nr+n+1 * d * on Ω r (M).
Exercise 15. Show that if the dimension of M is even and δ r is the adjoint of d :
Ω r−1 (M) → Ω r (M), then δ r = −⋆d⋆
on Ω r (M). Is this true if the dimension is odd?
Exercise 16. Show that d + δ maps Ω + (M) to Ω − (M).
Exercise 17. Show that if we write the Dirac operator for R 3
D = c 1 ∂ ∂x 1 + c 2 ∂ ∂x 2 + c 3 ∂ ∂x 3
using the Pauli spin matrices in geodesic polar coordinates
D = Z ∂ ∂r + D S ,
then ZD S restricts to a spin c Dirac operator on the unit sphere S 2 , and Z is Clifford multiplication by the vector ∂ ∂r . Exercise 18. Show that d+δ = n j=1 c (e j ) ∇ e j , with the definition of Clifford multiplication given in the notes.
Exercise 19. Show that the expression n j=1 c (e j ) ∇ e j for the Dirac operator is independent of the choice of orthonormal frame.
Properties of Dirac operators.
Here we describe some very important properties of Dirac operators.
First, Dirac operators are elliptic. Both the Laplacian and Dirac operators are examples of such operators. Very roughly, the word elliptic means that the operators differentiate in all possible directions. To state more precisely what this means, we need to discuss what is called the principal symbol of a differential (or pseudodifferential) operator.
Very roughly, the principal symbol is the set of matrix-valued leading order coefficients of the operator. If E → M, F → M are two vector bundles and P : Γ (E) → Γ (F ) is a differential operator of order k acting on sections, then in local coordinates of a local trivialization of the vector bundles, we may write
P = |α|=k s α (x) ∂ k ∂x α + lower order terms,
where the sum is over all possible multi-indices α = (α 1 , ..., α k ) of length |α| = k, and each s α (x) ∈ Hom (E x , F x ) is a linear transformation. If ξ = ξ j dx j ∈ T * x M is a nonzero covector at x, we define the principal symbol σ (P ) (ξ) of P at ξ to be
σ (P ) (ξ) = i k |α|=k s α (x) ξ α ∈ Hom (E x , F x ) ,
with ξ α = ξ α 1 ξ α 2 ...ξ α k (some people leave out the i k ). It turns out that by defining it this way, it is invariant under coordinate transformations. One coordinate-free definition of σ (P ) x : T * x (M) → Hom (E x , F x ) is as follows. For any ξ ∈ T * x (M), choose a locally-defined function f such that df x = ξ. Then we define the operator
σ m (P ) (ξ) = lim t→∞ 1 t m e −itf P e itf ,
where e −itf P e itf (u) = e −itf P e itf u . Then the order k of the operator and principal symbol are defined to be
k = sup {m : σ m (P ) (ξ) < ∞} σ (P ) (ξ) = σ k (P ) (ξ) .
With this definition, the principal symbol of any differential (or even pseudodifferential) operator can be found. Pseudodifferential operators are more general operators that can be defined locally using the Fourier transform and include such operators as the square root of the Laplacian.
An elliptic differential (or pseudodifferential) operator P on M is defined to be an operator such that its principal symbol σ (P ) (ξ) is invertible for all nonzero ξ ∈ T * M.
From the exercises at the end of this section, we see that the symbol of any Dirac operator
D = c (e j ) ∇ e j is σ (D) (ξ) = ic ξ # ,
and the symbol of the associated Laplacian D 2 is
σ D 2 = ic ξ # 2 = ξ 2 ,
which is clearly invertible for ξ = 0. Therefore both D and D 2 are elliptic. We say that an operator P is strongly elliptic if there exists c > 0 such that
σ (D) (ξ) ≥ c |ξ| 2
for all nonzero ξ ∈ T * M. The Laplacian and D 2 are strongly elliptic.
Following are important properties of elliptic operators P , which now apply to Dirac operators and their associated Laplacians:
• Elliptic regularity: if the coefficients of P are smooth, then if P u is smooth, then u is smooth. As a consequence, if the order of P is greater than zero, then the kernel and all other eigenspaces of P consist of smooth sections. • Elliptic operators are Fredholm when the correct Sobolev spaces of sections are used.
• Ellipticity implies that the spectrum of P consists of eigenvalues. Strong ellipticity implies that the spectrum is discrete and has the only limit point at infinity. In particular, the eigenspaces are finite-dimensional and consist of smooth sections. This now applies to any Dirac operator, because its square is strongly elliptic. • If P is a second order elliptic differential operator with no zero th order terms, strong ellipticity implies the maximum principle for the operator P . • Many inequalities for elliptic operators follow, like Gȧrding's inequality, the elliptic estimates, etc.
See [57], [53], and [54] for more information on elliptic differential and pseudodifferential operators on manifolds.
Next, any Dirac operator D : Γ (E) → Γ (E) is formally self-adjoint, meaning that when restricted to smooth compactly-supported sections u, v ∈ Γ (E) it satisfies
Du, v = u, Dv .
Since D is elliptic and if M is closed, it then follows that D is essentially self-adjoint, meaning that there is a Hilbert space H 1 (E) such that Γ (E) ⊂ H 1 (E) ⊂ L 2 (E) such that the closure of D in H 1 (E) is a truly self-adjoint operator defined on the whole space. In this particular case, H 1 (E) is an example of a Sobolev space, which is the closure of Γ (E) with respect to the norm u 1 = u + Du , where • denotes the ordinary L 2 -norm.
We now show the proof that D is formally self-adjoint. If the local bundle inner product on E is (•, •), we have
(Du, v) = c (e j ) ∇ e j u, v = − ∇ e j u, c (e j ) v = −e j (u, c (e j ) v) + u, ∇ e j (c (e j ) v) ,
since c (e j ) is skew-adjoint and ∇ is a metric connection. Using the compatibility of the connection, we have
(Du, v) = −e j (u, c (e j ) v) + u, c ∇ e j e j v + u, c (e j ) ∇ e j v .
Next, we use the fact that we are allowed to choose the local orthonormal frame in any way we wish. If we are evaluating this local inner product at a point x ∈ M, we choose the orthonormal frame (e i ) so that all covariant derivatives of e i vanish at x. Now, the middle term above vanishes, and
(Du, v) = (u, Dv) + −e j (u, c (e j ) v) .
Next, if ω denotes the one-form defined by ω (X) = (u, c (X) v) for X ∈ Γ (T M), then an exercise at the end of this section implies that
(δω) (x) = −e j (u, c (e j ) v) (x) ,
with our choice of orthonormal frame. Hence,
(Du, v) = (u, Dv) + δω,
which is a general formula now valid at all points of M. After integrating over M we have
Du, v = u, Dv + M δω dvol = u, Dv + M (d (1) , ω) dvol = u, Dv .
Thus, D is formally self-adjoint.
Exercise 20. Find the principal symbol of the wave operator ∂ 2 ∂t 2 − ∂ 2 ∂x 2 on R 2 , and determine if it is elliptic.
Exercise 21. If P 1 and P 2 are two differential operators such that the composition P 1 P 2 is defined, show that σ (P 1 P 2 ) (ξ) = σ (P 1 ) (ξ) σ (P 2 ) (ξ) .
ind (D) = M ch (σ (D)) ∧ Todd (T C M) = M α (x) dvol (x) ,
where ch (σ (D)) is a form representing the Chern character of the principal symbol σ (D), and Todd (T C M) is a form representing the Todd class of the complexified tangent bundle T C M; these forms are characteristic forms derived from the theory of characteristic classes and depend on geometric and topological data. The local expression for the relevant term of the integrand, which is a multiple of the volume form dvol (x), can be written in terms of curvature and the principal symbol and is denoted α (x) dvol (x).
Typical examples of this theorem are some classic theorems in global analysis. As in the earlier example, let D = d + δ from the space of even forms to the space of odd forms on the manifold M of dimension n, where as before δ denotes the L 2 -adjoint of the exterior derivative d. Then the elements of ker (d + δ) are the even harmonic forms, and the elements of the cokernel can be identified with odd harmonic forms. Moreover,
ind (d + δ) = dim H even (M) − dim H odd (M) = χ (M) , and M ch (σ (d + δ)) ∧ Todd (T C M) = 1 (2π) n M Pf ,
where Pf is the Pfaffian, which is, suitably interpreted, a characteristic form obtained using the square root of the determinant of the curvature matrix. In the case of 2-manifolds (n = 2), Pf is the Gauss curvature times the area form. Thus, in this case the Atiyah-Singer Index Theorem yields the generalized Gauss-Bonnet Theorem.
Another example is the operator D = d + d * on forms on an even-dimensional manifold, this time mapping the self-dual to the anti-self-dual forms. This time the Atiyah-Singer Index Theorem yields the equation (called the Hirzebruch Signature Theorem)
Sign (M) = M L,
where Sign(M) is signature of the manifold, and L is the Hirzebruch L-polynomial applied to the Pontryagin forms.
If a manifold is spin, then the index of the spin Dirac operator is the A genus ("Aroof" genus) of the manifold. Note that the spin Dirac operator is an example of a spin c Dirac operator where the spinor bundle is associated to a principal Spin (n) bundle. Such a structure exists when the second Stiefel-Whitney class is zero, a stronger condition than the spin c condition. The A genus is normally a rational number but must agree with the index when the manifold is spin.
Different examples of operators yield other classical theorems, such as the Hirzebruch-Riemann-Roch Theorem, which uses the Dolbeault operator.
All of the first order differential operators mentioned above are examples of Dirac operators. If M is spin c , then the Atiyah-Singer Index Theorem reduces to a calculation of the index of Dirac operators (twisted by a bundle). Because of this and the Thom isomorphism in K-theory, the Dirac operators and their symbols play a very important role in proofs of the Atiyah-Singer Index Theorem. For more information, see [6], [38].
Exercise 24. Prove that if L : H 1 → H 2 is a Fredholm operator between Hilbert spaces, then coker(L) ∼ = ker (L * ).
Exercise 25. Suppose that P : H → H is a self-adjoint linear operator, and H = H + ⊕ H − is an orthogonal decomposition. If P maps H + into H − and vice versa, prove that the adjoint of the restriction P : H + → H − is the restriction of P to H − . Also, find the adjoint of the operator P ′ : H + → H defined by P ′ (h) = P (h).
Exercise 26. If D is an elliptic operator and E λ is an eigenspace of D * D corresponding to the eigenvalue λ = 0, then D (E λ ) is the eigenspace of DD * corresponding to the eigenvalue λ. Conclude that the eigenspaces of D * D and DD * corresponding to nonzero eigenvalues have the same (finite) dimension.
Exercise 27. Let f : C → C be a smooth function, and let L : H → H be a self-adjoint operator with discrete spectrum. Let P λ : H → E λ be the orthogonal projection to the eigenspace corresponding to the eigenvalue λ. We define the operator f (L) to be
f (L) = λ f (λ) P λ ,
assuming the right hand side converges. Assuming f (L), g (L), and f (L) g (L) converge, prove that f (L) g (L) = g (L) f (L). Also, find the conditions on a function f such that f (L) = L.
Exercise 28. Show that if P is a self-adjoint Fredholm operator, then
ind (D) = tr (exp (−tD * D)) − tr (exp (−tDD * ))
for all t > 0, assuming that exp (−tD * D), exp (−tDD * ), and their traces converge.
Exercise 29. Find all homeomorphism types of surfaces S such that a metric g on S has Gauss curvature K g that satisfies −5 ≤ K g ≤ 0 and volume that satisfies 1 ≤Vol g (S) ≤ 4. Exercise 31. Suppose that on a certain manifold the A-genus is 3 4 . What does this imply about Stiefel-Whitney classes?
Transversal Dirac operators on distributions
This section contains some of the results in [45], joint work with I. Prokhorenkov. The main point of this section is to provide some ways to analyze operators that are not elliptic but behave in some ways like elliptic operators on sections that behave nicely with respect to a designated transverse subbundle Q ⊆ T M. A transversally elliptic differential (or pseudodifferential) operator P on M with respect to the transverse distribution Q ⊆ T M is defined to be an operator such that its principal symbol σ (P ) (ξ) is required to be invertible only for all nonzero ξ ∈ Q * ⊆ T * M. In later sections, we will be looking at operators that are transversally elliptic with respect to the orbits of a group action, and in this case Q is the normal bundle to the orbits, which may have different dimensions at different points of the manifold. In this section, we will restrict to the case where Q has constant rank. Now, let Q ⊂ T M be a smooth distribution, meaning that Q → M is a smooth subbundle of the tangent bundle. Assume that a Cl (Q)-module structure on a complex Hermitian vector bundle E is given, and we will now define transverse Dirac operators on sections of E. Similar to the above, M is a closed Riemannian manifold, c : Q → End (E) is the Clifford multiplication on E, and ∇ E is a Cl (Q) connection that is compatible with the metric on M; that is, Clifford multiplication by each vector is skew-Hermitian, and we require
∇ E X (c (V ) s) = c ∇ Q X V s + c (V ) ∇ E X s
for all X ∈ Γ (T M), V ∈ ΓQ, and s ∈ ΓE. Let L = Q ⊥ , let (f 1 , ..., f q ) be a local orthonormal frame for Q, and let π : T M → Q be the orthogonal projection. We define the Dirac operator A Q corresponding to the distribution Q as
A Q = q j=1 c (f j ) ∇ E f j . (2.1)
This definition is again independent of the choice of orthonormal frame; in fact it is the composition of the maps
Γ (E) ∇ E → Γ (T * M ⊗ E) ∼ = → Γ (T M ⊗ E) π → Γ (Q ⊗ E) c → Γ (E) .
We now calculate the formal adjoint of A Q , in precisely the same way that we showed the formal self-adjointness of the ordinary Dirac operator. Letting (s 1 , s 2 ) denote the pointwise inner product of sections of E, we have
(A Q s 1 , s 2 ) = q j=1 c (πf j ) ∇ E f j s 1 , s 2 = q j=1 − ∇ E f j s 1 , c (πf j ) s 2 .
Since∇ E is a metric connection,
(A Q s 1 , s 2 ) = −f j (s 1 , c (πf j ) s 2 ) + s 1 , ∇ E f j c (πf j ) s 2 = − f j (s 1 , c (πf j ) s 2 ) + s 1 , c (πf j ) ∇ E f j s 2 + s 1 , c π∇ M f j πf j s 2 , (2.2)
by the Cl (Q)-compatibility. Now, we do not have the freedom to choose the frame so that the covariant derivatives vanish at a certain point, because we know nothing about the distribution Q. Hence we define the vector fields
V = q j=1 π∇ M f j f j , H L = n j=q+1 π∇ M f j f j .
Note that H L is precisely the mean curvature of the distribution L = Q ⊥ . Further, letting ω be the one-form defined by
ω (X) = (s 1 , c (πX) s 2 ) ,
and letting (f 1 , ..., f q , f q+1 , ..., f n ) be an extension of the frame of Q to be an orthonormal frame of T M,
δω = − n j=1 i (f j ) ∇ f j ω = − n j=1 f j ω (f j ) − ω ∇ f j f j = n j=1 −f j (s 1 , c (πf j ) s 2 ) + s 1 , c π∇ M f j f j s 2 = s 1 , c V + H L s 2 + n j=1 (−f j (s 1 , c (πf j ) s 2 )) = s 1 , c V + H L s 2 + q j=1 (−f j (s 1 , c (πf j ) s 2 )) . From (2.2) we have (A Q s 1 , s 2 ) = δω − s 1 , c V + H L s 2 + (s 1 , A Q s 2 ) + (s 1 , c (V ) s 2 ) (A Q s 1 , s 2 ) = δω + (s 1 , A Q s 2 ) − s 1 , c H L s 2 .
Thus, by integrating over the manifold (which sends δω to zero), we see that the formal
L 2 -adjoint of A Q is A * Q = A Q − c H L . Since c H L is skew-adjoint, the new operator D Q = A Q − 1 2 c H L (2.3)
is formally self-adjoint.
A quick look at [15] yields the following. Remark 2.3. In general, the spectrum of D Q is not necessarily discrete. In the case of Riemannian foliations, we identify Q with the normal bundle of the foliation, and one typically restricts to the space of basic sections. In this case, the spectrum of D Q restricted to the basic sections is discrete.
Exercise 32. Let M = T 2 = R 2 Z 2 , and consider the distribution Q defined by the vectors parallel to (1, r) with r ∈ R. Calculate the operator D Q and its spectrum, where the Clifford multiplication is just complex number multiplication (on a trivial bundle E x = C). Does it make a difference if r is rational?
Exercise 33. With M and Q as in the last exercise, let E be the bundle ∧ * Q * . Now calculate D Q and its spectrum.
Exercise 34. Consider the radially symmetric Heisenberg distribution, defined as follows. Let α ∈ Ω 1 (R 3 ) be the differential form
α = dz − 1 2 r 2 dθ = dz − 1 2 (xdy − ydx) . Note that dα = −rdr ∧ dθ = −dx ∧ dy,
so that α is a contact form because
α ∧ dα = −dx ∧ dy ∧ dz = 0 at each point of H. The two-dimensional distribution Q ⊂ R 3 is defined as Q = ker α. Calculate the operator D Q . Exercise 35. Let (M, α) be a manifold of dimension 2n + 1 with contact form α; that is, α is a one-form such that α ∧ (dα) n
is everywhere nonsingular. The distribution Q = ker α is the contact distribution. Calculate the mean curvature of Q in terms of α.
A L = i∂ y , and D L = i ∂ y + 1 2 g ′ (y) .
Show that the spectrum σ (D L ) = Z is a set consisting of eigenvalues of infinite multiplicity.
Exercise 37. In the last example, show that the operator
D Q = ie −g(y) ∂ x has spectrum σ (D Q ) = n∈Z n [a, b] ,
where [a, b] is the range of e −g(y) .
Basic Dirac operators on Riemannian foliations
The results of this section are joint work with G. Habib and can be found in [25] and [26].
3.1.
Invariance of the spectrum of basic Dirac operators. Suppose a closed manifold M is endowed with the structure of a Riemannian foliation (M, F , g Q ). The word Riemannian means that there is a metric on the local space of leaves -a holonomy-invariant transverse metric g Q on the normal bundle Q = T M T F . The phrase holonomyinvariant means the transverse Lie derivative L X g Q is zero for all leafwise vector fields X ∈ Γ(T F ). We often assume that the manifold is endowed with the additional structure of a bundlelike metric [47], i.e. the metric g on M induces the metric on Q ≃ NF = (T F ) ⊥ . Every Riemannian foliation admits bundle-like metrics that are compatible with a given (M, F , g Q ) structure. There are many choices, since one may freely choose the metric along the leaves and also the transverse subbundle NF . We note that a bundle-like metric on a smooth foliation is exactly a metric on the manifold such that the leaves of the foliation are locally equidistant. There are topological restrictions to the existence of bundle-like metrics (and thus Riemannian foliations). Important examples of requirements for the existence of a Riemannian foliations may be found in [35], [31], [43], [56], [58], [55]. One geometric requirement is that, for any metric on the manifold, the orthogonal projection
P : L 2 (Ω (M)) → L 2 (Ω (M, F ))
must map the subspace of smooth forms onto the subspace of smooth basic forms ( [44]). Recall that basic forms are forms that depend only on the transverse variables. The space Ω (M, F ) of basic forms is defined invariantly as
Ω (M, F ) = {β ∈ Ω (M) : i (X) β = 0 and i (X) dβ = 0 for all X ∈ Γ (T F )} .
The basic forms Ω (M, F ) are preserved by the exterior derivative, and the resulting cohomology is called basic cohomology H * (M, F ) (see also Section ). It is known that the basic cohomology groups are finite-dimensional in the Riemannian foliation case. See [19], [20], [35], [32], [33], [21] for facts about basic cohomology and Riemannian foliations. For later use, the basic Euler characteristic is defined to be
χ (M, F ) = (−1) j dim H j (M, F ) .
We now discuss the construction of the basic Dirac operator, a construction which requires a choice of bundle-like metric. See [32], [33], [35], [18], [22], [45], [28], [29], [24], [25], [12], [13] for related results. Let (M, F ) be a Riemannian manifold endowed with a Riemannian foliation. Let E → M be a foliated vector bundle (see [31]) that is a bundle of Cl(Q) Clifford modules with compatible connection ∇ E . This means that foliation lifts to a horizontal foliation in T E. Another way of saying this is that connection is flat along the leaves of F . When this happens, it is always possible to choose a basic connection for E -that is, a connection for which the connection and curvature forms are actually (Lie algebra-valued) basic forms.
Let A N F and D N F be the associated transversal Dirac operators as in the previous section. The transversal Dirac operator A N F fixes the basic sections Γ
b (E) ⊂ Γ(E) (i.e. Γ b (E) = {s ∈ Γ(E) : ∇ E X s = 0 for all X ∈ Γ(T F )}) but is not symmetric on this subspace. Let P b : L 2 (Γ (E)) → L 2 (Γ b (E)
) be the orthogonal projection, which can be shown to map smooth sections to smooth basic sections. We define the basic Dirac operator to be
D b : = P b D N F P b = A N F − 1 2 c κ ♯ b : Γ b (E) → Γ b (E) P b A N F P b = A N F P b , P b c κ ♯ P b = c κ ♯ b P b
Here, κ b is the L 2 -orthogonal projection of κ onto the space of basic forms as explained above, and κ ♯ b is the corresponding basic vector field. Then D b is an essentially self-adjoint, transversally elliptic operator on Γ b (E). The local formula for D b is
D b s = q i=1 e i · ∇ E e i s − 1 2 κ ♯ b · s , where {e i } i=1,··· ,q is a local orthonormal frame of Q.
Then D b has discrete spectrum ( [22], [18], [13]). An example of the basic Dirac operator is as follows. Using the bundle ∧ * Q * as the Clifford bundle with Clifford action e· = e * ∧ −e * in analogy to the ordinary de Rham operator, we have
D b = d + δ b − 1 2 κ b − 1 2 κ b ∧ . = d + δ.
One might have incorrectly guessed that d + δ b is the basic de Rham operator in analogy to the ordinary de Rham operator, for this operator is essentially self-adjoint, and the associated basic Laplacian yields basic Hodge theory that can be used to compute the basic cohomology. The square D 2 b of this operator and the basic Laplacian ∆ b do have the same principal transverse symbol. In [25], we showed the invariance of the spectrum of D b with respect to a change of metric on M in any way that keeps the transverse metric on the normal bundle intact (this includes modifying the subbundle NF ⊂ T M, as one must do in order to make the mean curvature basic, for example). That is, We emphasize that the basic Dirac operator D b depends on the choice of bundle-like metric, not merely on the Clifford structure and Riemannian foliation structure, since both projections T * M → Q * and P depend on the leafwise metric. It is well-known that the eigenvalues of the basic Laplacian ∆ b (closely related to D 2 b ) depend on the choice of bundlelike metric; for example, in [52,Corollary 3.8], it is shown that the spectrum of the basic Laplacian on functions determines the L 2 -norm of the mean curvature on a transversally oriented foliation of codimension one. If the foliation were taut, then a bundle-like metric could be chosen so that the mean curvature is identically zero, and other metrics could be chosen where the mean curvature is nonzero. This is one reason why the invariance of the spectrum of the basic Dirac operator is a surprise.
| W = L * | W satisfies W = S ∩ L S ⊥ ⊥ S ,
where S ⊥ is the orthogonal complement of S in H and the superscript ⊥ S denotes the orthogonal complement in S.
Exercise 39. Prove that the metric on a Riemannian manifold M with a smooth foliation F is bundle-like if and only if the normal bundle NF with respect to that metric is totally geodesic.
Exercise 40. Let (M, F ) be a transversally oriented Riemannian foliation of codimension q with bundle-like metric, and let ν be the transversal volume form. The transversal Hodge star operator * :
∧ * Q * → ∧ * Q * is defined by α ∧ * β = (α, β) ν, for α, β ∈ Ω k (M, F ), so that * 1 = ν, * ν = 1. Let the transversal codifferential δ T : be defined by δ T = (−1) qk+q+1 * d * : Ω k (M, F ) → Ω k−1 (M, F ) . As above, let δ b be the adjoint of d with respect to L 2 (Ω (M, F )).
Prove the following identities:
• * 2 = (−1) k(q−k) on basic k-forms. • If β is a basic one-form, then (β ) = (−1) q(k+1) * (β∧) * as operators on basic k- forms. • δ b = δ T + κ b • δ b ν = * κ b • dκ b = 0 (Hint: compute δ 2 b ν.) • * d = ± δ * , with d = d − 1 2 κ b ∧, δ = δ b − 1 2 κ b . 3.2.
The basic de Rham operator. From the previous section, the basic de Rham operator is D b = d + δ acting on basic forms, where
d = d − 1 2 κ b ∧, δ = δ b − 1 2 κ b .
Unlike the ordinary and well-studied basic Laplacian, the eigenvalues of ∆ = D 2 b are invariants of the Riemannian foliation structure alone and independent of the choice of compatible bundle-like metric. The operators d and δ have following interesting properties.
Lemma 3.2. δ is the formal adjoint of d .Ω k (M, F ) = image d k−1 ⊕ image δ b k+1 ⊕ ker ∆ k ,
an L 2 -orthogonal direct sum. Also, ker ∆ k is finite-dimensional and consists of smooth forms.
We call ker ∆ the space ∆-harmonic forms. In the remainder of this section, we assume that the foliation is transversally oriented so that the transversal Hodge * operator is welldefined.
Definition 3.5. We define the basic d-cohomology H * (M, F ) by H k (M, F ) = ker d k image d k−1 .
The following proposition follows from standard arguments and the Hodge theorem (Theorem 3.4).
Proposition 3.6. The finite-dimensional vector spaces H k (M, F ) and ker ∆ k = ker d + δ k are naturally isomorphic.
We observe that for every choice of bundle-like metric, the differential d = d − 1 2 κ b ∧ changes, and thus the cohomology groups change. However, note that κ b is the only part that changes; for any two bundle-like metrics g M , g ′ M and associated κ b , κ ′ b compatible with (M, F , g Q ), we have κ ′ b = κ b + dh for some basic function h (see [2]). In the proof of the main theorem in [25], we essentially showed that the the basic de Rham operator D b is then transformed by D ′ b = e h/2 D b e −h/2 . Applying this to our situation, we see that the (ker D ′ b ) = e h/2 ker D b , and thus the cohomology groups are the same dimensions, independent of choices. To see this in our specific situation, note that if α ∈ Ω k (M, F ) satisfies dα = 0, then
d ′ e h/2 α = d − 1 2 κ b ∧ − 1 2 dh∧ e h/2 α = e h/2 dα + 1 2 e h/2 dh ∧ α − e h/2 2 κ b ∧ α − e h/2 2 dh ∧ α = e h/2 dα − e h/2 2 κ b ∧ α = e h/2 d − 1 2 κ b ∧ α = e h/2 dα = 0.
Similarly, as in [25] one may show ker δ ′ = e h/2 ker δ , through a slightly more difficult computation. Thus, we have [26].) This Riemannian foliation is the famous Carrière example from [14] in the 3-dimensional case. Let A be a matrix in SL 2 (Z) of trace strictly greater than 2. We denote respectively by V 1 and V 2 the eigenvectors associated with the eigenvalues λ and 1 λ of A with λ > 1 irrational. Let the hyperbolic torus T 3 A be the quotient of T 2 × R by the equivalence relation which identifies (m, t) to (A(m), t + 1). The flow generated by the vector field V 2 is a transversally Lie foliation of the affine group. We denote by K the holonomy subgroup. The affine group is the Lie group R 2 with multiplication (t, s).(t ′ , s ′ ) = (t + t ′ , λ t s ′ + s), and the subgroup K is
K = {(n, s), n ∈ Z, s ∈ R}.
We choose the bundle-like metric (letting (x, s, t) denote the local coordinates in the V 2 direction, V 1 direction, and R direction, respectively) as
g = λ −2t dx 2 + λ 2t ds 2 + dt 2 .
Prove that:
• The mean curvature of the flow is κ = κ b = log (λ) dt.
• The twisted basic cohomology groups are all trivial.
• The ordinary basic cohomology groups satisfy
H 0 (M, F ) ∼ = R , H 1 (M, F ) ∼ = R , H 2 (M, F ) ∼ = {0}. •
The flow is not taut.
Poincaré duality and consequences.
Theorem 3.9. (Poincaré duality for d-cohomology) Suppose that the Riemannian foliation (M, F , g Q ) is transversally oriented and is endowed with a bundle-like metric. For each k such that 0 ≤ k ≤ q and any compatible choice of bundle-like metric, the map * : Ω k (M, F ) → Ω q−k (M, F ) induces an isomorphism on the d-cohomology. Moreover, * maps the ker ∆ k isomorphically onto ker ∆ q−k , and it maps the λ-eigenspace of ∆ k isomorphically onto the λ-eigenspace of ∆ q−k , for all λ ≥ 0.
This resolves the problem of the failure of Poincaré duality to hold for standard basic cohomology (see [34], [56]). The following fact is a new result for ordinary basic cohomology of Riemannian foliations. Ordinary basic cohomology does not satisfy Poincaré duality; in fact, the top-dimensional basic cohomology group is zero if and only if the foliation is not taut. Also, leaf closures of a transversally oriented foliation can fail to be transversally oriented, so orientation is also a tricky issue. Proof. The basic Euler characteristic is the basic index of the operator D 0 = d + δ B : Ω even (M, F ) → Ω odd (M, F ). See [12], [18], [7], [19] for information on the basic index and basic Euler characteristic. The crucial property for us is that the basic index of D 0 is a Fredholm index and is invariant under perturbations of the operator through transversally elliptic operators that map the basic forms to themselves. In particular, the family of opera- Proof. Let H be the isotropy subgroup of a frame f ∈ F O . Then H also fixes p (f ) ∈ M, and since H fixes the frame, its differentials fix the entire tangent space at p (f ). Since it fixes the tangent space, every element of H also fixes every frame in p −1 (p (f )); thus every frame in a given fiber must have the same isotropy subgroup. Since the elements of H map geodesics to geodesics and preserve distance, a neighborhood of p (f ) is fixed by H. Thus, H is a subgroup of the isotropy subgroup at each point of that neighborhood. Conversely, if an element of G fixes a neighborhood of a point x in M, then it fixes all frames in p −1 (x), and thus all frames in the fibers above that neighborhood. Since M is connected, we may conclude that every point of F O has the same isotropy subgroup H, and H is the subgroup of G that fixes every point of M.
tors D t = d + δ b − t 2 κ b − t 2 κ b ∧ for 0 ≤ t ≤ 1
Remark 4.2.
Since this subgroup H is normal, we often reduce the group G to the group G/H so that our action is effective, in which case the isotropy subgroups on F O are all trivial.
Remark 4.3.
A similar idea is also useful in constructing the lifted foliation and the basic manifold associated to a Riemannian foliation (see [43]).
In any case, the G orbits on
E σ x = Γ p −1 (x) , E σ ,
where the superscript σ is defined for a O (n)-module Z by
Z σ = eval Hom O(n) (W σ , Z) ⊗ W σ , where eval : Hom O(n) (W σ , Z) ⊗ W σ → Z is the evaluation map φ ⊗ w → φ (w). The space Z σ isv x ,w x ∈ E σ x , we define v x , w x := p −1 (x) v x (y) , w x (y) y,E dµ x (y) ,
where dµ x is the measure on p −1 (x) induced from the metric on F O . See [12] for a similar construction. Similarly, we define the bundle T ρ → F O G by
T ρ y = Γ π −1 (y) , E ρ , and T ρ → F O G is a Hermitian O (n)-equivariant bundle of finite rank. The metric on T ρ is v z , w z := π −1 (y) v z (y) , w z (y) z,E dm z (y) ,
where dm z is the measure on π −1 (z) induced from the metric on F O . The vector spaces of sections Γ (M, E σ ) and Γ (F O , E) σ can be identified via the isomorphism
i σ : Γ (M, E σ ) → Γ (F O , E) σ ,
where for any section s ∈ Γ (M, E σ ), s (x) ∈ Γ (p −1 (x) , E) σ for each x ∈ M, and we let
i σ (s) (f x ) := s (x)| fx for every f x ∈ p −1 (x) ⊂ F O . Then i −1 σ : Γ (F O , E) σ → Γ (M, E σ ) is given by i −1 σ (u) (x) = u| p −1 (x) .
Observe that i σ : Γ (M, E σ ) → Γ (F O , E) σ extends to an L 2 isometry. Given u, v ∈ Γ (M, E σ ),
u, v M = M u x , v x dx = M p −1 (x) u x (y) , v x (y) y,E dµ x (y) dx = M p −1 (x) i σ (u) , i σ (v) E dµ x (y) dx = F O i σ (u) , i σ (v) E = i σ (u) , i σ (v) F O ,
where dx is the Riemannian measure on M; we have used the fact that p is a Riemannian submersion. Similarly, we let
j ρ : Γ (F O G, T ρ ) → Γ (F O , E) ρ
be the natural identification, which extends to an L 2 isometry.
Let Γ (M, E σ ) α = eval (Hom G (V α , Γ (M, E σ )) ⊗ V α ) . Similarly, let Γ (F O G, T ρ ) β = eval (Hom G (W β , Γ (F O G, T ρ )) ⊗ W β ) .
Theorem 4.4. For any irreducible representations
ρ : G → U (V ρ ) and σ : O (n) → U (W σ ), the map j −1 ρ • i σ : Γ (M, E σ ) ρ → Γ (F O G, T ρ ) σ is an isomorphism (with inverse i −1 σ • j ρ ) that extends to an L 2 -isometry.
Exercise 51. Prove that if M is a Riemannian manifold, then the orthonormal frame bundle of M has trivial tangent bundle.
Exercise 52. Suppose that a compact Lie group acts smoothly on a Riemannian manifold. Prove that there exists a metric on the manifold such that the Lie group acts isometrically. Exercise 53. Let Z 2 act on T 2 = R 2 Z 2 with an action generated by (x, y) = (−x, y) for x, y ∈ R Z.
• Find the quotient space T 2 Z 2 . Exercise 54. Suppose that M = S 2 is the unit sphere in R 3 . Let S 1 act on S 2 by rotations around the x 3 -axis.
• Show that the oriented orthonormal frame bundle F SO can be identified with SO (3), which in turn can be identified with RP 3 . • Show that the lifted S 1 action on F SO can be realized by the orbits of a left-invariant vector field on SO (3). • Find the quotient F SO S 1 .
Exercise 55. Suppose that a compact, connected Lie group acts by isometries on a Riemannian manifold. Show that all harmonic forms are invariant under pullbacks by the group action.
4.2.
Dirac-type operators on the frame bundle. Let E → F O be a Hermitian vector bundle of Cl (NF ) modules that is equivariant with respect to the G × O (n) action. With notation as in previous sections, we have the transversal Dirac operator A N F defined by the composition
Γ (F O , E) ∇ → Γ (F O , T * F O ⊗ E) proj → Γ (F O , N * F ⊗ E) c → Γ (F O , E) .
As explained previously, the operator
D N F = A N F − 1 2 c (H)
is a essentially self-adjoint G × O (n)-equivariant operator, where H is the mean curvature vector field of the G-orbits in F O . From D N F we now construct equivariant differential operators on M and F O G, as follows. We define the operators Thus, questions about the transversally elliptic operator D σ M can be reduced to questions about the elliptic operators D ρ F O G for each irreducible ρ : G → U (V ρ ). In particular, we are interested in the equivariant index, which we will explain in detail the next section. In the following theorem, ind G (·) denotes the virtual representation-valued index as explained in [3] and in Section 5.1; the result is a formal difference of finite-dimensional representations if the input is a symbol of an elliptic operator.
D σ M := i −1 σ • D N F • i σ : Γ (M, E σ ) → Γ (M, E σ ) , and D ρ F O G := j −1 ρ • D N F • j ρ : Γ (F O G, T ρ ) → Γ (F O G, T ρ ) . For an irreducible representation α : G → U (V α ), let (D σ M ) α : Γ (M, E σ ) α → Γ (M, E σ ) α be
Transverse index theory for G-manifolds and Riemannian foliations
The research content and some of the expository content in this section are joint work with J. Brüning and F. W. Kamber, from [12] and [13]. The group G acts on Γ (M, E ± ) by (gs) (x) = g · s (g −1 x), and the (possibly infinitedimensional) subspaces ker (D) and ker (D * ) are G-invariant subspaces. Let ρ : G → U (V ρ ) be an irreducible unitary representation of G, and let χ ρ = tr (ρ) denote its character. Let Γ (M, E ± ) ρ be the subspace of sections that is the direct sum of the irreducible Grepresentation subspaces of Γ (M, E ± ) that are unitarily equivalent to the ρ representation. The operators D ρ : Γ M, E + ρ → Γ M, E − ρ can be extended to be Fredholm operators, so that each irreducible representation of G appears with finite multiplicity in ker D ± . Let a ± ρ ∈ Z + be the multiplicity of ρ in ker (D ± ). The virtual representation-valued index of D (see [3]) is
ind G (D) := ρ a + ρ − a − ρ [ρ] ,
where [ρ] denotes the equivalence class of the irreducible representation ρ. The index multiplicity is
ind ρ (D) := a + ρ − a − ρ = 1 dim V ρ ind D| Γ(M,E + ) ρ →Γ(M,E − ) ρ .
In particular, if 1 is the trivial representation of G, then
ind 1 (D) = ind D| Γ(M,E + ) G →Γ(M,E − ) G ,
where the superscript G implies restriction to G-invariant sections.
There is a relationship between the index multiplicities and Atiyah's equivariant distributionvalued index ind g (D) (see [3]); the multiplicities determine the distributional index, and vice versa. Because the operator D| Γ(M,E + ) ρ →Γ(M,E − ) ρ is Fredholm, all of the indices ind G (D) , ind g (D), and ind ρ (D) depend only on the homotopy class of the principal transverse symbol of D.
The new equivariant index result (in [12])is stated in Theorem 5.13. A large body of work over the last twenty years has yielded theorems that express ind g (D) in terms of topological and geometric quantities (as in the Atiyah-Segal index theorem for elliptic operators or the Berline-Vergne Theorem for transversally elliptic operators -see [4], [8], [9]). However, until now there has been very little known about the problem of expressing ind ρ (D) in terms of topological or geometric quantities which are determined at the different strata
M ([H]) :=
Gx∈[H]
x of the G-manifold M. The special case when all of the isotropy groups are finite was solved by M. Atiyah in [3], and this result was utilized by T. Kawasaki to prove the Orbifold Index Theorem (see [37]). Our analysis is new in that the equivariant heat kernel related to the index is integrated first over the group and second over the manifold, and thus the invariants in our index theorem (Theorem 5.13) are very different from those seen in other equivariant index formulas. The explicit nature of the formula is demonstrated in Theorem 5.14, a special case where the equivariant Euler characteristic is computed in terms of invariants of the G-manifold strata.
One of the primary motivations for obtaining an explicit formula for ind ρ (D) was to use it to produce a basic index theorem for Riemannian foliations, thereby solving a problem that has been open since the 1980s (it is mentioned, for example, in [19]). In fact, the basic index theorem (Theorem 5.16) is a consequence of the equivariant index theorem. We note that a recent paper of Gorokhovsky and Lott addresses this transverse index question on Riemannian foliations. Using a different technique, they are able to prove a formula for the basic index of a basic Dirac operator that is distinct from our formula, in the case where all the infinitessimal holonomy groups of the foliation are connected tori and if Molino's commuting sheaf is abelian and has trivial holonomy (see [23]). Our result requires at most mild topological assumptions on the transverse structure of the strata of the Riemannian foliation and has a similar form to the formula above for ind 1 (D). In particular, the analogue for the Gauss-Bonnet Theorem for Riemannian foliations (Theorem 5.17) is a corollary and requires no assumptions on the structure of the Riemannian foliation.
There are several new techniques used in the proof of the equivariant index theorem that have not been explored previously, and we will briefly describe them in upcoming sections. First, the proof requires a modification of the equivariant structure. In Section 5.2, we describe the known structure of G-manifolds. In Section 5.3, we describe a process of blowing up, cutting, and reassembling the G-manifold into what is called the desingularization. The result is a G-manifold that has less intricate structure and for which the analysis is more simple. We note that our desingularization process and the equivariant index theorem were stated and announced in [50] and [51]; recently Albin and Melrose have taken it a step further in tracking the effects of the desingularization on equivariant cohomology and equivariant K-theory ( [1]).
Another crucial step in the proof of the equivariant index theorem is the decomposition of equivariant vector bundles over G-manifolds with one orbit type. We construct a subbundle of an equivariant bundle over a G-invariant part of a stratum that is the minimal G-bundle decomposition that consists of direct sums of isotypical components of the bundle. We call this decomposition the fine decomposition and define it in Section 5.4. More detailed accounts of this method are in [12], [30].
Exercise 60. Let Z : C ∞ (S 1 , C) → {0} denote the zero operator on complex-valued functions on the circle S 1 . If we consider Z to be an S 1 -equivariant operator on the circle, find ind ρ (Z) for every irreducible representation ρ : S 1 → U (1). (Important: the target bundle is the zero vector bundle). We are assuming that S 1 acts by rotations.
Exercise 61. In Exercise 60, instead calculate each ind ρ (Z), if Z : C) be an operator on complex-valued functions on the circle S 1 = e iθ : θ ∈ R 2πZ . Consider D to be a Z 2 -equivariant operator, where the action is generated by θ → θ +π. Find ind ρ (Z) for every irreducible representation ρ : Z 2 → U (1).
C ∞ (S 1 , C) → C ∞ (S 1 , C) is multiplication by zero. Exercise 62. Let D = i d dθ : C ∞ (S 1 , C) → C ∞ (S 1 ,
Exercise 63. Let D = i d dθ : C ∞ (S 1 , C) → C ∞ (S 1 , C) be an operator on complex-valued functions on the circle S 1 = e iθ : θ ∈ R 2πZ . Consider the Z 2 action generated by θ → −θ. Show that D is not Z 2 -equivariant.
Exercise 64. Let Z 2 act on T 2 = R 2 Z 2 with an action generated by (x, y) = (−x, y) for x, y ∈ R Z. Calculate ind ρ (D) for every irreducible representation ρ : Z 2 → U (1), where D is the standard Dirac operator on the trivial C 2 bundle.
5.2.
Stratifications of G-manifolds. In the following, we will describe some standard results from the theory of Lie group actions (see [11], [36]). As above, G is a compact Lie group acting on a smooth, connected, closed manifold M. We assume that the action is effective, meaning that no g ∈ G fixes all of M. (Otherwise, replace G with G {g ∈ G : gx = x for all x ∈ M}.) Choose a Riemannian metric for which G acts by isometries.
Given such an action and x ∈ M, the isotropy or stabilizer subgroup G x < G is defined to be {g ∈ G : gx = x}. The orbit O x of a point x is defined to be {gx : g ∈ G}.
Since G xg = gG x g −1 , the conjugacy class of the isotropy subgroup of a point is fixed along an orbit.
On any such G-manifold, the conjugacy class of the isotropy subgroups along an orbit is called the orbit type. On any such G-manifold, there are a finite number of orbit types, and there is a partial order on the set of orbit types. Given
such that [G j ] < [G k ] (equivalently, such that M k M j )
. It is known that each stratum is a G-invariant submanifold of M, and in fact a minimal stratum is a closed (but not necessarily connected) submanifold. Also, for each j, the submanifold M ≥j :=
[G k ]≥[G j ] M k is a closed, G-invariant submanifold.
Now, given a proper, G-invariant submanifold S of M and ε > 0, let T ε (S) denote the union of the images of the exponential map at s for s ∈ S restricted to the open ball of radius ε in the normal bundle at S. It follows that T ε (S) is also G -invariant. If M j is a stratum and ε is sufficiently small, then all orbits in
T ε (M j ) \ M j are of type [G k ], where [G k ] < [G j ]
. This implies that if j < k, M j ∩ M k = ∅, and M k M j , then M j and M k intersect at right angles, and their intersection consists of more singular strata (with isotropy groups containing conjugates of both G k and G j ).
Fix ε > 0. We now decompose M as a disjoint union of sets M ε 0 , . . . , M ε r . If there is only one isotropy type on M, then r = 0, and we let M ε 0 = Σ ε 0 = M 0 = M. Otherwise, for j = r, r − 1, ..., 0, let ε j = 2 j ε, and let
Σ ε j = M j \ k>j M ε k ; M ε j = T ε j (M j ) \ k>j M ε k .
Thus,
T ε Σ ε j ⊂ M ε j , Σ ε j ⊂ M j .
The following facts about this decomposition are contained in [36, pp. 203ff]:
Lemma 5.1. For sufficiently small ε > 0, we have, for every i ∈ {0, . . . , r}:
(1) M = r i=0 M ε i (disjoint union).
(2) M ε i is a union of G-orbits; Σ ε i is a union of G-orbits. (3) The manifold M ε i is diffeomorphic to the interior of a compact G -manifold with corners; the orbit space M ε i G is a smooth manifold that is isometric to the interior of a triangulable, compact manifold with corners. The same is true for each Σ ε i .
(4) If [G j ] is the isotropy type of an orbit in M ε i , then j ≤ i and [G j ] ≤ [G i ]. (5)
The distance between the submanifold M j and M ε i for j > i is at least ε. Exercise 65. Suppose G and M are as above. Show that if γ is a geodesic that is perpendicular at x ∈ M to the orbit O x through x, then γ is perpendicular to every orbit that intersects it.
Exercise 66. With G and M as above, suppose that γ is a geodesic that is orthogonal to a particular singular stratum Σ. Prove that each element g ∈ G maps γ to another geodesic with the same property. Exercise 69. Let Z 2 ×Z 2 act on S 2 ⊂ R 3 via (x, y, z) → (−x, y, z) and (x, y, z) → (x, −y, z). Determine the strata of this action and the isotropy types.
Exercise 70. Let O (2) act on S 2 ⊂ R 3 by rotations that fix the z-axis. Determine the strata of this action and the isotropy types.
Exercise 71. Let M be the rectangle [0, 1]×[−1, 1] along with identifications (s, 1) ∼ (s, −1) for 0 ≤ s ≤ 1, (0, x) ∼ 0, x + 1 2 for 0 ≤ x ≤ 1 2 , and (1, x) ∼ 1, x + 1 2 for 0 ≤ x ≤ 1 2 .
• Show that M is a smooth Riemannian manifold when endowed with the standard flat metric. • Find the topological type of the surface M.
• Suppose that S 1 = R 2Z acts on M via (s, x) → (s, x + t), with x, t ∈ R 2Z . Find the strata and the isotropy types of this action.
Equivariant desingularization.
Assume that G is a compact Lie group that acts on a Riemannian manifold M by isometries. We construct a new G-manifold N that has a single stratum (of type [G 0 ]) and that is a branched cover of M, branched over the singular strata. A distinguished fundamental domain of M 0 in N is called the desingularization of M and is denoted M. We also refer to [1] for their recent related explanation of this desingularization (which they call resolution).
A sequence of modifications is used to construct N and M ⊂ N. Let M j be a minimal stratum. Let T ε (M j ) denote a tubular neighborhood of radius ε around M j , with ε chosen sufficiently small so that all orbits in
T ε (M j ) \ M j are of type [G k ], where [G k ] < [G j ]. Let N 1 = (M \ T ε (M j )) ∪ ∂Tε(M j ) (M \ T ε (M j ))
be the manifold constructed by gluing two copies of (M \ T ε (M j )) smoothly along the boundary (the codimension one case should be treated in a slightly different way; see [12] for details). Since the T ε (M j ) is saturated (a union of G-orbits), the G-action lifts to N 1 . Note that the strata of the G-action on N 1 correspond to strata in
M \ T ε (M j ). If M k ∩ (M \ T ε (M j ))
is nontrivial, then the stratum corresponding to isotropy type [G k ] on N 1 is
N 1 k = (M k ∩ (M \ T ε (M j ))) ∪ (M k ∩∂Tε(M j )) (M k ∩ (M \ T ε (M j )))
. Thus, N 1 is a G-manifold with one fewer stratum than M, and M \ M j is diffeomorphic to one copy of (M \ T ε (M j )), denoted M 1 in N 1 . In fact, N 1 is a branched double cover of M, branched over M j . If N 1 has one orbit type, then we set N = N 1 and M = M 1 . If N 1 has more than one orbit type, we repeat the process with the G-manifold N 1 to produce a new G-manifold N 2 with two fewer orbit types than M and that is a 4-fold branched cover of M. Definition 5.2. We denote X α = GX H α , and X α is called a component of X relative to G.
Remark 5.3. The space X α is not necessarily connected, but it is the inverse image of a connected component of G X under the projection X → G X. Also, note that X α = X β if there exists n in the normalizer N = N (H) such that nX H α = X H β . If X is a closed manifold, then there are a finite number of components of X relative to G.
We now introduce a decomposition of a G-bundle E → X. Let E α be the restriction E| X H α . For any irreducible representation σ : H → U (W σ ), we define for n ∈ N the representation σ n : H → U (W σ ) by σ n (h) = σ (n −1 hn).
n : E [σ] α ∼ = −→ E [σ n ] n(α) ,
where n denotes the residue class class of n ∈ N in N N α, [σ] . Then a decomposition of E is obtained by 'inducing up' the isotypical components E If G X is not connected, one must construct the fine components separately over each X α . If E has finite rank, then E may be decomposed as a direct sum of distinct fine components over each X α . In any case, E N α,[σ] is a finite direct sum of isotypical components over each X H α . Definition 5.5. The direct sum decomposition of E| Xα into subbundles E b that are fine components E G α, [σ] for some [σ], written
[σ] α from N α,[σ] to N. That is, E N α,[σ] = N × N α,[σ] E [σ] α is a bundle containing E [σ] α X H α .E| Xα = b E b ,
is called the refined isotypical decomposition (or fine decomposition) of E| Xα .
We comment that if [σ,
W σ ] is an irreducible H-representation present in E x with x ∈ X H α , then E [σ]
x is a subspace of a distinct E b
x for some b. x have the same multiplicity and dimension if nX H α = X H α . Remark 5.7. The advantage of this decomposition over the isotypical decomposition is that each E b is a G-bundle defined over all of X α , and the isotypical decomposition may only be defined over X H α . Definition 5.8. Now, let E be a G-equivariant vector bundle over X, and let E b be a fine component as in Definition 5.4 corresponding to a specific component X α = GX H α of X relative to G. Suppose that another G-bundle W over X α has finite rank and has the property that the equivalence classes of G y -representations present in E b y , y ∈ X α exactly coincide with the equivalence classes of G y -representations present in W y , and that W has a single component in the fine decomposition. Then we say that W is adapted to E b . Lemma 5.9. In the definition above, if another G-bundle W over X α has finite rank and has the property that the equivalence classes of G y -representations present in E b y , y ∈ X α exactly coincide with the equivalence classes of G y -representations present in W y , then it follows that W has a single component in the fine decomposition and hence is adapted to E b . Thus, the last phrase in the corresponding sentence in the above definition is superfluous.
Exercise 75. Suppose that X is a G-manifold, H is an isotropy subgroup, and E ′ → X H is an N (H)-bundle over the fixed point set X H . Prove that E ′ uniquely determines a G-bundle E over X such that E| X H = E ′ .
Exercise 76. Prove Lemma 5.9.
Canonical isotropy G-bundles.
In what follows, we show that there are naturally defined finite-dimensional vector bundles that are adapted to any fine components. Once and for all, we enumerate the irreducible representations ρ j , V ρ j j=1,2,... of G. Let [σ, W σ ] be any irreducible H-representation. Let G × H W σ be the corresponding homogeneous vector bundle over the homogeneous space G H. Then the L 2 -sections of this vector bundle decompose into irreducible G-representations. In particular, let ρ j 0 , V ρ j 0 be the equivalence class of irreducible representations that is present in L 2 (G H, G × H W σ ) and that has the lowest index j 0 . Then Frobenius reciprocity implies so that the restriction of ρ j 0 to H contains the H-representation [σ]. Now, for a component X H α of X H , with X α = GX H α its component in X relative to G, the trivial bundle Exercise 78. Suppose G is a compact, connected Lie group, and T is a maximal torus. Let G act on left on the homogeneous space X = G T .
X α × V ρ j 0 is a G-bundle (with diagonal action) that contains a nontrivial fine component W α,[σ] con- taining X H α × V ρ j 0 [σ] .W σ , V ρ j 0 (the associated mul- tiplicity), independent of the choice of [σ, W σ ] present in W α,[σ],x , x ∈ X H α (see Remark 5.6). • d α,[σ] = dim W σ (the associated representation dimension), independent of the choice of [σ, W σ ] present in W α,[σ],x , x ∈ X H α . • n α,[σ] = rank(W α,[σ]) m α,[σ] d α,[σ] (the inequivalence number), the number of inequivalent repre- sentations present in W α,[σ],x , x ∈ X H α . Remark 5.11. Observe that W α,[σ] = W α ′ ,[σ ′ ] if [σ ′ ] = [σ n ] for some n ∈ N such that nX H α = X H α ′ .
• What is (G T ) T ?
• Let σ a be a fixed irreducible representation of T (on C), say σ a (t) = exp (2πi (a · t)) with a ∈ Z m , m = rank (T ). Let E = G × σa C → G T be the associated line bundle. Is E a canonical isotropy G-bundle associated to (·, [σ a ])? • Is it true that every complex G-bundle over G T is a direct sum of equivariant line bundles?
5.6. The equivariant index theorem. To evaluate ind ρ (D), we first perform the equivariant desingularization as described in Section 5.3, starting with a minimal stratum. In [12], we precisely determine the effect of the desingularization on the operators and bundles, and in turn the supertrace of the equivariant heat kernel. We obtain the following result. In what follows, if U denotes an open subset of a stratum of the action of G on M, U ′ denotes the equivariant desingularization of U, and U denotes the fundamental domain of U inside U ′ , as in Section 5.3. We also refer the reader to Definitions 5.2 and 5.10. transversally elliptic, G-equivariant differential operator. We assume that near each Σ α j , D is G-homotopic to the product D N * D α j , where D N is a G-equivariant, first order differential operator on B ε Σ that is elliptic and has constant coefficients on the fibers and D α j is a global transversally elliptic, G-equivariant, first order operator on the Σ α j . In polar coordinates
D N = Z j ∇ E ∂r + 1 r D S j ,
where r is the distance from Σ α j , where Z j is a local bundle isometry (dependent on the spherical parameter), the map D S j is a family of purely first order operators that differentiates in directions tangent to the unit normal bundle of Σ j . Then the equivariant index ind ρ (D) satisfies
ind ρ (D) = G M 0 A ρ 0 (x) |dx| + r j=1 β Σ α j , β Σ α j = 1 2 dim V ρ b∈B 1 n b rank W b −η D S+,b j + h D S+,b j G Σα j A ρ j,b (x) |dx| , where (1) A ρ 0 (x)
is the Atiyah-Singer integrand, the local supertrace of the ordinary heat kernel associated to the elliptic operator induced from D ′ (blown-up and doubled from D) on the quotient M ′ 0 G, where the bundle E is replaced by the finite rank bundle E ρ of sections of type ρ over the fibers.
(2) Similarly, A ρ i,b is the local supertrace of the ordinary heat kernel associated to the elliptic operator induced from (1 ⊗ D α j ) ′ (blown-up and doubled from 1 ⊗ D α j , the twist of D α j by the canonical isotropy bundle W b → Σ α j ) on the quotient Σ ′ α j G, where the bundle is replaced by the space of sections of type ρ over each orbit. x ; see [12]. This is constant on Σ α j . x , again constant on on Σ α j .
(5) n b is the number of different inequivalent G x -representation types present in each W b x , x ∈ Σ α j .
As an example, we immediately apply the result to the de Rham operator and in doing so obtain an interesting equation involving the equivariant Euler characteristic. In what follows, let L N j → Σ j be the orientation line bundle of the normal bundle to the singular stratum Σ j . The relative Euler characteristic is defined for X a closed subset of a manifold Y as χ (Y, X, V) = χ (Y, V) − χ (X, V), which is also the alternating sum of the dimensions of the relative de Rham cohomology groups with coefficients in a complex vector bundle V → Y . If V is an equivariant vector bundle, the equivariant Euler characteristic χ ρ (Y, V) associated to the representation ρ : G → U (V ρ ) is the alternating sum
χ ρ (Y, V) = j (−1) j dim H j (Y, V) ρ ,
where the superscript ρ refers to the restriction of these cohomology groups to forms of Grepresentation type [ρ]. An application of the equivariant index theorem yields the following result.
Theorem 5.14. (Equivariant Euler Characteristic Theorem, in [12]) Let M be a compact Gmanifold, with G a compact Lie group and principal isotropy subgroup H pr . Let M 0 denote the principal stratum, and let Σ α 1 ,...,Σ αr denote all the components of all singular strata relative to G. We use the notations for χ ρ (Y, X) and χ ρ (Y ) as in the discussion above. Then
χ ρ (M) = χ ρ (G H pr ) χ (G M, G singular strata) + j χ ρ G G j , L N j χ G Σ α j , G lower strata ,
where L N j is the orientation line bundle of normal bundle of the stratum component Σ α j .
Exercise 79. Let M = S n , let G = O (n) acting on latitude spheres (principal orbits, diffeomorphic to S n−1 ). Show that there are two strata, with the singular strata being the two poles. Show without using the theorem by identifying the harmonic forms that
χ ρ (S n ) = (−1) n if ρ = ξ 1 if ρ = 1 ,
where ξ is the induced one dimensional representation of O (n) on the volume forms.
Exercise 80. In the previous example, show that
χ ρ (G H pr ) = χ ρ S n−1 = (−1) n−1 if ρ = ξ 1 if ρ = 1 ,
and χ (G M, G singular strata) = −1. Show that at each pole,
χ ρ G G j , L N j = χ ρ (pt) = 1 if ρ = 1, 0 otherwise. ,
and χ G Σ α j , G lower strata = 1. Demonstrate that Theorem 5.14 produces the same result as in the previous exercise.
Exercise 81. If instead the group Z 2 acts on S n by the antipodal map, prove that
χ ρ (S n ) = 0 if ρ = 1
or ξ and n is odd 1 if ρ = 1 or ξ and n is even 0 otherwise , both by direct calculation and by using Theorem 5.14.
Exercise 82. Consider the action of Z 4 on the flat torus T 2 = R 2 Z 2 , where the action is generated by a π 2 rotation. Explicitly, k ∈ Z 4 acts on
y 1 y 2 by φ (k) y 1 y 2 = 0 −1 1 0 k y 1 y 2 .
Endow T 2 with the standard flat metric. Let ρ j be the irreducible character defined by k ∈ Z 4 → e ikjπ/2 . Prove that
χ 1 T 2 = 2, χ ρ 1 T 2 = χ ρ 3 T 2 = −1, χ ρ 2 T 2 = 0,
in two different ways. First, compute the dimensions of the spaces of harmonic forms to determine the equations. Second, use the Equivariant Euler Characteristic Theorem.
The basic index theorem for Riemannian foliations.
Suppose that E is a foliated Hermitian Cl (Q) module with metric basic Cl (Q) connection ∇ E over a Riemannian foliation (M, F ). Let D E b : Γ b E + → Γ b E − be the associated basic Dirac operator, as explained in Section 3.1.
In the formulas below, any lower order terms that preserve the basic sections may be added without changing the index. Note that
Definition 5.15. The analytic basic index of D E b is ind b D E b = dim ker D E b − dim ker D E b * .
As shown explicitly in [13], these dimensions are finite, and it is possible to identify
ind b D E b
with the invariant index of a first order, G-equivariant differential operator D over a vector bundle over a basic manifold W , where G is SO (q), O (q), or the product of one of these with a unitary group U (k). By applying the equivariant index theorem (Theorem 5.13) to the case of the trivial representation, we obtain the following formula for the index. In what follows, if U denotes an open subset of a stratum of (M, F ), U ′ denotes the desingularization of U very similar to that in Section 5.3, and U denotes the fundamental domain of U inside U ′ . (4) n τ is the number of different inequivalent G j -representation types present in a typical fiber of W τ .
An example of this result is the generalization of the Gauss-Bonnet Theorem to the basic Euler characteristic. Recall from Section 3.1 that the basic forms Ω (M, F ) are preserved by the exterior derivative, and the resulting cohomology is called basic cohomology H * (M, F ). The basic cohomology groups are finite-dimensional in the Riemannian foliation, and the basic Euler characteristic is defined to be
χ (M, F ) = (−1) j dim H j (M, F ) .
We have two independent proofs of the following Basic Gauss-Bonnet Theorem; one proof uses the result in [7], and the other proof is a direct consequence of the basic index theorem stated above (proved in [13]). We express the basic Euler characteristic in terms of the ordinary Euler characteristic, which in turn can be expressed in terms of an integral of curvature. We extend the Euler characteristic notation χ (Y ) for Y any open (noncompact without boundary) or closed (compact without boundary) manifold to mean
χ (Y ) = χ (Y ) if Y is closed χ (1-point compactification of Y ) − 1 if Y is open
Also, if L is a foliated line bundle over a Riemannian foliation (X, F ), we define the basic Euler characteristic χ (X, F , L) as before, using the basic cohomology groups with coefficients in the line bundle L. Remark 5.18. In [23, Corollary 1], they show that in special cases the only term that appears is one corresponding to a most singular stratum.
We now investigate some examples through exercises. The first example is a codimension 2 foliation on a 3-manifold. Here, O(2) acts on the basic manifold, which is homeomorphic to a sphere. In this case, the principal orbits have isotropy type ({e}), and the two fixed points obviously have isotropy type (O(2)). In this example, the isotropy types correspond precisely to the infinitesimal holonomy groups.
Exercise 83. (From [48], [52], and [13]) Consider the one dimensional foliation obtained by suspending an irrational rotation on the standard unit sphere S 2 . On S 2 we use the cylindrical coordinates (z, θ), related to the standard rectangular coordinates by x ′ = (1 − z 2 ) cos θ, y ′ = (1 − z 2 ) sin θ, z ′ = z. Let α be an irrational multiple of 2π, and let the three-manifold M = S 2 × [0, 1] / ∼, where (z, θ, 0) ∼ (z, θ + α, 1). Endow M with the product metric on T z,θ,t M ∼ = T z,θ S 2 × T t R. Let the foliation F be defined by the immersed submanifolds L z,θ = ∪ n∈Z {z} × {θ + α} × [0, 1] (not unique in θ). The leaf closures L z for |z| < 1 are two dimensional, and the closures corresponding to the poles (z = ±1) are one dimensional.
Show that χ (M, F ) = 2, using a direct calculation of the basic cohomology groups and also by using the Basic Gauss-Bonnet Theorem.
The next example is a codimension 3 Riemannian foliation for which all of the infinitesimal holonomy groups are trivial; moreover, the leaves are all simply connected. There are leaf closures of codimension 2 and codimension 1. The codimension 1 leaf closures correspond to isotropy type (e) on the basic manifold, and the codimension 2 leaf closures correspond to an isotropy type (O(2)) on the basic manifold. In some sense, the isotropy type measures the holonomy of the leaf closure in this case.
Exercise 84. (From [13]) This foliation is a suspension of an irrational rotation of S 1 composed with an irrational rotation of S 2 on the manifold S 1 × S 2 . As in Example 83, on S 2 we use the cylindrical coordinates (z, θ), related to the standard rectangular coordinates by x ′ = (1 − z 2 ) cos θ, y ′ = (1 − z 2 ) sin θ, z ′ = z. Let α be an irrational multiple of 2π, and let β be any irrational number. We consider the four-manifold M = S 2 × [0, 1] × [0, 1] / ∼, where (z, θ, 0, t) ∼ (z, θ, 1, t), (z, θ, s, 0) ∼ (z, θ + α, s + β mod 1, 1). Endow M with the product metric on T z,θ,s,t M ∼ = T z,θ S 2 × T s R × T t R. Let the foliation F be defined by the immersed submanifolds L z,θ,s = ∪ n∈Z {z} × {θ + α} × {s + β} × [0, 1] (not unique in θ or s). The leaf closures L z for |z| < 1 are three-dimensional, and the closures corresponding to the poles (z = ±1) are two-dimensional. By computing the basic forms of all degrees, verify that the basic Euler characteristic is zero. Next, use the Basic Gauss-Bonnet Theorem to see the same result.
The following example is a codimension two transversally oriented Riemannian foliation in which all the leaf closures have codimension one. The leaf closure foliation is not transversally orientable, and the basic manifold is a flat Klein bottle with an O(2)-action. The two leaf closures with Z 2 holonomy correspond to the two orbits of type (Z 2 ), and the other orbits have trivial isotropy.
Exercise 85. This foliation is the suspension of an irrational rotation of the flat torus and a Z 2 -action. Let X be any closed Riemannian manifold such that π 1 (X) = Z * Z , the free group on two generators {α, β}. We normalize the volume of X to be 1. Let X be the universal cover. We define M = X × S 1 × S 1 π 1 (X), where π 1 (X) acts by deck transformations on X and by α (θ, φ) = (2π − θ, 2π − φ) and β (θ, φ) = θ, φ + √ 2π on S 1 × S 1 . We use the standard product-type metric. The leaves of F are defined to be sets of the form (x, θ, φ) ∼ | x ∈ X . Note that the foliation is transversally oriented. Show that the basic Euler characteristic is 2, in two different ways.
The following example (from [14]) is a codimension two Riemannian foliation that is not taut.
Exercise 86. For the example in Exercise 44, show that the basic manifold is a torus, and the isotropy groups are all trivial. Verify that χ (M, F ) = 0 in two different ways.
1. 1 .
1The Laplacian. The Laplace operator (or simply, Laplacian) is the famous differential operator ∆ on R n ∈ C ∞ (R n )The solutions to the equation ∆h = 0 are the harmonic functions. This operator is present in both the heat equation and wave equations of physics.
Exercise 22 .
22Prove that if D = c (e j ) ∇ e j is a Dirac operator, then σ (D) (ξ) = ic ξ # and σ D 2 (ξ) = ξ 2 for all ξ ∈ T * M. Use the coordinate-free definition. Exercise 23. Show that if ω is a one-form on M, then (δω) (x) = − n j=1 e j (ω (e j )) (x) , if (e 1 , ..., e n ) is a local orthonormal frame of T M chosen so that ∇ e j e k (x) = 0 at x ∈ M, for every j, k ∈ {1, ..., n}. 1.4. The Atiyah-Singer Index Theorem. Given Banach spaces S and T , a bounded linear operator L : S → T is called Fredholm if its range is closed and its kernel and cokernel T L (S) are finite dimensional. The index of such an operator is defined to be ind (L) = dim ker (L) − dim coker (L) , and this index is constant on continuous families of such L. In the case where S and T are Hilbert spaces, this is the same as ind (L) = dim ker (L) − dim ker (L * ) . The index determines the connected component of L in the space of Fredholm operators. We will be specifically interested in this integer in the case where L is a Dirac operator. For the case of the de Rham operator, we have ker (d + δ) = ker (d + δ) 2 = ker ∆, so that H r (M) = ker ( (d + δ)| Ω r ) is the space of harmonic forms of degree r, which by the Hodge theorem is isomorphic to H r (M), the r th de Rham cohomology group. Therefore, index ind ( (d + δ)| Ω even ) : = dim ker ( (d + δ)| Ω even ) − dim ker ( (d + δ)| Ω even ) * = dim ker ( (d + δ)| Ω even ) − dim ker ( (d + δ)| Ω odd ) = χ (M) , the Euler characteristic of M. In general, suppose that D is an elliptic operator of order m on sections of a vector bundle E ± over a smooth, compact manifold M. Let H s (Γ (M, E ± )) denote the Sobolev s-norm completion of the space of sections Γ (M, E), with respect to a chosen metric. Then D can be extended to be a bounded linear operator D s : H s (Γ (M, E + )) → H s−m (Γ (M, E − )) that is Fredholm, and ind (D) := ind D s is well-defined and independent of s. In the 1960s, the researchers M. F. Atiyah and I. Singer proved that the index of an elliptic operator on sections of a vector bundle over a smooth manifold satisfies the following formula ([5], [6]):
Exercise 30 .
30Find an example of a smooth closed manifold M such that every possible metric on M must have nonzero L (the Hirzebruch L-polynomial applied to the Pontryagin forms).
Remark 2. 1 .
1For a given distribution Q ⊂ T M, it is always possible to obtain a bundle of Cl (Q)-modules with Clifford connection from a bundle of Cl (T M)-Clifford modules, but not all such Cl (Q) connections are of that type.
Theorem 2. 2 .
2(in[45]) For each distribution Q ⊂ T M and every bundle E of Cl (Q)modules, the transversally elliptic operator D Q defined by (2.1) and (2.3) is essentially selfadjoint.
Exercise 36 .
36(This example is in the paper[45].)We consider the torus M = (R 2πZ) 2 with the metric e 2g(y) dx 2 + dy 2 for some 2π-periodic smooth function g.Consider the orthogonal distributions L = span {∂ y } and Q = span {∂ x }. Let E be the trivial complex line bundle over M, and let Cl (Q) and Cl (L) both act on E via c (∂ y ) = i = c e −g(y) ∂ x . Show that the mean curvatures of these distributions are H Q = −g ′ (y) ∂ y and H L = 0 From formulas (2.1) and (2.3),
Theorem 3.1. (In[25] ) Let (M, F ) be a compact Riemannian manifold endowed with a Riemannian foliation and basic Clifford bundle E → M. The spectrum of the basic Dirac operator is the same for every possible choice of bundle-like metric that is associated to the transverse metric on the quotient bundle Q.
Exercise 38 .
38Suppose that S is a closed subspace of a Hilbert space H, and let L : H → H be a bounded linear map such that L (S) ⊆ S. Let L S denote the restriction L S : S → S defined by L S (v) = L (v) for all v ∈ S. Prove that the adjoint of L S satisfies L * S (v) = P S L * (v), where L * is the adjoint of L and P S is the orthogonal projection of H onto S. Show that the maximal subspace W ⊆ S such that L * S
Lemma 3. 3 .
3The maps d and δ are differentials; that is, d 2 = 0, δ 2 = 0. As a result, d and δ commute with ∆ = D 2 b , and ker d + δ = ker ∆ .LetΩ k (M, F ) denote the space of basic k-forms (either set of smooth forms or L 2completion thereof), let d k and δ b k be the restrictions of d and δ b to k-forms, and let ∆ k denote the restriction of D 2 b to basic k-forms. Proposition 3.4. (Hodge decomposition) We have
Theorem 3. 7 .
7(Conformal invariance of cohomology groups) Given a Riemannian foliation (M, F , g Q ) and two bundle-like metrics g M and g ′ M compatible with g Q , the d-cohomology groups H k (M, F ) are isomorphic, and that isomorphism is implemented by multiplication by a positive basic function. Further, the eigenvalues of the corresponding basic de Rham operators D b and D ′ b are identical, and the eigenspaces are isomorphic via multiplication by that same positive function.
Corollary 3. 8 .
8The dimensions of H k (M, F ) and the eigenvalues of D b (and thus of ∆ = D 2 b ) are invariants of the Riemannian foliation structure (M, F , g Q ), independent of choice of compatible bundle-like metric g M . Exercise 41. Show that if α is any closed form, then (d + α∧) 2 = 0.Exercise 42. Show that δ is the formal adjoint of d.
Exercise 43 .
43Show that a Riemannian foliation (M, F ) is taut if and only if H 0 (M, F ) is nonzero. (A Riemannian foliation is taut if there exists a bundle-like metric for which the leaves are minimal submanifolds and thus have zero mean curvature.) Exercise 44. (This example is contained in
Corollary 3. 10 .
10Let (M, F ) be a smooth transversally oriented foliation of odd codimension that admits a transverse Riemannian structure. Then the Euler characteristic associated to the H * (M, F ) vanishes.
Corollary 3 . 11 .
311Let (M, F ) be a smooth transversally oriented foliation of odd codimension that admits a transverse Riemannian structure. Then the Euler characteristic associated to the ordinary basic cohomology H * (M, F ) vanishes.
meets that criteria, and D 1 = D b is the basic de Rham operator D b : Ω even (M, F ) → Ω odd (M, F ). Thus, the basic Euler characteristic of the basic cohomology complex is the same as the basic Euler characteristic of the d-cohomology complex. The result follows from the previous corollary.Exercise 45. Prove that the twisted cohomology class [κ b ] is always trivial. Exercise 46. Prove that if (M, F ) is not taut, then the ordinary basic cohomology satisfies dim H 1 (M, F ) ≥ 1. Exercise 47. Prove that there exists a monomorphism from H 1 (M, F ) to H 1 (M). Exercise 48. True or false: dim H 2 (M, F ) ≤ dim H 2 (M).Exercise 49. Under what conditions is it true thatdim H 1 (M, F ) ≥ dim H 1 (M, F ) ?Exercise 50. (Hard) Find an example of a Riemannian foliation that is not taut and whose twisted basic cohomology is nontrivial. (If you give up, find an answer in[26].)4. Natural examples of transversal Dirac operators on G-manifoldsThe research content of this section is joint work with I. Prokhorenkov, from[45].4.1.Equivariant structure of the orthonormal frame bundle. We first make the important observation that if a Lie group acts effectively by isometries on a Riemannian manifold, then this action can be lifted to a free action on the orthonormal frame bundle. Given a complete, connected G-manifold, the action of g ∈ G on M induces an action of dg on T M, which in turn induces an action of G on the principal O (n)-bundle F O p → M of orthonormal frames over M.
Lemma 4. 1 .
1The action of G on F O is regular, and the isotropy subgroups corresponding to any two points of F O are the same.
F O are diffeomorphic and form a Riemannian fiber bundle, in the natural metric on F O defined as follows. The Levi-Civita connection on M determines the horizontal subbundle H of T F O . We construct the local product metric on F O using a biinvariant fiber metric and the pullback of the metric on M to H; with this metric, F O is a compact Riemannian G × O (n)-manifold. The lifted G-action commutes with the O (n)action. Let F denote the foliation of G-orbits on F O , and observe that F O π → F O G = F O F is a Riemannian submersion of compact O (n)-manifolds. Let E → F O be a Hermitian vector bundle that is equivariant with respect to the G×O (n) action. Let ρ : G → U (V ρ ) and σ : O (n) → U (W σ ) be irreducible unitary representations. We define the bundle E σ → M by
the vector subspace of Z on which O (n) acts as a direct sum of representations of type σ. The bundle E σ is a Hermitian G-vector bundle of finite rank over M. The metric on E σ is chosen as follows. For any
•
Find all the irreducible representations of Z 2 . (Hint: they are all homomorphisms ρ : Z 2 → U (1).) • Find the orthonormal frame bundle F O , and determine the induced action of Z 2 on F O . • Find the quotient space F O Z 2 , and determine the induced action of O (2) on this manifold.
the restriction of D σ M to sections of G-representation type [α]. Similarly, for an irreducible representation β : G → U (W β ), let D ρ F O G β : Γ (F O G, T ρ ) β → Γ (F O G, T ρ ) β be the restriction of D ρ F O G to sections of O (n)-representation type [β]. The proposition below follows from Theorem 4.4.
Proposition 4 . 5 .
45The operator D σ M is transversally elliptic and G-equivariant, and D ρ F O G is elliptic and O (n)-equivariant, and the closures of these operators are self-adjoint. The operators (D σ M ) ρ and D ρ F O G σ have identical discrete spectrum, and the corresponding eigenspaces are conjugate via Hilbert space isomorphisms.
Theorem 4 . 6 .
46Suppose that F O is G-transversally spin c . Then for every transversally elliptic symbol class [u] ∈ K cpt,G (T * G M), there exists an operator of type D 1 M such that ind G (u) = ind G (D 1 M ). Exercise 56. (continuation of Exercise 53) • Determine a Dirac operator on the trivial C 2 bundle over the three-dimensional F O . (Hint: use the Dirac operator from R 3 .) • Find the induced operator D 1 T 2 on T 2 , where 1 denotes the trivial representation 1 : O (2) → {1} ∈ U (1). This means the restriction of the Dirac operator of F O to sections that are invariant under the O (2) action. • Identify all irreducible unitary representations of O (2). (Hint: they are all homomorphisms σ : O (2) → U (1).) • Find ker D 1 T 2 and ker D 1 * T 2 , and decompose these vector spaces as direct sums of irreducible unitary representations of O (2). • For each irreducible unitary representation ρ : Z 2 → U (1) of Z 2 , determine the induced operator D ρ F O Z 2 . Exercise 57. (continuation of Exercise 54) • Starting with a transversal Dirac operator on the trivial C 2 bundle over SO (3), find the induced operator D 1 S 2 on S 2 . • Identify all irreducible unitary representations of S 1 . • Find ker D 1 S 2 and ker D 1 * S 2 , and decompose these vector spaces as direct sums of irreducible unitary representations of S 1 . Exercise 58. (continuation of Exercise 55) Suppose that a compact, connected Lie group G acts on a Riemannian manifold. Show that the ker (d + δ) is the same as G-invariant part ker (d + δ) 1 of ker (d + δ), and ker (d + δ) ρ = 0 for all other irreducible representations ρ : G → U (V ρ ).Exercise 59. Suppose that G = M acts freely on itself. Construct a transversal Dirac operator acting on a trivial spinor bundle on the orthonormal frame bundle. Determine the operator D 1 G for 1 being the trivial representation of O (n), and find ker D 1 G and ker D 1 * G .
5. 1 .
1Introduction: the equivariant index. Suppose that a compact Lie group G acts by isometries on a compact, connected Riemannian manifold M, and let E = E + ⊕ E − be a graded, G-equivariant Hermitian vector bundle over M. We consider a first order Gequivariant differential operator D = D + : Γ (M, E + ) → Γ (M, E − ) which is transversally elliptic (as explained at the beginning of Section 53). Let D − be the formal adjoint of D + .
Exercise 67 .
67Prove that if S is any set of isometries of a Riemannian manifold M, then the fixed point set M S := {x ∈ M : gx = x for every g ∈ S} is a totally geodesic submanifold of M. Exercise 68. Prove that if Σ is a stratum of the action of G on M corresponding to isotropy type [H], then the fixed point set Σ H is a principal N (H) H bundle over G Σ, where N (H) is the normalizer of the subgroup H.
Again, M 2 is a fundamental domain of M 1 \ {a minimal stratum}, which is a fundamental domain of M with two strata removed. We continue until N = N r is a G-manifold with all orbits of type [G 0 ] and is a 2 r -fold branched cover of M, branched over M \ M 0 . We set M = M r , which is a fundamental domain of M 0 in N.
Further, one may independently desingularize M ≥j , since this submanifold is itself a closed G-manifold. If M ≥j has more than one connected component, we may desingularize all components simultaneously. The isotropy type of all points of M ≥j is [G j ], and M ≥j G is a smooth (open) manifold.Exercise 72. Find the desingularization M j of each stratum M j for the G-manifold in Exercise 69. Exercise 73. Find the desingularization M j of each stratum M j for the G-manifold in Exercise 70. Exercise 74. Find the desingularization M j of each stratum M j for the G-manifold in Exercise 71. 5.4. The fine decomposition of an equivariant bundle. Let X H be the fixed point set of H in a G-manifold X with one orbit type [H]. For α ∈ π 0 X H , let X H α denote the corresponding connected component of X H .
Lemma 5 . 12 .
512Given any G-bundle E → X and any fine component E b of E over some X α = GX H α , there exists a canonical isotropy G-bundle W α,[σ] adapted to E b → X α . Exercise 77. Prove Lemma 5.12.
Theorem 5 .
513. (Equivariant Index Theorem, in [12]) Let M 0 be the principal stratum of the action of a compact Lie group G on the closed Riemannian M, and let Σ α 1 ,...,Σ αr denote all the components of all singular strata relative to G. Let E → M be a Hermitian vector bundle on which G acts by isometries. Let D : Γ (M, E + ) → Γ (M, E − ) be a first order,
eta invariant of the operator D S+ j induced on any unit normal sphere S x Σ α j , restricted to sections of isotropy representation types in W b
dimension of the kernel of D S+,b j , restricted to sections of isotropy representation types in W b
Theorem 5 .
516. (Basic Index Theorem for Riemannian foliations, in [13]) Let M 0 be the principal stratum of the Riemannian foliation (M, F ), and let M 1 , ... , M r denote all the components of all singular strata, corresponding to O (q)-isotropy types [G 1 ], ... ,[G r ] on the basic manifold. With notation as in the discussion above, we have b (x) |dx|, where the sum is over all components of singular strata and over all canonical isotropy bundles W τ , only a finite number of which yield nonzero A τ j,b , and where (1) A 0,b (x) is the Atiyah-Singer integrand, the local supertrace of the ordinary heat kernel associated to the elliptic operator induced from D E b (a desingularization of D E b ) on the quotient M 0 F, where the bundle E is replaced by the space of basic sections of over each leaf closure; in a similar way as in Theorem 5.13, using a decomposition D E b = D N * D M j at each singular stratum; (3) A τ j,b (x) is the local supertrace of the ordinary heat kernel associated to the elliptic operator induced from 1 ⊗ D M j ′ (blown-up and doubled from 1 ⊗ D M j , the twist of D M j by the canonical isotropy bundle W τ ) on the quotient M j F, where the bundle is replaced by the space of basic sections over each leaf closure; and
Theorem 5 .
517. (Basic Gauss-Bonnet Theorem, announced in [49], proved in [13]) Let (M, F ) be a Riemannian foliation. Let M 0 ,..., M r be the strata of the Riemannian foliation (M, F ), and let O M j F denote the orientation line bundle of the normal bundle to F in M j . Let L j denote a representative leaf closure in M j . With notation as above, the basic Euler characteristic satisfies χ (M, F ) = j χ M j F χ L j , F , O M j F .
subgroups H and K of G, we say that [H] ≤ [K] if H is conjugate to a subgroup of K, and we say [H] < [K] if [H] ≤ [K] and [H] = [K]. We may enumerate the conjugacy classes of isotropy subgroups as [G 0 ] , ..., [G r ] such that [G i ] ≤ [G j ] implies that i ≤ j. It is well-known that the union of the principal orbits (those with type [G 0 ]) form an open dense subset M 0 of the manifold M, and the other orbits are called singular. As a consequence, every isotropy subgroup H satisfies [G 0 ] ≤ [H]. Let M j denote the set of points of M of orbit type [G j ] for each j; the set M j is called the stratum corresponding to [G j ]. If [G j ] ≤ [G k ], it follows that the closure of M j contains the closure of M k . A stratum M j is called a minimal stratum if there does not exist a stratum M k
Let N [σ] = {n ∈ N : [σ n ] is equivalent to [σ] } . If the isotypical component E [σ] α is nontrivial, then it is invariant under the subgroup N α,[σ] ⊆ N [σ] that leaves in addition the connected component X H α invariant; again, this subgroup has finite index in N. The isotypical components transform under n ∈ N as
over the submanifold X α is called a fine component or the fine component of E → X associated to (α, [σ]).This is an N-bundle over NX H
α ⊆ X H , and a similar bundle
may be formed over each distinct NX H
β , with β ∈ π 0 X H . Further, observe that since each
bundle E N
α,[σ] is an N-bundle over NX H
α , it defines a unique G bundle E G
α,[σ] (see Exercise
75).
Definition 5.4. The G-bundle E G
α,[σ]
= Hom G V ρ j 0 , L 2 (G H, G × H W σ ) ∼ = Hom H V Res(ρ j 0 ) , W σ ,
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. E-Mail Address, K , Richardson: [email protected] address, K. Richardson: [email protected]
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[
"Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders",
"Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders"
]
| [
"Kiyoshi Kanazawa \nInstitute of Innovative Research\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan\n\nDepartment of Mathematical and Computing Science\nSchool of Computing\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan\n",
"Takumi Sueshige \nDepartment of Mathematical and Computing Science\nSchool of Computing\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan\n",
"Hideki Takayasu \nInstitute of Innovative Research\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan\n\nSony Computer Science Laboratories\n3-14-13 Higashi-Gotanda, Shinagawa-ku141-0022TokyoJapan\n",
"Misako Takayasu \nInstitute of Innovative Research\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan\n\nDepartment of Mathematical and Computing Science\nSchool of Computing\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan\n"
]
| [
"Institute of Innovative Research\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan",
"Department of Mathematical and Computing Science\nSchool of Computing\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan",
"Department of Mathematical and Computing Science\nSchool of Computing\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan",
"Institute of Innovative Research\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan",
"Sony Computer Science Laboratories\n3-14-13 Higashi-Gotanda, Shinagawa-ku141-0022TokyoJapan",
"Institute of Innovative Research\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan",
"Department of Mathematical and Computing Science\nSchool of Computing\nTokyo Institute of Technology\n4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan"
]
| []
| A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders' trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results. | 10.1103/physrevlett.120.138301 | [
"https://arxiv.org/pdf/1703.06739v3.pdf"
]
| 19,089,971 | 1703.06739 | a66c64c484964b454d38e1d4ac630bd115ff67b9 |
Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders
Kiyoshi Kanazawa
Institute of Innovative Research
Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan
Department of Mathematical and Computing Science
School of Computing
Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan
Takumi Sueshige
Department of Mathematical and Computing Science
School of Computing
Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan
Hideki Takayasu
Institute of Innovative Research
Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan
Sony Computer Science Laboratories
3-14-13 Higashi-Gotanda, Shinagawa-ku141-0022TokyoJapan
Misako Takayasu
Institute of Innovative Research
Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan
Department of Mathematical and Computing Science
School of Computing
Tokyo Institute of Technology
4259 Nagatsuta-cho, Midori-ku226-8502YokohamaJapan
Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders
A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders' trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.
Introduction.-In physics, the study of colloidal Brownian motion has a long history beginning with Einstein's famous work [1]; the understanding of its mechanism has been systematically developed in kinetic theory [2,3]. Specifically, from microscopic Newtonian dynamics, the Boltzmann and Langevin equations are derived for the mesoscopic and macroscopic dynamics, respectively. This framework is a rigid foundation for various nonequilibrium systems (e.g., active matter, granular gas, Feynman ratchets, and traffic flow [4][5][6][7][8][9][10]), and its direct experimental foundation has been revisited because of recent technological breakthroughs [11,12]. In light of this success, it is natural to apply this framework beyond physics to social science [13], such as finance. Indeed, the concept of random walks was historically invented for price dynamics by Bachelier earlier than Einstein [14], and its similarities to physical Brownian motion (e.g., the fluctuation-dissipation relation) are intensively studied by recent high-frequency data analysis [15]. As an idea in statistical physics, the dynamics of financial markets are expected to be clarified from first principles by extending kinetic theory.
Although this idea is attractive, the kinetic description has not been established for financial Brownian motion. Why has not this idea been realized yet? In our view, the biggest problem is the absence of established microscopic models; there exist empirical validations of mesoscopic [15][16][17][18][19][20][21] and macroscopic models [22][23][24][25][26][27][28], whereas no microscopic model has been validated by direct empirical analysis. Indeed, previous microscopic models [29][30][31][32][33] were purely theoretical and have no quantitative ev- * Corresponding author: [email protected] idence microscopically. To overcome this crucial problem as an empirical science, two missing links have to be connected: (i) establishment of the microscopic model by direct observation of traders' dynamics ( Fig. 1a) and (ii) construction of a kinetic theory to show its consistency with mesoscopic and macroscopic findings (i.e., the order-book and price dynamics (Fig. 1b, c)).
In this Letter, we present the corresponding solutions by direct observation of high-frequency trader (HFT) dynamics in a foreign exchange (FX) market: (i) a microscopic model of HFTs is established by direct microscopic evidence, and (ii) corresponding kinetic theory is developed to show its consistency with mesoscopic and macroscopic evidence. We analyzed order-book data with anonymized trader identifiers (IDs) to track trajectories of all individuals. We found an empirical law concerning trend following among HFTs, which has not been captured by previous order-book models. Remarkably, this property induces the collective motion of the order book and naturally leads the layered order-book structure [15]. We then introduce a corresponding microscopic model of trend-following HFTs. Starting from their "equations of motion," Boltzmann-like and Langevin-like equations are derived for the order-book and price dynamics. A quantitative agreement is finally shown with our empirical all findings. Our work opens the door to systematic descriptions of finance based on microscopic evidence.
Observed microscopic dynamics.-We analyzed the high-frequency FX data between the U.S. dollar (USD) and the Japanese Yen (JPY) on Electronic Broking Services for a week in June 2016 (see Appendix A 1). The currency unit used in this study is 0.001 yen, called the tenth pip (tpip). Here we particularly focused on the dynamics of HFTs [34], frequently submitting or canceling orders according to algorithms (see Appendix A 2). The typical trajectories of bid and ask quoted prices are illustrated in Fig. 2a-c for the top 3 HFTs. They modify their quoted prices by successive submission and cancellation at high speed typically within seconds; almost 99% of their submissions were finally canceled without transactions (see Appendix A 4). With the two-sided quotes they also play the role of liquidity providers [35,36] according to the market rule, keeping the balance between the bid and ask order book. Buy-sell spreads, the difference between the best bid and ask prices for a single HFT, were observed to fluctuate around certain time constants (see the insets for their distributions and Appendix A 5). We then report the empirical microscopic law for the trend-following strategy of individual traders. The bid and ask quoted prices of the top ith HFT are denoted by b i and a i (see Appendix A 6). We investigated the average movement of the trader's quoted midprice z i ≡ (b i +a i )/2 between transactions conditional on the previous market transacted price movement (Fig. 2d). Here we introduce the tick time T as an integer time incremented by every transaction. The mean transaction interval is 9.3 seconds during this week. Because typical HFTs frequently modify their price between transactions, we here study HFTs' trend following at one-tick precision. For the top 20 HFTs (Fig. 3), we found that the average and variance of movement
∆z i (T ) ≡ z i (T + 1) − z i (T ) obeyed ∆z i ∆p ≈ c i tanh ∆p ∆p * i , V ∆p [∆z i ] ≈ σ 2 i ,(1)
where the conditional average . . . ∆p is taken when the last price change is ∆p(T − 1) ≡ p(T ) − p(T − 1) and ∆z i = 0 (see Appendix A 6) and the conditional variance is defined by
V ∆p [∆z i ] ≡ (∆z i − ∆z i ∆p ) 2
∆p . Here, p(T ) is the market transacted price at the T tick, and c i , ∆p * i , σ 2 i are characteristic constants unique to the trader and independent of ∆p. Their typical values were found to be c i ≈ 6.0 tpip, ∆p * i ≈ 7.5 tpip, and σ i ≈ 14.5 tpip. Our finding (1) implies that the reaction of traders is linear for small trends but saturates for large trends, and quantifies the collective motion of HFTs. Remarkably, a similar behavior was reported from a price movement data analysis at one-month precision [37].
Microscopic model.-Here we introduce a minimal microscopic model of HFTs incorporating the above characters. We make four assumptions: (i) The number of traders is sufficiently large; (ii) traders always quote both bid and ask prices (for the ith trader, b i and a i ) simultaneously with a unit volume; (iii) buy-sell spreads are time constants unique to traders with distribution ρ(L). The trader dynamics are then characterized by the midprice z i ≡ (b i + a i )/2; and (iv) trend-following random walks are assumed in the microscopic dynamics ( Fig. 4a-c),
dz i (t) dt = c tanh ∆p(t) ∆p * + ση R i (t)(2)
with strength for trend following c, previous price movement ∆p, and white Gaussian noise ση R i with variance σ 2 . Here, c, ∆p * , and σ are assumed shared for all traders for simplicity. In this model, HFTs frequently modify their quoted price by successive submission and cancellation. Indeed, this model can be reformulated as a Poisson price modification process with high cancellation rate (see Appendix B 2). After transaction a j (t) = b i (t) (Fig. 4b), the updated market price and its corresponding movement are recorded as 4 -3 -2 -1 0 1 2 3 4 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th Scaling (a) Trend-following movement on average (hyperbolic function) (b) Standard deviation of random noise effect (independent of ) Scaling Scaling tpip tpip tpip tpip tpip 19th HFT 11th HFT 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th 4th and a requotation jump occurs (Fig. 4c),
p(t + 0) = b i (t), ∆p(t + 0) = b i (t) − p(t),(3)z i (t + 0) = z i (t) − L i 2 , z j (t + 0) = z j (t) + L j 2 .(4)
Here, t + 0 implies the time after transaction. A unique character of this model is the order-book collective motion due to trend following (Fig. 4d). For ∆p > 0, the bid (ask) volume change tends to be positive (negative) near the best price ( Fig. 4e), consistently with the layered order-book structure [15]. Kinetic formulation.-We next present an analytical solution to this model (2) according to kinetic theory [2,3]. Let us first introduce the relative distance r i ≡ z i − z c.m. from the "center of mass" z c.m. ≡ i z i /N (Fig. 4a), where the trend-following effect in Eq. (2) is absorbed into the dynamics of z c.m. . The dynamics of r i become simpler because trend-following effects disappear in this moving frame (see Appendix B 3). We next introduce the one-body (two-body) probability distribution as φ L (r) (φ LL (r, r )) conditional on traders' buy-sell spreads. From the microscopic model (2), the lowest-order hierarchy equation is derived as ∂φ L /∂t = (σ 2 /2)(∂ 2 φ L /∂r 2 )+N s=±1 dL ρ(L )[J s LL (r+sL/2)− J s LL ] with J s LL (r) ≡ (σ 2 /2)|∂ rr |φ LL (r, r ) r−r =s(L+L )/2 and |∂ rr |f ≡ |∂f /∂r| + |∂f /∂r | (see Appendix B 5). By assuming "molecular chaos"
φ LL (r, r ) ≈ φ L (r)φ L (r ),(5)
we derive the Boltzmann-like equation with collision integrals for the order book:
∂φ L ∂t ≈ σ 2 2 ∂ 2 φ L ∂r 2 +N s=±1 dL ρ(L )[J s LL (r+sL/2)−J s LL ](6)
withJ s LL (r) ≡ (σ 2 /2)|∂ rr |{φ L (r)φ L (r )} r−r =s(L+L )/2 . Here, s = +1 (s = −1) represents transactions as bidder (asker). Because traders exhibit collective motion arising from trend following, a Langevin-like equation is also derived as the macroscopic description of the model (2),
∆p(T + 1) = cτ (T ) tanh ∆p(T ) ∆p * + ζ(T ),(7)
where τ (T ) and ζ(T ) are transaction time interval and random noise at the T th tick time, respectively. The first trend-following term corresponds to the momentum inertia in the conventional Langevin equation. Equations (6) and (7) can be analytically assessed for N → ∞. We first set the buy-sell spread distribution as
ρ(L) = L 3 6L * 4 e −L/L *(8)
with decay length L * = 15.5 ± 0.2 tpip, empirically validated in our data set (Fig. 5a). The solution to Eq. (6) for N → ∞ is given by φ L (r) = (4/L 2 ) max{L/2−|r|, 0}. The average order-book profile f A (r) = dLρ(L)φ L (r − L/2) is then given for r > 0 by
f A (r) = 4e − 3r 2L * 3L * 2 + r L * sinh r 2L * − re − r 2L * 2L * .(9)
The statistics of τ (T ) in the macroscopic model (7) is derived from the mesoscopic model (6), and the tail of the price movement is approximately given by
P (≥ |∆p|; κ) ≈ e −|∆p|/κ (|∆p| → ∞)(10)
with decay length κ ≈ 2∆z * /3, average movement from trend following ∆z * ≡ cτ * , average transaction interval τ * ≈ 3L * 2 /N σ 2 , and complementary cumulative distribution function (CDF) P (≥ |∆p|; κ) (see also Appendices B 6 and B 7 for numerical validation). Mesoscopic and macroscopic data analysis.-We next investigated whether our microscopic model is consistent with our data set. The empirical daily profile was first studied for the average ask order book for the best prices of HFTs f A (r) (Fig. 5b). Surprisingly, we found a quantitative agreement with our theory (9) without any fitting parameters, which strongly supports the validity of our description.
The two-hourly segmented CDF for the price movement is also evaluated in one-tick precision P 2h (≥ |∆p|; κ) ( Fig. 5c), which obeys an exponential law that is qualitatively consistent with our theoretical prediction (10). The value of the two-hourly decay length κ fluctuates significantly during a week. To remove this nonstationary feature, we introduced the two-hourly scaled CDFP 2h (≥ |∆p|) ≡ P 2h (≥ κ|∆p|; κ)/Z with scaling parameters κ and Z (Fig. 5d), thereby incorporating the two-hourly exponential law for the whole week.
The price movements obey an exponential law for short periods but simultaneously obey a power law over long periods with exponent α = 3.6 ± 0.13 (Fig. 5e). This apparent discrepancy originates from the power-law nature of the decay length κ. Because κ approximately obeys a power-law CDF Q(≥ κ) ∼ κ −m over the week with m = 3.5 ± 0.13 ( Fig. 5f), the one-week CDF P w (≥ |∆p|) asymptotically obeys the power law as a superposition of the two-hourly segmented exponential CDF,
P w (≥ |∆p|) = ∞ 0 dκQ(κ)P 2h (≥ |∆p|; κ) ∝ |∆p| −m (11) with Q(κ) ≡ −dQ(≥ κ)/dκ, consistently with empirical exponent α ≈ m.
Our result is therefore consistent with the previous reported power law [24][25][26][27] as a nonstationary property of κ.
Since our trend-following HFT model exhibits the order-book collective motion ( Fig. 4d and e), this model can reproduce the layered order-book structure [15] (see Appendix B 8). Let us define c − r (c + r ) and a − r (a + r ) as the number of bid (ask) submission and cancellation between one tick at the relative distance r from the market midprice. We also define the number change
N − r = c − r − a − r (N + r = c + r − a + r )
at the distance r for the bid (ask) side. Fig. 5g between N − r (N + r ) and ∆p, showing positive and negative correlation in the inner (outer) and outer (inner) layers, respectively. We further show a linear correlation between the price movement ∆p and the total number change in the inner layer N inner ≡ γc −∞ dr(N − r − N + r ). The trend-following HFT model is thus qualitatively consistent with the previous findings [15] (see Appendix B 9 for data analyses), implying that the layered structure was the direct consequence of the collective motion.
Correlation coefficient C − r (C + r ) is plotted in
Discussion. -We have empirically studied the trend following of HFTs, inducing the collective motion of the order book. This property has not been captured in the previous order-book model [16][17][18][19][20][21] and was critical in reproducing our empirical findings. Indeed, none of our empirical findings, the order-book profile, the exponential price movement, and the layered order-book structure [15] were reproduced by the previous order-book model under realistic parameters in the absence of the collective motion (see Appendix B 10). We expect that introduction of this collective motion to order-book models would be the key to replicate these empirical findings.
Conclusion.-We have established both a microscopic model and a kinetic theory for FX traders by direct observation of the HFTs' dynamics, quantitatively agree-ing with empirical results under minimal assumptions. In the stream of econophysics, our model (2) is the first microscopic model directly supported by microscopic dynamical evidence and exhibiting agreement with mesoscopic and macroscopic findings. We expect that a new stream arises toward systematic descriptions of the financial market based on microscopic evidence. Interested readers are referred to Ref. [38] for more mathematical details. We analyzed high-frequency trading data in Electronic Broking Services (EBS), one of the biggest financial markets in the world. This market is continuously open except for weekends under few regulations. All trader activities were recorded for our data set with anonymized trader IDs and with one-millisecond time-precision from the 5th 18:00 to the 10th 22:00 GMT June 2016. The minimum price-precision was 0.005 yen for the USD/JPY pair at that time, and the currency unit used in this study is 0.001 yen, called the tenth pip (tpip). The minimum volume unit for transaction was one million USD, and the total monetary flow was about 68 billion USD during this week. The EBS market is a hybrid market combining both quote-driven and order-driven systems, where traders have three options: limit order, market order, and cancellation. A limit order is an order quoting price with a certain volume and the quoted price displayed on the order book. A market order is an order to buy or sell currencies immediately at the available best price.
Here we define terminology in this paper. The highest bid and lowest ask quoted prices are called the market best bid and ask prices (denoted by b M and a M ), respectively (see Fig. 6a). The average of the market bid and ask prices is called the market midprice (denoted by z M ). Also, the market transacted price p (or the market price for short) means the price at which a transaction occurs in the market.
We note a central trading rule regarding the mutual credit lines between traders [36]. All market participants are required to set credit lines to counterparties in advance, and they cannot transact with each other in the absence of mutual credit. Therefore, traders sometimes transact at the worse price than the best market price.
Definition of the high frequency traders
For this paper, a high frequency trader (HFT) is defined as a trader who submits more than 500 times a day on average (i.e., more than 2500 times for the week). This definition is similar to that introduced in Ref. [39]. As a few traders are unwilling to transact and often interrupt orders at the instant of submission, we excluded traders with live orders of less than 0.5% of the transaction time. With this definition, the number of HFTs was 134 during this week, whereas the total number of traders was 1015. We note that the total number of traders who submitted limit orders was 922; the other 93 traders submitted only market orders. We also note that the presence of HFTs has rapidly grown recently and 87.8% of the total orders were submitted by the HFTs in our data set.
Here we note a regulation on cancellations in this market, which is related to motivating HFTs to play the role of key liquidity providers (KLPs) [36]. For market stability, all traders are required not to cancel orders frequently; there is a threshold on the ratio between dealt quote and total number of quotations called the quote fill ratio (QFR). If the QFR of a trader is lower than a threshold, penalties are imposed on the trader in this market. However, there is a special rule to lower the threshold. If a trader maintains two-sided quotes continuously for a fixed time interval (called key liquidity hours), the trader qualifies as a KLP and is subject to a lower threshold QFR. Because HFTs tend to cancel orders frequently, they are typically KLPs as illustrated in Fig. 2a
-c.
We also note the typical number of HFTs related to snapshots of the order book. We took snapshots of the order book after every transaction and counted the total number of different trader identifiers (IDs) for both bid and ask sides. The counting weight for an HFT quoting both sides is 1 and that for an HFT quoting one side is 1/2. We then plotted the average of the number of trader IDs for both bid and ask sides every two hours in Fig. 6b, showing the periodic intraday activity pattern of HFTs (i.e., N tends to be small during 20:00-22:00 GMT). The typical number of HFT was about 35 in our data set with this definition. The number of total volumes quoted by HFTs is typically about 80. Admittedly, there is room for debate on which number is appropriate for the calibration of the total number of traders in our model; it remains a topic for future study.
Percentage of two-sided quotes
We calculated the percentage of two-sided quotes as follows; when a bid (ask) order is submitted by a trader, we check whether the corresponding ask (bid) orders exist. We then count the number of two-sided quotes for all traders at the submission of every order and finally divide it by the total number of submissions.
Cancellation ratio for individual traders
For each trader, we calculated the total number of canceled volumes over that of submitted volumes for the cancellation ratio of the trader. The cancellation ratio for the first, second, and third top HFTs were 98.59%, 99.93%, and 98.70%, respectively (or equivalently, their QFR were 1.41%, 0.07%, and 1.30%, respectively). The total cancellation ratio among all the HFTs was 94.42% (or equivalently the total QFR was 5.58%).
Buy-sell spread
The difference in the best bid and ask prices was studied as a buy-sell spread for an HFT. Samples where only both bid and ask prices exist are taken at one-second time-intervals for the insets in Fig. 2a-c and Fig. 5a. We plotted standard deviations of the averages as error bars for each point.
Trend-following effect
We explain the precise definition of the bid (ask) price of individual HFTs for the analysis of trend following. If a trader quotes both single-bid and single-ask orders at any time, the bid and ask prices are defined literally. In the presence of multiple bid or ask orders, we use the best value for the bid or ask orders as b i or a i . In the absence of any bid or ask or both orders, we use the most recent bid or ask price as b i or a i for interpolation.
Because of the discrete nature of this data analysis, the probability that traders do not move at all (i.e., ∆z i = 0) is estimated high. We therefore excluded the samples during an inactive time interval ∆z i = 0 for the calculation of representative values in the following. This exception handling does not have a big impact on the hyperbolic structure in Eq. (1). Exceptional samples for which the bid or ask price is far from the market price by 0.1 yen (0.02% of the total) are also excluded from the calculation of the conditional ensemble average . . . ∆p . In Fig. 3a, data points are plotted whose samples are over 100 in each bin. The standard deviations of the conditional averages are plotted for each point as error bars. Also, median values in the top 20 HFTs are given using c i ∼ 6.0 tpip/tick and ∆p * i ∼ 7.5 tpip, which are estimated by the least squares methods implemented in gnuplot.
We have also calculated the standard deviation of quoted price movements for individual traders at one-tick precision in Fig. 3b. For the ith top HFT, we calculated the conditional variance V ∆p [∆z i ] ≡ (∆z i − ∆z i ∆p ) 2 ∆p and took its square root. As can be seen from Fig. 3b, the standard deviation is approximately independent of ∆p for the top 20 HFTs. We note that the median value was σ i ∼ 14.5 tpip/tick. This observation is consistent with the assumption that only the drift term depends on ∆p but the random noise effect does not depend on ∆p in our microscopic model.
Average order-book profile
The daily average order-book profile is calculated for the best prices of the HFTs. We took snapshots of the order book for the best prices of the HFTs every second and we calculated its ensemble average every day. We also plotted standard deviations of the averages as error bars for each point.
Price movement distributions and decay length
The two-hourly segmented complementary cumulative distribution functions (CDFs) for the price movement ∆p are calculated in one-tick precision in Figs. 5c, d: ∆p(T ) ≡ p(T + 1) − p(T ) with market price p(T ) at tick time T . The decay length κ and its error were estimated by the least squares methods implemented in gnuplot (Figs. 5d, f) and the two-hourly scaled CDFs were plotted in Fig. 5d with the maximum samples excluded as outliers. The time-series of the estimated decay length κ is plotted in Fig. 6b, showing that κ was the longest just after the opening of the EBS market (the 5th 18:00-20:00 GMT). We conjectured that the decay length κ was related to the market activity, represented by such as the number of HFTs during the time region. Indeed, the number of HFTs was also the least during the 5th 18:00-20:00 GMT in the week.
Appendix B: Theoretical Analysis
Model dynamics
We explained the model dynamics as trend following random walks (2) with jump rules (3) and (4). These dynamics can be represented within the framework of Markovian stochastic processes using the δ-functions. The stochastic dynamics can be written as
dz i dt = c tanh ∆p ∆p * + ση R i + η T i , η T i ≡ ∞ k=1 j =i j ∆z ij δ(t − τ k;ij ), dp dt = ∞ k=1 i<j i,j (p post − p)δ(t − τ k;ij ), d∆p dt = ∞ k=1 i<j i,j (∆p post − ∆p)δ(t − τ k;ij ),(B1)
where we have used the Itô convention. Here, τ k;ij is the k-th collision time; jump size ∆z ij between traders i and j, post-collisional price p post , and price movement ∆p post are defined by with signature function sgn(x) defined by sgn(x) = x/|x| for x = 0 and sgn(0) = 0. Remarkably, the jump rule Eq. (B2) corresponds to the contact condition and momentum exchange in the conventional kinetic theory. In the following, we present effective descriptions of this model for mesoscopic and macroscopic hierarchies.
|z i (τ k;ij ) − z j (τ k;ij )| = L i + L j 2 =⇒ ∆z ij = − L i 2 sgn(z i − z j ), p post = z i + ∆z ij , ∆p post ≡ z i + ∆z ij − p(
Note on a Poisson price modification process
Since the Gaussian noise can be obtained by taking the high-frequent small jump limit for Poisson noises [40], the model (B1) can be reformulated as a Poisson price modification process with high-frequent cancellation rate. Here, let us focus on the quoted price dynamics for HFTs in the absence of transactions. As shown in Fig. 2a-c, HFTs tend to frequently and continuously modify their price by successive order cancellation and submission, possibly due to the market rule (i.e., they are required to maintain the continuous two-sided quote for a fixed time interval [36]). On the basis of these characters, we can consider a Poisson cancellation model corresponding to the model (B1). Let us introduce the order cancellation rate λ, which gives the cancellation probability during [t, t + dt] as λdt. The mean-cancellation interval is characterized by ∆t can ≡ 1/λ. After cancellation, we assume that HFTs instantaneously requote their price to maintain continuous limit orders. In the absence of transaction, the requoted price is assumed to be described by a discrete version of Eq. (B1) as
z i (t + dt) − z i (t) = 0 (Probability = 1 − λdt) c∆t can tanh ∆p ∆p * + σ √ ∆t can η R i (Probability = λdt) (B3)
with a standard Gaussian random number η R i according to our empirical finding (1). Transaction rule is also assumed the same as the continuous model (B1). Here, the infinitesimal time step dt is different from the mean-cancellation interval ∆t can . A schematic trajectory described by this Poisson dynamics is illustrated in Fig. 7. The continuous model (B1) is obtained in the high-frequent cancellation limit λ → ∞ for the discrete model (B3). The HFTs' nature on high-frequent price modifications is thus reflected in the continuous model (B1).
Introduction of the center of mass and the corresponding relative price
We here introduce the center of mass (c.m.) and the corresponding relative price (see Fig. 8 for a schematic):
z c.m. ≡ 1 N N i=1 z i , r i ≡ z i − z c.m. .(B4)
The dynamics of the c.m. and the relative price is given by Remarkably, trend following only appears in the dynamics of the c.m., but does not appear in that of the relative price. This is natural because trend following induces a collective behavior of traders, and can be absorbed into the dynamics of the c.m.. Furthermore, the contribution of ξ is much smaller than that of ση R i and η T i for N → ∞:
dz c.m. dt = c tanh ∆p ∆p * + ξ, dr i dt = ση R i + η T i − ξ, ξ ≡ 1 N N j=1 ση R j + η T j .(B5|ξ| |ση R i + η T i |.
In the moving frame of the c.m., the dynamics of the relative price r i is thus simplified and approximately obeys the following dynamical equation:
dr i dt ≈ ση R i + η T i . (B6)
BBGKY Hierarchical equation for two-body problem: N = 2
Before deriving the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchical equation for N 1, we first consider the two-body system of traders to specify the collision integrals. Extension to the many-body problem will be studied in the next subsection. Let us denote the relative midprices of the first and second traders by r 1 and r 2 with constant spreads L 1 and L 2 . The dynamics is given by
dr 1 dt = ση R 1;ε + ∞ k=1 ∆r 1 δ(t − τ k ), dr 2 dt = ση R 2;ε + ∞ k=1 ∆r 2 δ(t − τ k ) (B7)
with jump sizes ∆r 1 , ∆r 2 and k-th transaction time τ k . Here, η i;ε is the colored Gaussian noise satisfying η R i;ε (t)η R j;ε (s) = δ ij e −|t−s|/ε /2ε for i, j = 1, 2. Later, we shall take the ε → 0 limit, whereby colored Gaussian noise η R i;ε converges to white Gaussian noise as lim ε→0 η R i;ε (t)η R j;ε (s) = δ ij δ(t − s). The k-th transaction time τ k and the jump sizes ∆r 1 , ∆r 2 are determined using the collision rule,
|r 1 (τ k ) − r 2 (τ k )| = L 1 + L 2 2 =⇒ ∆r 1 = − L 1 2 sgn(r 1 − r 2 ), ∆r 2 = − L 2 2 sgn(r 2 − r 1 ).(B8)
We first derive the master equation for this system. For the two-body probability distribution function (PDF) P 12 (r 1 , r 2 ), we exactly obtain a time-evolution equation
∂P 12 ∂t = i=1,2 σ 2 2 ∂ 2 P 12 ∂r 2 i + s=±1 σ 2 2 δ(r 1 −r 2 )|∂ 12 |P 12 r 1 + sL 1 2 , r 2 − sL 2 2 −δ r 1 − r 2 − s L 1 + L 2 2 |∂ 12 |P 12 , (B9)
where |∂ 12 |g(r 1 , r 2 ) ≡ |∂g(r 1 , r 2 )/∂r 1 | + |∂g(r 1 , r 2 )/∂r 2 | is the sum of the absolute value of the partial derivatives for arbitrary g(r 1 , r 2 ). This equation can be derived as follows. For an arbitrary function f (r 1 , r 2 ), we obtain an identity
df (r 1 , r 2 ) dt = i=1,2 ση R i;ε ∂f (r 1 , r 2 ) ∂r i + ∞ k=1 [f (r 1 + ∆r 1 , r 2 + ∆r 2 ) − f (r 1 , r 2 )] δ(t − τ k ) = i=1,2 ση R i;ε ∂f (r 1 , r 2 ) ∂r i + σ [f (r 1 + ∆r 1 , r 2 + ∆r 2 ) − f (r 1 , r 2 )] δ |r 1 − r 2 | − L 1 + L 2 2 |η R 1;ε − η R 2;ε |,(B10)
where we have used the expansion of the δ-function: δ(g(t)) = ∞ k=0 δ(t − τ k )/|g (τ k )| with the k-th zero points, such that g(τ k ) = 0 and τ k < τ k+1 . Here we consider the direction of the collision; that is, η 1;ε − η 2;ε must be positive just before the collision r 1 − r 2 = (L 1 + L 2 )/2. Inversely, η 1;ε − η 2;ε must be negative just before the collision r 1 − r 2 = −(L 1 + L 2 )/2. We thus obtain
df dt = i=1,2 ση R i;ε ∂f ∂r i + s=±1 sσ f r 1 − sL 1 2 , r 2 + sL 2 2 − f δ r 1 − r 2 − s L 1 + L 2 2 (η R 1;ε − η R 2;ε ). (B11)
We take the ensemble average of both sides to obtain
df dt = i=1,2 σ η R i;ε ∂f ∂r i + s=±1 sσ f r 1 − sL 1 2 , r 2 + sL 2 2 − f δ r 1 − r 2 − s L 1 + L 2 2 (η R 1;ε − η R 2;ε ) . (B12)
Here the two-body PDF P 12 (x 1 , x 2 ) characterizes the probability of
r 1 ∈ [x 1 , x 1 + dx 1 ] and r 2 ∈ [x 2 , x 2 + dx 2 ] as P 12 (x 1 , x 2 )dx 1 dx 2 . By substituting f (r 1 , r 2 ) = δ(r 1 − x 1 )δ(r 1 − x 2 )
, we obtain the master equation
∂P 12 ∂t = i=1,2 σ 2 2 ∂ 2 P 12 ∂x 2 i + s=±1 sσ 2 2 −δ(x 1 − x 2 )∂ 12 P 12 x 1 + sL 1 2 , x 2 − sL 2 2 + δ x 1 − x 2 − s L 1 + L 2 2 ∂ 12 P 12
(B13) where the abbreviation symbol involving the derivatives is defined as∂ 12 ≡ ∂/∂x 1 − ∂/∂x 2 , using the Novikov's theorem [41] for an arbitrary function g(r 1 , r 2 ) as
lim ε→0 η R i;ε (t)g(r 1 (t), r 2 (t)) = lim ε→0 t 0 ds η R i;ε (t)η R i;ε (s) δg(r 1 (t), r 2 (t)) δη R i;ε (s) = σ 2 ∂g(r 1 , r 2 ) ∂r i . (B14)
We note that∂ 12 is a slightly different symbol from |∂ 12 | in terms of signatures (see Eq. (B15) for their relation). We comment on the signature of the derivatives. Considering that P 12 (x 1 , x 2 ) ≥ 0 for all x 1 , x 2 and P 12 (x 1 , x 2 ) = 0 for
x 1 −x 2 > (L 1 +L 2 )/2, we obtain (∂P 12 (x 1 , x 2 )/∂x 1 )| x1−x2=(L1+L2)/2 ≤ 0 and (∂P 12 (x 1 , x 2 )/∂x 2 )| x1−x2=(L1+L2)/2 ≥ 0.
We also obtain (∂P 12 (x 1 ,
x 2 )/∂x 1 )| x1−x2=−(L1+L2)/2 ≥ 0 and (∂P 12 (x 1 , x 2 )/∂x 2 )| x1−x2=−(L1+L2)/2 ≤ 0.
In summary, we have
s∂ 12 P 12 (x 1 , x 2 ) x1−x2=s(L1+L2)/2 = −|∂ 12 |P 12 (x 1 , x 2 ) x1−x2=s(L1+L2)/2 . (B15)
By a change of notation x 1 → r 1 and x 2 → r 2 , we obtain Eq. (B9). By integrating over r 2 on both sides, we obtain a hierarchical equation for the one-body PDF P 1 (r 1 ) ≡ dr 2 P 12 (r 1 , r 2 ) as
∂P 1 (r 1 ) ∂t = σ 2 2 ∂ 2 P 1 (r 1 ) ∂r 2 1 + s=±1 [J s 12 (r 1 + sL 1 /2) − J s 12 (r 1 )] , J s 12 (r) ≡ σ 2 2 |∂ 12 |P 12 (r 1 , r 2 ) r1−r2=s(L1+L2)/2 ,(B16)
where J s 12 (r 1 ) is the transaction probability per unit time as bidder (s = +1) or asker (s = −1). The first and second terms on the right-hand side account for the self-diffusion and collision terms, respectively. This is a lowest-order BBGKY hierarchical equation for the special case of N = 2. Remarkably, the collision term has a quite similar mathematical structure to the collision integral in the conventional Boltzmann equation.
BBGKY hierarchical equation for many-body problem: N 1
We have derived the hierarchical equation for the one-body PDF for the special case N = 2. Here we extend the hierarchical equation for the many-body problem with N 1. We first assume that the number of traders N is sufficiently large that the spread distribution ρ(L) can be approximated as a continuous function. The one-body and two-body PDFs conditional on buy-sell spread L and L are denoted by φ L (r) and φ LL (r, r ), respectively. We note the relations P i (r i ) = φ Li (r i ) and P ij (r i , r j ) = φ LiLj (r i , r j ) hold for the one-body and two-body PDFs P i (r i ) and P ij (r i , r j ) for the traders i and j, considering the symmetry between traders. Within the spirit of the Boltzmann equation, the dynamical equation for the one-body distribution φ L (r) can be decomposed into two parts: with the self-diffusion term (σ 2 /2)(∂ 2 φ L /∂r 2 ) and the collision integral C(φ LL ). By extending the collision term in Eq. (B16) for large N 1, we can specify the collision integral as
∂φ L (r) ∂t = σ 2 2 ∂ 2 φ L (r) ∂r 2 + C(φ LL )(C(φ LL ) = N s=±1 dL ρ(L ) [J s LL (r + sL/2) − J s LL (r)] , J s LL (r) = σ * 2 2 |∂ rr |φ LL (r, r ) r−r =s(L+L )/2 (B18)
with the collision probability per unit time as bidder (s = +1) or asker (s = −1) against a trader with spread L . This is the Boltzmann-like equation, Eq. (6). We note that this BBGKY hierarchical equation can be systematically derived via the pseudo-Liouville equation. The derivation will be given in another technical paper in preparation [38].
Boltzmann-like equation for finance
We next derive a closed equation for the one-body distribution function φ L by assuming a mean-field approximation. Let us truncate the two-body correlation (i.e., molecular chaos in kinetic theory),
φ LL (r, r ) ≈ φ L (r)φ L (r ). (B19)
A closed mean-field equation for the one-body distribution φ L is thereby obtained,
∂φ L (r) ∂t = σ 2 2 ∂ 2 φ L (r) ∂r 2 + N s=±1 dL ρ(L ) J s LL (r + sL/2) −J s LL (r) (B20)
with the mean-field collision probability per unit time as bidder (s = +1) or asker (s = −1)
J s LL (r) = σ 2 2 |∂ rr | {φ L (r)φ L (r )} r−r =s(L+L )/2 .(B21)
Equation (B20) is a closed equation for the one-body distribution function, and corresponds to the Boltzmann equation in molecular kinetic theory. Equation (B20) can be analytically solved for N → ∞, and the steady solution ψ L (r) is given by the tent function,
ψ L (r) ≡ lim t→∞ lim N →0 φ L (r; t) = 4 L 2 max L 2 − |r|, 0 .(B22)
Here, a technicality on the appropriate boundary condition will be summarized in another technical paper in preparation [38]. Note that the tent function (B22) for the traders' midprice order book implies the tent functions for both bid and ask order books in shifted coordinates (see Fig. 9 for a schematic). The average order-book profile for the ask side f A (r) is then given by convolution with the tent function,
f A (r) = dLρ(L)ψ L (r − L/2).(B23)
We discuss here the intuitive meaning of the mean-field solution (B23). The mean-field solution (B23) is exactly zero at r = ±L/2 as ψ L (+L/2) = ψ L (−L/2) = 0, implying that the edge points r = ±L/2 effectively play the role of hopping barriers at which the particle hops into r = 0. Indeed, Eq. (B22) gives exactly the same solution to the problem of the Brownian motion confined by hopping barriers, as shown in Sec. C 2. This is a reasonable result for the N → ∞ limit, where the market is sufficiently liquid and most of the transactions occur just around r = ±L/2. If the spreads are distributed in accordance with the γ-distribution, as empirically studied in the main text, the average order-book profile is given by
ρ(L) = L 3 e −L/L * 6L * 4 =⇒ f A (r) = 1 L * f r L * ,f (r) ≡ 4 3 e − 3r 2 (2 +r) sinhr 2 −r 2 e −r 2 .(B24)
To check the validity of this formula, we performed Monte Carlo simulations of the microscopic model (B1) (Fig. 10a), where the theoretical formula (B24) works for various N . In the figure, we denote the relative price by r c.m. to stress that it is defined from the c.m. as r c.m. ≡ z − z c.m. . (r c.m. ) is theoretically more tractable than f mid A (r mid ). Here r mid ≡ a i − z M is the relative distance from the market midprice z M for the ask price a i of the ith trader. Fortunately, they are asymptotically equivalent for the large N limit and the above formulation is sufficient in understanding the average order-book f mid A (r mid ) from the market midprice:
f c.m. A (r c.m. ) ≈ f mid A (r mid ) (N → ∞).(B25)
To validate this asymptotic equivalence, we numerically demonstrate the average order-book profile f mid A (r mid ) from the market midprice z mid in Fig. 10b. This figure numerically shows that the average order-book formula (B24) is valid even for the order-book from the market midprice.
c. Statistics of transaction interval
We comment on the statistics of the transaction interval τ . In the mean-field approximation, the average of the transaction interval is given by
τ * ≡ τ ≈ 1 2N σ 2 L −2 ρ(L)dL + O(N −2 ) = 3L * 2 N σ 2 + O(N −2 ),(B26)
which is phenomenologically derived in Sec. C 1 and is numerically validated in Fig. 10c. Note that this formula can be derived from the pseudo-Liouville equation more systematically [38]. Based on the average transaction interval (B26), the CDF for the transaction interval P (≥ τ ) is approximately given by the phenomenological formula,
P (≥ τ ) ≡ ∞ τ dτ P (τ ) ≈ 1 − (1 − e −3τ /2τ * ) 2(B27)
with transaction interval PDF P (τ ). This formula is derived in Sec. C 2 and is numerically validated in Fig. 10d.
Langevin-like equation for finance
Here we derive phenomenologically the fundamental equation for the financial Brownian motion, which corresponds to the conventional Langevin equation. Let us denote the T th transaction price by p(T ) and the T th price movement by ∆p(T ) ≡ p(T + 1) − p(T ). Here we focus on the effects of trend following obtained from Eq. (B5), which induces inertia-like collective motion of the macroscopic dynamics. The dynamical equation for the price movement is thus given by
∆p(T + 1) = c * τ (T ) tanh ∆p(T ) ∆p * + ζ(T ),(B28)
where τ (T ) is the time interval between the T th and (T + 1)th transaction. The first and second terms originate from the trend following and random noise, respectively. Note that the statistics of τ (T ) is derived from the mesoscopic model (B20) as Eqs. We next study the price movement distribution using the financial Langevin equation (B28), which is however a stochastic difference equation that cannot be solved exactly. Nonetheless, its qualitative behavior can be assessed approximately by making the following two assumptions.
(i) The effect of trend following is sufficiently large compared with random noise: |c * τ (T )| |ζ(T )|.
(ii) The average movement by trend following ∆z * ≡ cτ * is much larger than the saturation threshold: ∆z * ∆p * .
Under condition (i), the qualitative behavior is governed by the statistics of the transaction interval τ (T ). Under condition (ii), furthermore, the hyperbolic function in Eq. (1) can be approximated for large fluctuations as tanh(∆p/∆p * ) ≈ sgn(∆p) and the term ζ(T ) is irrelevant for the tail. Based on Eq. (B27) for the transaction interval τ , the price distribution P (≥ |∆p|) is approximately obtained for |∆p| → ∞ as
P (≥ |∆p|) ≈ e −3|∆p|/2∆z * = e −|∆p|/κ ,(B30)
with an estimated decay length of κ ≈ 2∆z * /3. The validity of this formula was numerically checked in Fig. 11.
8. Numerical analysis of the layered order-book structure for the HFT model
Here we show the detailed analysis to study the layered order-book structure for the HFT model according to the method in Ref. [15]. the instant of an order submission and cancellation for the bid (ask) side, with bin-width of 1 tpip we measured the relative depth r from the market midprice z M , defined as r ≡ z M − b i (r ≡ a i − z M ), and we incremented the numbers of c − r (c + r ) and a − r (a + r ) by one, respectively. We then accumulated their numbers between T and T + 1 tick to obtain one sample of N −
r (T ) ≡ c − r (T ) − a − r (T ) (N + r (T ) ≡ c + r (T ) − a + r (T ))
. We also study the movement of the market price ∆p(T ) ≡ p(T + 1) − p(T ) and calculated Pearson's correlation coefficient C − r (C + r ) between ∆p(T ) and N − r (T ) (N + r (T )) as shown in Fig. 5g. The crossover point was estimated to be γ c ≈ 16.5 tpip in the numerical simulation. We next study the linear correlation between the number change N inner in the inner layer and the price movement ∆p. Between T and T + 1 tick, we take a sample of both N − r (T ) and N + r (T ), and calculated their integral in the inner layer as N inner (T ) ≡ We show the empirical layered structure of the HFTs' order book in our data set. According to the essentially same method in Sec. B 8, we have calculated the layered structure as shown in Fig. 12a and b. The volume change in the inner layer N inner (T ) has a significant correlation of Pearson's coefficient 0.616 with the price movement ∆p(T ).
For consistency throughout this Letter, we have focused on the best prices of HFTs for the correlation analysis in Fig. 12a and b. In other words, we incremented c − r and c + r when the newly quoted price was the best price of the trader. Also, we incremented a − r and a + r when the price of the canceled order was the best price of the trader.
Numerical comparison with the zero-intelligence order-book models
Here we compare our empirical findings with the zero-intelligence order-book (ZI-OB) model [18][19][20]. The basic ZI-OB model is the uniform decomposition order-book model introduced in Ref. [18,19], where both submission and cancellation are assumed to obey the homogeneous Poisson processes. To understand real average order-book profiles, in Ref. [20], the uniform submission rate was replaced with a real nonuniform submission rate obeying a power law. Here we study the improved ZI-OB model in Ref. [20] from the viewpoint of the consistency with our empirical findings. The inputs to the ZI-OB model are the following three components.
1. Submission rate density µ(r mid ): limit order submissions are assumed to obey the inhomogeneous Poisson process characterized by the submission rate µ(r mid ) with relative depth r mid from the market midprice. In other words, a new limit order is submitted in the range [r mid , r mid + dr mid ] between time interval [t, t + dt] with probability of µ(r mid )dr mid dt.
The empirical submission histogram in our data set is depicted in Fig. 12c, showing a power-law tail with exponent 2.9. For our numerical implementation, the limit order submission rate is directly fixed from the [18][19][20] to examine its consistency with the empirical findings in the main text. (a-d) We first studied a simulation under a realistic parameter set satisfying (N vol , QFR) ≈ (100, 5%). The price movement obeyed the Gaussian law (Fig. a), which is contradictory to the exponential law in our data set. The average order-book profile did not quantitatively fit the real order book during this week (Fig. b). The layered order-book structure was not also observed ( Figs. c and d). (e-h) By adjusting the market order rate ω, we attempted to fit the real order-book profile by the ZI-OB model. Though the average order-book profile was replicated by the ZI-OB model by parameter adjustment (Fig. f), neither the price movement statistics nor the layered order-book structure were consistent with our data set instead (i.e., Fig. e shows the power-law statistics for ∆p and Figs. g and h shows the absence of the layered structure). We also note that the adjusted parameter implies QFR ≈ 75%, which is over ten-times larger than the real QFR. In this sense, the ZI-OB model was not consistent with the empirical findings under realistic parameters. empirical submission histogram for r mid ≥ 0. The total submission rate is given by
µ tot ≡ ∞ 0 dr mid µ(r mid ),(B32)
characterizing the frequency of total submissions. The gender of order (i.e., buy or sell) is randomly selected with equal probability.
2. Cancellation rate λ: any order is assumed to be cancel according to the homogeneous Poisson process with intensity λ. In other words, an order is canceled between time interval [t, t + dt] with probability of λdt.
3. Market order rate ω: market orders are assumed to obey the Poisson process with intensity ω. In other words, a buy or sell market order is submitted between interval [t, t + dt] with probability of ωdt.
The gender of order is randomly selected with equal probability.
These parameters characterize the order-book dynamics in the steady state. For example, the average total order-book volume N vol in both sides and the QFR (i.e., the probability for an order to be transacted finally) are given by
N vol ≈ µ tot − ω λ , QFR ≈ ω µ tot ,(B35)
respectively. These relations are deduced from the conservation of order flux in the steady state: µ tot ≈ λN vol + ω.
a. Numerical simulation with realistic parameters
We first consider a numerical simulation based on realistic parameters in our data set with minimum price precision of 1 tpip. The submission rate density µ(r mid ) is directly fixed from the empirical submission histogram in our data set (Fig. 12c). The cancellation and market order rates are fixed as λ/µ tot = 9.5 × 10 −3 and ω/µ tot = 0.05 to satisfy N vol ≈ 100 and QFR ≈ 5%. Though these parameters were realistic in our data set, the numerical results in Fig. 13a-d were not consistent with the empirical findings. Indeed, the price movement in the ZI-OB model obeyed the Gaussian statistics (Fig. 13a), which is different from the empirical exponential law in our data set. The numerical average order-book profile did not quantitatively fit the real order-book profile (Fig. 13b). In addition, the layered structure of the order book did not emerge from the ZI-OB model (Fig. 13c and d). To replicate these empirical findings, in particular the layered order-book structure, we conjectured that the collective motion of the limit order book (i.e., the microscopic trend-following behavior) needs to be incorporated with conventional order-book models.
b. Numerical simulation with adjusted parameters
In Ref. [20], the possibility of the ZI-OB model was studied to fit realistic order-book profiles by adjusting parameters. In the same way, we here seek the possibility to adjust the model parameters in replicating the real order-book profile in our data set. By fixing the average order-book volume as N vol ≈ 100, we adjusted the market order rate ω as a fitting parameter to replicate the real order-book profile in our data set (see Fig. 13e-h). By inputting ω/µ tot = 0.75 (or equivalently QFR ≈ 75%), the ZI-OB model replicated the real order-book profile as shown in Fig. 13f. Instead, however, other numerical results of the ZI-OB model were not consistent with the exponential price movement statistics (Fig. 13e for the power-law price movement) nor the layered order-book structure ( Fig. 13g and h). In addition, the parameter adjustment implied QFR ≈ 75%, which was over ten-times larger than the real QFR (see Sec. A 4). We thus conclude that our empirical results were not consistently replicated by the ZI-OB model under realistic parameters, at least in our data set.
Appendix C: Technical Issues for Derivation
Brownian motion confined by hopping barriers
In this subsection, we study the Brownian motion confined by the hopping barriers at r = ±L/2 (see Fig. 14 for a schematic). Let us assume that a particle moves randomly in the absence of collision for r ∈ (−L/2, L/2). We then place hopping barriers at r = ±L/2, and we assume that the particle moves to the origin r = 0 after collisions. The particle's position r(t) then obeys the dynamical equation
dr dt = ση R +η T + +η T − ,η T + = − L 2 ∞ k=1 δ(t − τ + k ),η T − = + L 2 ∞ k=1 δ(t − τ − k ),(C1)
whereη R is the white Gaussian noise with unit variance, andη T + andη T − are respectively the jump terms originating from the hopping barriers at r = ±L/2. Here, the kth collision times τ + k and τ − k at the barriers r = ±L/2 satisfy the relation r(τ ± k ) = ±L/2. In a parallel calculation to that in Sec. B 4, the dynamical equation for the probability distribution function P (r) is given by
∂P (r) ∂t = σ 2 2 ∂ 2 ∂r 2 P (r) + s=±1
[J s (r − sL/2) − J s (r)], J s (r) = σ 2 2 δ(r + sL/2)|∂ s P (r)|. (C2)
Collision & Hopping FIG. 14. Schematic of Brownian motion confined by the hopping barriers at r = ±L/2. When the Brownian particle collides with the hopping barriers, the particle hops to the origin r = 0.
during the time interval T yields n L = T /(L 2 /4σ 2 ), when T is sufficiently large. The total number of collisions n tot is then given by
n tot = N i=1 T L 2 i /4σ 2 ≈ N dLρ(L)T L 2 /4σ 2 ,(C3)
where there are duplicate counts because any transaction occurs as a binary collision. Considering the duplicate counts, the mean transaction interval τ * for the whole system is given by τ * = T /(n tot /2), which implies Eq. (B26).
Transaction interval distribution
The phenomenological estimation of the cumulative distribution for transaction interval (B27) is presented here. Let us assume that the arrival-time intervals of a bidder and an asker at the center of mass obey the Poisson statistics:
P A (≥ τ A ) = ∞ τA P A (τ A )dτ A = e −τA/a , P B (≥ τ B ) = ∞ τB P B (τ B )dτ B = e −τB/a (C4)
with the characteristic time interval a. P A (τ A ) (P B (τ B )) and P A (≥ τ A ) (P B (≥ τ B )) are the PDFs and CDFs of arrival time intervals for an asker (a bidder), respectively. We also assume that the transaction occurs when both bidder and asker arrive at the center of mass. This picture implies that the transaction interval τ is approximately given by
τ ≈ max{τ A , τ B } =⇒ P (≥ τ ) = 1 − (1 − e −τ /a ) 2 ,(C5)
where we have used a formula for the order statistics [42]. Considering the consistency between Eq. (C5) and the mean transaction interval (B26), we obtain the self-consistent condition τ * = 3a/2. Equation (B27) then follows.
FIG. 2 .
2(a-c) Lifetimes of orders are plotted as trajectories for the top 3 HFTs. Typical traders tend towards continuous two-sided quotes, with the buy-sell spread fluctuating around a time constant unique to the trader. The percentage of two-sided quotes among HFTs was 48.4% (see Appendix A 3). (d) Quantification of trend following for individual traders, where ∆p and ∆zi are the movements of the market price and the midprice of the ith trader, respectively.
14th FIG. 3 .FIG. 4 .
14th34(a) Average ∆zi assuming previous price movements of ∆p and an active trader with ∆zi = 0. The behavior can be fitted by the master curve (1) for the top 20 HFTs by introducing scaling parameters ∆p * i and ci. (b) Conditional standard deviation on price movement ∆p, showing that the randomness associated with trend following is independent of ∆p. Schematic of the microscopic model (2). (a) Midprice of each trader obeys trend-following random walks. (b) Transaction takes place after price matching bi = aj with market transacted price p and its movement ∆p updated. (c) Pair of traders requote their bid and ask prices simultaneously after transactions. (d) Order-book collective motion induced by trend following. (e) Volume change in the bid (ask) order book is positive (negative) near the best price on average for ∆p > 0.
FIG. 5 .
5(a) Daily distribution of buy-sell spreads for HFTs with the empirical master curve (8) (see Appendix A 5). (b) Daily average order-book profile for the best prices of HFTs, agreeing with our theoretical line (9) without fitting parameters (see Appendix A 7). Here the relative depth is measured from the market midprice instead of the c.m. for simplicity, which did not cause big numerical difference in comparing with our theory (see AppendixB 6). (c) Two-hourly segmented CDFs for the price movement in one-tick precision for the three typical time regions (see Appendix A 8). The CDFs are exponential, consistently with our theoretical prediction (10). (d) Two-hourly segmented CDFs are scaled into the single exponential master curve every 2 hour (62 time regions). (e) Price movement CDF over the whole week obeys a power law of exponent α. (f) Decay length κ obeys a power law Q(≥ κ) ∼ κ −m . (g) Order-book layered structure by our HFT model. Pearson's coefficient C − r (C + r ) are numerically plotted between N − r (N + r ) and ∆p with crossover point γc ≈ 16.5 tpip. (h) Linear correlation between the total number change Ninner in the inner layer and the price movement ∆p with correlation coefficient of 0.63.
ACKNOWLEDGMENTS
We greatly appreciate NEX for their provision of the EBS data. We also appreciate M. Katori, H.Hayakawa, S. Ichiki, K. Yamada, S. Ogawa, F. van Wijland, D. Sornette, M. Sano, T.G. Sano, and T. Ito for fruitful discussions. This work was supported by JSPS KAKENHI (Grants No. 16K16016 and No. 17J10781) and JST, Strategic International Collaborative Research Program (SICORP) on the topic of "ICT for a Resilient Society" by Japan and Israel. We thank Richard Haase, Ph.D, from Edanz Group for editing a draft of this manuscript.
FIG. 6 .
6(a) Schematic for the market best bid price bM, the market best ask price aM, and the market midprice zM. (b) Time series of the decay length κ and the inverse number of HFTs 1/N during this week, showing that both values of κ and 1/N were largest during the 5th 18:00 -20:00 GMT June (an inactive hour just after the opening of the EBS market).
FIG. 7 .
7Continuous trend-following random walk model (B1) can be reformulated as the Poisson price modification process (B3) with high-frequent cancellation rate λ → ∞. (a) A typical trajectory of the Poisson price modification process (B3) with finite cancellation rate λ. Here the mean interval between price modification by a single trader is set to be ∆tcan ≡ 1/λ = τ * /4 with the mean transaction interval τ * . Other parameters are given by L * = 15 tpip, ∆p * = 6.75 tpip, ∆z * = 3.6 tpip, and N = 25. (b) A typical trajectory of the Poisson price modification process (B3) with high cancellation rate λ = 400/τ * , where the Poisson model (B3) asymptotically reduces to the continuous model (B1) for λ → ∞.
FIG. 8 .
8Relative price ri ≡ zi − zc.m. from the c.m.. (a) The quoted bid and ask prices (bi, ai) of an individual trader are plotted with the c.m. and market price (zc.m., p). (b) The trend-following effect is removed in the moving frame ri. The trajectories were obtained from a Monte Carlo simulation of the microscopic model (B1) with time unit L * 2 /σ 2 , discretized time step ∆t = 4.0 × 10 −4 L * 2 /σ 2 , L * = 15 tpip, ∆p * = 6.75 tpip, ∆z * = 3.6 tpip, and N = 25.
FIG. 9 .
9From the trader's midprice order book to the traders' ask order book, with the coordinate shifted by L/2.
FIG. 10 .
10Numerical plots obtained by Monte Carlo simulations of the microscopic model (B1) The parameter settings used for the simulation are: ∆t = 1.0 × 10 −2 L * 2 /N σ 2 , L * = 15 tpip, ∆p * = 4.5 tpip, ∆z * = 3.15 tpip for various N . (a) Numerical average order-book profiles f c.m. A (rc.m.) from the c.m. and the theoretical guideline (B24). (b) Numerical average order-book profiles f mid A (r mid ) from the the market midprice, showing the asymptotic equivalence f c.m. A (rc.m.) ≈ f mid A (r mid ). (c) Numerical mean transaction interval and the theoretical guideline (B26). (d) Numerical CDF for transaction interval and the theoretical guideline (B27).a. Average order-book profile from the center of mass
b. Average order-book profile from the market midprice Technically, we have studied the average order-book profile f c.m. A (r c.m. ) from the c.m. instead of that from the market midprice f mid A (r mid ), because f c.m. A
FIG. 11 .
11Numerical plots obtained by Monte Carlo simulations of the microscopic model (B1) for ∆t = 1.0 × 10 −4 L * 2 /σ 2 , L * = 15 tpip, ∆p * = 3.0 tpip, ∆z * = 7.2 tpip for N = 100.
(B26) and (B27). Equation (B28) governs the macroscopic dynamics of the financial Brownian motion, corresponding to the conventional Langevin equation. For small trend |∆p| ∆p * , we indeed obtain a formal expression similar to the conventional Langevin equation, T ) ≡ (1 − cτ (T ))/∆T ,ζ(T ) ≡ ζ(T )/(∆T ) 2 , and ∆ 2 p(T ) ≡ ∆p(T + 1) − ∆p(T ).
FIG. 12 .
12The numerical simulation was performed for the Poisson price modification process (B3) under the parameter set of L * = 15.5 tpip, ∆p * = 3.65 tpip, ∆z * = 4.56 tpip, ∆t can ≡ 1/λ = 4/τ * , and N = 50. At (a) The empirical layered structure of the HFT's order book in the data set, showing the crossover point γc ≈ 26 tpip between the inner and outer layers. (b) The linear correlation between the total number change in the inner layer Ninner and the market transacted price movement ∆p. Pearson's correlation coefficient is given by 0.616 between Ninner and ∆p. (c) The distribution of limit order submissions from the market midprice, showing a power-law tail with exponent 2.9 (see the inset for the log-log plot).
γc −∞ dr{N − r (T ) − N + r (T )}. We then calculated the correlation between N inner (T ) and ∆p(T ) for Fig. 5h. We have plotted the average of ∆p(T ) conditional on N inner (T ) with errorbars representing the conditional variance. The figure shows their significant linear correlation of Pearson's coefficient 0.63. 9. Empirical analysis of the layered structure of the order book in the data set
FIG. 13 .
13Average order-book distribution (c) Layered structure of order book (d) Correlation between (a) Price movement CDF (Gaussian) (f) Average order-book distribution (g) Layered structure of order book (h) Correlation between (e) Price movement CDF (Numerical study on the ZI-OB model
The steady solution is then given by the tent function, P SS (r) ≡ lim t→∞ P (r) = ψ L (r), which is the same as the mean-field solution (B22) for N → ∞. This implies that the mean-field description corresponds to Brownian motion confined by the hopping barriers in the limit N → ∞.From the above picture, we can derive Eq. (B26) for the mean transaction interval asymptotically in terms of N . Because a single particle in our model behaves as the Brownian motion confined by the hopping barriers for N → ∞, the mean transaction interval for a single particle can be derived by considering the survival rate problem for the model (C1). According to Ref.[33], the mean transaction interval is given by L 2 /4σ 2 for a single particle. We next derive the mean transaction interval for the whole system. A count of the number of collisions n L for the spread L
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| []
|
[
"Chabauty limits of simple groups acting on trees",
"Chabauty limits of simple groups acting on trees"
]
| [
"Pierre-Emmanuel Caprace \nUCLouvain\n1348Louvain-la-NeuveBelgium\n",
"Nicolas Radu \nUCLouvain\n1348Louvain-la-NeuveBelgium\n"
]
| [
"UCLouvain\n1348Louvain-la-NeuveBelgium",
"UCLouvain\n1348Louvain-la-NeuveBelgium"
]
| []
| Let T be a locally finite tree without vertices of degree 1. We show that among the closed subgroups of Aut(T ) acting with a bounded number of orbits, the Chabautyclosure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of T have degree ≥ 3, then the set of isomorphism classes of topologically simple closed subgroups of Aut(T ) acting doubly transitively on ∂T carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented. | 10.1017/s1474748018000348 | [
"https://arxiv.org/pdf/1608.00461v3.pdf"
]
| 119,683,942 | 1608.00461 | 957d234565eb5b2804aae783d9fa8a684ae16f88 |
Chabauty limits of simple groups acting on trees
4 Jul 2018 July 4, 2018
Pierre-Emmanuel Caprace
UCLouvain
1348Louvain-la-NeuveBelgium
Nicolas Radu
UCLouvain
1348Louvain-la-NeuveBelgium
Chabauty limits of simple groups acting on trees
4 Jul 2018 July 4, 2018arXiv:1608.00461v3 [math.GR]
Let T be a locally finite tree without vertices of degree 1. We show that among the closed subgroups of Aut(T ) acting with a bounded number of orbits, the Chabautyclosure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of T have degree ≥ 3, then the set of isomorphism classes of topologically simple closed subgroups of Aut(T ) acting doubly transitively on ∂T carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.
Introduction
Hêtre c'est mon identité (Jacques Prévert, Arbres, 1976) Beyond algebraic groups over local fields, groups acting on trees provide the largest (and historically the first) known source of examples of non-discrete compactly generated locally compact groups that are topologically simple, i.e. whose only closed normal subgroups are the trivial ones. Since the automorphism group of a given locally finite tree T may host many pairwise non-isomorphic topologically simple closed subgroups, it is natural to consider those collectively, by viewing them as a subset of the space Sub(Aut(T )) of all closed subgroups of Aut(T ), endowed with the Chabauty topology, which is compact. The starting point of this work is the following basic question: what is the Chabauty-closure of the set of topologically simple closed subgroups of Aut(T )? In order to stay in the realm of compactly generated groups, we will frequently impose that the groups under consideration act with a bounded number of orbits. Assuming the weaker condition that the groups act cocompactly on T is sufficient to guarantee that they are compactly generated, but that condition is not Chabauty-closed. To facilitate the statements of our results, we introduce the following notation. For a given number C > 0, we denote by Sub(Aut(T )) ≤C the set of closed subgroups of Aut(T ) acting with at most C orbits of vertices. It is a clopen subset of Sub(Aut(T )) (see Proposition 2.6 (3)).
Theorem 1.1. Let T be a locally finite tree all of whose vertices have degree ≥ 2. For any C > 0, the Chabauty-closure of the set of topologically simple groups in Sub(Aut(T )) ≤C is the set of groups in Sub(Aut(T )) ≤C without proper open subgroup of finite index.
The conclusion of Theorem 1.1 may fail if the tree T is allowed to have vertices of degree 1, see Lemma 5.14 below. We remark that if C = 1 then the set {H ∈ Sub(Aut(T )) ≤C | H = H (∞) } is empty, while if C ≥ 2 and T is semi-regular (i.e. Aut(T ) is edge-transitive), that set contains at least one group, namely the group Aut(T ) + of type-preserving automorphisms, which is simple by [Tit70]. For a general tree T and an arbitrarily large C, it may be the case that Sub(Aut(T )) ≤C contains only discrete, hence virtually free, groups (see [BT]), so that the set {H ∈ Sub(Aut(T )) ≤C | H = H (∞) } is also empty in that case.
It is important to note that a Chabauty limit of topologically simple groups need not be simple. Indeed, explicit examples of non-simple closed subgroups H of Aut(T ) that are edge-transitive (indeed locally 2-transitive) and satisfy H = H (∞) are provided by Burger and Mozes in [BM00, Example 1.2.1] (see also Remark 5.13 below). Thus the set of topologically simple edge-transitive closed subgroups is not closed in Sub(Aut(T )). Nevertheless, that situation changes if one considers the subset of groups acting doubly transitively on the set of ends of a thick tree T (which is automatically contained in Sub(Aut(T )) ≤2 , see [BM00, Lemma 3.1.1]). Recall that T is thick if all its vertices have degree ≥ 3, and remark that Sub(Aut(T )) ≤2 is non-empty only when T is semi-regular.
Theorem 1.2. Let T be a locally finite thick semi-regular tree. The set of topologically simple closed subgroups of Aut(T ) acting 2-transitively on ∂T is Chabauty-closed.
Moreover, the isomorphism relation within that set has closed classes, and the set S T of isomorphism classes of topologically simple groups acting continuously and properly on T and 2-transitively on ∂T , endowed with the quotient topology, is compact Hausdorff.
Theorem 1.2 has several consequences. First of all, it can be interpreted as providing qualitative information on the complexity of the isomorphism relation within topologically simple boundary-2-transitive closed subgroups of Aut(T ). Indeed, Theorem 1.2 implies that that relation is smooth in the sense of [Gao09,Definition 5.4.1], which means that it comes at the bottom of the hierarchy of complexity of classification problems in the formalism established by invariant descriptive set theory (see [Gao09,Chapter 15]). In fact, it is tantalizing to believe that for a given tree T , the set S T of isomorphism classes as above can be described exhaustively. This has actually recently been accomplished by the second-named author for all semi-regular trees whose vertex degrees are ≥ 6 and such that the only finite 2-transitive groups of those degrees are the full symmetric or alternating groups, see [Rad17] and Appendix A below. For all those trees, the set S T happens to be countable. Moreover, the second Cantor-Bendixson derivative of S T is reduced to the singleton consisting of the isomorphism class of the group Aut(T ) + (see Proposition A.1 and Remark A.3 below). However, the classification problem remains open for semi-regular trees T whose vertex degrees are the degrees of smaller finite 2-transitive groups, like Lietype groups or affine groups. In particular, we do not know whether there exists a tree T such that S T is uncountable. The case of the trivalent tree is especially intriguing.
The compactness of S T asserted by Theorem 1.2 also fosters less ambitious hope than a full classification of S T . Indeed, it opens up the possibility to find new isomorphism types of simple groups by taking limits of known ones. Implementing this idea requires to have at hand an infinite family of pairwise non-isomorphic topologically simple groups acting boundary-2-transitively on the same locally finite tree T . Rank one simple algebraic groups over p-adic fields provide examples of such families. However, in all cases where it could be verified, any limit of (classes of) such groups in S T happens to be a rank one simple algebraic groups over a local field of positive characteristic. Indeed, T. Stulemeijer has proved that if T is the regular tree of degree p + 1 with p prime, then the set of isomorphism classes of algebraic groups in S T , denoted by S alg T , is closed. Moreover the non-isolated points are precisely the isomorphism classes of the simple algebraic groups over local fields of positive characteristic. That set is finite (of cardinality 2) if p > 2 and infinite if p = 2. We refer to [Stu16] for general results and full details.
Another potential source of examples for the implementation of that idea is the class of complete Kac-Moody groups of rank two over finite fields. In that class, the tree T is determined by the finite ground field. Letting the defining generalized Cartan matrix run over the infinite set of possibilities in rank two, one obtains a countable family of topologically simple boundary-2-transitive groups in Sub(Aut(T )). The difficulty arising here is that we do not know whether those groups are pairwise non-isomorphic: we do not even know whether they form infinitely many isomorphism classes. A discussion of this rather subtle question, and partial answers, may be found in [Mar15,Theorem F and §6].
An important tool in the proofs of the results above is provided by the notion of kclosures recently introduced by Banks-Elder-Willis [BEW14], some of whose properties are reviewed in §3 below. We establish a key relation between Chabauty convergence and k-closures in the general context of automorphism groups of locally finite graphs, see Proposition 3.2. We deduce the following statement, which is the main intermediate step in the proof of Theorem 1.1. Theorem 1.3. Let Λ be a locally finite connected (simple, undirected) graph and Γ ≤ Aut(Λ) act cocompactly on Λ. Let H n → H be a converging sequence in Sub(Aut(Λ)). Suppose that for each n ≥ 1, there exists τ n ∈ Aut(Λ) such that τ n Γτ −1 n ≤ H n . Then we have
[H : H (∞) ] ≤ lim sup n→∞ [H n : H (∞) n ].
In particular, the set
{H ∈ Sub(Aut(Λ)) | H ≥ Γ and H = H (∞) } is Chabauty-closed.
The condition that all groups H n contain a conjugate of a fixed group Γ acting cocompactly may be viewed as a strengthening of the condition bounding the number of orbits, which was imposed in Theorem 1.1. Classical results by Bass [Bas93] and Bass-Kulkarni [BK90] ensure that when Λ is a tree, both conditions are equivalent (see §5.1 below). Building upon this, we tighten the relation between Chabauty convergence of unimodular cocompact subgroups of Aut(Λ) and k-closures (see Corollary 5.6) and deduce that the algebraic properties of local pro-π-ness and local torsion-freeness are both Chabauty-open in that context, see Propositions 5.19 and 5.22.
Taking advantage of the rather flexible hypotheses of Theorem 1.3, we include applications to groups acting on buildings that are not necessarily trees, see Corollary 6.3. We are not aware of families of graphs other than trees where analogues of the aforementioned results by Bass-Kulkarni hold. However, we note that chamber-transitive buildings whose Weyl group is virtually free all admit a canonical continuous proper cocompact action on a tree (see Lemma 6.6), so that the condition that the groups under consideration all contain a conjugate of a fixed group Γ also becomes redundant in that context, see Corollary 6.8.
Acknowledgement
We thank the anonymous referee for constructive comments and suggestions.
The Chabauty space
Given a locally compact group G, we denote by Sub(G) the set of closed subgroups of G equipped with the Chabauty topology, which is compact Hausdorff (see [Bou63, Chapitre VIII, §5, no. 3, Théorème 1]). Recall that a base of neighborhoods of H ∈ Sub(G) in the Chabauty topology is given by the sets
V K,U (H) := {J ∈ Sub(G) | J ∩ K ⊆ HU and H ∩ K ⊆ JU },
where K ranges over compact subsets of G and U over non-empty open subsets of G.
Assume that G is second countable. In that case, the compact space Sub(G) is also second countable. In particular Sub(G) is metrizable by Urysohn's Metrization Theorem (alternatively, one may directly define a compatible metric on Sub(G), see [Gel18, Exercise 2]). The locally compact groups appearing in this paper will mostly be automorphism groups of connected locally finite graphs: endowed with the compact open topology, those are second countable (totally disconnected) locally compact groups.
Lemma 2.1. Let G be a second countable locally compact group. A sequence (H n ) in Sub(G) converges to H ∈ Sub(G) if and only if the two conditions below are satisfied:
(i) Let (H k(n) ) be a subsequence of (H n ) and let (h k(n) ) be a sequence in G such that h k(n) ∈ H k(n) for each n ≥ 1. If (h k(n) ) converges to h ∈ G, then h ∈ H.
(ii) Any h ∈ H is the limit of a sequence (h n ) with h n ∈ H n for each n ≥ 1.
Proof. See [GR06, Lemma 2].
The following results are then immediate.
Lemma 2.2. Let G be a second countable locally compact group. The conjugation action of G on Sub(G) is jointly continuous, i.e. if g n → g is a converging sequence in G and H n → H is a converging sequence in Sub(G), then g n H n g −1 n → gHg −1 . Proof. This is an easy consequence of Lemma 2.1.
Lemma 2.3. Let G be a second countable locally compact group.
(1) If (H n ) is a descending chain in Sub(G), then H n → i≥1 H i ;
(2) If (H n ) is an ascending chain in Sub(G), then H n → i≥1 H i .
Proof. We prove (1), the proof of (2) being similar. Let us check (i) and (ii) in Lemma 2.1. Any h ∈ i≥1 H i is the limit of the constant sequence (h), so (ii) is clear. Now in order to prove (i), let h k(n) → h be a converging sequence in G such that h k(n) ∈ H k(n) for each n ≥ 1. For each i ≥ 1, the sequence (h k(n) ) k(n)≥i is contained in H i . Since H i is closed and h k(n) → h, we get h ∈ H i . This being true for any i ≥ 1, we have h ∈ i≥1 H i .
We also record the following essential result for the sake of future references.
Theorem 2.4. Let G be a locally compact group. The set Sub(G) 0 of unimodular closed subgroups of G is closed in Sub(G).
Proof. See [Bou63, Chapitre VIII, §5, no. 3, Théorème 1].
The next basic lemma plays a key role in the proof of Proposition 2.6.
Lemma 2.5. Let G be a locally compact group and C be a compact open subset of G (e.g. C is a coset of a compact open subgroup). Then the set
{H ∈ Sub(G) | H ∩ C = ∅} is clopen in Sub(G).
Proof. Since C is compact, we have U CU −1 = C, where the intersection is taken over all open, relatively compact, identity neighbourhoods U in G. Since C is also open, it follows that there exists an open, relatively compact, identity neighborhood U in G such that CU −1 = C. For any H ∈ Sub(G), we then consider the basic Chabauty-neighborhood
V C,U (H) = {J ∈ Sub(G) | J ∩ C ⊆ HU and H ∩ C ⊆ JU } of H. We observe that, for any J ∈ V C,U (H), we have J ∩ C = ∅ if and only if H ∩ C = ∅. Thus the set {H ∈ Sub(G) | H ∩ C = ∅} and its complement {H ∈ Sub(G) | H ∩ C = ∅} are both open.
In the following proposition, as well as in the rest of the paper, we adopt the terminology from [Bas93, §1] concerning graphs. Given a graph Λ and a group H ≤ Aut(Λ) acting without inversion on Λ, one can form the quotient graph H\Λ and the canonical projection p : Λ → H\Λ. We recall from [BL01, §2.5] that the quotient graph Q = H\Λ is an edge-indexed graph, i.e. it comes equipped with the map i associating to each oriented edge e of Q (where a geometric edge is seen as a pair of oriented edges) the cardinal i(e) = #{a ∈ E(Λ) | p(a) = e and x is the origin of a}, where x is any vertex in Λ such that p(x) is the origin vertex of e. Since H always acts without inversion on the first barycentric subdivision Λ (1) of Λ, it follows that any group H ≤ Aut(Λ) yields a well defined edge-indexed quotient graph H\Λ (1) .
We also need to define a coloring of a graph Λ as a map c : V (Λ) → C, where C is any set. We write Λ c for Λ considered with its coloring c and Aut(Λ c ) for the group of all automorphisms of Λ preserving c.
Proposition 2.6. Let Λ be a locally finite connected graph and (Q, i) be a finite edgeindexed graph. Then the following assertions hold.
(1) Let c : V (Λ) → C and c ′ : V (Q) → C be colorings of the graphs Λ and Q respectively.
For any closed subset H ⊆ Sub(Aut(Λ c )) consisting of groups acting without inversion, the set
{H ∈ H | H\Λ c ∼ = (Q c ′ , i)}
is clopen in H.
(2) Let c :
V (Λ (1) ) → {0, 1} be the coloring of Λ (1) defined by setting c(v) = 0 if v is a vertex of Λ and c(v) = 1 if v the midpoint of a geometric edge of Λ. Let c ′ : V (Q) → {0, 1} be any coloring of Q. The set {H ∈ Sub(Aut(Λ)) | H\(Λ (1) ) c ∼ = (Q c ′ , i)}
is clopen in Sub(Aut(Λ)).
(3) For any C > 0, the set
Sub(Aut(Λ)) ≤C := {H ∈ Sub(Aut(Λ)) | #V (H\Λ) ≤ C} is clopen in Sub(Aut(Λ)). Proof. (1). Let F ⊆ V (Λ) ∪ E(Λ)
be a finite set of vertices and edges of Λ. We denote by
H co-F the set of those H ∈ H such that HF = V (Λ) ∪ E(Λ), i.e.
those H ∈ H such that F meets every H-orbit of vertices and every H-orbit of edges in Λ.
Claim 1. The set H co-F is clopen in H.
Proof of the claim: Define F as the set consisting of all vertices that are adjacent to a vertex in F or incident to an edge in F. Since F is finite and Λ is locally finite, we infer that F is finite. Define the set
J = {J ∈ H | ∀x ∈ F, ∃j ∈ J : j(x) ∈ F}.
It is clear that H co-F ⊆ J , and we claim that H co-F = J . Indeed, let J ∈ J . Observe that X = JF is a subset of V (Λ) ∪ E(Λ) satisfying the property that for any vertex x in X, all edges of Λ incident to x and all vertices of Λ adjacent to x are also in X. Since Λ is connected, we deduce that X = V (Λ) ∪ E(Λ) and hence that J ∈ H co-F . Now remark that
H co-F = J = x∈F {J ∈ H | J ∩ C x = ∅},
where C x is the compact open subset of Aut(Λ) consisting of the elements h with h(x) ∈ F. As F is finite, Lemma 2.5 ensures that H co-F is clopen in H.
Claim 2. The set
V (Q c ′ ,i),F := {H ∈ H co-F | H\Λ c ∼ = (Q c ′ , i)} is clopen in H.
Proof of the claim: For each H ∈ H co-F , the isomorphism type of the edge-indexed (colored) quotient graph H\Λ c is completely determined by the following finite subset of F × (V (Λ) ∪ E(Λ)):
S H := {(x, y) ∈ F × (V (Λ) ∪ E(Λ)) | ∃h ∈ H : hx = y and d(y, F) ≤ 1}. Moreover, it is clear from Lemma 2.1 that if H n → H in H co-F then S Hn = S H for sufficiently large n. Consequently, the set V (Q c ′ ,i),F is clopen in H co-F .
As H co-F is itself clopen in H by Claim 1, the conclusion follows.
We now finish the proof as follows. We must show that the set
V (Q c ′ ,i) := {H ∈ H | H\Λ c ∼ = (Q c ′ , i)} is clopen in H.
We may assume that it is nonempty. Fix a base vertex v 0 ∈ V (Λ). For any group H ∈ V (Q c ′ ,i) , we can find a set of representatives F 0 of the H-orbits of vertices and edges in Λ, in such a way that v 0 ∈ F 0 and that F 0 is connected. Notice that there are only finitely many connected subsets F ⊆ V (Λ) ∪ E(Λ) containing v 0 and such that #V (F) = #V (Q) and #E(F) = #E(Q). Let us enumerate all of them,
namely F 0 , F 1 , . . . , F m . We have V (Q c ′ ,i) = m j=0 V (Q c ′ ,i),F j . Each V (Q c ′ ,i),F j is clopen by Claim 2, hence V (Q c ′ ,i) is clopen as well.
(2). We may identify Aut(Λ) with Aut((Λ (1) ) c ), which acts without inversion on Λ (1) . The desired assertion then follows from (1).
Lemma 3.1. Let Λ be a locally finite connected graph. For any k ≥ 0 and J ≤ Aut(Λ), k J is a closed subgroup of Aut(Λ). Moreover we have
J = k≥0 k J.
Proof. The proofs when Λ is a locally finite tree are given in [BEW14, Proposition 3.4], but they are independent from the tree structure and thus also work for any locally finite connected graph Λ.
In view of Lemma 2.3 (1), the previous lemma implies that
k J → J in Sub(Aut(Λ)).
The next result is then a key tool for the proof of Theorem 1.3. In order to facilitate its statement, we introduce the following notation. Given a group Γ ≤ Aut(Λ), we write
Sub(Aut(Λ)) ≥Γ = {H ∈ Sub(Aut(Λ)) | H ≥ τ Γτ −1 for some τ ∈ Aut(Λ)}.
Observe that if the normalizer of Γ in Aut(Λ) is cocompact, then Sub(Aut(Λ)) ≥Γ is Chabauty-closed. Given a group H ≤ Aut(Λ), a vertex v ∈ V (Λ) and an integer r ≥ 0, we also write H
[r]
v for the pointwise stabilizer of the ball B(v, r) in H.
Proposition 3.2. Let Λ be a locally finite connected graph, Γ ≤ Aut(Λ) act cocompactly on Λ and H ∈ Sub(Aut(Λ)) ≥Γ . Fix v 0 ∈ V (Λ). Then for each k ≥ 0, the set V k := {J ∈ Sub(Aut(Λ)) ≥Γ | σJσ −1 ≤ k H for some σ ∈ Aut(Λ) [k] v 0 } is a neighborhood of H in Sub(Aut(Λ)) ≥Γ .
Proof. Consider a sequence H n → H in Sub(Aut(Λ)) ≥Γ and let us show that H n ∈ V k for sufficiently large n. Let X ⊂ Λ be a compact fundamental domain for the action of Γ on Λ. For each n, let τ n ∈ Aut(Λ) be such that H n ≥ τ n Γτ −1 n . We may assume, up to precomposing τ n with an adequate element of Γ, that τ n sends v 0 to a vertex in X. Since X is compact, the sequence (τ n ) is bounded and we can further assume (by passing to a subsequence) that (τ n ) converges to some τ ∈ Aut(Λ). Define σ n := τ τ −1 n for each n ≥ 1 so that σ n → id. In this way, we have
Γ ′ := τ Γτ −1 ≤ τ τ −1 n H n τ n τ −1 = σ n H n σ −1 n =: H ′ n for each n ≥ 1,
where Γ ′ also acts cocompactly on Λ with X ′ := τ (X) as a fundamental domain. Moreover, as σ n → id, we have H ′ n → H by Lemma 2.2 and in particular Γ ′ ≤ H. In order to conclude, it suffices to find N ≥ 1 such that H ′ n ≤ k H for each n ≥ N . Let D be the diameter of X ′ and set
K := {g ∈ Aut(Λ) | d(g(v 0 ), v 0 ) ≤ 2D} and U := Aut(Λ) [k+D] v 0 .
The set K is compact and the set U is open, so there exists N ≥ 1 such that H ′ n ∈ V K,U (H) for each n ≥ N . In particular, we have H ′ n ∩ K ⊆ HU for each n ≥ N . This exactly means that, for any
g ∈ H ′ n (with n ≥ N ) satisfying d(g(v 0 ), v 0 ) ≤ 2D, there exists h ∈ H such that g| B(v 0 ,k+D) = h| B(v 0 ,k+D) . We need to show that H ′ n ≤ k H for each n ≥ N . In order to do so, consider g ∈ H ′ n with n ≥ N and v ∈ V (Λ). Let γ 1 ∈ Γ ′ be such that d(γ 1 g(v), v 0 ) ≤ D and γ 2 ∈ Γ ′ be such that d(γ 2 (v 0 ), v) ≤ D.
Those elements exist because D is the diameter of the fundamental domain X ′ for the action of Γ ′ . The two previous inequalities imply that
d(γ 1 gγ 2 (v 0 ), v 0 ) ≤ 2D. Hence, by definition of N there exists h ∈ H with γ 1 gγ 2 | B(v 0 ,k+D) = h| B(v 0 ,k+D) ,
which is equivalent to saying that
g| B(γ 2 (v 0 ),k+D) = γ −1 1 hγ −1 2 | B(γ 2 (v 0 ),k+D) . But d(γ 2 (v 0 ), v) ≤ D, so B(γ 2 (v 0 ), k + D) ⊇ B(v, k) and g| B(v,k) = γ −1 1 hγ −1 2 | B(v,k) , which is sufficient to conclude since γ −1 1 hγ −1 2 ∈ H.
The following observation describes a local algebraic property that is preserved when taking the k-closure (with a sufficiently large k).
Proposition 3.3. Let Λ be a locally finite connected graph and H ∈ Sub(Aut(Λ)). Let also π be a set of primes and r ≥ 0. Suppose that H
[r]
v is a pro-π group for all v ∈ V (Λ). Then for each k ≥ r + 1, the group ( k H)
[r]
v is a pro-π group for all v ∈ V (Λ). In particular, if H acts cocompactly on Λ and has an open pro-π subgroup, then so does k H for all sufficiently large k.
Proof. Since k H ≤ ℓ H for all k ≥ ℓ and since a closed subgroup of a pro-π group is pro-π, it suffices to consider G = r+1 H. We show that for each n ≥ r and each v ∈ V (Λ), the
finite group G [n] v G [n+1] v
is a π-group. This assertion implies the required conclusion.
Fix n ≥ r and v ∈ V (Λ) and assume for a contradiction that G
[n] v G [n+1] v
is not a π-group. Then it contains an element g of prime order p, with p ∈ π. There exists a vertex z with d(v, z) = n such that the restriction g| B(z,1) contains a p-cycle. Let x be a vertex on a geodesic path from v to z such that d(
x, z) = r. Thus g fixes B(x, r) ⊆ B(v, n) pointwise. Since g ∈ G = r+1 H, there is h ∈ H such that g| B(x,r+1) = h| B(x,r+1) . Hence h belongs to H [r]
x and the image of h modulo H
[r+1] x is of order p ∈ π. This contradicts the hypothesis that H x is pro-π. Now suppose H acts cocompactly on Λ and has an open pro-π subgroup U . Since U is open, there exists v 0 ∈ V (Λ) and r ≥ 0 such that H
[r] v 0 ⊆ U . Let X ∋ v 0 be a compact
fundamental domain for the action of H on Λ, and denote by D its diameter. For each vertex x ∈ X, we have
H [r+D] x ⊆ H [r] v 0 ⊆ U, so H [r+D] x
is a pro-π group. Since X is a fundamental domain for the action of H on Λ, we even have that H
[r+D] v
is a pro-π group for all v ∈ V (Λ). By the previous assertion, this implies that k H has an open pro-π subgroup for each k ≥ r + D + 1.
Applying the previous proposition in the case of the empty set of primes, we obtain the following corollary for discrete groups.
Corollary 3.4. Let Λ be a locally finite connected graph and H be a discrete subgroup of Aut(Λ) acting cocompactly on Λ. Then H = k H for all sufficiently large k.
Proof. Applying Proposition 3.3 to the empty set π = ∅, we obtain that k H is discrete for each sufficiently large k. Since H acts cocompactly on Λ, so does k H for any k.
Fixing k 0 such that k 0 H is discrete, we deduce that the index of H in k 0 H is finite. Since H ≤ k+1 H ≤ k H ≤ k 0 H for each k ≥ k 0 ,
Finite quotients of groups acting on graphs
The goal of this section is to prove Theorem 1.3. We first recall that, for a topological group G, the symbol G (∞) denotes the intersection of all open subgroups of finite index of G. The following lemma is classical. The notation P ≤ ofi G means that P is an open subgroup of finite index of G.
H = m i=1 h i P for some h 1 , . . . , h m ∈ H.
Since H acts cocompactly on Λ, the action of P on Λ is also cocompact. Let X ⊆ Λ be a compact fundamental domain for the action of P and denote by D the diameter of X.
The fact that P is an open subgroup of H implies that there exists R ≥ 0 with
H [R] v 0 ⊆ P.
We claim that [ k H : k P ] ≤ m for each k ≥ R + D + 1. To prove the claim, we fix k ≥ R + D + 1 and show that
k H = m i=1 h i k P .
Take g ∈ k H and v ∈ V (Λ). There exists i ∈ {1, . . . , m} and x ∈ P such that g| B(v,k) =
h i x| B(v,k) , which is equivalent to saying that h −1 i g| B(v,k) = x| B(v,k)
. If we prove that i is independent of the choice of v, then we will get h −1 i g ∈ k P which will end the proof. Since Λ is connected, it suffices to show that the value of i is the same for any two adjacent vertices. Fix v and v ′ two neighboring vertices of Λ and suppose that
g| B(v,k) = h i x| B(v,k) and g| B(v ′ ,k) = h j y| B(v ′ ,k)
for some x, y ∈ P and some i, j ∈ {1, . . . , m}. It follows that
h i x| B(v,k−1) = h j y| B(v,k−1) or equivalently that h −1 j h i xy −1 | B(y(v),k−1) = id | B(y(v),k−1) . The element e := h −1 j h i xy −1 is thus such that e ∈ H [k−1] y(v)
. As X is a fundamental domain (with diameter D) for the action of P on Λ, there exists p ∈ P such that p(y(v)) ∈ B(v 0 , D). Hence, the element pep −1 satisfies
pep −1 ∈ H [k−1] p(y(v)) ⊆ H [k−1−D] v 0 ⊆ H [R] v 0 ⊆ P.
We get h −1 j h i xy −1 = e ∈ P and thus h −1 j h i ∈ P , which implies that i = j as desired.
Before proving Theorem 1.3, we still need a technical lemma.
Lemma 4.3. Let Λ be a locally finite connected graph and H n → H, L n → L be two converging sequences in Sub(Aut(Λ)) such that L n ≤ H n for each n ≥ 1. Assume that there exists C > 0 such that H n ∈ Sub(Aut(Λ)) ≤C for each n ≥ 1. Suppose also that there exists S ≥ 1 such that [H n : L n ] ≤ S for each n ≥ 1. Then L ≤ H and [H : L] ≤ S.
Proof. The fact that L ≤ H is clear. For each n ≥ 1, let F n ⊆ Aut(Λ) be such that H n = L n F n and |F n | ≤ S. We directly get that L n ∈ Sub(Aut(Λ)) ≤CS for each n ≥ 1.
If v 0 ∈ V (Λ) is a fixed vertex, for each n ≥ 1 and f ∈ F n we can thus assume that d(f (v 0 ), v 0 ) ≤ CS. By adding elements to F n if necessary, we can also suppose that |F n | = S and write F n = {f . . , f S }. It is clear that F ⊆ H and we claim that H = LF . Take h ∈ H. By Lemma 2.1, there exists a converging sequence h n → h with h n ∈ H n for each n ≥ 1. As H n = L n F n , we can write h n = ℓ n f (n) in with ℓ n ∈ L n and i n ∈ {1, . . . , S}. There is a subsequence (i k(n) ) of (i n ) which is constant, say
equal to j ∈ {1, . . . , S}. Then h k(n) = ℓ k(n) f (n) j and hence ℓ k(n) = h k(n) (f (n) j ) −1 → hf −1
j . This limit belongs to L, so hf −1 j = ℓ ∈ L and h = ℓf j .
The proof of Theorem 1.3 is now an easy combination of the previous results. Given a tree T and two groups H, H ′ ≤ Aut(T ) acting without inversion on T , we write the equality H\T = H ′ \T whenever H and H ′ have the same orbits on T . The latter condition, which means that the canonical projections p : T → H\T and p ′ : T → H ′ \T coincide, implies in particular that the quotient graphs H\T and H ′ \T are isomorphic as edge-indexed graphs, since the edge-indexing function of the quotient graph is completely determined by the projection map. The following basic fact clarifies the difference between isomorphism and equality of quotients.
Lemma 5.2. Let T be a tree and H, H ′ ≤ Aut(T ) act without inversion on T . If H\T and H ′ \T are isomorphic as edge-indexed graphs, then there exists g ∈ Aut(T ) such that
H\T = H ′′ \T , where H ′′ = gH ′ g −1 .
Proof. This is a particular case of [MSW02, Lemma 13 (1)]. Corollary 5.4. Let T be a locally finite tree and H ∈ Sub(Aut(T )). Suppose that H is unimodular and acts cocompactly on T (these conditions hold, for instance, if H is edgetransitive and type-preserving). Then H has a subgroup Γ acting freely and cocompactly on T such that Sub(Aut(T )) ≥Γ is a neighborhood of H in Sub(Aut(T )).
Proof. Upon replacing T by its first barycentric subdivision, we may assume that H acts without inversion. By Proposition 5.1, there exists Γ ≤ H acting cocompactly and freely on T . Consider a converging sequence H n → H in Sub(Aut(T )). By Proposition 2.6 (3) H n acts with at most C orbits of vertices for all sufficiently large n, where C = #V (H\T ). We may then deduce from Lemma 2.5 that H n acts without inversion for all sufficiently large n because H does. Moreover Proposition 2.6 (1) ensures that, for sufficiently large n, the quotient graphs H n \T and H\T are isomorphic as edge-indexed graphs. Hence, by Lemma 5.2 and Proposition 5.3, for sufficiently large n, there exists τ n ∈ Aut(T ) such that τ n Γτ −1 n ≤ H n , i.e. H n ∈ Sub(Aut(T )) ≥Γ .
We record the following result for its own interest. It shows that, in Corollary 5.4, the choice of Γ can be made uniform, i.e. independent of the choice of H. In order to make this precise, we define Sub(Aut(T )) 0 ≤C := {H ∈ Sub(Aut(T )) ≤C | H is unimodular}.
Corollary 5.5. Let T be a locally finite tree. For each C > 0, the set Sub(Aut(T )) 0 ≤C is clopen in Sub(Aut(T )). Moreover there exists a subgroup Γ ≤ Aut(T ) acting freely and cocompactly on T such that
Sub(Aut(T )) 0 ≤C ⊆ Sub(Aut(T )) ≥Γ .
Proof. We already know by Proposition 2.6 (3) that Sub(Aut(T )) ≤C is a clopen subset of Sub(Aut(T )). Moreover the set Sub(Aut(T )) 0 of unimodular subgroups is closed by Theorem 2.4. In particular Sub(Aut(T )) 0 ≤C is closed. For each H ∈ Sub(Aut(T )) 0 ≤C , Corollary 5.4 yields a discrete cocompact group Γ such that Sub(Aut(T )) ≥Γ is a neighborhood of H. Since every locally compact group containing a lattice is unimodular, this implies that Sub(Aut(T )) 0
≤C is also open. Let us now partition the set Sub(Aut(T )) 0 ≤C into subsets V 1 , . . . , V m in such a way that H, H ′ ∈ V i if and only if H\T ∼ = H ′ \T as edge-indexed graphs. By Lemma 5.2 and Proposition 5.3, for each i there exists a discrete cocompact group Γ i such that V i ⊆ Sub(Aut(T )) ≥Γ i . In particular
Sub(Aut(T )) 0 ≤C ⊆ m i=1
Sub(Aut(T )) ≥Γ i .
By [BK90, Commensurability Theorem], upon replacing each Γ i by a conjugate, we may assume that they are pairwise commensurate, i.e. the index of Γ i ∩ Γ j is of finite index in Γ i for all i and j. It follows that Γ = m i=1 Γ i is itself a cocompact lattice in Aut(T ). The required assertion follows since Sub(Aut(T )) ≥Γ i ⊆ Sub(Aut(T )) ≥Γ for all i.
We then deduce the following corollary from Proposition 3.2.
Corollary 5.6. Let T be a locally finite tree and H ∈ Sub(Aut(T )). Suppose that H is unimodular and acts cocompactly on T . Fix v 0 ∈ V (T ). Then for each k ≥ 0, the set
{J ∈ Sub(Aut(T )) | σJσ −1 ≤ k H for some σ ∈ Aut(T ) [k] v 0 } is a neighborhood of H in Sub(Aut(T )).
Proof. This is the combination of Corollary 5.4 and Proposition 3.2.
Arguing similarly, we obtain the following consequence of Theorem 1.3.
Corollary 5.7. Let T be a locally finite tree and H n → H be a converging sequence in Sub(Aut(T )). Suppose that H is unimodular and acts cocompactly on T . Then we have Proof. Let Γ be the subgroup of H given by Corollary 5.4. It acts cocompactly on T and is such that Sub(Aut(T )) ≥Γ is a neighborhood of H. We thus have H n ∈ Sub(Aut(T )) ≥Γ for all sufficiently large n, and the conclusion follows from Theorem 1.3.
Limits of simple groups acting on trees
The goal of this section is to prove the next theorem, which is a stronger version of Theorem 1.1.
Theorem 5.8. Let T be a locally finite tree all of whose vertices have degree ≥ 2. For any C > 0, the Chabauty-closure of the set of abstractly simple groups in Sub(Aut(T )) ≤C is the set of groups in Sub(Aut(T )) ≤C without proper open subgroup of finite index.
We start by proving the following.
Proposition 5.9. Let T be a locally finite tree and let C > 0. The set
Sub(Aut(T )) (∞) ≤C := {H ∈ Sub(Aut(T )) ≤C | H = H (∞) } is closed in Sub(Aut(T )).
Proof. Let H n → H be a converging sequence in Sub(Aut(T )) with H n ∈ Sub(Aut(T )) (∞) ≤C for each n. We already know by Proposition 2.6 (3) that H ∈ Sub(Aut(T )) ≤C . For each n ≥ 1, H n acts cocompactly on T and is thus compactly generated. Therefore, the image of the modular character of H n is a finitely generated subgroup of R, which is thus residually finite. In particular, the condition that H n = H (∞) n implies that H n is unimodular. By Theorem 2.4, H is also unimodular and we can apply Corollary 5.7 to get H = H (∞) , as required. This confirms that Sub(Aut(T )) (∞) ≤C is closed.
There remains to show that any group in Sub(Aut(T )) (∞) ≤C is a limit of abstractly simple groups in that same set. Before proving this we need two more technical results.
Lemma 5.10. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and H ≤ Aut(T ) be a closed subgroup without any infinite cyclic discrete quotient (e.g. H = H (∞) ). If H acts cocompactly on T , then it does not preserve any proper non-empty subtree and does not fix any end of T .
Proof. Since all vertices of T have degree ≥ 2 and H acts cocompactly on T , we deduce from [Tit70, Lemme 4.1] that H does not preserve any non-empty subtree of T .
Suppose now for a contradiction that H fixes some end b ∈ ∂T . Let (v n ) be the sequence of vertices on a ray in T toward b. Then the map φ : H → Z defined by φ(h) := lim n→∞ d(h(v n ), v n ) is a group homomorphism and has infinite image (because H acts cocompactly on T ), which contradicts the fact that H has no infinite cyclic discrete quotient.
In the following proposition and as in [BEW14], given J ≤ Aut(T ) and k > 0, the symbol J + k denotes the subgroup of J generated by the pointwise stabilizers of (k−1)-balls around edges of T .
Proposition 5.11. Let T be a locally finite tree and G ≤ Aut(T ) be a non-discrete group which acts cocompactly on T , does not preserve any proper non-empty subtree and does not fix any end of T . Suppose that G = k G for some k ≥ 0. Then G + k is abstractly simple and G/G + k is virtually free.
Proof. From [BEW14, Theorem 7.3] we know that G + k is abstractly simple or trivial. Also, it is clear from the definition that G + k is an open normal subgroup of G. Since G is non-discrete, G + k is non-discrete and in particular non-trivial (hence simple). The discrete quotient group G/G + k acts cocompactly on the quotient graph G + k \T . Bass-Serre theory ensures that G + k is the fundamental group of a graph of groups, whose underlying graph is nothing but G + k \T (see [Ser77,§I.5.4, Théorème 13]). By definition G + k is generated by pointwise stabilizers of edges. In particular it is generated by vertex stabilizers. It then follows that the quotient graph G + k \T is a tree (see [Ser77,§I.5.4, Corollaire 1]). We next observe that the G/G + k -action on the tree G + k \T is proper. Indeed, a coset gG + k stabilizes a vertex in G + k \T if and only if gv ∈ G + k v for some v ∈ V (T ). This is equivalent to the requirement that g ∈ G + k U , where U is the stabilizer of v in G, which is compact. This confirms that the stabilizer of a vertex of G + k \T in the discrete quotient group G/G + k is indeed compact, hence finite. Therefore G/G + k is a discrete group acting properly and cocompactly on a tree. It is thus virtually free.
Proposition 5.12. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and let C > 0. In Sub(Aut(T )) (∞) ≤C , the subset consisting of the abstractly simple groups is dense.
Proof. Pick any H ∈ Sub(Aut(T )) (∞) ≤C . We must show that H is a limit of abstractly simple groups contained in Sub(Aut(T )) where T is regular of degree p 2 + p + 1 (p being an arbitrary prime). In particular, the set of topologically simple locally 2-transitive closed subgroups of Aut(T ) is generally not Chabauty-closed.
(∞) ≤C . For each k > 0, set H k = ( k H) + k .
The following result shows that the conclusion of Theorem 1.1 may fail if the tree T is allowed to have vertices of degree 1.
Lemma 5.14. Let T be the universal covering tree of the graph on 7 vertices depicted in the figure below. Let V 1 , V 3 and V 8 denote the set of vertices of T of degree 1, 3 and 8 respectively. Let X be the subtree of T which is the convex hull of V 3 . Thus X is isomorphic to the trivalent tree. Its vertex set is V 3 ∪ V 8 , and those two sets V 3 and V 8 are the two parts in the canonical bipartition of X. The following assertions hold.
(1) Aut(T ) has a closed subgroup H isomorphic to v∈V 8
Alt(5) ⋊ Aut(X) + .
(2) H = H (∞) .
(3) H is not a Chabauty limit of topologically simple closed subgroups of Aut(T ).
Proof.
(1) The subtree X is Aut(T )-invariant. Thus we have a canonical continuous homomorphism Aut(T ) → Aut(X). Its kernel is compact and isomorphic to v∈V 8 Sym(5). It contains a characteristic subgroup K isomorphic to v∈V 8 Alt(5). Moreover Aut(T ) has a closed subgroup S isomorphic to Aut(X) + . The requested subgroup H can be defined as H = KS.
(2) Let N be an open normal subgroup of finite index in H = KS. Then N ∩ K is an open normal subgroup of K, and thus contains all but finitely many factors of v∈V 8 Alt(5) (i.e. N ∩ K ⊇ v∈I Alt(5) for some cofinite set I ⊆ V 8 ). Since the conjugation action of H is transitive on those factors, we infer that N contains them all. Hence K ≤ N . Thus the quotient map H → H/N factors through H/K ∼ = S, which is simple by [Tit70]. Hence H/N is trivial, which confirms that H = H (∞) .
(3) Any infinite topologically simple subgroup of Aut(T ) acts faithfully on X. On the other hand, the group H contains an element h fixing a vertex v ∈ V 8 and permuting cyclically the 5 neighbors of v with degree 1. Any closed subgroup J of Aut(T ) which is sufficiently close to H in the Chabauty topology also contains elements fixing v with the same action on its neighbors. In particular the stabilizer J v has a non-trivial 5-Sylow subgroup. Since every vertex stabilizer in Aut(X) + is a pro-{2, 3} group, we deduce that J does not act faithfully on X and is thus not topologically simple.
Boundary-2-transitive automorphism groups of trees
Recall that the monolith Mon(G) of a topological group G is defined to be the (possibly trivial) intersection of all its non-trivial closed normal subgroups. It is clear from Lemma 4.1 that, when G is infinite, Mon(G) ≤ G (∞) (because an open subgroup is always closed). If moreover G is totally disconnected and locally compact, then it appears that Mon(G) = G (∞) as soon as Mon(G) is cocompact in G.
Lemma 5.15. Let G be a totally disconnected locally compact group. If G/ Mon(G) is compact, then G (∞) ≤ Mon(G).
Proof. By Lemma 4.1, we have
G (∞) = N ofi G N.
The group G/ Mon(G) is compact by hypothesis and totally disconnected (as a quotient of a totally disconnected locally compact group by a closed subgroup), so it is profinite. In The previous lemma can be applied when G is a boundary-2-transitive automorphism group of a tree, as the following result (due to M. Burger and S. Mozes) shows. Proof. This follows from [BM00, Propositions 1.2.1 and 3.1.2, Lemma 3.1.1] (see also Proposition 6.2 below).
Corollary 5.17. Let T be a locally finite thick semi-regular tree and H n → H be a converging sequence in Sub(Aut(T )) whose limit H acts 2-transitively on ∂T . Then we have Proof. This follows by assembling Corollary 5.7 and Proposition 5.16.
Corollary 5.18. Let T be a locally finite thick semi-regular tree. The set of topologically simple closed subgroups of Aut(T ) acting 2-transitively on ∂T is closed in Sub(Aut(T )).
Proof. It follows easily from [BM00, Lemma 3.1.1] that the set of boundary-2-transitive groups is closed in Sub(Aut(T )) (and is contained in Sub(Aut(T )) ≤2 ). Within that set, the subset of topologically simple groups is closed in view of Corollary 5.17.
Recall that, for a locally finite thick tree T , we defined the space S T by S T := {H ∈ Sub(Aut(T )) | H is topologically simple and 2-transitive on ∂T } /∼ = , where ∼ = is the relation of topological isomorphism. In our context, it actually appears that two groups are topologically isomorphic if and only if they are conjugate in Aut(T ) (see [Rad17,Proposition A.1]). This equivalence enables us to show Theorem 1.2.
Proof of Theorem 1.2. Let C = {H ∈ Sub(Aut(T )) | H is topologically simple and 2-transitive on ∂T }.
By Corollary 5.18, the set C is closed in Sub(Aut(T )) and hence compact Hausdorff.
The set C /∼ = endowed with the quotient topology is then also compact. In order to show that this space is Hausdorff, we can simply prove that the quotient map q : C → C /∼ = is open and that the set
D = {(H, H ′ ) ∈ C × C | H ∼ = H ′ } is closed in C × C.
By H ′ n = σ n H n σ −1 n for some σ n ∈ Aut(T ). As H n is edge-transitive, we can assume that σ n sends a fixed vertex v 0 to a vertex at distance ≤ 1 for all n ≥ 1. Hence, (σ n ) subconverges to some σ ∈ Aut(T ) and H ′ = σHσ −1 by Lemma 2.2. So D is closed in C × C.
Finally, the fact that D is closed in C × C also implies that ∼ = has closed classes.
Local prime content and local torsion-freeness
Let T be a locally finite tree. In this section, we provide applications of Corollary 5.6 by highlighting two algebraic properties that define open subsets of the space Sub(Aut(T )) 0 ≤C := {H ∈ Sub(Aut(T )) ≤C | H is unimodular}. Let π be a set of primes. A totally disconnected locally compact group is called locally pro-π if it has an open pro-π subgroup. If G is the full automorphism group of a regular rooted tree, then the set of locally pro-π subgroups is generally neither open nor closed in the Chabauty space Sub(G). The following result shows that this situation changes if one considers closed subgroups of bounded covolume in Aut(T ).
Proposition 5.19. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and let C > 0. Then for any set of primes π, the set of locally pro-π groups is open in Sub(Aut(T )) 0 ≤C . In particular the set of discrete subgroups is open in Sub(Aut(T )) 0 ≤C . Proof. Let H be a locally pro-π group in Sub(Aut(T )) 0 ≤C . By Proposition 3.3, there exists K ≥ 0 such that K H is also locally pro-π. We also know from Corollary 5.6 that the set
{J ∈ Sub(Aut(T )) | σJσ −1 ≤ K H for some σ ∈ Aut(T )}
is a neighborhood of H in Sub(Aut(T )). This set only contains locally pro-π groups, so the conclusion follows.
Remark 5.20. We emphasize that the set of locally pro-π groups in Sub(Aut(T )) 0 ≤C need not be closed in general. In order to see that, let T be the d-regular tree, with d ≥ 3. By [BK90, Theorem 7.1 (a)], the group Aut(T ) contains a properly ascending chain of cocompact lattices Γ 1 < Γ 2 < . . . . Denoting by C the number of Γ 1 -orbits of vertices, we have Γ i ∈ Sub(Aut(T )) 0 ≤C for all i. Let H = i≥1 Γ i . Since Γ 1 is a lattice in H, it follows that H is unimodular, so that H ∈ Sub(Aut(T )) 0 ≤C . If H were discrete, it would be a cocompact lattice in Aut(T ), and the chain of inclusions Γ 1 ≤ H ≤ Aut(T ) would force the index [H : Γ 1 ] to be finite, contradicting the properly ascending property of the chain Γ 1 < Γ 2 < . . . . We infer that H is non-discrete. In particular H is not locally pro-∅. On the other hand, we have H = lim i→∞ Γ i by Lemma 2.3 (2). Hence we have constructed a converging sequence of locally pro-∅ groups in Sub(Aut(T )) 0 ≤C , whose limit is not locally pro-∅. This confirms that the set of locally pro-π groups in Sub(Aut(T )) 0 ≤C is not closed in general. We shall need the following basic fact. Proof. Suppose the contrary. Then there exist m ≥ 0, a sequence of integers (k n ) tending to infinity with n, and a sequence (u n ) in U such that u p n ∈ U kn and u n ∈ U m . Upon extracting, we may assume without loss of generality that (u n ) converges to some u ∈ U . Since U m is open and u n ∈ U m for all n, we also have u ∈ U m . In particular u = 1. On the other hand, we have u p n ∈ U kn , so that u p = (lim n u n ) p = lim n u p n = 1. Hence u is a non-trivial torsion element of U , a contradiction.
Proposition 5.22. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and let C > 0. Then the set of locally torsion-free groups is open in Sub(Aut(T )) 0 ≤C .
Proof. We can suppose that Sub(Aut(T )) 0 ≤C is non-empty. It follows that T is of bounded degree. We define the finite set of primes π = {p prime | p ≤ deg(v) ∀v ∈ V (T )} and observe that the stabilizer Aut(T ) v of any vertex v ∈ V (T ) is a pro-π group.
Let H ∈ Sub(Aut(T )) 0 ≤C be locally torsion-free. We must show that H has a neighborhood in Sub(Aut(T )) 0 ≤C that consists of locally torsion-free groups.
Claim 1. There exist integers M > n > 0 such that for all v ∈ V (T ), p ∈ π and h ∈ H
[n] v , if h p ∈ H [M ] v then h ∈ H [n+1] v .
Proof of the claim: Since H is locally torsion-free, there exists n 0 ≥ 0 and v 0 ∈ V (T ) such that H
[n 0 ]
v 0 is torsion-free. As H acts cocompactly on T , there exists n ≥ n 0 such that H Let us now fix a prime p ∈ π and a vertex v ∈ V (T ) and let us apply Lemma 5.21 to the torsion-free profinite groups U k = H
[n+k] v and the integer m = 1. This yields a constant M (p, v) ≥ n such that for all h ∈ H
[n] v , if h p ∈ H [M (p,v)] v then h ∈ H [n+1] v .
We next define M as the supremum of M (p, v) taken over all p ∈ π and all vertices v in a (necessarily finite) fundamental domain for the H-action on V (T ). Then the required property holds (and we can assume that M > n). v is torsion-free for all v ∈ V (T ). In particular G is locally torsion-free. Proof of the claim: Suppose for a contradiction that for some v ∈ V (T ), there exists a non-trivial torsion element in G v . By the definition of π, every non-trivial torsion element of Aut(T ) has a power which is a non-trivial element of order p for some p ∈ π. We may thus assume that G
[n]
v contains a non-trivial element g of prime order p ∈ π. Let then k ≥ n be the largest integer such that g ∈ G v . Then there exists a vertex x ∈ B(v, k) fixed by g, such that g does not fix B(x, 1) pointwise. It follows that there exists a vertex y on the geodesic segment joining v to x such that g ∈ G
{J ∈ Sub(Aut(T )) | σJσ −1 ≤ M H for some σ ∈ Aut(T )},
which is a neighborhood of H in Sub(Aut(T )) by Corollary 5.6, only contains locally torsion-free groups
Remark 5.23. We emphasize that the set of locally torsion-free groups in Sub(Aut(T )) 0
≤C
need not be closed in general. An excellent illustration of that fact is provided by the main results of [Stu16], showing that some simple algebraic groups over local fields of positive characteristic (which are not locally torsion-free) are Chabauty limits of simple algebraic groups over p-adic fields (which are p-adic analytic, hence locally torsion-free). The following result, which is a straightforward adaptation of [CM11, Corollary 3.1] dealing with strongly transitive actions, shows that monolithic groups naturally appear in the context of Weyl-transitive automorphism groups of buildings. It may be seen as a generalization of Proposition 5.16. Proof. We follow the proof of [CM11, Corollary 3.1]. In generalizing from strongly transitive to Weyl-transitive actions, the point requiring a supplementary check is that Tits' transitivity lemma, which was originally stated for strongly transitive actions, holds more generally for Weyl-transitive action. This is indeed the case by [AB08, Lemma 6.61]. We are thus ensured that any non-trivial normal subgroup of H is transitive on the set of chambers of ∆. Therefore any non-trivial closed normal subgroup of H is cocompact. Since H is Weyl-transitive on ∆, it is chamber-transitive, hence compactly generated. We may then invoke [CM11, Theorem E], and conclude the proof word-by-word as in [CM11, Corollary 3.1]. The argument can be summarized as follows. We know from [CM11, Theorem E] that the monolith of H is a quasi-product of topologically simple groups. However, there can be only one simple factor using that the building ∆ has locally compact CAT(0) metric realization. The desired assertions follow. The next corollary is then a direct consequence of Theorem 1.3. Corollary 6.3. Let ∆ be an infinite irreducible locally finite thick building and Γ ≤ Aut(∆) act cocompactly on ∆. Let H n → H be a converging sequence in Sub(Aut(∆)) whose limit H is Weyl-transitive. Suppose that for each n ≥ 1, there exists τ n ∈ Aut(∆) such that τ n Γτ −1 n ≤ H n . Then we have Proof. This follows from Proposition 6.2 and Theorem 1.3, since a locally finite building can be seen as a locally finite connected graph whose vertices are the chambers and whose edges are the pairs of adjacent chambers.
Remark 6.4. If ∆ is a tree, then a closed Weyl-transitive subgroup of Aut(∆) is 2transitive on the set of ends ∂∆. Thus Corollary 5.17 can be deduced from Corollary 6.3.
Remark 6.5. If ∆ is a locally finite Euclidean building of dimension ≥ 2, it can be seen that there is a unique topologically simple closed subgroup of Aut(∆) acting Weyltransitively: namely the simple algebraic group to which ∆ is associated via Bruhat-Tits theory. This is of course not the case for trees. For higher-dimensional more exotic buildings (e.g. Bourdon buildings), there can be a much larger collection of simple groups acting Weyl-transitively, whose variety might potentially be comparable to one encountered in the case of trees (see [DMSS16]).
Buildings of virtually free type
We have seen in §5.1 that, for trees, the condition about the common cocompact group Γ was always fulfilled. It appears that, more generally, it is possible to drop the hypothesis about Γ in the context of buildings whose associated Coxeter group is virtually free. The reason is the existence of a strong relation between such buildings and trees.
Lemma 6.6. Let ∆ be an infinite irreducible locally finite thick building whose Weyl group W is virtually free. Suppose that Aut(∆) is chamber-transitive. Then there exists a locally finite tree T on which Aut(∆) acts continuously, properly, faithfully and cocompactly.
Proof. By [Dav08, Proposition 8.8.5], W is virtually free if and only if W has a tree of groups decomposition where each vertex group is a spherical special subgroup. If X is the tree of groups, then we write X for the underlying tree and denote W = π 1 (X ). Since Aut(∆) is chamber-transitive, we have by [Tit86, Proposition 2] that Aut(∆) = π 1 (X 0 ) where X 0 has the same underlying tree X as X and has adequate residue stabilizers as vertex groups and edge groups. By [Ser77, §I.4.5, Théorème 9], we deduce that Aut(∆) acts on a locally finite tree T in such a way that Aut(∆)\T = X. Moreover, the stabilizer of a vertex of T in Aut(∆) corresponds to a stabilizer of a spherical residue of ∆ and hence is compact and open. This implies that the action of Aut(∆) on T is continuous and proper. Finally, the kernel K ≤ Aut(∆) of this action on T stabilizes all residues of ∆ of a fixed spherical type. Since ∆ is infinite and irreducible, this implies that K is trivial (see [AB10,Main Theorem]). The action is thus faithful.
Remark 6.7. The tree of group decomposition of W is generally not unique. In particular, the tree T and the Aut(∆)-action on T afforded by Lemma 6.6 are not canonical.
Corollary 6.8. Let ∆ be an infinite irreducible locally finite thick building of virtually free type W . Let H n → H be a converging sequence in Sub(Aut(∆)) whose limit H is Weyl-transitive. Then we have In particular, if H n has no proper open subgroup of finite index for each n ≥ 1 then H is topologically simple.
Proof. Let T be the locally finite tree given by Lemma 6.6. The fact that Aut(∆) acts continuously, properly and faithfully on T means that there is a map i : Aut(∆) → Aut(T ) for each n ≥ 1. Then, by Fact 4 and Corollary 5.6, there exists (σ n ) with σ n ∈ Aut(T )
Following
Burger-Mozes [BM00], it is customary to denote the intersection of all open subgroups of finite index in a given locally compact group H by H (∞) . We also denote by Mon(H) the monolith of H, i.e. the (possibly trivial) intersection of all non-trivial closed normal subgroups of H. Notice that H is topologically simple if and only if H = Mon(H). With these notations at hand, the statement of Theorem 1.1 can be epitomized by the following equality: {H ∈ Sub(Aut(T )) ≤C | H = Mon(H)} = {H ∈ Sub(Aut(T )) ≤C | H = H (∞) }.
Lemma 4 . 1 .
41Let G be a topological group and P ≤ ofi G. There exists R ≤ P such that R ofi G. In particular, G (∞) coincides with the intersection of all open normal subgroups of finite index of G.Proof. It suffices to take for R the kernel of the natural action of G on G/P . The next result shows, in the context of automorphism groups of graphs, how k-closures preserve open subgroups of finite index.
Lemma 4 . 2 .
42Let Λ be a locally finite connected graph and H ∈ Sub(Aut(Λ)) act cocompactly on Λ. If P ≤ ofi H, then [ k H : k P ] ≤ [H : P ] for all sufficiently large k. Proof. Fix v 0 ∈ V (Λ) and let m = [H : P ]. We can write
S
}. Since the set {g ∈ Aut(Λ) | d(g(v 0 ), v 0 ) ≤ CS} is compact, we can finally assume by passing to subsequences that (f(n) i ) converges to some f i ∈ Aut(Λ) for each i ∈ {1, . . . , S}. Define F := {f 1 , .
Proposition 5. 3 .
3Let T be a tree and H, H ′ ≤ Aut(T ) act without inversion on T . Suppose that H\T = H ′ \T . If Γ ≤ H acts freely on T , then there exists τ ∈ Aut(T ) such that τ Γτ −1 ≤ H ′ . Proof. See [Bas93, Corollary 5.3].
[
H : H (∞) ] ≤ lim sup n→∞ [H n : H (∞) n ]. In particular, if H n has no proper open subgroup of finite index for each n ≥ 1 then H has no proper open subgroup of finite index.
particular, the open (and hence finite index) normal subgroups of G/ Mon(G) form a base of neighborhood of the identity. Their intersection is thus trivial, which implies that the intersection of all open normal subgroups of finite index of G is contained in Mon(G).
Proposition 5 . 16 .
516Let T be a locally finite thick semi-regular tree and H ∈ Sub(Aut(T )) act 2-transitively on ∂T . Then H/ Mon(H) is compact and Mon(H) is topologically simple. In particular Mon(H) = H (∞) .
[H : Mon(H)] ≤ lim sup n→∞ [H n : H (∞) n ].In particular, if H n has no proper open subgroup of finite index for each n ≥ 1 then H is topologically simple.
[ Rad17 ,
Rad17Proposition A.1], we have H ∼ = H ′ with H, H ′ ∈ C if and only if H and H ′ are conjugate in Aut(T ). Let U be an open subset of C. We first need to show that q(U ) is open, i.e. that q −1 (q(U )) is open. We have q −1 (q(U )) = σ∈Aut(T ) σU σ −1 , and σU σ −1 is clearly open for each σ ∈ Aut(T ), so q is an open map as wanted. Now consider two sequences H n → H and H ′ n → H ′ in C with H n ∼ = H ′ n for all n ≥ 1, i.e.
A totally disconnected locally compact group is called locally torsion-free if it has an open torsion-free subgroup. Typical examples are provided by p-adic analytic groups (see [DdSMS99, Theorems 4.5 and 8.1]).
Lemma 5 . 21 .
521Let U be a torsion-free profinite group and U = U 0 ≥ U 1 ≥ . . . be a descending chain of open subgroups of U with trivial intersection. Let also p be a positive integer. For each m ≥ 0, there exists M such that for all u ∈ U , if u p ∈ U M then u ∈ U m .
-free for all v ∈ V (T ).
Claim 2 .
2Let M > n be the constants afforded by Claim 1 and let G = M H.
Proposition 6 . 2 .
62Let ∆ be an infinite irreducible locally finite thick building and H ∈ Sub(Aut(∆)) be Weyl-transitive. Then Mon(H) is topologically simple and transitive on the set of chambers of ∆. In particular, H/ Mon(H) is compact and Mon(H) = H (∞) .
[
H : Mon(H)] ≤ lim sup n→∞ [H n : H (∞) n ].In particular, if H n has no proper open subgroup of finite index for each n ≥ 1 then H is topologically simple.
[
H : Mon(H)] ≤ lim sup n→∞ [H n : H (∞) n ].
the conclusion follows from Lemma 3.1.
Proof of Theorem 1.3. Let S = lim sup n→∞ [H n : H n ] ≤ S for each n ≥ 1. By Proposition 3.2, we may further assume that for eachk ≥ 0, there exists N (k) ≥ 1 such that H n ≤ k H for each n ≥ N (k). P ] ≤ S.An open subgroup is always closed, so P = P and the conclusion follows.Proposition 5.1. Let T be a locally finite tree. Let H ≤ Aut(T ) act cocompactly on T and suppose that H is unimodular. Then H contains a free uniform lattice, i.e. there exists a discrete subgroup Γ ≤ H acting freely and cocompactly on T . Proof. See [BK90, Existence Theorem].(∞)
n ]. Without loss of generality, we
may assume that [H n : H
(∞)
In order to prove that [H : H (∞) ] ≤ S, it suffices to prove that [H : P ] ≤ S for each
P ≤ ofi H. By Lemma 4.2, there exists K ≥ 0 such that
k P ≤ ofi
k H for any k ≥ K. Let us
temporarily fix k ≥ K. For each n ≥ N (k), we have H n ≤
k H and hence
k P ∩ H n ≤ ofi H n .
By hypothesis, this means that [H n :
k P ∩ H n ] ≤ S. Letting n tend to infinity, we obtain
with Lemma 4.3 that [H :
k P ∩ H] ≤ S for each k ≥ K. Now letting k tend to infinity and
because
k P → P (see Lemmas 2.3 (1) and 3.1), we get [H : 5 Trees
5.1 Existence and conjugation of tree lattices
When Λ is a locally finite tree, Propositions 5.1 and 5.3 below (which come from [BK90]
and [Bas93] respectively) can be used to drop the hypothesis about Γ in Theorem 1.3.
Since k H/H k is virtually free, it is residually finite. Recalling now that H has no finite discrete quotient other than the trivial one, we infer that H has trivial image in k H/H k , so that H ≤ H k ≤ k H. Since k H → H (by Lemma 3.1 and Lemma 2.3 (1)), we also get that H k → H, thereby completing the proof.First
note that H is not discrete, otherwise it would be virtually free, hence residually finite,
contradicting H = H (∞) . We can therefore invoke Lemma 5.10 and Proposition 5.11
(applied to
k H) to get that H k is abstractly simple and
k H/H k is virtually free.
Proof of Theorem 5.8. Follows by assembling Propositions 5.9 and 5.12.
Remark 5.13. It is important to note that the set
{H ∈ Sub(Aut(T )) | H is locally 2-transitive and H = H (∞) }
(in the terminology of [BM00]) may contain groups that are not topologically simple.
Explicit examples of such H are constructed in [BM00, Example 1.2.1],
[ n ]
ny and g ∈ G [n+1] y . Since G = M H, there exists h ∈ H such that g| B(y,M ) = h| B(y,M ) . The properties that g ∈ G [n] y , that g p = 1 and that g ∈ G respectively imply that h ∈ H [n] y , that h p ∈ H [M ] y and that h ∈ H [n+1] y . This contradicts Claim 1. From Claim 2 we know that M H is locally torsion-free. Hence, the set[n+1]
y
Let ∆ be a locally finite thick building. A subgroup H of Aut(∆) is said to be Weyltransitive if, for all w ∈ W , the action of H on the ordered pairs (c 1 , c 2 ) of chambers such that δ(c 1 , c 2 ) = w is transitive, where δ : Ch(∆) × Ch(∆) → W is the Weyl-distance.Remark 6.1. If H ≤ Aut(∆) is strongly transitive on ∆ (i.e. transitive on pairs (A, c) consisting of an apartment A and a chamber c ∈ A), then it is Weyl-transitive. The converse holds if ∆ is spherical, but not in general: see [AB08, Proposition 6.14]. If ∆ is of affine type (e.g. ∆ is a tree) and H is closed, it may be seen that if H is Weyltransitive, then it is strongly transitive on the spherical building at infinity of ∆, hence strongly transitive on ∆ by [CC15, Theorem 1.1]. For ∆ arbitrary (e.g. hyperbolic), the existence of Weyl-transitive but non-strongly transitive closed subgroups H ≤ Aut(∆) is likely, but currently we do not know explicit examples.6 Buildings
6.1 Weyl-transitive automorphism groups of buildings
(3). Let c : V (Λ (1) ) → {0, 1} be the coloring of Λ (1) as defined in (2). For any C > 0, there are finitely many edge-indexed (colored) graphs (Q c ′ , i) that can be isomorphic to H\(Λ (1) ) c for some H ∈ Sub(Aut(Λ)) ≤C . Moreover, given such a (Q c ′ , i), if H ′ ∈ Sub(Aut(Λ)) satisfies H ′ \(Λ (1) ) c ∼ = (Q c ′ , i) then #V (H ′ \Λ) = #V (H\Λ) ≤ C.Indeed, #V (H ′ \Λ) is equal to the number of vertices v of Q with c ′ (v) = 0. The conclusion then follows from (2).3 The k-closure of a graph automorphism group Let Λ be a locally finite connected graph. We define the k-closure k J of an automorphism group J ≤ Aut(Λ) byk J = {g ∈ Aut(Λ) | ∀v ∈ V (Λ), ∃h ∈ J : g| B(v,k) = h| B(v,k) },where B(v, k) is the ball centered at v and of radius k in Λ. That notion was first introduced and studied by Banks-Elder-Willis in[BEW14], in the case where Λ is a tree, even though they used the notation J (k) instead of k J.It is clear from the definition that k J ⊇ ℓ J ⊇ J for any k ≤ ℓ. Other basic properties of k-closures, due to Banks-Elder-Willis, are collected in the following lemma.
) and (τ n ) with τ n ∈ Aut(T )such that τ n G + (i) (Y (n) 0 , Y (n) 1 )τ −1 n → G + (i) (Y 0 , Y 1 ) and with [G + (i) (Y (n) 0 , Y (n) 1 )] = [G + (i) (Y 0 , Y 1 )]
which is an isomorphism onto its image, the latter being closed in Aut(T ). We thus have the converging sequence i(H n ) → i(H) in Sub(Aut(T )), such that i(H) acts cocompactly on T and is unimodular (because it is generated by compact subgroups). The conclusion then follows from Corollary 5.7, since Mon(H) = H (∞) (see Proposition 6.2).A The clopen subset S Alt T ⊆ S T Let T be a locally finite thick semi-regular tree. As before, we denote by S T the set of isomorphism classes of groups in Sub(Aut(T )) which are topologically simple and 2transitive on ∂T . By Theorem 1.2, this set carries a compact Hausdorff topology induced from the Chabauty topology on Sub(Aut(T )). The goal of this appendix is to provide supplementary information on that compact space. First recall that, when H ∈ Sub(Aut(T )) is 2-transitive on ∂T , the action of the stabilizer H v of a vertex v ∈ V (T ) in H is 2-transitive on the set of neighbors of v (see [BM00, Lemma 3.1.1]). In particular, H must be edge-transitive.Recall also that S T contains the isomorphism class of Aut(T ) + (which is simple by[Tit70]). In[Rad17], the second-named author restricted his attention to the groups H which locally contain the full alternating group, i.e. such that the action of H v on its set of d neighbors contains Alt(d) for each v ∈ V (T ). Let us denote by S Alt T the subset of S T consisting of the isomorphism classes of all those groups. An exhaustive description of the set S Alt T when the vertices of T have degree ≥ 6 is given in[Rad17]. Below we summarize some of its properties. Proof. The first assertion is clear. In order to prove the second one, we freely use the terminology and notation from[Rad17]without repeating all the definitions in full details. In that paper, a legal coloring i of T is fixed and, given two possibly empty finite subsetsLet us describe their properties which will be needed here. We assume henceforth that d 0 , d 1 ≥ 6. The group Gis the semiregular analog of the universal locally alternating group of Burger-Mozes[BM00].In fact, the groups G + (i) (Y 0 , Y 1 ) are not pairwise distinct, but this is not important for the following discussion. We now give some other properties of these groups. In order to shorten the statements, we adopt the convention max(∅) := +∞.. Proof of the fact: When X is a non-empty finite subset of Z ≥0 and Y is a finite sub-. Then h n → h because max X (n) → +∞, which proves (ii). The reasoning is exactly the same to obtain that G +If σ ∈ Aut(T )\Aut(T ) + , then there exists a particular element ν ∈ Aut(T )\Aut(T ) + such that νG + (i) (Z 0 , Z 1 )ν −1 = G + (i) (Z 1 , Z 0 ) and the conclusion follows.Table 1]), which suffices to conclude.Fact 4. For all finite subsets Y 0 , Y 1 ⊂ Z ≥0 , there exists an integer K ≥ 0 such thatLet us now compute the Cantor-Bendixson derivatives of S Alt T . In S Alt T , the points(Facts 2 and 3). We claim that the points [G + (i) (Y 0 , Y 1 )] with Y 0 and Y 1 non-empty are isolated. Suppose for a contradiction that there exists sequences (Yfor all sufficiently large n. But there are only finitely many, so we cannot have the supposed convergence. We just proved that the first Cantor-Bendixson derivative of S AltWith the exact same reasoning, we then obtain that is infinite also in that case. Indeed, the definition of the groups G + (i) (Y 0 , Y 1 ) (where Y 0 , Y 1 are finite subsets of Z ≥0 ) from loc. cit. makes sense for all d 0 , d 1 ≥ 3. For these groups to be boundary-2-transitive, one however needs to require Y 0 = {0} (resp. Y 1 = {0}) when d 0 = 3 (resp. d 1 = 3). Under the latter hypothesis, it is then possible to adapt the ideas from[Rad17,§4]and show that these groups are abstractly simple and that they represent infinitely many isomorphism classes. In the specific case of the trivalent tree T 3 , the infiniteness of S T 3 can alternatively be established using rank one simple algebraic groups over local fields with residue field of order 2. An exhaustive description of the subset of S T 3 consisting of (isomorphism classes of) algebraic groups may be found in[Stu16].
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|
[
"Influence of Dzialoshinskii-Moriya interaction on static and dynamic properties of a transverse domain wall",
"Influence of Dzialoshinskii-Moriya interaction on static and dynamic properties of a transverse domain wall"
]
| [
"Volodymyr P Kravchuk \nBogolyubov Institute for Theoretical Physics\n14-b, Metrologichna str03680KievUkraine\n"
]
| [
"Bogolyubov Institute for Theoretical Physics\n14-b, Metrologichna str03680KievUkraine"
]
| []
| It is shown that the Dzialoshinskii-Moriya interaction leads to asymmetrical deformation of the transverse domain wall profile in one-dimensional biaxial magnet. Amplitude of the deformation is linear with respect to the Dzialoshinskii constant D. Corrections caused by the Dzialoshinskii-Moriya interaction are obtained for the number of the domain wall parameters: energy density, Döring mass, Walker field. The modified q-Φ model with an additional pair of conjugated collective variables is proposed for studying the dynamical properties of the wall with taking into account the internal degrees of freedom. | 10.1016/j.jmmm.2014.04.073 | [
"https://arxiv.org/pdf/1402.3238v2.pdf"
]
| 119,230,043 | 1402.3238 | d2673f82994f0805e3d4777c40fe93690a84299a |
Influence of Dzialoshinskii-Moriya interaction on static and dynamic properties of a transverse domain wall
Volodymyr P Kravchuk
Bogolyubov Institute for Theoretical Physics
14-b, Metrologichna str03680KievUkraine
Influence of Dzialoshinskii-Moriya interaction on static and dynamic properties of a transverse domain wall
Landau-Lifshitz equationDzialoshinskii-Moriya interactiontransverse domain wallq-Φ model
It is shown that the Dzialoshinskii-Moriya interaction leads to asymmetrical deformation of the transverse domain wall profile in one-dimensional biaxial magnet. Amplitude of the deformation is linear with respect to the Dzialoshinskii constant D. Corrections caused by the Dzialoshinskii-Moriya interaction are obtained for the number of the domain wall parameters: energy density, Döring mass, Walker field. The modified q-Φ model with an additional pair of conjugated collective variables is proposed for studying the dynamical properties of the wall with taking into account the internal degrees of freedom.
Introduction
The influence of Dzialoshinskii-Moriya interaction [1] (DMI) on properties of domain walls (DW) is of high interest during the few last years. A especially drastic effect of DMI was recently reported for perpendicularly magnetized ultrathin films: a large tilting of the DW surface was predicted under magnetic field or a spin polarized current driving [2], asymmetric expansion of the circular DW under field driving was demonstrated [3]. Moreover, new types of DW can appear, e.g. Dzialoshinskii DW [4,5] and chiral DW [6,7,8]. The last one demonstrates very high mobility under spin-transfer torques induced by spin Hall effect, this feature undoubtedly can be used for improvement of current controlled magnetic memory devices [9]. This paper has two objectives: (i) to study the DMI influence on static and dynamical properties of a transverse DW in in-plane magnetized narrow stripe, (ii) to propose the modified q-Φ model with an additional pair of canonically conjugated collective variables which enables one to study dynamics of the DW with taking into account the internal degrees of freedom.
First the collective variable approach was used to describe the dynamics of one-dimensional DW more then 40 tears ago [10,11]. Up to now the collective variable model proposed by Slonczewski [10,12] is widely used for different types of DWs and different drivings [13,14,15,16,17,18,19,4,20,21] and it is often called "q-Φ model". In the frames of this model the DW dynamics is described by a pair of conjugated collective variables: q determines position of the DW and angle Φ determines the magnetization orientation in the DW center. This model allows one to describe general properties of motion of the DW as a localized object, but it does not take into account internal degrees of freedom of the DW. However one should note a number of papers where the DW width ∆ was considered as a third collective variable [13,14,19,16]. But as it was shown in these works the width ∆ is a slaved variable and its independent introduction to the model does not allow to describe the internal DW dynamics.
In this paper we show that due to the DMI the DW gains a deformation whose amplitude κ can be used as a collective variable conjugated to ∆. In such a modified model the variables (∆, κ) are not slaved and reflect a rich dynamics of the DW internal degrees of freedom.
The paper is organized as follows. In Section 2 we introduce the model of long and thin stripe and discuss the considered interactions in the system. In Section 3 the static DW structure is obtained and deformations caused by DMI are considered. In the Section 4 the modified q-Φ model is introduced and DMI influence on the dynamical properties of the DW are studied. Results of the paper are summarized in the Section 5.
Model
We consider here a case of thin and narrow ferromagnetic stripe whose thickness and width are small enough to ensure the magnetization one-dimensionality, and the stripe length much exceeds the lateral dimensions. Thus the magnetization is described by continuous and normalized function m = m(t, z), where z-axis is orientated along the stripe and t denotes time. Since |m| = 1 it is convenient to proceed to the angular representation m = (m x , m y , m z ) = (sin θ cos φ, sin θ sin φ, cos θ), (1) where angle θ describes the deviation of magnetization from the magnet axis and angle φ determines orientation of vector m in plane perpendicular to z-axis. These notations are also explained in the Fig. 1. To describe the magnetization dynamics we use the phenomenological Landau-Lifshitz-Gilbert equations which in terms of angular variables have the form
− sin θθ = δE δφ + η sin 2 θφ,(2a)sin θφ = δE δθ + ηθ.(2b)
Here the overdot indicates the derivative with respect to the dimensionless time τ = tω 0 , where ω 0 = 4πγM s with γ being the gyromagnetic ratio and M s being the saturation magnetization and E = E/(4πM 2 s S) is the normalized energy with S being the area of the stripe cross-section. η indicates the Gilbert damping. Typical scale of the physical quantities can be illustrated by the example of Permalloy: ω 0 ≈ 30.3 GHz, 4πM 2 s ≈ 0.93 MJ/m 3 , and η ≈ 0.005 − 0.01.
We assume that energy of the system has the following form
E = ∞ −∞ [E ex + E an + E DM + E z ]dz,(3a)
The first term in (3a) denotes the exchange energy
E ex = 2 2 (∂ z m) 2 = 2 2 [θ 2 + sin 2 θφ 2 ](3b)
with being the exchange length and prime denotes derivative with respect to the spatial coordinate z, e.g. θ ≡ ∂ z θ. The second term describes the anisotropy
E an = − k a 2 m 2 z + k p 2 m 2 y = (3c) = k a 2 sin 2 θ + k p 2 sin 2 θ sin 2 φ + const,
where k a > 0 and k p > 0 are constants of the easy-axis and easy-plane anisotropies respectively. The easy-axis is orientated along the stripe (z-axis) whereas the easyplane coincides with the stripe plane (x0z), see Fig. 1.
The anisotropy is chosen in such a form because for the thin and narrow stripe the expression (3c) approximately models the stray-field contribution originated from the surface magnetostatic charges [14,17]. Also the anisotropy term (3c) allows the head-to-head (tail-to-tail) transverse DW [16,22] which is the subject of this paper. Energy of the Dzialoshinskii-Moriya interaction (DMI) is taken in the form typical for cubic crystals with T symmetry [23,24]
E DM = d 2 m · [∇ × m] = − d 2 sin 2 θφ ,(3d)
where the normalized Dzialoshinskii constant d = D/4πM 2 s is measured in units of length, therefore the comparison of d and is equivalent to the comparison of strengths of the DMI and exchange contributions. The expression (3d) can be written in the equivalent form
E DM = d 2 [m × ∂ z m]
where the Dzialoshinskii vector d = −dẑ is collinear with the stripe; the case when the Dzialoshinskii vector is perpendicular to the one-dimensional magnet is considered in detail in Ref. [25].
The last term
E z = −h · m = −h cos θ (3e)
corresponds to interaction with the external magnetic field h =ẑH/4πM s , which is measured in units of the saturation field and is applied along the stripe.
Static domain wall
In this section the no-driving case h = 0 is considered and all the following analysis is based on the static form of equations (2):
2 (sin 2 θφ ) = k p sin 2 θ sin φ cos φ + d sin θ cos θθ (4a) 2 θ = sin θ cos θ 2 φ 2 + k a + k p sin 2 φ − dφ .(4b)
Without DMI (d = 0) the ground state of the system with energy (3) is doubly degenerated: θ = 0 and θ = π. The transition between domains of different ground states forms a DW. Structure of the DW can be found as a solution of (4) with boundary conditions θ(−∞) = 0 and θ(+∞) = π (case of head-to-head DW). This solution is well known [26] θ dw (z) = 2 arctan e z/∆ ,
φ dw = 0,(5)
where ∆ = / √ k a is width of the static DW 1 . Nevertheless the DW solution (5) does not satisfy the equations (4) in case d = 0. In the following we obtain deformation of the DW solution (5) induced by the DMI, considering the DMI as a small perturbation d/ 1. With this purpose we introduce small deviations from the non-perturbed solution (5)
θ = θ dw + ϑ, φ = φ dw + ϕ.(6)
Substituting now (6) into (4a) and linearizing the obtained equation with respect to the deviations one obtains the following equation for the deviation ϕ
ϕ (ζ) − 2 tanh ζ ϕ (ζ) − αϕ(ζ) + β tanh ζ = 0, ζ = z ∆ , α = k p k a , β = d √ k a .(7)
The corresponding equation for ϑ originated from (4b) is homogenous one ϑ (ζ) + (2 sech 2 ζ − 1)ϑ(ζ) = 0, and besides the trivial solution 2 ϑ = 0 it also has a solution ϑ = dθ dw /dζ = (cosh ζ) −1 which corresponds to the translational shift of the DW (5). Physically important solution of (7) must be bounded one. Such a solution for certain values of parameters is plotted in the Fig. 2. It exponentially approaches the horizontal asymptotes ϕ = ±β/α and it is linear in the neighborhood of the origin
ϕ ≈ βf ( √ 1 + α)ζ, ζ → 0,(8)
where function f is determined in (A.5b). See Appendix A for details and the exact solution. It should be noted that though the correction ϕ is not localized function the corresponding corrections for Cartesian components of the magnetization m i are localized within the DW width, for small ϕ
m x ≈ m dw x , m y ≈ ϕ cosh ζ , m z ≈ m dw z ,(9)
where m dw i is the magnetization component of nonperturbed DW. Therefore the behavior of ϕ at neighborhood of the origin (8) only matters.
Structure of the DW, where the deformation ϕ(ζ) is taken into account, is shown in the Fig. 3. One should note the appearance of the out-of-plane component m y which is absent for the static transverse DW without DMI. 2 The trivial solution means that correction ϑ is of higher order of smallness than linear one with respect to the perturbation:
ϑ = o(d/ ).
Domain wall dynamics
To study dynamics of the DW we use the collective variable approach based on the Lagrangian formalism. The equations of motion (2) can be treated as Euler-Lagrange equations
δL δξ i − d dt δL δξ i = δF δξ i , ξ = θ, φ(10a)
with the Lagrange function 3 [27]
L = − ∞ −∞ φ sin θθdz − E,(10b)
and dissipative function
F = η 2 ∞ −∞ (θ) 2 + sin 2 θ(φ) 2 dz.(10c)
Basing on the static solution (5) and taking into account the form of deformation (8) we propose the following traveling wave Ansatz
θ(z, t) = 2 arctan exp z − q(t) ∆(t) , φ(z, t) = Φ(t) + κ(t) z − q(t) ∆(t) ,(11)
where (q, Φ) and (∆, κ) are two pairs of time dependent collective variables. Pair (q, Φ) determines the general properties of motion of the DW as a localized object, whereas the pair (∆, κ) describes the internal degrees of freedom. Substituting the Ansatz (11) into (10b) and performing the integration over z one obtains the effective Lagrangian in form
L = 2(Φq + cκ∆) − E,(12)
where c = π 2 /12. Accordingly to (12) the amplitude of the DW asymmetry κ is canonically conjugated momentum to the DW width ∆, as well as DW phase Φ is canonically conjugated momentum to the DW position q. The effective energy in (12) reads
E = 2 ∆ (1 + κ 2 ) exchange + ∆ k a + k p sin 2 Φ + ck p κ 2 cos 2Φ anisotropy −d κ DMI −2h q Zeeman .(13)
Here we assume that κ 1 and therefore only terms linear and quadratic with respect to κ are considered. The dissipative function can be obtained in the similar way by substituting the Ansatz (11) into (10c):
F = η ∆ q 2 + (qκ −Φ∆) 2 + c ∆ 2 + (κ∆ −∆κ) 2(14)
Lagrangian (12) and dissipative function (14) produce a set of equations of motion which in the lowest approximation with respect to κ can be written in form
q/∆ Φ = M k p sin Φ cos Φ h , (15a) 2c ∆ /∆ κ = M 2κ 2 ∆ 2 + ck p cos 2Φ − d ∆ 2 ∆ 2 − k a − k p sin 2 Φ ,(15b)
where matrix M reads
M = 1 1 + η 2 1 − ηκ η −η 1 + ηκ .(16)
Equations (15) have the solution in form of translational motion q = V t, Φ = Φ 0 , ∆ = ∆ 0 and κ = κ 0 . In a low damping case (one neglects terms η 2 , ηκ and ηd) the parameters of this translational motion are the following
Φ 0 = 1 2 arcsin h h 0 w , h ≤ h 0 w = ηk p /2, ∆ 0 = (k a + k p sin 2 Φ 0 ) −1/2 , V = h∆ 0 /η, κ 0 = d∆ 0 2( 2 + ck p ∆ 2 0 cos 2Φ 0 ) .(17)
Thus in linear approximation with respect to κ the translational motion of the DW is the typical one for a biaxial magnet [12,16] except appearance of the asymmetrical deformation with amplitude κ 0 . Taking into account the neglected damping terms we were able to obtain the following correction for the Walker field
h w ≈ h 0 w 1 − ηd/ 4k a + 2k p .(18)
Since the obtained correction is linear with respect to d and the Dzialoshinskii vector d is aligned along the magnet the Walker fields are expected to be slightly different for the opposite directions of the DW motion. The relative difference of the Walker fields is of order of magnitude ∼ ηd. It should be noted that the simplified form (15) Using (17) one can write the DW parameters Φ 0 , ∆ 0 , κ 0 as functions of the DW velocity. Then the substitution of this parameters into (13) enable us to series the DW energy in V :
E ≈ E 0 1 − d 2 2 ε 0 + M 1 − d 2 2 µ 0 V 2 2 ,(19a)
where
E 0 = 2 k a (19b)
is energy of the static DM without DMI,
M = 2 √ k a k p (19c)
is the Döring mass [27] of the DM without DMI. The DMI reduces values of these quantities and the corresponding corrections are proportional to d 2 . In contrast, the corresponding correction of energy of a Bloch wall is linear with respect to the DMI [28,29,5]. Values of the energy and mass corrections are determined by constants ε 0 = 1/[8(k a + ck p )] and µ 0 = [k a (4c − 1) + ck p ]/[8(k a + ck p ) 2 ].
To study dynamics of the DW the regime h > h w (non-translational motion) we analyse the system (15) numerically. First of all it should be noted that in the low damping case the Eqs. (15a) coincide with the well known equations of the q-Φ model [12,13]. Therefore the general properties of the DW motion are close to ones obtained from the conventional q-Φ model: above the Walker breakdown the DW demonstrates oscillation motion with non-zero averaged velocity, at the same time the magnetization angle Φ precesses non-uniformly in time, for details see e.g. [12,16].
A new feature of the proposed modification of the q-Φ model is internal DW dynamics which is described by the pair of conjugated variables (∆, κ), it is illustrated in the corresponding phase diagram, see Fig. 4. If the applied field is lower than the Walker breakdown h < h w then the single stationary point with coordinates (∆ 0 , κ 0 ) appears in the phase diagram. As the field increases from the value h = 0 to the value h = h w the stationary point moves along the trajectory which is shown by dashed line in the Fig. 4. Above the Walker breakdown h > h w the limit circle appears instead of the stationary point. The narrow limit circle appears abruptly with the finite size when the field overcomes the critical value h = h w . The further field increasing leads to the limit circle broadening, see Fig. 4a. When the applied field exceeds some critical value (for parameters used in the Fig. 4 it is h c ≈ 17h w ) the amplitude κ of the DMI deformation changes its sign during the dynamics. At the same time the limit circle gains significant deformations with necks and loops, see Fig. 4. It indicates the possibility of chaotic dynamics for the large applied fields. However, this issue is beyond the scope of the current paper and it should be studied in a separate work.
Conclusions
It is shown that the DMI results in asymmetrical deformation of the profile of transverse DW. Amplitude of the deformation is linear with respect to the Dzialoshinskii constant d. To study dynamical properties of the DW the q-Φ model was modified by adding new pair of the conjugated collective variables (DW width, amplitude of the DMI deformation). That enables one to study the dynamics of internal degrees of freedom of DW and to find out caused by DMI corrections for the dynamical properties: Walker field gains the relative correction linear with respect to d, see (18); Döring mass and energy of the static DW gain the corrections quadratic with respect to d, see (19a). The possibility of chaotic dynamics for high applied fields is indicated.
where C > 0. This asymptotics can be easily obtained as a solution of the simple linear ODE which appears from (7) after replacement tanh ζ → 1. Basing on these two statements we can restrict ourselves with the solution of the Eq. (7) for interval 0 ≤ ζ < ∞ with the following boundary conditions
ϕ(0) = 0, ϕ(∞) = β/α. (A.2)
Transition to the new function u = ϕ/ cosh ζ allow us to exclude the first derivative in the Eq. (7) and finally we proceed to the problem
d 2 u dζ 2 + 2 cosh 2 ζ − µ 2 u + β sinh ζ cosh 2 ζ = 0, (A.3a) u(0) = 0, u(∞) = 0, (A.3b)
where µ = √ 1 + α. Since the change of variable ξ = tanh ζ transforms the Eq. (A.3a) to an inhomogeneous Legendre equation
(1 − ξ 2 ) d 2 u dξ 2 − 2ξ du dξ + 2 − µ 2 1 − ξ 2 u = −β ξ 1 − ξ 2
the general solution of (A.3a) can be written as
u(ζ) = C 1 P µ 1 (tanh ζ) + C 2 Q µ 1 (tanh ζ) + β µ − 1 µ + 1 Γ(1 − µ) µΓ(µ) × × ζ 0 [Q µ 1 (tanh ζ)P µ 1 (tanh t) − P µ 1 (tanh ζ)Q µ 1 (tanh t)] sinh t dt cosh 2 t , (A.4)
where P µ 1 (x) and Q µ 1 (x) are Legendre functions of the first and second kinds respectively [30], Γ(µ) denotes the Gamma function and C 1 and C 2 are constants of integration. Applying now the boundary conditions (A.3b) and using the corresponding asymptotic properties of the Legendre functions [30] we obtain the following values of the integration constants
C 1 =
βπ cot µπ 2 f (µ) 2(µ + 1)Γ(µ)
, C 2 = − βf (µ) (µ + 1)Γ(µ)
,
(A.5a) f (µ) = 1 2 − µ − 1 µ 4 ψ 1 − µ 4 − ψ 3 − µ 4 + 2π cos µπ 2 , (A.5b)
where ψ(x) = Γ (x)/Γ(x) is digamma function.
So solution of the Eq. (7) reds ϕ(ζ) = u(ζ) cosh ζ, where function u(ζ) is defined in (A.4) and (A.5). It is easy tosee now that ϕ (0) = u (0) = βf (µ), and therefore the asymptotic behavior (8) takes place.
Appendix B. Exact form of the equations of motion
Substituting the Ansatz (11) into (3) and performing the integration over z one obtains the effective energy in form which in harmonic approximation with respect to κ coincides with (13). Lagrangian (12) and dissipative function (14) produce the following set of equations of motioṅ which in linear approximation with respect to κ coincides with (15).
q ∆ − k p πκ sinh(πκ) sin Φ cos Φ = η Φ − κq ∆ , (B.2a) Φ − h = η κΦ −q ∆ (1 + κ 2 ) , (B.2b) c∆ ∆ − 2 ∆ 2 κ + d 2∆ + π 4 k p cos(2Φ) sinh(πκ) − πκ cosh(πκ) sinh 2 (πκ) (B.2c) = ηc κ − κ∆ ∆ ,
Figure 1 :
1Geometry and notations of the problem. The 1D biaxial ferromagnet with the easy-axis (e.a.) z and easy plane (e.p) z0x is considered. Angles θ and φ determine orientation of the magnetization vector m.
Figure 2 :
2The bounded solution of the Eq. (7) for parameters values α = 1, β = 0.5. The shaded central region (green) shows size of the DW width.
Figure 3 :
3Structure of the transverse DW deformed by the DMI. The top inset demonstrates the changing of the magnetization components along the stripe. Structure of the DM is shown below, distribution of the component my is shown both by the stripe color and by 3D line. Parameters are the same as in the Fig. 2
of the equations of motion is valid only under the condition d/( √ k a ) 1, otherwise one should use an exact form of the equations of motion, see Appendix B.
Figure 4 :
4Phase diagram for the conjugated pair (∆, κ). Dashed line is a locus of stationary points when the field is changing in the interval 0 ≤ h < hw. The limit circles which appear for fields h > hw are shown by solid lines. Parameters are the following: kp = 1, ka = 0.5, η = 0.01, d = 0.1 .
The case of tail-to-tail DW θ dw (z) = 2 arctan e −z/∆ which originates from the opposite boundary conditions θ(−∞) = π and θ(+∞) = 0 , as well as case φ dw = π is absolutely analogous to the considered one.
Kinetic part of the Lagrange function is chosen in form convenient for the further integration with the Ansatz(11).
AcknowledgementsThe author is grateful to Prof. Yuri Gaididei and Prof. Franz Mertens for fruitful discussions.Appendix A. Solution for the correction ϕLet us start from definition of the boundary conditions for the Eq.(7). First of all, note that the Eq.(7)has an odd solution: ϕ(−ζ) = −ϕ(ζ). Along with this, the bounded solution of the Eq. (7) has the following asymptotic behavior
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| []
|
[
"MILNOR-HAMM SPHERE FIBRATIONS AND THE EQUIVALENCE PROBLEM",
"MILNOR-HAMM SPHERE FIBRATIONS AND THE EQUIVALENCE PROBLEM"
]
| [
"Raimundo N Araújo ",
"Dos Santos ",
"ANDMaico F Ribeiro ",
"Mihai Tibăr "
]
| []
| []
| We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor-Hamm tube fibration. | 10.2969/jmsj/82278227 | [
"https://arxiv.org/pdf/1809.08384v4.pdf"
]
| 119,304,931 | 1809.08384 | ec7b9b21369c6a773e3b3020239f09bb69762be0 |
MILNOR-HAMM SPHERE FIBRATIONS AND THE EQUIVALENCE PROBLEM
Raimundo N Araújo
Dos Santos
ANDMaico F Ribeiro
Mihai Tibăr
MILNOR-HAMM SPHERE FIBRATIONS AND THE EQUIVALENCE PROBLEM
We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor-Hamm tube fibration.
Introduction
Let G : (R m , 0) → (R p , 0) be a non-constant real analytic map germ. Under the condition that G has isolated singularity at 0 ∈ R m , it was shown by Milnor [Mi] that there exists a tube fibration and a sphere fibration. Together they contribute to the definition of a higher open book structure as explained in [AT1,AT2,ACT] in the more general case when the singular set Sing G is non-isolated but still included in the central fibre G −1 (0). Milnor [Mi] construction of a sphere fibration by the method of blowing away the tube fibration holds under certain conditions (see the discussion in [AT2,§2]), providing a topology equivalence between the empty tube fibration and the sphere fibration.
In case of holomorphic functions f : (C n , 0) → (C, 0), Milnor [Mi] proved that the sphere fibration is induced by the special map f / f . However this particular construction does not extend to real map germs, as already suggested by Milnor in [Mi,§11]. There have been several successful tries to add up supplementary conditions so that the map G/ G defines a fibration, e.g. [Ja,RSV,RA,AT1,AT2,CSS,Ar] etc. In case this fibration exists, there remains the question if it is equivalent to the empty tube fibration. This is not solved even in the case when G has isolated singularity 1 , a setting in which Milnor [Mi,§9] gave already several related results. We formulate this problem here as the Equivalence Conjecture 4.4.
Recently one started to enrich this landscape by treating the case of a positive dimensional discriminant Disc G. As a matter of fact this is the natural general setting for map germs with p > 1. As one can easily see, Disc G = {0} remains a very special situation, which for instance never happens in case of maps (C n , 0) → (C p , 0) with p > 1 defining isolated complete intersection singularities.
The tube fibration for positive dimensional discriminant, predicted in [ACT], has been introduced recently in [ART] under the name singular Milnor-Hamm tube fibration, and several new classes of singular map germs with such fibration have been presented. This means that over each connected component of the complement of Disc G there is a welldefined locally trivial fibration, and there are finitely many such components 2 . According to Milnor's program [Mi] detailed in [AT2], there are two more steps in order to define open book structures with singularities. Our paper is devoted to this task in the most reasonable setting of a radial discriminant. We introduce the Milnor-Hamm sphere fibration, we give natural sufficient conditions such that this exists, and we exhibit several such classes of singular maps.
We then state the problem of the equivalence with the corresponding Milnor-Hamm empty tube fibration and we show how to solve it in our general setting under natural supplementary conditions. Several conditions for the existence of the Milnor vector field are presented in [AR], and further details can be found in [Ri].
2. The singular tube fibration 2.1. Nice map germs. Given a non-constant analytic map germ G : (R m , 0) → (R p , 0), m ≥ p > 0, the set germ Sing G is well defined on the source space but the images G(Sing G) and Im G are in general not well-defined as germs of subanalytic sets, see [ART]. If they are, then we say that G is nice.
A simple example of a non-nice map is (x, xy) : (C 2 , 0) → (C 2 , 0), for which Im G is not a germ. Under these notations, one has the following results about the existence of nice maps: [ART,Lemma 2.4] If Sing G∩G −1 (0) G −1 (0) then Im G contains an open neighbourhood of the origin. [ART,Theorem 2.6] Let f, g : (C n , 0) → (C, 0) be complex polynomials with no common factor of order ≥ 1. Then fḡ : (C n , 0) → (C, 0) is a nice map germ.
As introduced in [ART], we shall call discriminant of a nice map germ G the following set:
(1)
Disc G := G(Sing G) ∪ ∂Im G
where the boundary ∂Im G := Im G \ int(Im G) is a closed subanalytic proper subset of R p and well-defined as a set germ, where intA :=Å denotes the p-dimensional interior of a semianalytic set A ⊂ R p (hence it is empty whenever dim A < p), and A denotes the closure of it. It follows from the definition that Disc G is a closed subanalytic set of dimension strictly less than p, well-defined as a germ.
The Milnor-Hamm fibration.
Definition 2.1. [ART,Definition 2.1] Let G : (R m , 0) → (R p , 0) be a non-constant nice analytic map germ. We say that G has a Milnor-Hamm tube fibration if for any ε > 0 small enough, there exists 0 < η ε such that the restriction:
(2)
G | : B m ε ∩ G −1 (B p η \ Disc G) → B p η \ Disc G
is a locally trivial smooth fibration which is independent, up to diffeomorphisms, of the choices of small enough ε and η.
We then also say that the restriction of (2) over a small enough circle (still denoted by S p η but keeping in mind that the radius is slightly smaller than the η in (2)):
(3) G | : B m ε ∩ G −1 (S p η \ Disc G) → S p η \ Disc G is a Milnor-Hamm empty tube fibration.
One defines in [ART] a more general notion of stratified tube fibration called singular Milnor tube fibration by considering in addition all singular fibres over the stratified discriminant. In all cases, the tube fibration is a collection of finitely many fibrations over path-connected subanalytic sets.
2.3. ρ-regularity of map germs. Let U ⊂ R m be an open set, 0 ∈ U , and let ρ E : U → R ≥0 be the Euclidean distance squared. We recall the following definition from [ART]:
Definition 2.2. Let G : (R m , 0) → (R p , 0) be a non-constant nice analytic map germ. The set germ at the origin:
M (G) := {x ∈ U | ρ x G}
is called the set of ρ-nonregular points of G, or the Milnor set of G.
The following inclusion of set germs at the origin will play an important role:
(4) M (G) \ G −1 (Disc (G)) ∩ V G ⊆ {0}.
Condition (4) is a direct extension of the condition used in [AT1,AT2,Ma,ACT] in case Disc (G) = {0}; it was shown that it is implied by the Thom regularity condition, in loc.cit. and several other papers. The reciprocal is however not true, counterexamples are provided in [Ti,ACT,Oka3,PT]. Condition (4) enables the following existence result proved in [ART,Lemma 3.3]: Let G : (R m , 0) → (R p , 0) be a non-constant nice analytic map germ. If G satisfies condition (4), then G has a Milnor-Hamm tube fibration (2).
The Milnor-Hamm sphere fibration
We introduce a natural condition under which one may define sphere fibrations whenever G is nice and Disc G is positive dimensional.
Definition 3.1. Let G : (R m , 0) → (R p , 0) be a nice real analytic map germ. We say that its discriminant Disc G is radial if, as a set germ at the origin, it is a union of real half-lines or just the origin.
Example 3.2. Let f, g : (C n , 0) → (C, 0) be holomorphic function germs and let G := fḡ : (C n , 0) → (C, 0) where f and g do not have any common factor of order > 0. Then G is nice and Disc G is radial, by [ART,Theorem 2.6] based on the radiality of the discriminant proved in [PT,Theorem 2.3].
Example 3.3. Let f : (R m , 0) → (R p , 0) be a nice real analytic map germ and let g : (R, 0) → (R, 0) be an analytic invertible germ, such that f and g are in separate variables. Then the map germ G :
= (f, g) : (R m × R, 0) → (R p × R, 0) has Sing G = Sing f × R and it is nice. If moreover Disc f is radial, then Disc G is radial. Let G : U → R p be a representative of the nice map germ G in some open set U 0.
We consider the map
(5) Ψ G := G G : U \ V G → S p−1 1 .
If G is nice and Disc G is radial, then it follows from the definitions that the restriction:
(6) Ψ G| : S m−1 ε \ G −1 (Disc G) → S p−1 1 \ Disc G is well defined for any ε > 0 small enough.
Definition 3.4. We say that the nice map germ G : (R m , 0) → (R p , 0) with radial discriminant has a Milnor-Hamm sphere fibration if the restriction map (6) is a locally trivial smooth fibration which is independent, up to diffeomorphisms, of the choice of ε > 0 provided it is small enough. Let M (Ψ G ) be the Milnor set of the map (5), i.e. the germ at the origin of the ρnonregular points of Ψ G , cf Definition 2.2. We say that Ψ G is ρ-regular if: Theorem 3.5. Let G : (R m , 0) → (R p , 0), m > p ≥ 2, be a non-constant nice analytic map germ with radial discriminant, satisfying the condition (4). If Ψ G is ρ-regular then G has a Milnor-Hamm sphere fibration.
(7) M (Ψ G ) ⊂ G −1 (Disc G).
Proof. The condition (7) controls the topology of the map Ψ G on the complementary of a tubular neighbourhood of V G , while the condition (4) controls the behaviour of the map Ψ G close to V G . Both conditions are essential, as one can see in many examples.
Step 1. Under the condition (4), by [ART,Lemma 3.3], the restriction
(8) G | : S m−1 ε ∩ G −1 (B p η \ Disc G) → B p η \ Disc G is a locally trivial fibration for any small enough 0 < η ε. Since Disc G is radial, for π := s/ s we have that (9) π : B p η \ Disc G → S p−1 1 \ Disc G
is a trivial fibration and by (6), we have the inclusion π(S p−1 η ∩ Disc G) = S p−1 1 ∩ Disc G. Composing the maps (8) and (9) one concludes that
(10) Ψ G| : S m−1 ε ∩ G −1 (B p η \ Disc G) → S p−1 1 \ Disc G
is a locally trivial fibration, and its restriction to the boundary of the empty tube S m−1 ε ∩ G −1 (S p−1 η \ Disc G) coincides with the restriction of G:
(11) G | : S m−1 ε ∩ G −1 (S p−1 η \ Disc G) → S p−1 η \ Disc G.
More precisely, in our case of a radial discriminant, the bases of the fibrations (3) and (10) can be identified with (π S p−1
η \Disc G ) −1 : S p−1 1 \ Disc G → S p−1 η \ Disc G, which is the multiplication by η.
Step 2. The condition (7) is equivalent to the fact that the map Ψ G :
S m−1 ε \G −1 (Disc G) → S p−1 1
\ Disc G is a submersion (over its image) for any small enough ε. Consequently, the restriction
(12) Ψ G| : S m−1 ε \ {G −1 (Disc G) ∪ G −1 (B p η )} → S p−1 1 \ Disc G is a submersion. It coincides with the fibration (11) on S m−1 ε ∩ G −1 (S p−1 η \ Disc (G)). Moreover, the map (12) is proper since the restriction Ψ G| : S m−1 ε \ G −1 (B p η ) → S p−1 1
is a proper map, and by using (6).
Finally, the fibrations (10) and (12) may be glued together along the fibration (11) to induce the locally trivial smooth fibration
Ψ G| : S m−1 ε \ G −1 (Disc G) → S p−1 1 \ Disc G which is independent of the small enough ε > 0.
Remark 3.6. One would like to have a more general existence result for a sphere fibration, namely without the radiality condition, like one can prove for isolated singularities [Mi] and more generally for a point discriminant [ACT,Theorem 2.1]. In order to do that we need a blow-away vector field which is tangent to Sing G. (Note that we do not ask that the vector field is tangent to the fibres of Ψ G ; this condition is investigated in the Equivalence section.) We conjecture that this can be done for any Sing G:
Conjecture 3.7. Under the conditions of Theorem 3.5 but without the radiality of the discriminant, there is a singular open book structure on the sphere S m−1 ε \ G −1 (Disc G), independent of the small enough ε.
Example 3.8. Let G : R n → R 2 , G(x 1 , . . . , x n ) = (x 1 , x 2 2 + · · · + x 2 n−1 − x 2 n ). One has V G = {x 1 = 0} ∩ {x 2 2 + · · · + x 2 n−1 − x 2 n = 0}
and Sing G = {x 2 = · · · = x n = 0}, thus Disc G = R × {0} is radial, and the map G is nice since it verifies [ART,Lemma 2.4] quoted in §2.1. See also Example 3.3 for an alternate argument.
Since (4) is satisfied (see also [ART,Corollary 2.4]). By straightforward computations one gets that M (Ψ) = Sing G, thus M (Ψ) \ G −1 (Disc (G)) = ∅, thus Ψ is ρ-regular. By Theorem 3.5 it follows that G has a Milnor-Hamm sphere fibration.
M (G) ∩ V G ⊂ Sing G ∩ V G = {0}, the condition
Equivalence of Milnor-Hamm tube and sphere fibrations
Let G : (R m , 0) → (R p , 0), m > p ≥ 2, be a nice analytic map germ. The equivalence problem 4.1 makes sense in the case of a radial discriminant since the bases of the fibrations (3) and (6) can be identified via multiplication by η, as we have explained in last part of Step 1 of the proof of Theorem 3.5.
Milnor [Mi] introduced the method of "blowing away the tube to the sphere" which uses integration of a special vector field in order to prove the equivalence of these two fibrations in the case of non-constant holomorphic function germs (C n , 0) → (C, 0). As pointed out by Milnor [Mi] and explained in all details in [ACT,§2], this method may be applied in the real setting under certain conditions, still referring to a point-discriminant. We now formulate the properties of such a vector field in the more general context of a radial discriminant Disc G:
Definition 4.2. One calls Milnor vector field for G, abbreviated MVF, a vector field ν which satisfies the following conditions for any
x ∈ B m ε \ G −1 (Disc G): (c1) ν(x) is tangent to the fibre Ψ −1 G (Ψ G (x)), (c2) ν(x), ∇ρ(x) > 0, (c3) ν(x), ∇ G(x) 2 > 0.
We then have the following general equivalence theorem: 0), m > p ≥ 2 be a nice analytic map germ with radial discriminant, such that the Milnor-Hamm tube fibration (2) and sphere fibration (6) do exist. If there is a MVF for G, then the fibrations (3) and (6) are equivalent.
Theorem 4.3. Let G : (R m , 0) → (R p ,
Proof. The proof follows Milnor's pattern [Mi] explained in detail in [ACT]. The Milnor vector field is by definition radial pointing to the exterior of the spheres. It is well-defined and non-zero by definition outside the inverse image of the discriminant, and the excepted set Disc G is itself radial by assumption (Definition 3.1), thus it is a collection of radii in the target. Therefore the Milnor vector field produces a flow on B m ε \ G −1 (Disc G), the projection by G of which is radial in the target.
Let us remark here that the flow does not cross the set G −1 (Disc G) precisely because the vector field is tangent to the fibres of Ψ G (i.e. condition (c1)) and that the discriminant is radial. This flow yields an isotopy from the Milnor-Hamm tube fibration to the Milnor-Hamm sphere fibration.
In the real setting, the MVF existence problem appeared first in case Sing G = {0} in Jacquemart's work [Ja] and later in [Ar,Oka2] etc. The more general case Disc G = {0} has been addressed later in [AT1,AT2,Ar,ACT,Han,Oka3]. The authors produced sufficient conditions in each setting, but there is yet no valid proof of the existence of a MVF without conditions 3 even in the simplest case Sing G = {0}.
Equivalence Conjecture 4.4. Let G : (R m , 0) → (R p , 0), m > p ≥ 2 be a nice analytic map germ with radial discriminant and such that Ψ G is ρ-regular. If both fibrations (2) and (6) exist, then the fibrations (3) and (6) are equivalent.
Let us remind that the complement S p−1 η \ Disc G may have several connected components and thus two types of fibrations over each such component.
Definition 4.5. We say that the fibrations (3) and (6) are fibre-equivalent if the corresponding fibres over each connected component of S p−1 η \ Disc G are isotopic.
We show here that Conjecture 4.4 is true at the level of fibres.
Theorem 4.6. Let G = (G 1 , . . . , G p ) : (R m , 0) → (R p , 0) be a nice analytic map germ with radial discriminant such that Ψ G is ρ-regular. If the Milnor-Hamm fibrations (2) and (6) exist, then:
(a) the fibrations (3) and (6) are fibre-equivalent, (b) the fibrations (3) and (6) are equivalent over any contractible component of S p−1 \ Disc G.
Proof of Theorem 4.6. Since (b) is a simple consequence of (a), we stick to the proof of (a). For any vector field ω on B m ε \ G −1 (Disc G), we denote by proj T (ω(x)) the orthogonal projection of ω(x) to some linear subspace T ⊂ T x B m ε = R m . Milnor [Mi] proved the following result by using the Curve Selection Lemma, see also [Io,Lemma 1,p. 343].
Lemma 4.7. [Mi] Let X ⊂ R m be an analytic manifold such that 0 ∈ X. Let f and g be analytic functions on R m such that f (0) = g(0) = 0 and that f |X\{0} > 0 and g |X\{0} > 0. Then there exists ε > 0 such that for all x ∈ X ∩ B m ε the vectors proj TxX (∇f (x)) and proj TxX (∇g(x)) cannot have opposite direction whenever both are non-zero.
We apply this lemma to the following situation. Let X y := Ψ −1 G (y) be the fibre over some fixed value y ∈ S p−1 1 \ Disc G. We may assume, without lost of generality, that x belongs to the open set {G 1 (x) = 0}. The normal space N x X y of the fibre X y in R m is spanned by {Ω 2 (x), . . . , Ω p (x)} where Ω k = G 1 ∇G k − G k ∇G 1 , for k = 2, . . . , p.
Consider the following vector fields on B m ε \ G −1 (Disc G): v 1 (x) := proj TxXy (∇ G(x) 2 ) v 2 (x) := proj TxXy (∇ρ(x)).
The vector field v 1 has no zeros since the tube G(x) 2 = const. is transversal to X y , for y ∈ Disc G. The second vector field v 2 has no zeros on B m ε \ G −1 (Disc G) if and only if M (Ψ G ) \ G −1 (Disc G) = ∅, i.e. Ψ G is ρ-regular, which is our assumption.
By Lemma 4.7, there is ε > 0 such that for any x ∈ X y ∩ B ε either the vectors v 1 (x) and v 2 (x) are linearly independent, or they are linearly dependent but cannot have opposite direction.
Let ε 0 > 0 such that both fibrations (2) and (6) exist, for any 0 < ε < ε 0 and 0 < η ε ≤ ε 0 . Let us fix some y ∈ S p−1 1 \ Disc G. By the above Lemma and discussion, there is 0 < ε < ε 0 such that v 1 (x) and v 2 (x) cannot have strictly opposite direction on B m ε ∩ X y . Using Milnor's original idea, we may consider the bisector vector field:
(13) ν(x) = v 1 (x) v 1 (x) + v 2 (x) v 2 (x)
defined on B m ε \G −1 (Disc G). It has no zeros on X y ∩(B m ε \G −1 (Disc G)) precisely because v 1 and v 2 do not point in opposite directions. This is thus a MVF on B m ε ∩ X y and we may now apply Theorem 4.3 to the restriction of G to the space B m ε ∩ X y in order to prove the isotopy of the fibres over y of the two fibrations.
Varying the point y ∈ S p−1 1 \ Disc G we get the isotopy of the corresponding fibres over any connected component.
4.1. The importance of the Milnor set M (G). We tacitly assume throughout this subsection that G is a nice analytic map with radial discriminant and that the Milnor-Hamm fibrations (2) and (6) exist. Let us show that the obstruction to the existence of a MVF is the Milnor set. Proof. For x ∈ B ε \ G −1 (Disc G) we consider the decomposition:
(14) T x X y = T x G −1 (G(x)) ⊕ R v 1 (x) ,
where, by its definition, the vector v 1 (x) is orthogonal to T
x G −1 (G(x)) in T x X y . One writes v 1 (x) = v 1 1 (x) + v 2 1 (x) and v 2 (x) = v 1 2 (x) + v 2 2 (x) according to the decom- position (14). From the definitions, we have v 1 1 (x) = 0 for all x ∈ B ε \ G −1 (Disc G), and v 1 2 (x) = 0 if and only if x ∈ M (G)
. This proves the first claim, and the second is an easy consequence.
For x ∈ M (G) \ G −1 (Disc G) one has that v 1 (x) and v 2 (x) are collinear, which mounts to the relation:
(15) ∇ρ(x) = a(x)∇ G(x) 2 + p j=2 b j (x)Ω j (x), where (16) a(x) = ∇ρ(x), v 1 (x) v 1 (x) 2 .
This proves in particular that
x ∈ M (Ψ G ) \ G −1 (Disc G) ⇔ a(x) = 0.
With these notations one may characterise the existence of a MVF for G as follows, extending the particular case Disc G = {0} of [Han,Theorem 3.3.1].
Theorem 4.9. Let G : (R m , 0) → (R p , 0), m ≥ p ≥ 2 be a nice analytic map with radial discriminant such that the Milnor-Hamm fibrations (2) and (6) exist. There exists a MVF for G on B ε \ G −1 (Disc G), for some enough small ε > 0 if and only if a(x) > 0 for any x ∈ M (G) \ G −1 (Disc G).
Proof. The implication "⇒" follows from the definition (13) of the vector field ν(x). In fact, by condition (c1) one has that ν(x), Ω j (x) = 0 for any j = 2, . . . , p. Therefore, ∇ρ(x), ν(x) = a(x) ∇ G(x) 2 , ν(x) , which by (c2) and (c3) implies that a(x) > 0.
Reciprocally, if a(x) > 0, it follows from Lemma 4.8 and (15) that the vector field ν(x) has no zeroes on B m ε \ V G , hence it is a MVF for G.
There are several other criteria for the existence of a MVF for G; we discuss some of them in a forthcoming paper [AR].
Proposition 4.10. The image of the restriction G | : B m ε ∩ M (G) \ G −1 (Disc G) → R p contains B p η ∩ Im G \ Disc G for some small enough ball B p η centred at the origin. Proof. Let y ∈ B p η ∩ Im G \ Disc G, for some η > 0 which fits in the Milnor-Hamm tube fibration, and such that the fibre G −1 (y) is not empty. The distance function to the origin ρ | : B p η ∩ G −1 (y) → R ≥0 has at least one local minimum point x y in the interior B p η ∩ G −1 (y). This implies that x y ∈ M (G).
Let β M β be the decomposition into connected components of the subanalytic set germ at the origin M (G) \ G −1 (Disc G). See Example 5.4 where we have 8 connected components.
Corollary 4.11. If one of the following conditions holds:
(a) M (G) \ G −1 (Disc G) is connected, (b) for any β there is y = y β ∈ S p−1 1
\ Disc G such that the germ at 0 of M β ∩ X y has dimension > 0, then there exists a MVF for G.
Proof. (a). Proposition 4.10 implies that dim M (G) ∩ X y > 0 for any y ∈ S p−1 1 \ Disc G, thus (a) is a particular case of (b). (b). For some fixed β we apply Milnor's Lemma 4.7 to that X y for which dim M β ∩X y > 0. This shows that the vectors v 1 (x) and v 2 (x) point in the same direction for all x ∈ M β ∩X y and hence for all x ∈ M β since this is connected. Since this is true for any β it is then true for the whole set M (G) \ G −1 (Disc G), thus the vector field ν has no zeroes on it. Finally we may apply Lemma 4.8 to conclude that ν is a MVF.
Classes of maps with equivalent Milnor-Hamm fibrations
5.1. Mixed functions and the equivalence problem.
Definition 5.1. The mixed function F : C n → C is called polar weighted-homogeneous of degree k if there are non-zero integers p 1 , . . . , p n and k > 0, such that gcd(p 1 , . . . p n ) = 1 and n j=1 p j (ν j −µ j ) = k, for any monomial of the expansion F (z) = F (z,z) = ν,µ c ν,µ z νzµ .
The corresponding S 1 -action on C n is, for λ ∈ S 1 : λ · (z, z) = (λ p 1 z 1 , . . . , λ pn z n , λ −p 1 z 1 , . . . , λ −pn z n ).
Theorem 5.2. Let F : C n → C be a polar weighted homogeneous mixed function. Then F is a nice map germ with radial discriminant, the Milnor fibrations (2) and (6) exist, and the fibrations (3) and (6) are equivalent.
Proof. From [ACT, §4.1] it follows (due to the S 1 -action) that polar weighted-homogeneous maps are nice, more precisely because the image F (B m ε ) contains a small neighbourhood of the origin for any ε > 0, and necessarily Disc F = {0}. The discriminant is thus trivially radial. It was proved in [PT,Theorem 5.2] that F has tube fibration, and in [ACT,Theorem 1.4] that F has a sphere fibration. It therefore remains to show that they are equivalent.
Each component M β of the decomposition into connected components β M β of the subanalytic set germ at the origin M (F ) \ V F is invariant under the S 1 -action, more precisely one has that λ M β = M β for any λ ∈ S 1 . It follows that this verifies the hypothesis (b) of Corollary 4.11 and thus our claim follows. 5.2. Maps with radial action. Let t·x := (t q 1 x 1 , . . . , t qm x m ) for t ∈ R + and q 1 , . . . , q m ∈ N * relatively prime positive integers. One says that the map G = (G 1 , · · · , G p ) : R m → R p is radial weighted-homogeneous (or radial, for short) of weights (q 1 , . . . , q m ) and of degree
d > 0, if G(t · x) = t d G(x).
Theorem 5.3. Let G : (R m , 0) → (R p , 0) be a radial weighted homogeneous map germ, and satisfying the condition (4). Then G is nice, with radial discriminant, has Milnor-Hamm tube and sphere fibrations, and the fibrations (3) and (6) are equivalent.
Proof. The image of G is a real cone and this cone is stable as a germ, in the sense that G(B m ε ) and G(B m ε ) have the same germs at the origin, for any 0 < ε < ε. Moreover, the boundary ∂Im G is also a conical set germ at the origin.
The image by G of any analytic germ X ⊂ R m which is invariant under the R + -action is a conical germ, and Sing G is such an invariant set germ. It follows that Disc G is well-defined as a germ, and it is radial. These show that G is a nice map germ, without using the hypothesis about condition (4).
The assumed condition (4) insures now the existence of the Milnor-Hamm tube fibration via [ART,Lemma 3.3]. Let us see that the Milnor-Hamm sphere fibration exists too. It was proved in [ACT,Prop. 3.2] by using the Euler vector field γ(x) := m j=1 q j x j (∂/∂x j ) that the spheres are transversal to the fibres of the map Ψ G . In our setting this implies that Ψ G is ρ-regular, thus our claim follows by Theorem 3.5.
The existence of a MVF follows by noting that any connected component M β of M (F )\ V F is also invariant under the R + -action, and thus we may apply Corollary 4.11(b).
Example 5.4. [ART,Example 5.6] Let G : (R 3 , 0) → (R 2 , 0) given by G(x, y, z) = (xy, z 2 ) is radial homogeneous. One has V G = {x = z = 0}∪{y = z = 0}, Sing G = {z = 0}∪{x = y = 0}, Disc G = {(0, β) | β ≥ 0} ∪ {(λ, 0) | λ ∈ R}, and G −1 (Disc G) = {x = 0} ∪ {y = 0} ∪ {z = 0}. We see that Disc G is radial, as predicted by Theorem 5.3.
By further computations one gets M (G) = {x = ±y} ∪ {z = 0}. To check that G satisfies the condition (4), let us consider p 0 = (x 0 , y 0 , z 0 ) ∈ M (G) \ G −1 (Disc G) ∩ V G . Then there is a sequence p n := (x n , y n , z n ) ∈ M (G) \ G −1 (Disc G) such that p n → p 0 with p 0 ∈ V G . Consequently, z 0 = 0 and x n = ±y n = 0 since p n ∈ G −1 (Disc G). Thus x 0 = lim x n = ± lim y n = y 0 = 0, and therefore p 0 = (0, 0, 0).
Then by Theorem 5.3 the map germ G has Milnor-Hamm tube and sphere fibration, and the fibrations (3) and (6) Corollary 5.5. Let (f, g) be a holomorphic map germ which is Thom regular at V (f,g) , and such that f and g do not have common factor of order > 0.
If fḡ is a radial weighted homogeneous function, then fḡ has Milnor-Hamm tube and sphere fibrations, and the fibrations (3) and (6) are equivalent.
Proof. The Thom regularity of (f, g) implies the Thom regularity of fḡ by [ART,Theorem 4.3] which extends [PT,Theorem 3.1], and thus condition (4) is verified and we may apply the above Theorem 5.3 to conclude.
Corollary 5.6. Let f and g be holomorphic, radial weighted-homogeneous such that the map germ (f, g) is an ICIS. Then the map germ fḡ : (C n , 0) → (C, 0) has Milnor-Hamm tube and sphere fibrations, and the fibrations (3) and (6) are equivalent.
Proof. Since (f, g) is an ICIS, it follows that the map germ fḡ is nice, Thom regular and has a Milnor-Hamm tube fibration, by [ART,Theorem 4.3(a)]. If we add up the R + -action then we get, as in Theorem 5.3 above, the existence of a Milnor-Hamm sphere fibration and thus the equivalence of the fibrations.
Example 5.7. f, g : (C 2 , 0) → (C, 0), f (x, y) = x 2 + y 2 and g(x, y) = x 2 − y 2 , verify the assumptions of Corollary 5.6. Notice that Disc fḡ is positive dimensional.
3. 1 .
1Existence of Milnor-Hamm sphere fibrations. The following existence criterion extends the case Disc G = {0} considered in [ACT, Theorem 1.3].
Problem 4.1 (The Equivalence Problem). Assuming that the Milnor-Hamm tube fibration and sphere fibration exist, under what conditions they are equivalent, in the sense that the fibrations (3) and (6) are equivalent?
Lemma 4 . 8 .
48Let x ∈ B ε \ G −1 (Disc G). The vectors v 1 and v 2 are linearly dependent if and only if x ∈ M (G). In particular the vector field ν from (13) is a MVF on B m ε \ (M (G) ∪ G −1 (Disc G)).
are equivalent.
Figure 1 .
1Milnor-Hamm fibration for G = (xy, z 2 )
a subanalytic set has locally finitely many connected components, see e.g.[BM].
the existence proof in[CSS] appears to contain a non-removable gap.
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. Universidade Icmc, Paulo De São, Av. Trabalhador São-Carlense. 400CP BoxICMC, Universidade de São Paulo, Av. Trabalhador São-Carlense, 400 -CP Box 668, 13560-970 São Carlos, São Paulo, Brazil E-mail address: [email protected]
. Univ, Cnrs Lille, Paul Umr 8524 -Laboratoire, Painlevé, [email protected], France E-mail addressUniv. Lille, CNRS, UMR 8524 -Laboratoire Paul Painlevé, F-59000 Lille, France E-mail address: [email protected]
| []
|
[
"An explicit representation of Verblunsky coefficients",
"An explicit representation of Verblunsky coefficients"
]
| [
"N H Bingham \nDepartment of Mathematics\nImperial College London\nSW7 2AZLondonUK\n",
"Akihiko Inoue \nDepartment of Mathematics\nHiroshima University\nHigashi-Hiroshima739-8526Japan\n",
"Yukio Kasahara \nDepartment of Mathematics\nHokkaido University\n060-0810SapporoJapan\n"
]
| [
"Department of Mathematics\nImperial College London\nSW7 2AZLondonUK",
"Department of Mathematics\nHiroshima University\nHigashi-Hiroshima739-8526Japan",
"Department of Mathematics\nHokkaido University\n060-0810SapporoJapan"
]
| []
| We prove a representation of the partial autocorrelation function (PACF) of a stationary process, or of the Verblunsky coefficients of its normalized spectral measure, in terms of the Fourier coefficients of the phase function. It is not of fractional form, whence simpler than the existing one obtained by the second author. We apply it to show a general estimate on the Verblunsky coefficients for short-memory processes as well as the precise asymptotic behaviour, with remainder term, of those for FARIMA processes. | 10.1016/j.spl.2011.11.004 | [
"https://arxiv.org/pdf/1109.4513v1.pdf"
]
| 119,689,141 | 1109.4513 | 7a4d50cd2906bee50201e7538f3f7cc4926b94ba |
An explicit representation of Verblunsky coefficients
21 Sep 2011
N H Bingham
Department of Mathematics
Imperial College London
SW7 2AZLondonUK
Akihiko Inoue
Department of Mathematics
Hiroshima University
Higashi-Hiroshima739-8526Japan
Yukio Kasahara
Department of Mathematics
Hokkaido University
060-0810SapporoJapan
An explicit representation of Verblunsky coefficients
21 Sep 2011Verblunsky coefficientsPartial autocorrelation functionsPhase functionsFARIMA processesLong memory 2000 MSC: 62M1042C0560G10
We prove a representation of the partial autocorrelation function (PACF) of a stationary process, or of the Verblunsky coefficients of its normalized spectral measure, in terms of the Fourier coefficients of the phase function. It is not of fractional form, whence simpler than the existing one obtained by the second author. We apply it to show a general estimate on the Verblunsky coefficients for short-memory processes as well as the precise asymptotic behaviour, with remainder term, of those for FARIMA processes.
Introduction
Let {X n : n ∈ Z} be a real, zero-mean, weakly stationary process, defined on a probability space (Ω, F , P), with spectral measure not of finite support, which we shall simply call a stationary process. Here the spectral measure is the finite measure µ on (−π, π] in the spectral representation γ(n) = (π,π] e inθ µ(dθ) of the autocovariance function γ(n) := E[X n X 0 ], n ∈ Z. For {X n }, we have another sequence {α(n)} ∞ n=1 called the partial autocorrelation function (PACF); see (2.1) below for the definition. In the theory of orthogonal polynomials on the unit circle (OPUC), however, the PACF {α(n)} ∞ n=1 appears as the sequence of Verblunsky coefficients of the normalized spectral measurẽ µ := (µ(−π, π]) −1 µ. Notice that (−π, π] can be identified with the unit circle T := {z ∈ C | |z| = 1} by the map θ → e iθ , whence µ orμ with a measure on T. For a survey of OPUC, see Simon (2005aSimon ( , 2005bSimon ( , 2005cSimon ( , 2011.
The Verblunsky coefficients {α(n)} ∞ n=1 give an unrestricted parametrization of the normalized spectral measureμ of {X n }, in that the only inequalities restricting the α(n) are α(n) ∈ [−1, 1], or α(n) ∈ (−1, 1) in the non-degenerate case relevant here. This result is due to Barndorff-Nielsen and Schou (1973), Ramsey (1974) in the time-series context. However, in OPUC, the result dates back to Verblunsky (1935Verblunsky ( , 1936. See, e.g., Simon (2005bSimon ( , 2005c and Bingham (2011) for background.
The aim of this paper is to prove an explicit representation of the Verblunsky coefficients {α(n)} ∞ n=1 in terms of another sequence {β n } ∞ n=0 defined by
β n := ∞ v=0
c v a v+n , n = 0, 1, . . . , ). We notice that β n correspond to the Fourier coefficients of the phase function of the process (see Remark 1 in §2). The proof of the representation of {α(n)} is based on the result of Inoue and Kasahara (2006) on the explicit representation of finite predictor coefficients as well as the Levinson (or Levinson-Durbin) algorithm or the Szegö recursion. The algorithm is due to Szegö (1939), Levinson (1947), and Durbin (1960); for a textbook account, see Pourahmadi (2001, Section 7.2).
We notice that, in Inoue (2008), the second author already proved a representation of {α(n)} ∞ n=1 in terms of {β n } ∞ n=0 . However, the representation of {α(n)} ∞ n=1 in the present paper is much simpler than that in Inoue (2008), in that the latter is of fractional form while the former not. We apply the result to show a general estimate of α(n) for shortmemory processes as well as the precise asymptotic behaviour, with remainder, of α(n) for FARIMA processes. The FARIMA model is a popular parametric model with long memory, and was introduced independently by Granger and Joyeux (1980) and Hosking (1981). See Brockwell and Davis (1991, Section 9) for textbook treatment. The long memory of the FARIMA model comes from the singularity at zero of its spectral density.
In §2, we state the main result, i.e., the representation of the Verblunsky coefficients. Its proof is given in §3. In §4, we apply the main result to both short-memory and FARIMA processes.
Main result
Let H be the real Hilbert space spanned by {X k : k ∈ Z} in L 2 (Ω, F , P), which has inner product (Y 1 ,
Y 2 ) := E[Y 1 Y 2 ]
and norm Y := (Y, Y) 1 Our main result, i.e., Theorem 2.1 below, is an explicit representation of {α(n)} ∞ n=1 . To state it, we need some notation. A stationary process {X n } is said to be purely nondeterministic (PND) if ∩ ∞ n=−∞ H (−∞,n] = {0}, or, equivalently, there exists a positive even and integrable function ∆ on (−π, π] such that π −π | log ∆(θ)|dθ < ∞ and γ n = π −π e inθ ∆(θ)dθ for n ∈ Z; see Brockwell and Davis (1991, Section 5.7), Rozanov (1967, Chapter II) and Grenander and Szegö (1958, Chapter 10). We call ∆ the spectral density of {X n }. Using ∆, we define the Szegö function h by
h(z) := √ 2πexp 1 4π π −π e iθ + z e iθ − z log ∆(θ)dθ , z ∈ C, |z| < 1. (2.2)
The function h(z) is an outer function in the Hardy space H 2 of class 2 over the unit disk |z| < 1. Using h, we define the MA coefficients c n and the AR coefficients a n , respectively, by (A1) {X n } is PND, and both ∞ n=0 |a n | < ∞ and ∞ n=0 |c n | < ∞ hold as a standard one for processes with short memory, and (A2) {X n } is PND, and, for some d ∈ (0, 1/2) and ℓ ∈ R 0 , {c n } and {a n } satisfy, respectively,
h(z) = ∞ n=0 c n z n , − 1 h(z) = ∞c n ∼ n −(1−d) ℓ(n), a n ∼ n −(1+d) 1 ℓ(n) · d sin(πd) π , n → ∞
as a standard one for processes with long memory. Here p n ∼ q n as n → ∞ means lim n→∞ p n /q n = 1. Recall β n from (1.1). Notice that the sum in (1.1) converges absolutely under either (A1) or (A2). For n ∈ N ∪ {0}, we define α 1 (n) := β n and, for k = 3, 5, 7, . . . ,
α k (n) := ∞ v 1 =0 · · · ∞ v k−1 =0 β n+v 1 β n+1+v 1 +v 2 · · · β n+1+v k−2 +v k−1 β n+1+v k−1 .
As in the case of d k (n, j) in Inoue and Kasahara (2006, Section 2.3), the sums converge absolutely. We write ∞− to indicate that the sum does not necessarily converge absolutely, i.e., ∞− k=m := lim M→∞ M k=m . Here is the main result of this paper.
Theorem 2.1. We assume either (A1) or (A2). Then α(n) = ∞− k=1 α 2k−1 (n) for n = 2, 3, . . . .
The proof of Theorem 2.1 is given in Section 3.
Remark 1.
We assume {a n } ∈ ℓ 2 . As usual, we identify h with its boundary-value function h(e iθ ) = lim r↑1 h(re iθ ).
Then, since h(e iθ ) = ∞ k=0 c k e ikθ and 1/h(e iθ ) = − ∞ k=0 e ikθ a k e ikθ , Parseval's identity yields π −π e −inθ {h(e iθ )/h(e iθ )} dθ 2π = − ∞ k=0 c k a k+n = −β n , n = 0, 1, . . . .
Here notice that, in our set-up, {c n } is real. Thus β n (or, more precisely, −β n ) is the n-th Fourier coefficient ofh/h. The functionh/h is called the phase function of the process. See Peller (2003, p. 405); see also Dym and McKean (1976) for its continuous-time analogue.
Proof of Theorem 2.1
In this section, we assume either (A1) or (A2). For n ∈ N, we can express P [−n,−1] X 0 uniquely in the form
P [−n,−1] X 0 = n j=1 φ n, j X − j .
We call φ n, j the finite predictor coefficients. The proof of Theorem 2.1 is based on the explicit representation of φ n, j , i.e., (3.5) below, and the following Szegö recursion (or the Levinson-Durbin algorithm):
φ n, j − φ n+1, j = φ n,n+1− j α(n + 1), j = 1, . . . ,d 0 (n, j) = δ j0 , d 1 (n, j) = β n+ j , d 2 (n, j) = ∞ v 1 =0 β n+ j+v 1 β n+v 1 , and d k (n, j) = ∞ v 1 =0 · · · ∞ v k−1 =0 β n+ j+v k−1 β n+v k−1 +v k−2 · · · β n+v 2 +v 1 β n+v 1 , k ≥ 3,
the sums converging absolutely. These satisfy the following recursion: for n, j ∈ N ∪ {0},
d 0 (n, j) = δ j0 , d k+1 (n, j) = ∞ v=0 β n+ j+v d k (n, v), k ≥ 0. (3.2)
From the definition of α k (n) above, we also have
α 2k+1 (n) = ∞ v=0 β n+v d 2k (n + 1, v), n, k ∈ N ∪ {0}. (3.3)
The next proposition is the key to the proof of Theorem 2.1. Proof. Let n, j ∈ N ∪ {0}. We use mathematical induction on k. First, since α 1 (n) = β n and d 1 (n, j) = β n+ j , we
have d 2 (n, j) = ∞ v=n β v β v+ j = ∞ v=n α 1 (v)d 1 (v, j)
, which implies (3.4) with k = 1. Next, we assume that (3.4) holds for k ∈ N. Then, by (3.2) and the Fubini-Tonelli theorem, we have d 2k+2 (n, j)
= ∞ v 2 =0 [ ∞ v 1 =n β v 2 +v 1 β j+v 1 ]d 2k (n, v 2 ), whence d 2k+2 (n, j) − d 2k+2 (n + 1, j) = I + II, where I := ∞ v 2 =0 ∞ v 1 =0 β n+v 2 +v 1 β n+ j+v 1 [d 2k (n, v 2 ) − d 2k (n + 1, v 2 )] , II := ∞ v 2 =0 β n+v 2 β n+ j d 2k (n + 1, v 2 ).
By (3.2), (3.4), and the Fubini-Tonelli theorem,
I = ∞ v 2 =0 ∞ v 1 =0 β n+v 2 +v 1 β n+ j+v 1 k l=1 α 2k−2l+1 (n)d 2l−1 (n, v 2 ) = k l=1 α 2k−2l+1 (n) ∞ v 1 =0 β n+ j+v 1 ∞ v 2 =0 β n+v 1 +v 2 d 2l−1 (n, v 2 ) = k l=1 α 2k−2l+1 (n)d 2l+1 (n, j) = k+1 l=2 α 2(k+1)−2l+1 (n)d 2l−1 (n, j), while, by (3.3), we have II = [ ∞ v 2 =0 β n+v 2 d 2k (n + 1, v 2 )]β n+ j = α 2k+1 (n)d 1 (n, j).
Thus we obtain (3.4) with k replaced by k + 1, as desired.
For n ∈ N and j = 1, . . . , n, Theorem 2.9 in Inoue and Kasahara (2006) asserts the representation
φ n, j = ∞− k=1 {b 2k−1 (n, j) + b 2k (n, n + 1 − j)} ,(3.b 2k+1 (n, j) − b 2k+1 (n + 1, j) = c 0 ∞ u=0 a j+u {d 2k (n + 1, u) − d 2k (n + 2, u)} = c 0 ∞ u=0 a j+u k l=1 α 2k−2l+1 (n + 1)d 2l−1 (n + 1, u) = k l=1 α 2k−2l+1 (n + 1) c 0 ∞ u=0 a j+u d 2l−1 (n + 1, u) = k l=1 α 2k−2l+1 (n + 1)b 2l (n, j).
We turn to (3.8). Since d 1 (n + 1, u) = α 1 (n + 1 + u) and b 1 (n, j) = c 0 a j , it follows from (3.6) that
b 2 (n, n + 1 − j) = c 0 ∞ u=0 a n+1− j+u d 1 (n + 1, u) = ∞ u=n+1 α 1 (u)b 1 (n, u − j).
Similarly, b 2 (n + 1, n + 2 − j) = c 0 ∞ u=0 a n+2− j+u d 1 (n + 2, u) = ∞ u=n+2 α 1 (u)b 1 (n, u − j). Thus (3.8) holds for k = 1. If k ≥ 2, then, by (3.2) and (3.6), we have
b 2k (n, n + 1 − j) = c 0 ∞ u=0 a n+1− j+u ∞ v=0 β n+1+u+v d 2(k−1) (n + 1, v), b 2k (n + 1, n + 2 − j) = c 0 ∞ u=1 a n+1− j+u ∞ v=0 β n+1+u+v d 2(k−1) (n + 2, v), whence b 2k (n, n + 1 − j) − b 2k (n + 1, n + 2 − j) = I + II with I := c 0 ∞ u=0 a n+1− j+u ∞ v=0 β n+1+u+v d 2(k−1) (n + 1, v) − d 2(k−1) (n + 2, v) , II := c 0 a n+1− j ∞ v=0 β n+1+v d 2(k−1) (n + 2, v).
By (3.2), (3.4) and (3.6),
I = c 0 ∞ u=0 a n+1− j+u ∞ v=0 β n+1+u+v k−1 l=1 α 2(k−1)−2l+1 (n + 1)d 2l−1 (n + 1, v) = k−1 l=1 α 2(k−1)−2l+1 (n + 1) c 0 ∞ u=0 a n+1− j+u ∞ u=0 β n+1+u+v d 2l−1 (n + 1, v) = k−1 l=1 α 2(k−1)−2l+1 (n + 1)b 2l+1 (n, n + 1 − j) = k l=2 α 2k−2l+1 (n + 1)b 2l−1 (n, n + 1 − j),
while, by (3.3) and b 1 (n, n + 1 − j) = ca n+1− j , we have II = α 2k−1 (n + 1)b 1 (n, n + 1 − j). Thus (3.8) follows.
We are now ready to prove Theorem 2.1.
Proof (of Theorem 2.1). For n ∈ N and j = 1, . . . , n, we have b 1 (n, j) − b 1 (n + 1, j) = c 0 a j − c 0 a j = 0. This, together with (3.5), (3.7) and (3.8), yields
φ n, j − φ n+1, j = ∞− k=1 {b 2k+1 (n, j) − b 2k+1 (n + 1, j) + b 2k (n, n + 1 − j) − b 2k (n + 1, n + 2 − j)} = ∞− k=1 k l=1 α 2k−2l+1 (n + 1) {b 2l (n, j) + b 2l−1 (n, n + 1 − j)} = ∞− l=1 {b 2l (n, j) + b 2l−1 (n, n + 1 − j)} ∞− k=l α 2k−2l+1 (n + 1) = ∞− k=1 α 2k−1 (n + 1) φ n,n+1− j .
Since P [−n,−1] X 0 0, we have (φ n,1 , . . . , φ n,n ) 0. Combining these and (3.1), we obtain the theorem.
Applications
Short memory processes
In this subsection, we apply Theorem 2.1 to the Verblunsky coefficients of short-memory processes. We define
F( j) := ∞ v=0 |c v | ∞ u= j |a u | , j = 0, 1, . . . .
Then F( j) decreases to zero as j → ∞ under (A1). Recall d k (n, j) from Section 3.
Lemma 4.1.
We assume (A1). Then ∞ u=0 |d k (n, u)| ≤ F(n) k for k, n ∈ N.
Proof. Let n ∈ N. We use induction on k. Since d 1 (n, u) = β n+u , we have
∞ u=0 |d 1 (n, u)| = ∞ u=0 |β n+u | ≤ ∞ v=0 |c v | ∞ u=0 |a n+u+v | ≤ F(n).
We assume ∞ u=0 |d k (n, u)| ≤ F(n) k for k ∈ N. Then, by (3.2),
∞ u=0 |d k+1 (n, u)| ≤ ∞ v=0 |d k (n, v)| ∞ u=0 |β n+v+u | ≤ F(n) ∞ v=0 |d k (n, v)| ≤ F(n) k+1 .
Thus the inequality also holds for k + 1.
Notice that {a n } ∈ ℓ 1 implies a n → 0 as n → ∞.
Theorem 4.2.
We assume (A1). Then, for N ∈ N such that F(N + 1) < 1, the Verblunsky coefficients {α(n)} satisfy
|α(n)| ≤ ∞ v=0 |c v | 1 − F(n + 1) 2 max j≥n |a j | , n ≥ N.
Proof. Recall α 2k+1 (n) from §2. We have |α 1 (n)| = |β n | ≤ max j≥n |a j | ∞ v=0 |c v |. By (3.3) and Lemma 4.1, we also have |α 2k+1 (n)| ≤ ∞ v=0 |β n+v d 2k (n + 1, v)| ≤ F(n + 1) 2k max j≥n |a j | ∞ v=0 |c v | for n, k ∈ N. Choose N ∈ N so that F(N + 1) < 1. Then, combining the estimates above with Theorem 2.1, we see that, for n ≥ N,
|α(n)| ≤ max j≥n |a j | ∞ v=0 |c v | ∞ k=0 F(n + 1) 2k = ∞ v=0 |c v | 1 − F(n + 1) 2 max j≥n |a j | .
Thus the theorem follows.
For example, if, in addition to (A1), a n = O(n −p ) as n → ∞ for some p > 1, then max j≥n |a j | = O(n −p ), whence, by Theorem 4.2, α(n) = O(n −p ) as n → ∞.
The FARIMA model
For d ∈ (−1/2, 1/2) and p, q ∈ N ∪ {0}, a stationary process {X n } is said to be a FARIMA(p, d, q) (or fractional ARIMA(p, d, q)) process if it has a spectral density ∆ of the form
∆(θ) = 1 2π |Θ(e iθ )| 2 |Φ(e iθ )| 2 |1 − e iθ | −2d , −π < θ ≤ π,
where Φ(z) and Θ(z) are polynomials with real coefficients of degrees p, q, respectively, satisfying the following condition: Φ(z) and Θ(z) have no common zeros, and have no zeros in the closed unit disk {z ∈ C | |z| ≤ 1}.
In what follows, we assume that {X n } is a FARIMA(p, d, q) process with 0 < d < 1/2. Then {X n } satisfies (A2) for some constant function ℓ (cf. Inoue (2002, Corollary 3.1)). Let {α(n)} be the Verblunsky coefficients of {X n }. The aim of this subsection is to apply Theorem 2.1 to {α(n)} to prove the next theorem.
a ′ n = Γ(n − d)d Γ(n + 1)Γ(1 − d) , c ′ n = Γ(n + d) Γ(n + 1)Γ(d)
, n = 0, 1, . . .
(see, e.g., Brockwell and Davis (1991, Section 13.2)). Notice that c ′ n > 0 for n ≥ 0 and a ′ n > 0 for n ≥ 1. Put Proof. Using the hypergeometric function, we have, for n ≥ 0,
β ′ n := ∞ v=0 c ′ v a ′ n+v , n = 0, 1, . . . .β ′ n = Γ(n − d)d Γ(n + 1)Γ(1 − d) ∞ v=0 Γ(d + v) Γ(d) · Γ(n − d + v) Γ(n − d) · Γ(n + 1) Γ(n + 1 + v) · 1 v! = Γ(n − d)d Γ(n + 1)Γ(1 − d) F(d, n − d; n + 1; 1) = Γ(n − d)d Γ(n + 1)Γ(1 − d) · Γ(n + 1) Γ(n + 1 − d)Γ(1 + d) = sin(πd) π · 1 n − d ,
as desired.
Proposition 4.5.
There exist a real sequence {δ n } ∞ n=1 and M ∈ (0, ∞) such that β n = β ′ n {1 + δ n } and |δ n | ≤ Mn −d for n ∈ N.
Proof. By Inoue (2002, Lemma 2.2), we have, as n → ∞,
c n n d−1 = K 1 Γ(d) + O n −1 , a n n −d−1 = − 1 K 1 Γ(−d) + O n −1 , c ′ n n d−1 = 1 Γ(d) + O n −1 , a ′ n n d−1 = − 1 Γ(−d) + O n −1 ,
where K 1 := θ(1)/φ(1) > 0. Hence we may write c n = {K 1 + s n }c ′ n for n ≥ 0 and a n = {(1/K 1 ) + t n }a ′ n for n ≥ 1, where {s n } and {t n } are sequences satisfying |s n | ≤ L/(n + 1) for n ≥ 0 and |t n | ≤ L/n for n ≥ 1, for some L ∈ (0, ∞).
We have, for n = 1, 2, . . . ,
|β n − β ′ n | ≤ ∞ v=0 |s v |c ′ v a ′ n+v + K 1 ∞ v=0 |t n+v |c ′ v a ′ n+v + (1/K 1 ) ∞ v=0 |s v t n+v |c ′ v a ′ n+v . From c ′ n /(n + 1) ∼ 1/{n 2−d Γ(d)} as n → ∞, we see that ∞ v=0 |s v |c ′ v a ′ n+v ≤ L ∞ v=0 c ′ v v + 1 a ′ n+v ∼ a ′ n L ∞ v=0 c ′ v v + 1 , n → ∞. Hence, using a ′ n ∼ constant · n −(1+d) as n → ∞, we get ∞ v=0 |s v |c ′ v a ′ n+v = O n −(1+d) as n → ∞. Similarly, as n → ∞, ∞ v=0 |t n+v |c ′ v a ′ n+v = O n −(2+d) , ∞ v=0 |s v t n+v |c ′ v a ′ n+v = O n −(2+d) .
Combining these and β ′ n ∼ π −1 sin(πd) n −1 , we obtain the proposition. Proof (of Theorem 4.3). By Theorem 2.1, the Verblunsky coefficients {α(n)} of {X n } and {α ′ (n)} of {X ′ n } admit the representations α(n) = ∞− k=1 α 2k−1 (n) and α ′ (n) = ∞− k=1 α ′ 2k−1 (n), respectively, where α 2k−1 (·) are those defined for {X n } in §2, while α ′ 2k−1 (·) are their counterparts defined for {X ′ n }, that is, α ′ 1 (n) = β ′ n and, for k = 3, 5, . . . ,
α ′ k (n) = ∞ m 1 =0 · · · ∞ m k−1 =0 β ′ n+1+m 1 β ′ n+1+m 1 +m 2 · · · β ′ n+1+m k−2 +m k−1 β ′ n+m k−1 .
For k ∈ N, we define τ 2k−1 ∈ (0, ∞) by τ 2k−1 := (2k − 2)!/[π2 2k−2 ((k − 1)!) 2 (2k − 1)], or by By Proposition 4.5, we have |β n+v | ≤ 1 + Mn −d β ′ n+v and |β n+v − β ′ n+v | ≤ Mn −d β ′ n+v for n ≥ 1 and v ≥ 0. We also have (1 + x) k − 1 ≤ kx(1 + x) k for x ≥ 0. Hence, for n ≥ N,
|α 3 (n) − α ′ 3 (n)| ≤ ∞ m 1 =0 ∞ m 2 =0 |β n+1+m 1 − β ′ n+1+m 1 | · |β n+1+m 1 +m 2 | · |β n+m 2 | + ∞ m 1 =0 ∞ m 2 =0 β ′ n+1+m 1 |β n+1+m 1 +m 2 − β ′ n+1+m 1 +m 2 | · |β n+m 2 | + ∞ m 1 =0 ∞ m 2 =0 β ′ n+1+m 1 β ′ n+1+m 1 +m 2 |β n+m 2 − β ′ n+m 2 | ≤ Mn −d (1 + Mn −d ) 2 + (1 + Mn −d ) + 1 α ′ 3 (n) = (1 + Mn −d ) 3 − 1 α ′ 3 (n) ≤ 3Mn −d (1 + Mn −d ) 3 α ′ 3 (n) ≤ 3Mn −(d+1) {r 2 sin(πd)} 3 τ 3 .
In the same way, |α 2k−1 (n) − α ′ 2k−1 (n)| ≤ (2k − 1)Mn −(d+1) {r 2 sin(πd)} 2k−1 τ 2k−1 for k = 1, 2, . . . and n ≥ N. Since α ′ (n) = d/(n − d) (see Hosking (1981, Theorem 1)), it follows that
α(n) − d n − d ≤ ∞ k=1 |α 2k−1 (n) − α ′ 2k−1 (n)| ≤ n −(d+1) M ∞ k=1
(2k − 1)τ 2k−1 {r 2 sin(πd)} 2k−1 .
By (4.1), we have ∞ k=1 (2k − 1)τ 2k−1 {r 2 sin(πd)} 2k−1 < ∞, so that
α(n) = d n − d + O n −(d+1) = d n + O n −(d+1) , n → ∞.
Thus the theorem follows.
( 2000 ,
2000Section 4) and Inoue and Kasahara (2006, Section 2.2) for background. Both {c n } and {a n } are real sequences, and {c n } is in l 2 . We write R 0 for the class of slowly varying functions at infinity: the class of positive, measurable ℓ, defined on some neighborhood [A, ∞) of infinity, such that lim x→∞ ℓ(λx)/ℓ(x) = 1 for all λ > 0; see Bingham et al. (1989, Chapter 1) for background. Among several possible choices of assumption on {X n }, as in Inoue and Kasahara (2006), we consider
Proposition 3. 1 .
1For n, j ∈ N ∪ {0} and k ∈ N, we have d 2k (n, j) − d 2k (n + 1, j)
d k−1 (n + 1, u), k = 1, 2, . . . . (3.6) Using Proposition 3.1, we derive two kinds of difference equations for b k (n, j). Proposition 3.2. For n, k ∈ N and j = 1, . . . , n, we have b 2k+1 (n, j) − b 2k+1 (n + 1, j) = k l=1 α 2k−2l+1 (n + 1)b 2l (n, j), (3.7)b 2k (n, n + 1 − j) − b 2k (n + 1, n + 2 − j) = k l=1 α 2k−2l+1 (n + 1)b 2l−1 (n, n + 1 − j).(3.8) Proof. From (3.4) and (3.6), we easily obtain (3.7) in the following way:
Theorem 4. 3 .
3We have nα(n) = d + O(n −d ) as n → ∞.
Theorem 4.3 is more precise than Inoue (2008, Theorem 2.5) with 0 < d < 1/2, in that the former gives an estimate on the remainder term. The rest of this subsection is devoted to the proof of Theorem 4.3.As before, we denote by {c n } and {a n } the MA and AR coefficients, respectively, of {X n }. We also consider a FARIMA(0, d, 0)process {X ′ n } satisfying E[(X ′ n ) 2 ] = Γ(1−2d)/Γ(1−d) 2 . The AR coefficients {a ′ n } and MA coefficients {c ′ n } of {X ′ n } are given by
Lemma 4. 4 .
4We have β ′ n = sin(πd)/{π(n − d)} for n = 0, 1, . . . .
∞ k=1 τ
k=12k−1 x 2k−1 = π −1 arcsin x, |x| < 1 (4.1) (see Inoue and Kasahara (2006, Lemma 3.1) and Inoue (2008, Section 5)). Let M be as in Proposition 4.5 and let r > 1 be chosen so that r 2 sin(πd) < 1. Then, as in the proof of Inoue and Kasahara (2006, Proposition 3.2), there exists an integer N independent of k such that 1 + (M/n d ) ≤ r, α ′ 2k−1 (n) ≤ 1 n {r sin(πd)} 2k−1 τ 2k−1 , n ≥ N, k ≥ 1.
/2 . For an interval I ⊂ Z, we write H I for the closed subspace of H spanned by {X k : k ∈ I} and Verblunsky coefficients of the normalized spectral measureμ. In what follows, we also call {α(n)} ∞ n=1 the Verblunsky coefficients of {X n }.H ⊥
I for the orthogonal complement of H I in H. Let P I and P ⊥
I be the orthogonal projection operators of H onto H I and
H ⊥
I , respectively. The projection P I Y stands for the best linear predictor of Y based on the observations {X k : k ∈ I},
and P ⊥
I Y for its prediction error.
The PACF {α(n)} ∞
n=1 of {X n } is defined by
α(1) :=
γ(1)
γ(0)
,
α(n) :=
(P ⊥
[1,n−1] X n , P ⊥
[1,n−1] X 0 )
P ⊥
[1,n−1] X n 2
,
n = 2, 3, . . .
(2.1)
(cf. Brockwell and Davis (1991, Sections 3.4 and 5.2)). As stated in §1, The PACF {α(n)} ∞
n=1 coincides with the
See, e.g.,(5.2.4) inBrockwell and Davis (1991) for the latter.As in Inoue and Kasahara (2006, Section 2.3), we define, for n, j ∈ N ∪ {0},n.
(3.1)
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| []
|
[
"A novel camera type for very high energy gamma-ray astronomy based on Geiger-mode avalanche photodiodes *",
"A novel camera type for very high energy gamma-ray astronomy based on Geiger-mode avalanche photodiodes *"
]
| [
"H Anderhub \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"M Backes \nTU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany\n",
"A Biland \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"A Boller \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"I Braun \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"T Bretz \nUniversity of Würzburg\nAm Hubland97074WürzburgGermany\n",
"S Commichau \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"V Commichau ",
"D Dorner \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n\nISDC Data Center for Astrophysics\nUniversity of Geneva\nChemin d'Ecogia 161290VersoixSwitzerland\n",
"A Gendotti \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"O Grimm \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"H Von Gunten \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"D Hildebrand \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"U Horisberger \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"T Krähenbühl \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"D Kranich \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"E Lorenz \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n\nMax Planck Institute for Physics\nFöhringer Ring 680805MunichGermany\n",
"W Lustermann \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"K Mannheim \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n\nUniversity of Würzburg\nAm Hubland97074WürzburgGermany\n",
"D Neise \nTU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany\n",
"F Pauss \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"D Renker \nPaul Scherrer Institute\n5232Villigen PSISwitzerland\n",
"W Rhode \nTU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany\n",
"M Rissi \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"U Röser \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"S Rollke \nTU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany\n",
"L S Stark \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"J.-P Stucki \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"G Viertel \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"P Vogler \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n",
"Q Weitzel \nInstitute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland\n"
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"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"TU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"University of Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"ISDC Data Center for Astrophysics\nUniversity of Geneva\nChemin d'Ecogia 161290VersoixSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Max Planck Institute for Physics\nFöhringer Ring 680805MunichGermany",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"University of Würzburg\nAm Hubland97074WürzburgGermany",
"TU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Paul Scherrer Institute\n5232Villigen PSISwitzerland",
"TU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"TU Dortmund University\nOtto-Hahn-Str. 444227DortmundGermany",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland",
"Institute for Particle Physics\nETH Zurich\nSchafmattstr. 208093ZurichSwitzerland"
]
| [
"JINST"
]
| Geiger-mode avalanche photodiodes (G-APD) are promising new sensors for light detection in atmospheric Cherenkov telescopes. In this paper, the design and commissioning of a 36-pixel G-APD prototype camera is presented. The data acquisition is based on the Domino Ring Sampling (DRS2) chip. A sub-nanosecond time resolution has been achieved. Cosmic-ray induced air showers have been recorded using an imaging mirror setup, in a self-triggered mode. This is the first time that such measurements have been carried out with a complete G-APD camera. | 10.1088/1748-0221/4/10/p10010 | [
"https://arxiv.org/pdf/0911.4920v1.pdf"
]
| 119,290,616 | 0911.4920 | b23bfba4450c4ee0703cdb710dfaf43e3f2d5646 |
A novel camera type for very high energy gamma-ray astronomy based on Geiger-mode avalanche photodiodes *
2009
H Anderhub
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
M Backes
TU Dortmund University
Otto-Hahn-Str. 444227DortmundGermany
A Biland
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
A Boller
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
I Braun
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
T Bretz
University of Würzburg
Am Hubland97074WürzburgGermany
S Commichau
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
V Commichau
D Dorner
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
ISDC Data Center for Astrophysics
University of Geneva
Chemin d'Ecogia 161290VersoixSwitzerland
A Gendotti
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
O Grimm
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
H Von Gunten
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
D Hildebrand
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
U Horisberger
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
T Krähenbühl
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
D Kranich
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
E Lorenz
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
Max Planck Institute for Physics
Föhringer Ring 680805MunichGermany
W Lustermann
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
K Mannheim
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
University of Würzburg
Am Hubland97074WürzburgGermany
D Neise
TU Dortmund University
Otto-Hahn-Str. 444227DortmundGermany
F Pauss
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
D Renker
Paul Scherrer Institute
5232Villigen PSISwitzerland
W Rhode
TU Dortmund University
Otto-Hahn-Str. 444227DortmundGermany
M Rissi
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
U Röser
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
S Rollke
TU Dortmund University
Otto-Hahn-Str. 444227DortmundGermany
L S Stark
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
J.-P Stucki
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
G Viertel
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
P Vogler
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
Q Weitzel
Institute for Particle Physics
ETH Zurich
Schafmattstr. 208093ZurichSwitzerland
A novel camera type for very high energy gamma-ray astronomy based on Geiger-mode avalanche photodiodes *
JINST
4100102009* This is an author-created, un-copyedited version of an article published in Journal of Instrumentation. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at† Corresponding author; Electronic address: qweitzel@physethzch 2
Geiger-mode avalanche photodiodes (G-APD) are promising new sensors for light detection in atmospheric Cherenkov telescopes. In this paper, the design and commissioning of a 36-pixel G-APD prototype camera is presented. The data acquisition is based on the Domino Ring Sampling (DRS2) chip. A sub-nanosecond time resolution has been achieved. Cosmic-ray induced air showers have been recorded using an imaging mirror setup, in a self-triggered mode. This is the first time that such measurements have been carried out with a complete G-APD camera.
Geiger-mode avalanche photodiodes (G-APD) are promising new sensors for light detection in atmospheric Cherenkov telescopes. In this paper, the design and commissioning of a 36-pixel G-APD prototype camera is presented. The data acquisition is based on the Domino Ring Sampling (DRS2) chip. A sub-nanosecond time resolution has been achieved. Cosmic-ray induced air showers have been recorded using an imaging mirror setup, in a self-triggered mode. This is the first time that such measurements have been carried out with a complete G-APD camera. * This is an author-created, un-copyedited version of an article published in Journal of Instrumentation.
IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at http://www.iop.org/EJ/abstract/1748-0221/4/10/P10010.
I. INTRODUCTION
Imaging atmospheric Cherenkov telescopes (IACT) have been very successful in detecting very high energy (∼ 0.1-30 TeV) γ-rays from cosmic sources [1]. The key component is a pixelated camera which has to resolve flashes of Cherenkov light from air showers (duration 1-5 ns for γ-induced air showers, main wavelength range 300-650 nm). High-sensitivity photo-sensors are needed, even using light-collecting optics, since e. g. a 1 TeV primary photon hitting the atmosphere only results in about one hundred Cherenkov photons per square meter. Until now, matrices of photomultiplier tubes (PMT) have always been employed for this task. This is a well-known technology which, however, comes with some intrinsic disadvantages for telescope applications. PMTs are rather heavy and bulky, but at the same time fragile. They require high voltages of several 100 V or even kV and are damaged when exposed to sunlight. Typical PMTs furthermore have photon detection efficiencies (PDE) of only 20-30%.
Since a few years, a new type of semiconductor light-sensor is being developed, the socalled Geiger-mode avalanche photodiode (G-APD 1 ) [2]. These light-weight devices are built up from multiple APD cells operated in Geiger-mode. All the cells are connected in parallel and the overall signal is the sum of all simultaneously fired cells. They need bias voltages of 50-100 V, usually 1-5 V above the breakdown voltage. A high gain similar to PMTs is reached (10 5 -10 6 ) and, potentially, higher PDEs of up to 50%. The market for G-APDs is continuously growing, and several manufacturers are working on their improvement. For an IACT application under realistic ambient conditions, several technical challenges have to be met. This mostly concerns the necessity to compensate for gain variations due to changes in temperature or night-sky background light (NSB). While the former is an intrinsic feature of G-APDs, the latter is a general problem and applies to other photo-sensors as well. An advantage of G-APDs is that they can be operated at NSB rates up to several GHz per sensor, thus allowing measurements under twilight and moonlight conditions. First tests to detect Cherenkov light with small G-APD arrays have been performed in the past [3]. In order to develop a complete G-APD camera and find solutions for the technical challenges, the FACT project (First G-APD Camera Test) [4] has been launched. The prototype presented in this paper marks the first step towards a large camera with a field of view of about 5 • (0.1-0.2 • per pixel). This device is foreseen for the DWARF telescope (Dedicated multi-Wavelength AGN Research Facility) [5] which will perform monitoring of strong and varying γ-ray sources.
II. CAMERA LAYOUT AND FRONT-END ELECTRONICS
The prototype module comprises 144 G-APDs of type Hamamatsu MPPC S10362-33-50C [6]. Each has a sensitive area of 3 mm × 3 mm, covered by 3600 cells of 50 ñm × 50 ñm size. The operating bias voltage ranges from 71.15 V to 71.55 V for the 144 sensors (at 25 • C), corresponding to a gain of 7.5 · 10 5 . The dark count rate is below 5 MHz. A non-imaging light collector is placed on top of each G-APD to compensate for dead spaces due to the diode-chip packaging (see figure 1, top part). Open aluminum cones with an affixed reflecting foil (ESR Vikuitiç by 3M) and a quadratic base are in use (bot- tom side: 2.8 mm × 2.8 mm, top side: 7.2 mm × 7.2 mm, height: 17.5 mm, effective solid angle: 0.55 sr). With such collectors, the NSB rate during the darkest nights at e.g. the Observatorio Roque de los Muchachos, La Palma, is ∼ 10 MHz per sensor [7].
The signals from four G-APDs are summed up in front-end electronics boards (FEB) [8], resulting in 36 quadratic pixels of 14.4 mm × 14.4 mm size. These boards also perform a signal shaping and amplification on the analog level (4 mV voltage output per ñA current input, pulse decay times < 10 ns). Each FEB comprises twelve channels, with a power consumption of 150 mW per channel. Figure 2 (left) shows a photograph of the camera module during assembly, including the FEBs. Also 16 of the 144 G-APDs are visible, and a block of four light collectors which are usually mounted on top of the sensors. Such a block corresponds to one pixel (readout channel).
In order to keep the gain of the G-APDs stable in case of changes of the ambient temperature and NSB conditions 2 , a bias voltage feedback system has been foreseen. A calibration signal from a pulsed and temperature-stabilized light emitting diode can be monitored continuously and, if a change in amplitude is detected on a certain channel, the voltage of the corresponding pixel is modified to readjust the amplitude. Special power supply modules have been developed for this task [9], which can communicate with the camera control software (see also section III) via an USB interface and allow the bias voltage of each pixel to be regulated individually 3 . In addition, a water cooling system has been installed. An external cooling unit provides water of an adjustable temperature, which is pumped through copper tubes soldered onto a copper plate. The G-APDs are coupled to this plate by a thermally 2 The NSB light produces a permanent current in a serial resistor of the readout electronics and thereby reduces the effective bias voltage, which is supplied through the FEBs. 3 Since four G-APDs are combined in one pixel, they have been selected such that their nominal operating voltages are within ±10 mV. This translates into gain variations of a few percent [6]. conductive but electrically insulating filling material. Several temperature sensors and one humidity sensor are used to monitor the conditions inside the box and on the cooling plate, as can be seen from figure 2 (middle). The cooling system is not obligatory to operate the sensors, but can be used optionally. The whole camera module is mounted inside a watertight box (see figure 2, right), which has a protection window, a separate shutter (not shown in the photograph) and connectors for the signal and voltage supply cables.
III. TRIGGER AND DATA ACQUISITION SYSTEM
The analog signals are transferred from the camera box to the counting room by means of 20 m long coaxial cables. They are fed into linear fan-out modules (NIM standard) [10]. One module comprises twelve sub-units, each with one input channel and two normal and one inverted output channels. One of the (positive) outputs is connected to the data acquisition (DAQ) system, while the inverted (negative) signal is used for triggering purposes (see figure 1). The trigger logic consists of a CAEN V812 VME board where a majority coincidence of the innermost 16 pixels is formed. Individual trigger thresholds can be set via the VME bus.
The DAQ itself is based on the DRS2 (Domino Ring Sampling) chip [11] containing ten analog pipelines of 1024 capacitive cells. Signal sampling is performed with a frequency of 0.5-4.5 GHz, generated on-chip by a series of inverters. Such high sampling frequencies are desired for an IACT, since excellent timing significantly improves the reconstruction of the properties of the primary particle [12]. Each pipeline is read out at 40 MHz and externally digitized by a multiplexed 12 bit flash ADC (analog-to-digital converter). The DRS2 chips are mounted on mezzanine cards, which are hosted by VME boards (two chips per card and two cards per VME board). In this design, eight channels of each chip are available for external signals (see also [11]). A single-board computer is used as VME controller, with an attached hard disk for data storage. The DAQ and bias voltage control software are also running on this computer. For a trigger rate of 10 Hz, typical data rates are of the order of 1 MByte/s. Sampling the signal with the ring sampler at frequencies in the GHz range allows a precise photon arrival time measurement, provided the so-called fixed-pattern aperture jitter of the DRS2 chip is corrected. This relatively large, but systematic and gradual spread in its bin widths results from the manufacturing process. As the occurrence of a trigger is random with respect to the physical pipeline, the time measured between the trigger and the signal, being the sum of the widths of the involved bins, is also randomized to some degree. At 2 GHz sampling, this jitter can amount to about 5 ns and can have a complicated distribution, depending on the distance between trigger and signal. It can be corrected by calibrating with a high-frequency signal from a precision generator. The procedure involves measuring the period of this signal with the DRS2, and stretching or shrinking of the bin widths within that particular period to match the generator output. As the revolution frequency of the domino wave is phase locked to a stable oscillator, the revolution time (the sum over all 1024 individual bin widths) itself is fixed. Therefore, the inverse correction is applied to all bins outside the current period. Doing this for all periods of a sampled waveform and iteratively for many waveforms with random phase relative to the domino wave converges towards the required correction values. The effectiveness of the jitter correction is shown in figure 3 (left). The histograms show the time distributions of the rising edge of a pulse relative to the trigger signal with and without the correction at 2 GHz sampling frequency. Both the (square) test-pulse and the trigger were derived from a single output of a pulse generator.
The correction values were determined in this case using a 200 MHz rectangular signal. After applying them, the pulse time distribution has a width of 390 ps root-mean-square.
IV. MEASUREMENT SETUP AND DATA TAKING PARAMETERS
In figure 3 (right) the experimental setup used for the detection of cosmic air showers is shown. The camera was mounted in the focus of a zenith-pointing spherical mirror (90 cm diameter). The focal length of the mirror was 80 cm which, taking into account the pixel dimensions, translates into a field of view of 1.0 • per pixel. The measurements were performed during the night of July 2, 2009, near the city center of Zurich. Thus the NSB from surrounding buildings and also from partial moonlight was rather high. It was determined with an external sky quality meter (Unihedron) to be 300 MHz per G-APD (1.2 GHz per pixel). During the measurements, which took about an hour, the outdoor temperature was around 22 • C. The G-APD plane was cooled to 18 • C and the bias voltage for all pixels set to the nominal values for this temperature (∼ 400 mV lower than for operation at 25 • C). Under these stable conditions, no bias voltage feedback had to be used.
A sampling frequency of 2 GHz was set for the DRS2 chips. The trigger threshold was adjusted for each of the innermost 16 pixels, such that single pixel trigger rates of 1-3 kHz were obtained for an opened camera shutter. This was achieved by applying discriminator thresholds of 40 ± 4 mV to the analog signals 4 . A majority coincidence of four channels above threshold within a time window of 20 ns was furthermore required. Under these circumstances, the DAQ system recorded data with an average trigger rate of 0.02 Hz (also including noise triggers). The latency of the whole system was about 350 ns.
V. RESULTS AND DISCUSSION
For the offline analysis, events corresponding to Cherenkov light flashes have been filtered out from the recorded data. Several neighboring channels containing a clear signal, at the time position expected from the trigger latency, have been required. This is demonstrated in figure 4, which presents the raw data for a certain pixel (event #16 of run #207). The large plot shows the full DRS2 pipeline with a Cherenkov-light signal at about 160 ns, while the inset presents a zoom to the data between 0 and 150 ns. In the latter, the fluctuating signals from NSB photons are visible, which are very frequent and therefore pile up. The red line indicates the trigger threshold 5 . From the data analysis it has been estimated that the trigger rate for air shower events was about 0.01 Hz.
In figure 5 (left) the intensity distribution for the whole event introduced above is plotted. The maximum sample amplitude is shown on each of the 36 pixels, searched for within a time window of 100 ns around the expected signal position. Each amplitude has been corrected for the DRS2 baseline, evaluated event-by-event from the data samples recorded before the signal arrival (see figure 4). On the right side of figure 5, the jitter corrected (cf. section III) arrival time of the signals is presented 6 . Only pixels with a signal amplitude above 60 mV have been used for the timing calculation, the others are displayed in white color. A clear shower development is apparent for this event, starting in the top left corner of the display and extending to the bottom right corner within almost 20 ns. Figure 6 shows an event with a less extended time distribution (event #14 of run #206). Especially the core pixels with the largest amplitudes are within a few ns. The two outer pixels with a time offset of 3-4 ns compared to the core pixels are likely due to a sub-shower. 5 Translated to the digitized sample amplitudes. Please note that the signal is attenuated in front of the DRS2 chip to match its dynamic range. 6 Defined as the time of the last sample where the signal is below 50% of its amplitude (no interpolation). Taking into account that the camera covers a field of view of 6 • in both dimensions (see also section IV), such events certainly come from air showers induced by very high energetic cosmic-ray particles. Dedicated simulations have been carried out 7 , concluding that the experimental setup described in this paper has an energy threshold of several TeV. Primary protons with off-axis angles up to 5 • have been simulated 8 , and photon arrival time distributions consistent with the event presented in figure 6 have been observed. The event shown in figure 5 most probably corresponds to a primary particle with an off-axis angle of 10 • or larger. Because of the comparatively long trigger coincidence window, it was possible to trigger such events.
VI. CONCLUSION
A 36-pixel prototype G-APD camera for Cherenkov astronomy has successfully been constructed and commissioned. A DAQ system based on the DRS2 chip has been set up. In a self-triggered mode, images of air showers induced by cosmic rays have been recorded. For the first time, this has been achieved with a complete G-APD camera. Stable operation at room temperature and under high NSB light conditions is possible. In summary, these photo-sensors have proven to fulfill the requirements of IACT applications, with the potential to replace or complement PMTs for future projects like the planned Cherenkov Telescope Array (CTA) [15]. 7 The air shower simulation using CORSIKA [13], the detector simulation and data analysis using the MARS package (CheObs edition) [14]. 8 Protons compose the largest fraction of the cosmic rays. Since in this case the mirror axis points up vertically to the sky, the off-axis angle coincides with the zenith angle.
FIG. 1 :
1Schematic overview of the G-APD camera layout (upper part) and the DAQ system (lower part); signal flow of one pixel shown. The camera is mounted inside a water-tight box, while the DAQ and trigger components are located in a counting room.
FIG. 2 :
2Photographs taken during the camera assembly. Left: carrier PCB with 16 G-APDs ( = 4 pixel) attached to the front, and the three front-end electronics boards attached to the back; a block of four light collectors ( = 1 pixel) is mounted separately for demonstration purposes. Middle: fully assembled module including cooling plate. Right: integration into water-tight box; signal and voltage supply cables attached.
FIG. 3 :
3Left: time distribution of square pulses sampled at 2 GHz with the DRS2 chip without (red filled histogram) and with jitter correction (blue shaded). The root-mean-square of the latter is 390 ns. Right: photograph of the experimental setup to record cosmic-ray induced air showers (see also text).
FIG. 4 :
4Raw data of one pixel recorded with a sampling frequency of 2 GHz; full DRS2 pipeline shown. A Cherenkov-light signal can be seen at about 160 ns. The red solid line indicates the trigger threshold. The inset shows a zoom to the data from 0-150 ns as marked by the dashed-line box.
FIG. 5 :FIG. 6 :
56Air shower image recorded with the G-APD camera (event #16 of run #207); the signal shown infigure 4belongs to the pixel in the second column from the left, third row from the top. Left: intensity distribution over the 6 pixels × 6 pixels. Right: corresponding signal arrival time distribution (see text). Further air shower image recorded with the G-APD camera (event #14 of run #206). Left: intensity distribution over the 6 pixels × 6 pixels. Right: corresponding signal arrival time distribution (see text). The color scales are the same as infigure 5.
Also referred to as silicon photomultiplier (SiPM) or multi-pixel photon counter (MPPC).
The thresholds had to be exceeded for at least 4 ns to fulfill the trigger requirement. A laboratory calibration, though without NSB light, showed that the analog readout features 7 mV signal amplitude per fired G-APD cell (approximately equivalent to the number of incoming photons per pixel multiplied with the PDE of the G-APDs).
AcknowledgmentsThis work is supported by ETH research grants 0-43391-08 and 0-20486-08.
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First avalanche-photodiode camera test (FACT): a novel camera using G-APDs for the observation of very high-energy gamma-rays with Cherenkov telescopes. I Braun, proceedings of 5th International Conference on New Developments in Photodetection. 5th International Conference on New Developments in PhotodetectionAix-les-Bains, Francein press atI. Braun et al., First avalanche-photodiode camera test (FACT): a novel camera using G-APDs for the observation of very high-energy gamma-rays with Cherenkov telescopes, in proceedings of 5th International Conference on New Developments in Photodetection, June, 15-20, 2008, Aix-les-Bains, France, in press at Nucl. Instrum. Meth. A.
Long-term monitoring of bright blazars with a dedicated Cherenkov telescope. T Bretz, proceedings of 4th International Meeting on High Energy Gamma-Ray Astronomy. 4th International Meeting on High Energy Gamma-Ray AstronomyHeidelberg, Germany1085850T. Bretz et al., Long-term monitoring of bright blazars with a dedicated Cherenkov telescope, in proceedings of 4th International Meeting on High Energy Gamma-Ray Astronomy, July, 7-11, 2008, Heidelberg, Germany, AIP Conf. Proc. 1085 (2008) 850.
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Improving the performance of the single-dish Cherenkov telescope MAGIC through the use of signal timing. E Aliu, Astropart. Phys. 30293E. Aliu et al., Improving the performance of the single-dish Cherenkov telescope MAGIC through the use of signal timing, Astropart. Phys. 30 (2009) 293.
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Conceptional design report. in preparationCTA consortium, http://www.cta-observatory.org, Conceptional design report in preparation.
| []
|
[
"Approximation of the potential in scalar field dark energy models",
"Approximation of the potential in scalar field dark energy models"
]
| [
"Richard A Battye *[email protected] \nJodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nThe University of Manchester\nM13 9PLManchesterUnited Kingdom\n",
"Francesco Pace \nJodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nThe University of Manchester\nM13 9PLManchesterUnited Kingdom\n"
]
| [
"Jodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nThe University of Manchester\nM13 9PLManchesterUnited Kingdom",
"Jodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nThe University of Manchester\nM13 9PLManchesterUnited Kingdom"
]
| []
| We study the nature of potentials in scalar field based models for dark energy -with both canonical and noncanonical kinetic terms. We calculate numerically, and using an analytic approximation around a ≈ 1, potentials for models with constant equation-of-state parameter, w φ . We find that for a wide range of models with canonical and noncanonical kinetic terms there is a simple approximation for the potential that holds when the scale factor is in the range 0.6 a 1.4. We discuss how this form of the potential can also be used to represent models with nonconstant w φ and, hence, how it could be used in reconstruction from cosmological data. 95.36.+x | 10.1103/physrevd.94.063513 | [
"https://arxiv.org/pdf/1607.01720v2.pdf"
]
| 55,130,729 | 1607.01720 | 04f7963aafbf3f8b9d83a11ec394fe9641271955 |
Approximation of the potential in scalar field dark energy models
1 Oct 2016
Richard A Battye *[email protected]
Jodrell Bank Centre for Astrophysics
School of Physics and Astronomy
The University of Manchester
M13 9PLManchesterUnited Kingdom
Francesco Pace
Jodrell Bank Centre for Astrophysics
School of Physics and Astronomy
The University of Manchester
M13 9PLManchesterUnited Kingdom
Approximation of the potential in scalar field dark energy models
1 Oct 2016(Dated: October 4, 2016)arXiv:1607.01720v2 [astro-ph.CO] 2Cosmologyscalar fieldk-essencedark energyequation of state
We study the nature of potentials in scalar field based models for dark energy -with both canonical and noncanonical kinetic terms. We calculate numerically, and using an analytic approximation around a ≈ 1, potentials for models with constant equation-of-state parameter, w φ . We find that for a wide range of models with canonical and noncanonical kinetic terms there is a simple approximation for the potential that holds when the scale factor is in the range 0.6 a 1.4. We discuss how this form of the potential can also be used to represent models with nonconstant w φ and, hence, how it could be used in reconstruction from cosmological data. 95.36.+x
I. INTRODUCTION
The origin of the cosmic acceleration is one of the most significant open questions in cosmology and fundamental physics. A cosmological constant is still very much consistent with the data [1, 2], but in order to either refute or confirm this simple hypothesis one needs to consider alternative models to explain the observations. One very simple idea is to postulate a dark energy component dominated by a scalar field either with a canonical or noncanonical kinetic term. Such models are known as quintessence models [3][4][5][6][7][8][9][10] and k-essence models [11][12][13], respectively.
The standard approach when constraining cosmological models with a dark energy component that is not the cosmological constant is to define an equation-of-state parameter w φ = P φ /ρ φ = −1, where P φ is the pressure of dark energy and ρ φ is its density, making no assumption as to the origin of the dark energy. In principle this is a general function of time, but it is often considered to be either constant, or to be represented by a specific functional form, for example [14,15]. At the moment the data barely constrain anything beyond a constant w φ , but this is likely to change in the near future as more observations probing the equation of state become available, such as Euclid 1 [16,17], LSST 2 [18] and SKA 3 [19][20][21][22]. Various ideas have been put forward to extend to time varying situations. These include various limited functional forms [14,15,[23][24][25], the Om diagnostic [26,27], the state-finder approach [28,29] and even using principal component analysis on general piecewise linear parametrizations of w φ [30]. For a review of the parametric and nonparametric methods to reconstruct the dark energy equation-of-state parameter, we refer to [31]. Since many of the observations are sensitive to perturbations in the dark energy it is also necessary to make some assumptions about the perturbations, but we will not consider this here.
An alternative is to presume that the origin of the dark energy is a model based on a scalar field. However, such models usually involve one or more arbitrary functions which would need to be specified before any model prediction could be made. One of these is the potential V (φ) of the scalar field which one might try to reconstruct from observations. One obvious suggestion [32], which extends the approach of [33] for inflation, is to represent the potential as a Taylor series expanded around the present-day value of the field φ 0
V (φ) = V 0 + V 1 (φ − φ 0 ) + V 2 (φ − φ 0 ) 2 + . . . ,(1)
and attempt to fit for the coefficients V i . However, it is not clear where to truncate this series in a controlled way. Similar and complementary methods have been proposed by [34][35][36][37]. Other reconstruction methods are valid in the slow-roll regime, that is, when 1 + w φ ≈ 0. For quintessence models, a one-parameter [38] or two-parameter [39][40][41] formula has been used and for k-essence models we refer to works by [11,42].
In this paper we first calculate potentials for a range of minimally coupled scalar field models with canonical (section II) and noncanonical (section III) kinetic terms assuming initially that w φ is constant. It is possible to derive an analytic solution for the potential in Quintessence models, but this is not possible in general for the case of k-essence models and therefore we resort to numerical calculations and an analytic approximation around the present day which is valid for 0.6 a 1. 4. Based on this analytic approximation we suggest a form of a potential with just four parameters which we demonstrate can lead to a wide range of behaviour for w φ as a function of time (section IV) and, by design, includes models with constant w φ . Of course, this functional form will not include every possible behaviour in a general model, but it does provide more physical insights and it is useful for models which are not significantly different from a linearly evolving equation-of-state parameter. We conclude and discuss our results in section V.
In the following, we will use natural units with c = = 1, the Planck mass is M pl = G −1/2 and we assume a metric with signature (−, +, +, +).
II. MINIMALLY COUPLED SCALAR FIELDS WITH CONSTANT w φ
The Lagrangian for minimally coupled scalar fields is
L = − 1 2 ηg µν ∇ µ φ∇ ν φ − V (φ) ,(2)
and its corresponding stress-energy tensor
T µν = g µν L + η∇ µ φ∇ ν φ = η∇ µ φ∇ ν φ − g µν 1 2 ηg αβ ∇ α φ∇ β φ + V (φ) .(3)
The constant η distinguishes between the Quintessence case (η = +1, −1 < w φ < 1) and the phantom case (η = −1, w φ < −1) [43]. Density and pressure are given by
ρ φ = T 0 0 = 1 2 ηφ 2 + V (φ) , P φ = 1 3 T i i = 1 2 ηφ 2 − V (φ) ,(4)
and the conservation equationρ φ +3H(ρ φ +P φ ) = 0 gives rise to the Klein-Gordon equation, which describes the time evolution of the scalar fieldφ
+ 3Hφ + η dV dφ = 0 .(5)
To achieve an accelerated expansion, we require w φ < −1/3. In fact, observations require w φ ≃ −1 (due to the cosmological constant case) [1], hence we can evaluate deviations of w φ from −1 with the help of (5)
1 + w φ = V 2 φ 9H 2 (ξ s + 1) 2 ρ φ ,(6)
with ξ s =φ/(3Hφ) [17]. Note that in a pure slow-roll approximation, ξ s = 0. By using Friedmann equations, we can determine the time evolution of the scalar field and its potential for a given w φ (a) [44] φ
(a) − φ 0 M pl = ± 3Ω de 8π a 1 η[1 + w φ (x)]g(x) xE(x) dx ,(7)V (a) = 3H 2 0 M 2 pl Ω de [1 − w φ (a)]g(a) 16π ,(8)
where Ω de is the dark energy density parameter today, H 0 the Hubble constant and φ 0 the value of the scalar field at a = 1. Finally, g(a) represents the time evolution of the dark energy component
g(a) = exp −3 a 1 1 + w φ (x) x dx .(9)
Assuming a flat geometry, the Hubble parameter is given by
H = H 0 E(a) = H 0 Ω m a 3 + Ω de g(a) 1 2 ,(10)
with Ω m the matter density parameter today. For a constant equation of state w φ , integral (7) can be evaluated as
φ − φ 0 M pl = ∓ 2 3w φ 3η(1 + w φ ) 8π sinh −1 Ω de Ω m a − 3w φ 2 − sinh −1 Ω de Ω m ,(11)
and its inverse gives an expression for the scale factor in terms of the scalar field
a(φ) = Ω m Ω de − 1 3w φ sinh ∓ 3 2 w φ 8ηπ 3(1 + w φ ) φ − φ 0 M pl + sinh −1 Ω de Ω m − 2 3w φ .(12)
With these relations in hand, we can deduce the full expression for the potential V (φ)
V (φ) = 3H 2 0 M 2 pl Ω de (1 − w φ ) 16π Ω m Ω de 1+w φ w φ sinh ∓ 3 2 w φ 8ηπ 3(1 + w φ ) φ − φ 0 M pl + sinh −1 Ω de Ω m 2(1+w φ ) w φ .(13)
Since φ 0 just shifts the potential in the φ-direction, we can make the choice
φ 0 = ± 2 3w φ M pl 3η(1 + w φ ) 8π sinh −1 Ω de Ω m ,(14)
that simplifies the form of the potential
V (φ) = 3H 2 0 M 2 pl Ω de (1 − w φ ) 16π Ω m Ω de 1+w φ w φ sinh 2(1+w φ ) w φ ∓ 3w φ 2 8ηπ 3(1 + w φ ) φ M pl ,(15)
in agreement with [31,45]. An analytic solution of (7) can be found also when dark matter has a constant equation-of-state parameter w m = 0, as shown in [46]. While it is possible to find exact solutions for the potential of Quintessence models with a constant equation of state, this is not the case when w φ is a function of time or for more general scalar field models, such k-essence models. Moreover, the expressions for the potential at early and late times are not very useful from an observational point of view, since they assume one of the component to be dominant and are not relevant for modelling late-time observations. It is, therefore, worthwhile to find approximate solutions valid for a ≈ 1 that can be probed with data. To do this, we expand in series dφ da for a ≈ 1, but a priori it is not clear where to truncate the series. We have checked that a first-order expansion is a very good approximation, leading to a scalar field evolving quadratically with respect to the scale factor.
The differential equation describing the approximate evolution of the scalar field for a ≈ 1 is
dφ da = ± 3η(1 + w φ )M 2 pl Ω de 8π 1 − 1 2 (2 + 3Ω m w φ ) (a − 1) ,(16)
which implies the following approximate evolution for the scalar field
φ − φ 0 M pl = ± 3η(1 + w φ )Ω de 8π a − 1 − 1 4 (2 + 3Ω m w φ ) (a − 1) 2 .(17)
By inverting this, we can find a relation between the scale factor and the scalar field
a = 1 + 2 2 + 3Ω m w φ 1 − 1 ∓ [2 + 3Ω m w φ ] 8ηπ 3Ω de [1 + w φ ] φ − φ 0 M pl ,(18)
which leads to the following functional form for the approximate potential
V (φ) = AH 2 0 M 2 pl 1 − B + C φ − φ 0 M pl D ,(19)
where the coefficients A, B, C and D are dimensionless constants depending on the cosmological parameters characterising the model. For minimally coupled models with constant w φ , the four coefficients assume the following values (20) depend on two quantities, Ω de and w φ , therefore, we can express two of them (A and C) in terms of B and D:
A = 3Ω de (1 − w φ ) 16π 4 + 3Ω m w φ 2 + 3Ω m w φ −3(1+w φ ) , B = 4 (4 + 3Ω m w φ ) 2 , C = ∓4 2 + 3Ω m w φ (4 + 3Ω m w φ ) 2 8ηπ 3Ω de (1 + w φ ) , D = −3(1 + w φ ) .(20)A = (D + 6)[ √ B(D + 1) − 2] 16π √ B(D − 3)(1 − √ B) D , C = ∓2 8ηπ √ B(3 − D) D[ √ B(D + 1) − 2] √ B(1 − √ B) .(21)
To see how good our approximation is for a ≈ 1, we compare the approximate expression for the potential to the exact solution for different values of the equation-of-state parameter w φ in the top left panel of Figure 1. We assumed the following cosmological parameters: Ω m = 0.3, Ω de = 0.7 and H 0 = 70 km s −1 Mpc −1 . The value of φ at a = 1 ranges from φ/M pl ≈ 0.02 for w φ = −0.99 to φ/M pl ≈ 0.2 for w φ = −0.7. The approximate solution agrees very well with the analytic one over a range of values centred on a = 1 (by construction) and it deviates from it at both low and high values of the scale factor (corresponding to low and high values of the scalar field, respectively). In particular, by inspecting the top right panel of Figure 1 we find an excellent agreement for 0.7 a 1.2. We also note that a better agreement occurs when the equation-ofstate parameter is not substantially different from w φ = −1: this is due to the fact that for the cosmological constant the scalar field and the potential are constant in time. If we require a tolerance of 1% in the equation of state derived from the approximate potential, then the confidence interval is 0.5 a 1.5.
In the bottom panel of Figure 1 we show the evolution of the scalar field with respect to the scale factor for w φ = −0.9. We show the time evolution of the scalar field rather than that of the potential because by construction, the latter evolves as a −3(1+w φ ) . Note how the two expressions for the scalar field agree remarkably well over a range 0.5 a 1.7. For values outside this range the approximate solution underestimates the exact one and it becomes negative for a 0.2. This range is largely in agreement with what we found for the reconstructed equation-of-state parameter.
III. K-ESSENCE WITH CONSTANT w φ
A straightforward extension of minimally coupled scalar fields is given by models with a noncanonical kinetic term. These models are described by a Lagrangian of the form L = L(φ, χ) [12] where χ = − 1 2 g µν ∇ µ φ∇ ν φ is the canonical kinetic energy term. These models have been extensively used to describe dark energy scenario [12,13,[47][48][49][50] and several works studied their dynamics and stability [51][52][53][54][55]. These models are dubbed "k-essence" models because the kinetic term χ can be responsible for the cosmic acceleration. A wide variety of models have been proposed and studied in different contests, such as low-energy effective string theory [56], tachyon models [49,57], ghost condensates [52,[58][59][60], Dirac-Born-Infeld (DBI) theories [61][62][63].
The density and pressure are given by ρ = 2χL χ − L and P = L, respectively, where L χ = ∂L ∂χ and we will also use L χχ = ∂ 2 L ∂χ 2 and L χφ = ∂ 2 L ∂χ∂φ . The sound speed for sub-horizon modes is
α = P χ ρ χ = 1 + 2χL χχ L χ −1 .(22)
Using P φ = w φ (a)ρ φ , we can deduce that
2χL χ = 1 + w φ (a) w φ (a) L .(23)
From (23) we see that k-essence models can achieve w φ ≈ −1 without χ ≈ 0. This means that such models need not be in the slow-roll regime to act as a dark energy component. The energy-momentum tensor of k-essence is that of a perfect fluid
T µν = L χ ∇ µ φ∇ ν φ + Lg µν = (ρ + P )u µ u ν + P g µν ,(24)
and velocity u µ = ∇ µ φ/ √ 2χ. The equation of motion for the scalar field iṡ
χ (L χ + 2χL χχ ) + 2χ (2χL χφ − L φ ) + 6HχL χ = 0 ,(25)
and by rearranging the terms in (25), the equation of motion reads [17] H
µν ∇ µ ∇ ν φ + 2χL χφ − L φ = H µν ∇ µ ∇ ν φ − L χφ g µν ∇ µ φ∇ ν φ − L φ = 0 ,(26)
where
H µν = L χχ ∇ µ φ∇ ν φ − L χ g µν .(27)
By inspecting (26), we notice that the equation of motion can be written in a very compact form as
∇ µ J µ = −L φ , with J µ = L χ ∇ µ φ.
Many of the k-essence models proposed in literature fall into one of the following types:
(A) Models of type A are given by [13,[64][65][66]]
L = M 4 F (χ) − V (φ) ,(28)
where M has dimensions of mass and F is a dimensionless function. In the following, it is helpful to consider F (χ) to be a power-law F (χ) = χ M 4 n , for n constant. Setting n = 1 implies L = χ − V (φ) which corresponds to the Quintessence case discussed in section II.
(B) Models of type B are given by [11,12,59,67]
L = G(χ)V (φ) ,(29)
where G is a dimensionless function. Common Lagrangians proposed in literature, which are mainly for purely kinetic k-essence
model (i.e. V (φ) = M 4 =constant), are [50, 68-73] 1. G(χ) = − 1 + 2η χ M 4 , 2. G(χ) = 2 χ M 4 n − 1 1 2n , 3. G(χ) = − 1 + 2η χ M 4 n 1 2n , 4. G(χ) = A 1 χ M 4 − A 2 χ M 4 α , 5. G(χ) = − 1 − 2 χ M 4 β , 6. G(χ) = χ M 4 − χ M 4 ,
where n, α, β, A 1 and A 2 are constant and η = ±1. Typically it is not possible to transform between η = +1 and η = −1 via a simple redefinition of the scalar field. An interesting Lagrangian to consider is the ghost condensate model [59]
L = K(φ)χ + L(φ) χ 2 M 4 ,(30)
where K(φ) < 0 and L(φ) are dimensionless potentials and M , again, has dimensions of mass. If one defines the scalar field ψ by
dψ dφ 2 = L |K| ,(31)
and write X = − 1 2 g µν ∇ µ ψ∇ ν ψ, then
L = V (φ) − X M 4 + X 2 M 8 ,(32)
where
V (φ) = [K(φ)] 2 /L(φ) if K < 0. Hence, this can be considered as a model of type B with G(χ) = −χ/M 4 + χ 2 /M 8 .
(C) Models of type C are given by [60]
L = −χ − N (χ)V (φ) ,(33)
where
N = χ M 4
n is a dimensionless function. The model represents a generalization of the dilatonic ghost condensate model and it is a special case of (30
), where K(φ) = −1, L(φ) = V (φ) M 4 and N (χ) = χ 2 .
III.1. Type A models with constant w φ
For models of type A, with F (χ) being a power-law, we have χFχ F = n and χFχχ Fχ = n − 1 and therefore α = (2n − 1) −1 , constant. For a general w φ (a) we have
V (φ) = 3H 2 0 M 2 pl Ω de [1 − (2n − 1)w φ (a)]g(a) 16πn ,(34)dφ da = √ 2 M 2 H 0 3H 2 0 M 2 pl Ω de 16πnM 4 1 2n {[1 + w φ (a)]g(a)} 1 2n aE(a) ,(35)φ − φ 0 M pl = √ 2 M 2 H 0 M pl 3H 2 0 M 2 pl Ω de 16πnM 4 1 2n a 1 {[1 + w φ (x)]g(x)} 1 2n xE(x) dx .(36)
It is, therefore, possible in principle to find φ(a), at least numerically. Note that n = 0, otherwise the potential and the scalar field diverge. Our general results are consistent with [74] if we set M 4 = 1 and F (χ) = χ 2 and with [75]. If w φ is constant, we can recover analogous results to the Quintessence case. In this case, (36) becomes
φ − φ 0 M pl = − 2 √ 2M 2 3w φ H 0 M pl Ω 1/2 m 3H 2 0 M 2 pl Ω de (1 + w φ ) 16πnM 4 1 2n Ω m Ω de 1+w φ −n 2nw φ sinh −1 Ω de Ωm a − 3w φ 2 sinh −1 Ω de Ωm dx sinh (1+w φ )(1−n)
nw φ
x .
(37) It is not possible to compute this integral analytically for general w φ and n, but it at least illustrates that a solution exists and the solution can be computed numerically. It is also important to notice that for a given equation-of-state parameter, the scalar field and its potential are not uniquely determined since for a given w φ , these two quantities depend also on n. Note also that equations (34), (36) and (37) reduce to the Quintessence case for n = 1. At early and late times, the potential is given by
V E (φ) ∝ φ − 2n(1+w φ ) [n−(1+w φ )] and V L (φ) ∝ φ − 2n
(n−1) , respectively, which are in agreement with [75].
Since analytical solutions are not possible, we find it useful to derive approximated expressions also for a ≈ 1. In this case, (35) is approximated by
dφ da = √ 2 M 2 H 0 3H 2 0 M 2 pl Ω de (1 + w φ ) 16πnM 4 1 2n 1 + n − 3 + 3(Ω de n − 1)w φ 2n (a − 1) ,(38)
which leads to
φ − φ 0 M pl = √ 2 M 2 H 0 M pl 3H 2 0 M 2 pl Ω de (1 + w φ ) 16πnM 4 1 2n a − 1 + n − 3 + 3(Ω de n − 1)w φ 4n (a − 1) 2 .(39)
By inverting this expression to find a(φ), the potential can be written with the same functional form as (19), with the following coefficients:
A = 3Ω de [1 − (2n − 1)w φ ] 16πn 3 + n − 3(Ω de n − 1)w φ 3 − n − 3(Ω de n − 1)w φ −3(1+w φ ) , B = 4n 2 [3 + n − 3(Ω de n − 1)w φ ] 2 , C = −2 √ 2n H 0 M pl M 2 3 − n − 3(Ω de n − 1)w φ [3 + n − 3(Ω de n − 1)w φ ] 2 16πnM 4 3H 2 0 M 2 pl Ω de (1 + w φ ) 1 2n , D = −3(1 + w φ ) .(40)
When n = 1, this reverts to the coefficients presented in (20).
In the top left panel of Figure 2 we make a comparison between exact numerically generated solutions and the approximation around a ≈ 1. As in the case of Quintessence there is a good agreement between the two. We also show in the right panel that the potential for n > 2 quickly asymptotes to the n → ∞ solution. This should be expected from the form of (37). In the lower panels we show the range of validity of the approximated scalar field (left) and potential (right) for the approximated expression found and described by the four coefficients listed above. As for Quintessence models, the approximate potential recovers the exact one only for a limited range in the scale factor, hence also the reconstructed equation of state will be limited to the range of validity of the approximate potential.
We found that the behaviour of the reconstructed equation of state is very similar to the Quintessence case. More quantitatively, we match the true w φ with ∆w φ = 0.01 for 0.6 a 1.5; a range that is largely in agreement with Quintessence.
III.2. Type B models with constant w φ
Type B models behave quite differently from type A models and it is not possible to make a direct comparison with Quintessence or phantom models. These models are commonly studied in literature because the kinetic term is completely factorized from the potential term, making the calculations relatively easy.
The evolution of the potential, the kinetic term and the sound speed are given by
V (a) = 3H 2 0 M 2 pl Ω de w φ (a)g(a) 8πG(χ) ,(41)χG χ G = 1 + w φ (a) 2w φ (a) ,(42)α = 1 + 2 χG χχ G χ −1 .(43)
One approach would be to solve for G(χ) from (42). When w φ is constant it is given by
G(χ) = χ M 4 1+w φ 2w φ .(44)
However, if we do this then we find that α = w φ which would mean that perturbations would be unstable if w φ < 0. This is, therefore, not the correct approach for deducing a potential from constant w φ . The alternative is to specify G(χ) and consider (42) as a constraint on χ which will be constant. Letχ be the constant value which solves (42) for a specific choice of G(χ), then whose solution is
dφ da = 1 H 0 2χa Ω m + Ω de a −3w φ 1/2 ,(45)φ − φ 0 M pl = − 2 3wΩ 1/2 m √ 2χ H 0 M pl Ω de Ω m 1 2w φ sinh −1 Ω de Ωm a − 3w φ 2 sinh −1 Ω de Ωm dx sinh − 1+w φ w φ x .(46)
Again this at least proves the existence of an φ(a), and hence a V (φ), which gives rise to constant w φ , although there is no analytic solution for general w φ . Note that this expression is equivalent to (37) in the limit n → ∞ and with √χ = M 2 . We are able to find useful approximations at early and late times. In particular we find V E (φ) ∝ φ −2(1+w φ ) at early times and V L (φ) ∝ φ −2 at late times, respectively. Let us now consider the specific case of
G(χ) = − χ M 4 + χ 2 M 8 .(47)
from which we can deduce thatχ
M 4 = 1−w φ 1−3w φ . Hence, we find that α = 1−2χ M 4 1−6χ M 4 = 1+w φ 5−3w φ . Note that 0 ≤ α ≤ 1 implies that −1 ≤ w φ ≤ 1. From (30), one finds that χL(φ) M 4 K(φ) = w φ − 1 1 − 3w φ ,(48)
and the corresponding sound speed is which is what should be expected from the general discussion about the generalised ghost condensate model. Note that in type B models with constant w φ , the sound speed becomes a function the of equation of state α = α(w φ ); such models have been also studied in [76] where the authors used shear and CMB lensing data to constrain dark energy perturbations. For a ≈ 1, the differential equation governing the evolution of the scalar field φ as a function of the scale factor a is
α = 1 + 2 L K χ M 4 1 + 6 L K χ M 4 = 1 + w φ 5 − 3w φ ,(49)dφ da = √ 2χ H 0 1 + 1 2 (1 + 3Ω de w φ )(a − 1) ,(50)
which leads to
φ − φ 0 M pl = √ 2χ H 0 M pl a − 1 + 1 4 (1 + 3Ω de w φ ) (a − 1) 2 .(51)
The relation between the scale factor and the scalar field is then
a = 1 − 2 1 + 3Ω de w φ 1 − 1 + (1 + 3Ω de w φ ) H 0 √ 2χ (φ − φ 0 ) .(52)
Hence, also for type B models we will have an approximate potential of the form of (19) with dimensionless coefficients
A = 3Ω de w φ 8πG(χ) 3Ω de w φ − 1 3Ω de w φ + 1 −3(1+w φ ) , B = 4 (1 − 3Ω de w φ ) 2 , C = 4 1 + 3Ω de w φ (1 − 3Ω de w φ ) 2 H 0 M pl √ 2χ , D = −3(1 + w φ ) .(53)
In the left panel of Figure 3 we show the validity of the approximation for the scalar field potential for a ≈ 1 for different constant equation-of-state parameters, as described in the caption, for a model with G(χ) = − χ M 4 + χ 2 M 8 . We obtain a similar level of agreement as for Quintessence and type A models. The accuracy increases with the decrease of w φ and is limited to an epoch centred on a = 1. In the right and middle panels of Figure 3 we compare the approximated expression for the potential and the corresponding scalar field evolution with the exact solution. Note that since (46) is a limiting case of (37), the range of agreement of the equation of state for type B models is similar to what found for type A models.
In the general discussion of type B models, we showed that α = α(w φ ) andχ =χ(w φ ); each model will have, therefore, its own particular functional form and a range of values for the parameters ensuring their stability. In Table I we show the specific functional form ofχ and α for several forms of G(χ) proposed in literature and determine when their perturbations are stable (α ≥ 0) and subluminal (α ≤ 1).
G(χ)χ (w φ ) M 4 α(w φ ) Stability χ M 4 1+w φ 2w φ - w φ w φ ≥ 0 − χ M 4 + χ 2 M 8 1−w φ 1−3w φ 1+w φ 5−3w φ −1 ≤ w φ ≤ 1 − χ M 4 + χ M 4 n 1−w φ 1−(2n−1)w φ 1 n−1 1+w φ (2n+1)−(2n−1)w φ −1 ≤ w φ ≤ 1 for n > 0 − 1 + 2η χ M 4 −η 1+w φ 2 −w φ −1 ≤ w φ ≤ 0 2 χ M 4 n − 1 1 2n 1+w φ 2 1 n − 1 2n−1 w φ 1 − 2n ≤ w φ ≤ 0 for n > 1 2 0 ≤ w φ ≤ 1 − 2n for n < 1 2 − 1 + 2η χ M 4 n 1 2n −η 1+w φ 2 1 n − 1 2n−1 w φ 1 − 2n ≤ w φ ≤ 0 for n > 1 2 0 ≤ w φ ≤ 1 − 2n for n < 1 2 χ M 4 − χ M 4 1 (1−w φ ) 2 1+w φ 2 −1 ≤ w φ ≤ 1 − 1 − 2 χ M 4 β 1+w φ 2[1+(1−2β)w φ ] βw φ β−1+(2β−1)w φ w φ ≤ −1 or w φ ≥ 0 for β < 0 or β > 1 −1 ≤ w φ ≤ 0 for 0 < β < 1 A1 χ M 2 − A2 χ M 2 α A 2 A 1 [1 − (2α − 1)w φ ] 2 1−2α 1+w φ 2α −1 ≤ w φ ≤ 2α − 1 for α > 0 2α − 1 ≤ w φ ≤ −1 for α < 0
III.3. Type C models with constant w φ
Type C models resemble phantom models discussed in section II. They reduce to phantom models when the function N (χ) is constant, that is, n = 0 for the power-law choice of N (χ). Despite the apparent complexity, type C models, in contrast to type A and B models, have a general analytical solution when N (χ) is a power-law. Key equations for general n > 0 and w φ (a) are
V (a) = − [1 − w(a)]M 4n [1 − (2n − 1)w φ (a)] n (n − 1) n−1 3H 2 0 M 2 pl Ω de g(a) 16π 1−n ,(54)dφ da = 3M 2 pl Ω de [1 − (2n − 1)w φ (a)]g(a) 8π(n − 1)a 2 E 2 (a) 1 2 ,(55)α = 1 + w φ 2n + 1 − (2n − 1)w φ ,(56)
for the evolution of the potential, of the scalar field and of the sound speed, respectively. Note in particular that for n = 0 we recover results for the phantom models. The evolution of the scalar field as a function of the scale factor is and the corresponding potential is
φ − φ 0 M pl = − 2 3w φ 3[1 + (1 − 2n)w φ ] 8π(n − 1) sinh −1 Ω de Ω m a − 3w φ 2 − sinh −1 Ω de Ω m ,(57)V (φ) = k sinh − 2(n−1)(1+w φ ) w φ − 3w φ 2 8π(n − 1) 3[1 + (1 − 2n)w φ ] φ M pl ,(58)
with the constant k
k = 3H 2 0 M 2 pl Ω de (1 − w φ ) 16π(1 − n) 3H 2 0 M 2 pl Ω de [1 − (2n − 1)w φ ] 16π(n − 1)M 4 −n Ω m Ω de − (n−1)(1+w φ ) w φ ,(59)
when we choose as we did in the Quintessence case
φ 0 M pl = − 2 3w φ 3[1 + (1 − 2n)w φ ] 8π(n − 1) sinh −1 Ω de Ω m .(60)
These expressions are as expected similar to those for Quintessence models. Given the general form of the potential for type C models, we can expect a similar behaviour to Quintessence models for a ≪ 1, a ≈ 1 and a ≫ 1. In particular, at early times
V E (φ) ∝ φ 2(1−n)(1+w φ ) w φ
and at late times we recover the usual exponential behaviour V L (φ) ∝ exp (φ/M pl ) typical of the minimally coupled models.
For a ≈ 1, the scalar field is
φ − φ 0 M pl = 3Ω de [1 − (2n − 1)w φ ] 8π(n − 1) a − 1 − 1 4 (2 + 3Ω m w φ ) (a − 1) 2 ,(61)
and the corresponding potential can be once again written in the general approximated form of (19), with coefficients:
A = 3Ω de (1 − w φ ) 16π(1 − n) 3H 2 0 M 2 pl Ω de [1 + (1 − 2n)w φ ] 16π(n − 1)M 4 −n 4 + 3Ω m w φ 2 + 3Ω m w φ −3(1−n)(1+w φ ) , B = 4 (4 + 3Ω m w φ ) 2 , C = −4 2 + 3Ω m w φ (4 + 3Ω m w φ ) 2 8π(n − 1) 3Ω de [1 + (1 − 2n)w φ ] , D = −3(1 − n)(1 + w φ ) .(62)
As with Quintessence, Type A and B models, we show a comparison between the approximation and the exact solutions in Figure 4. The picture is similar to the previous ones but the range of scale factor where the approximation is good is more restricted, 0.8 a 1.2. This is due to the fact that for type C models, the potential has a stronger dependence on the scale factor with respect to the other models, given by the 1 − n power of g(a) in (54).
IV. POTENTIAL FOR NON-CONSTANT w φ
In the previous sections we have shown that a potential of the form (19) is a good approximation to that for scalar field models with constant w φ -for both minimal and non-minimal kinetic terms -for some choice of the parameters A, B, C and D over a range of the scale factors around a ≈ 1.
In Figure 5 we have varied the parameters around their values for a specific constant w φ = −0.9 model with a minimal kinetic term. We do this by keeping three of the parameters fixed and vary the fourth one by ±30% while requiring that φ 0 coincides with the exact solution. We see that a wide range of behaviour of the actual w(a) can be achieved in these models suggesting that this parametrization of the potential could be used as a proxy for a significant range of models, albeit with some restrictions.
We can attempt to generalise the set of coefficients of (19) to models with a non-constant equation of state. It is not possible to adapt the exact method used for constant w φ because usually there is not a general expression for g(a). However, we have been able to make some progress by realising that (1 + αx) β ≈ 1 + αβx for x ≪ 1 and performing an expansion around φ ≈ φ 0 in our set up. Expanding (19) to first order and matching the coefficients with a similar expansion derived from V (a) and φ(a), we can determine a new set of parameters A-D. As before, they will depend on the background cosmological parameters, Ω m , w φ and in this case, also on its derivative with respect to the scale factor, w ′ φ , evaluated at a = 1. In the appendix we report the explicit expression for the four coefficients for minimally coupled, type A and type C models, respectively. To understand them note that the relation between the scale factor and the scalar field is now, for Quintessence models,
a = 1 + 2 2 + 3Ω m w φ (1) − w ′ φ (1) 1+w φ (1) 1 − 1 − 2 + 3Ω m w φ (1) − w ′ φ (1) 1 + w φ (1) 8ηπ 3Ω de [1 + w φ (1)] φ − φ 0 M pl .(63)
Also note that for w ′ φ (1) = 0, we recover the result in (18) for Quintessence models. Similar expressions hold for type A and type C models.
To see how well this new parametrization performs we use the Chevallier-Polarski-Linder (CPL) parametrization [14,15],
w φ (a) = w 0 + w a (1 − a) ,(64)
where w 0 and w a are constants. In Figure 6 we show the comparison between the true and the approximated equation of state evaluated from the potential in (19) with the set of coefficients given in the appendix for minimally coupled models and type A k-essence models. We use two different sets of coefficients (w 0 , w a ): one with a very gentle slope, w φ (a) = −0.9 + 0.02(1 − a), and one with a more pronounced variation, w φ (a) = −0.9 + 0.15(1 − a). At early times we find a better agreement for models not differing too much from a constant equation of state, while at late times, the agreement is better for models with w a = 0.15.
As it can be seen in Figure 6, this is due to the fact that φ ′ shows fluctuations around the true value. Note that if we limit ourselves to a sub-percent agreement between the true and the reconstructed equation of state, then the agreement is much more limited with respect to the case of constant w φ . This is because we poorly approximate the function g(a): for a CPL model, it consists of two elements: a power-law and an exponential and we only include the power-law. When the exponential behaviour dominates, our proposed potential is a less good fit to the true behaviour. Note also that the range of agreement is similar for both Quintessence and type A models. Deviations in type A models are suppressed with respect to Quintessence models thanks to a higher value of n (2 in the example). From a quantitative point of view, for w a = 0.02 (w a = 0.15), for quintessence models we reach a 1% agreement for 0.5 a 1.4 (0.5 a 1.7), while for type A models we have 0.5 a 1.4 and 0.7 a 1.7, respectively. This is similar to what found before for a model with constant w φ = −0.9. One caveat to our approach is that the only knowledge of the evolution of the equation of state is given by its value and its time derivative, both evaluated at a = 1. Therefore nothing is known about its general time evolution and as consequence, nothing is known about the functional form of g(a). This implies that our approach would work well for models with a monotonic equation of state (and hence a monotonic g(a)), but we expect it to fail and not be a good representation for the true potential for oscillating dark energy models, [see e.g. 77, for a recent study of their properties and comparison with observations].
V. CONCLUSIONS
Scalar fields are an important field of research in cosmology and are one of the most studied candidates used to explain and describe the accelerated expansion of the Universe. In this work, we consider two main classes of models: minimally coupled models (both quintessence and phantom) and k-essence models. For this second class, we specialise the Lagrangian to assume three particular functional forms, dubbed type A, type B and type C models. In each case, we have shown that specifying the scalar field potential V (φ), one can determine the evolution of the scalar field and the corresponding equation of state w φ (a).
This is true generally but in order to make it clear, we have assumed the equation of state to be known and we calculated explicitly the time evolution of the scalar field and of the potential in some cases. We showed that it is possible to obtain an exact analytic solution for minimally coupled and the type C models with constant equation of state. This is not possible for more general k-essence models or for models with a time-varying w φ (a), but we have solutions for φ(a) as definite integrals and these can be used to establish the potentials, V (φ) numerically.
We have also derived useful approximate forms of the potential which are valid in different epochs, corresponding to the domination of one cosmic fluid. In particular we deduce the form of the potential at early times (a ≪ 1, corresponding to the matter dominated epoch) and at late times (a ≫ 1, corresponding to the scalar field dominated regime), showing that in general the potential is often very well approximated by a power-law.
From an observational point of view, the most important regime to understand the potential is around a ≈ 1. Assuming initially a constant equation of state w φ , we showed that the scalar field potential can be approximated by the expression given in (19). This expression depends only on four parameters and with the appropriate choice of coefficients can cover all the classes of models studied in this work. In section IV we discussed how this expression might be applied to dynamical dark energy models, by appropriately choosing a new set of parameters which reduces to the correct expression in the limit of constant w φ . Note that this can not be done for type B models, since our formalism only works for constant equations of state.
In some respect our approach is similar to the work of [32]. To derive our expression in (19), we performed a Taylor expansion of the scalar field evolution, so the same critique could be applied: where to stop the series? Our approximate potential, for a ≈ 1 (φ ≈ φ 0 ) can be expanded in powers of φ−φ 0 leading to the same form of the potential proposed by [32]. In contrast to that work, our proposed potential has well motivated coefficients and in the regime of interest it would be possible to map the V i coefficients of (1) in terms of our four parameters. For example, at zeroth order, we can write V 0 in (1) as
V 0 = A(1 − √ B)H 2 0 M 2 pl . w ′ φ (1) 1+w φ (1) 4 + 3Ω m w φ (1) − w ′ φ (1) 1+w φ (1) 2 8ηπ 3Ω de [1 + w φ (1)] , D = − w ′ φ (1) 1 − w φ (1) + 3[1 + w φ (1)] .(65)
For models of type A we find
A = 3Ω de [1 − (2n − 1)w φ (1)] 16πn 3 + n − 3(Ω de n − 1)w φ (1) − w ′ φ (1) 1+w φ (1) 3 − n − 3(Ω de n − 1)w φ (1) − w ′ φ (1) 1+w φ (1) − (2n−1)w ′ φ (1) 1−(2n−1)w φ (1) +3[1+w φ (1)] , B = 4n 2 3 + n − 3(Ω de n − 1)w φ (1) − w ′ φ (1) 1+w φ (1) 2 , C = −2 √ 2n 3 − n − 3(Ω de n − 1)w φ (1) − w ′ φ (1) 1+w φ (1) 3 + n − 3(Ω de n − 1)w φ (1) − w ′ φ (1) 1+w φ (1) 2 H 0 M pl M 2 16πnM 4 3H 2 0 M 2 pl Ω de (1 + w φ ) 1 2n , D = − (2n − 1)w ′ φ (1) 1 − (2n − 1)w φ (1) + 3[1 + w φ (1)] .(66)
For models of type C we find
A = 3Ω de (1 − w φ ) 16π(1 − n) 3H 2 0 M 2 pl Ω de [1 + (1 − 2n)w φ ] 16π(n − 1)M 4 −n × 4 + 3Ω m w φ − (1−2n)w ′ φ (1) 1+(1−2n)w φ (1) 2 + 3Ω m w φ − (1−2n)w ′ φ (1) 1+(1−2n)w φ (1) −(1−n) [1+2n+(1−2n)w φ (1)]w ′ φ (1) [1+(1−2n)w φ (1)][1−w φ (1)] +3[1+w φ (1)] , B = 4 4 + 3Ω m w φ − (1−2n)w ′ φ (1) 1+(1−2n)w φ (1) 2 , C = −4 2 + 3Ω m w φ − (1−2n)w ′ φ (1) 1+(1−2n)w φ (1) 4 + 3Ω m w φ − (1−2n)w ′ φ (1) 1+(1−2n)w φ (1) 2 , D = −(1 − n) [1 + 2n + (1 − 2n)w φ (1)]w ′ φ (1) [1 + (1 − 2n)w φ (1)][1 − w φ (1)] + 3[1 + w φ (1)] .
FIG. 1 .
1Top left panel: Comparison between the exact solution for a constant equation of state for the potential (solid line) and its approximate expression (dashed line), for a ≈ 1. Different colours refer to different values of w φ . From top to bottom: the black, red, blue, yellow and violet lines correspond to w φ = −0.7, −0.8, −0.9, −0.95 and −0.99, respectively. Top right panel: Equation of state for the approximated potential of (19) for w φ = −0.9. The subscripts a and e represent the approximated (blue dashed line) and the exact (black solid line) solutions, respectively. Black horizontal dashed lines show differences of 1% with respect to the exact value. Bottom panel: Comparison between the exact solution for the scalar field with a constant equation of state w φ = −0.9 (solid line) and the approximate expression (dashed line), for a ≈ 1.
FIG. 2 .
2Top left panel: Comparison between the full solution, generated numerically, and the approximation for a ≈ 1 for the scalar field potential for type A model with n = 2. Solid lines represent the full solution, dashed lines the approximate solution. Different colours show different equations of state, labelled as in Figure 1. Top right panel: scalar field potential for w φ = −0.9 for different values of n. From top to bottom we show n = 1, 2, 3, 4. We see that as n increases, the shape of the potential quickly asymptotes to that of n → ∞. Bottom left (right) panel: Scalar field (potential) for the approximated solution compared with the exact expression (Equation 34 together with Equation 36) for w φ = −0.9.
FIG. 3 .
3Left panel: Approximation of the scalar field potential for a ≈ 1 for different constant equation-of-state parameters w φ . Solid lines represent the exact numerical solution, while the dashed line show the approximated solution. Colours are as in Figure 1. Middle (right) panel: scalar field (potential) for the approximated solution compared with the exact expression (41) together with (46) for w φ = −0.9. In all the panels we assume G(χ) = − χ M 4 + χ 2 M 8 .
FIG. 4 .
4Top Left panel: Comparison between the exact solution for constant equation of state for the absolute value of the potential (solid line) and its approximate expression, for a ≈ 1 (dashed line) for type C models. Line styles and colours are as in Figure 1. Top right (middle) panel: scalar field (absolute value of the potential) for the approximated solution compared with the exact expression in (57) [(58)] for a model with w φ = −0.9. The subscripts a and e represent the approximated (blue dashed line) and the exact (black solid line) solutions, respectively.
FIG. 5 .
5Effect of the variations of the parameters A, B, C and D on the equation of state w φ . In each panel, three of the coefficients are held constant at their exact value for w φ = −0.9 while the fourth one is varied by ±30% -left negative and right positive. The black curve shows the equation of state using the exact values of the coefficients as represented in Figure 1. Effects of variations of A, B, C and D are shown with the red dashed, blue short dashed, brown dotted and green dot-dashed curve, respectively.
-
FIG. 6 .
6Left (right) panel: Equation of state for the approximated potential of (19) for a Quintessence (type A) model described by a CPL equation of state. Red (blue) curve represents a model with w φ (a) = −0.9 + 0.02(1 − a) (w φ (a) = −0.9 + 0.15(1 − a)). Dotted lines represent a 1% difference from the true equation of state. Solid (dashed) lines show the true (approximated) equation of state.
TABLE I .
IDependence ofχ and α on the constant equation-of-state parameter w φ and stability conditions for the model.
http://www.euclid-ec.org/ 2 http://www.lsst.org 3 https://www.skatelescope.org/
ACKNOWLEDGEMENTSWe would like to thank Stefano Camera and Robert Reischke for useful comments. The work for this article was funded by an STFC postdoctoral fellowship.APPENDIX: COEFFICIENTS FOR DYNAMICAL DARK ENERGY MODELSIn this section we write explicitly the generalization of the set of coefficients A-D for non-constant equations of state using the approach discussed in the text. For minimally coupled models we find
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| []
|
[
"Applying clock comparison methods to pulsar timing observations",
"Applying clock comparison methods to pulsar timing observations"
]
| [
"Siyuan Chen \nStation de Radioastronomie de Nançay\nObservatoire de Paris\nPSL University\nCNRS\nUniversité d'Orléans\n18330NançayFrance\n\nFEMTO-ST\nDepartment of Time and Frequency\nUBFC and CNRS\n25030BesançonFrance\n\nLaboratoire de Physique et Chimie de l'Environnement et de l'Espace\nUniversité d'Orléans\nCNRS\nLPC2E UMR7328, 45071OrléansFrance\n",
"François Vernotte \nFEMTO-ST\nDepartment of Time and Frequency\nUBFC and CNRS\n25030BesançonFrance\n\nObservatoire des Sciences de l'Univers THETA\nUBFC and CNRS\n20510BesançonFrance\n",
"Enrico Rubiola \nFEMTO-ST\nDepartment of Time and Frequency\nUBFC and CNRS\n25030BesançonFrance\n\nObservatoire des Sciences de l'Univers THETA\nUBFC and CNRS\n20510BesançonFrance\n\nDivsion of Quantum Metrology and Nanotechnology\nIstituto Nazionale di Ricerca Metrologica INRiM\n10135TorinoItaly\n"
]
| [
"Station de Radioastronomie de Nançay\nObservatoire de Paris\nPSL University\nCNRS\nUniversité d'Orléans\n18330NançayFrance",
"FEMTO-ST\nDepartment of Time and Frequency\nUBFC and CNRS\n25030BesançonFrance",
"Laboratoire de Physique et Chimie de l'Environnement et de l'Espace\nUniversité d'Orléans\nCNRS\nLPC2E UMR7328, 45071OrléansFrance",
"FEMTO-ST\nDepartment of Time and Frequency\nUBFC and CNRS\n25030BesançonFrance",
"Observatoire des Sciences de l'Univers THETA\nUBFC and CNRS\n20510BesançonFrance",
"FEMTO-ST\nDepartment of Time and Frequency\nUBFC and CNRS\n25030BesançonFrance",
"Observatoire des Sciences de l'Univers THETA\nUBFC and CNRS\n20510BesançonFrance",
"Divsion of Quantum Metrology and Nanotechnology\nIstituto Nazionale di Ricerca Metrologica INRiM\n10135TorinoItaly"
]
| [
"MNRAS"
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| Frequency metrology outperforms any other branch of metrology in accuracy (parts in 10 −16 ) and small fluctuations (< 10 −17 ). In turn, among celestial bodies, the rotation speed of millisecond pulsars (MSP) is by far the most stable (< 10 −18 ). Therefore, the precise measurement of the time of arrival (TOA) of pulsar signals is expected to disclose information about cosmological phenomena, and to enlarge our astrophysical knowledge. Related to this topic, Pulsar Timing Array (PTA) projects have been developed and operated for the last decades. The TOAs from a pulsar can be affected by local emission and environmental effects, in the direction of the propagation through the interstellar medium or universally by gravitational waves from super massive black hole binaries. These effects (signals) can manifest as a low-frequency fluctuation over time, phenomenologically similar to a red noise. While the remaining pulsar intrinsic and instrumental background (noise) are white. This article focuses on the frequency metrology of pulsars. From our standpoint, the pulsar is an accurate clock, to be measured simultaneously with several telescopes in order to reject the uncorrelated white noise. We apply the modern statistical methods of time-and-frequency metrology to simulated pulsar data, and we show the detection limit of the correlated red noise signal between telescopes. | 10.1093/mnras/stab742 | [
"https://arxiv.org/pdf/2011.01912v3.pdf"
]
| 226,237,637 | 2011.01912 | 63d450b5ef3036627776d8d1928e71ccae1c7660 |
Applying clock comparison methods to pulsar timing observations
2020
Siyuan Chen
Station de Radioastronomie de Nançay
Observatoire de Paris
PSL University
CNRS
Université d'Orléans
18330NançayFrance
FEMTO-ST
Department of Time and Frequency
UBFC and CNRS
25030BesançonFrance
Laboratoire de Physique et Chimie de l'Environnement et de l'Espace
Université d'Orléans
CNRS
LPC2E UMR7328, 45071OrléansFrance
François Vernotte
FEMTO-ST
Department of Time and Frequency
UBFC and CNRS
25030BesançonFrance
Observatoire des Sciences de l'Univers THETA
UBFC and CNRS
20510BesançonFrance
Enrico Rubiola
FEMTO-ST
Department of Time and Frequency
UBFC and CNRS
25030BesançonFrance
Observatoire des Sciences de l'Univers THETA
UBFC and CNRS
20510BesançonFrance
Divsion of Quantum Metrology and Nanotechnology
Istituto Nazionale di Ricerca Metrologica INRiM
10135TorinoItaly
Applying clock comparison methods to pulsar timing observations
MNRAS
0002020Preprint 11 March 2021 Compiled using MNRAS L A T E X style file v3.0 Accepted . . . Received . . . ; in original form . . .pulsars: general -methods: data analysis
Frequency metrology outperforms any other branch of metrology in accuracy (parts in 10 −16 ) and small fluctuations (< 10 −17 ). In turn, among celestial bodies, the rotation speed of millisecond pulsars (MSP) is by far the most stable (< 10 −18 ). Therefore, the precise measurement of the time of arrival (TOA) of pulsar signals is expected to disclose information about cosmological phenomena, and to enlarge our astrophysical knowledge. Related to this topic, Pulsar Timing Array (PTA) projects have been developed and operated for the last decades. The TOAs from a pulsar can be affected by local emission and environmental effects, in the direction of the propagation through the interstellar medium or universally by gravitational waves from super massive black hole binaries. These effects (signals) can manifest as a low-frequency fluctuation over time, phenomenologically similar to a red noise. While the remaining pulsar intrinsic and instrumental background (noise) are white. This article focuses on the frequency metrology of pulsars. From our standpoint, the pulsar is an accurate clock, to be measured simultaneously with several telescopes in order to reject the uncorrelated white noise. We apply the modern statistical methods of time-and-frequency metrology to simulated pulsar data, and we show the detection limit of the correlated red noise signal between telescopes.
INTRODUCTION
Millisecond pulsars (MSP) are considered extremely stable astronomical clocks because of their high energy and momentum in a small size (Verbiest et al. 2009), albeit the observations can be affected by large observational white noise. Such noise can be due to the low signal to noise ratio of the MSP observations, and depends on the radio-telescope (RT). On the other hand, red noise could originate from the MSP, the propagation through the interstellar medium or from gravitational waves (GW) on the line of sight, hence it is common to all the RTs (Hellings & Downs 1983;Jenet & Romano 2015). The characterization of the spectral signature of GWs is a major scientific challenge because it is expected to disclose information about the astrophysical sources. As pulsars have been precisely timed for decades with up to weekly cadence, the frequency range that can be probed by Pulsar Timing Array (PTA) (Foster & Backer 1990) projects is f ∈ [10 −9 , 10 −6 ] Hz, which is a window out of reach for the LIGO/VIRGO interferometers and at the edge of the LISA band. The most interesting and likely to be detected source is a cosmic population of super massive black hole E-mail: [email protected] binaries (Taylor et al. (2016) and Perrodin & Sesana (2018) for a recent review), whose interaction with the MSP signals introduces a correlated red noise in the Time of arrival (TOA) series, with a phase power spectral density (PSD) proportional to 1/ f 13/3 (for a circular population, see eg. Phinney (2001); Chen et al. (2017)). The choice of the algorithm is therefore of paramount importance to disentangle the pulsar specific small red noise (signal) from the effects of GWs in the presence of large white noise (background) in the shortest observation time. In this regard, PTAs (Desvignes et al. 2016;Alam et al. 2020;Kerr et al. 2020;Perera et al. 2019) enable the simultaneous measurement of the same pulsar with several RTs, especially the Large European Array for Pulsars (LEAP) project (Bassa et al. 2016).
The extraction of small signals from noise exploiting simultaneous measurements is a well-known method for the measurement of oscillators and atomic clocks. Specific tools have been developed over more than 50 years using the PSD and wavelet variances (Barnes et al. 1971;Rutman 1978;Gray & Allan 1974;Allan & Barnes 1981). Among them, the cross-spectrum method (Rubiola & Vernotte 2010) deserves separate mention in view of our application. Under certain assumptions, this method rejects the background noise (uncorrelated), and converges to the oscillator noise even if it is significantly lower than the background. Wavelet variances are commonly used in the time domain, the most known of which is the 2-sample Allan variance (AVAR), after the pioneering work of Allan and Barnes (Allan 1966). Such variances enable to distinguish different types of noise defined by their exponent in the PSD, e.g. f 0 for white phase, 1/ f for flicker phase modulation, 1/ f 2 for white frequency modulation, 1/ f 3 for flicker frequency modulation, 1/ f 4 for random walk frequency modulation noise, etc. The parabolic variance (PVAR) (Benkler et al. 2015;Vernotte et al. 2016) extends this concept, and exhibits (i) the highest rejection of white noise, and (ii) the efficient detection red noise underneath the background with the shortest data record. Moreover, the wavelet covariance (Fest et al. 1983;Lantz et al. 2019), i.e., Allan covariance (ACOV) or parabolic covariance (PCOV), enables the rejection of the background using two (or more) uncorrelated instruments.
This article is intended to port the time-and-frequency metrology methods to pulsar astronomy. In this respect, the pulsar is the clock under test, observed simultaneously with two or more RTs playing the role of phase meters. We compare the results of different methods applied to simulated time series representing the TOAs of millisecond pulsars. In section 1 we summarize the concept of pulsar timing as well as the spectral and variance methods from the time-and-frequency metrology. Section 2 describes the simulation process of the TOA time series, followed by the statistical methods in section 3. Results and the conclusions are presented in sections 4 and 5 respectively.
Pulsar timing
Pulsars are spinning neutron stars that emit radio waves from the region above their magnetic poles. If these regions are misaligned with the rotation axis and happen to point towards Earth as they sweep space, we receive one radio pulse each rotation. The most stable pulsars have rotation period of milliseconds and are stable on a very long timescale. Over the decades that they have been timed, very small variations have been detected, which translate into low frequency red noise in the Fourier domain. A detailed review about pulsars can be found in Lorimer & Kramer (2012).
In general, the time of arrival (TOA) of the pulses can be described precisely by the following timing model (Hobbs et al. 2006)
t obs = t model + t red + t white ,(1)
where t obs is the observed and t model is the predicted TOA considering all known pulsar properties and propagation effects. For this study we also include the effects of dispersion measure variation over time into the perfect model. Ideally, we should be able to precisely predict the TOAs if all model parameters are perfectly known and there are no unknown sources of noise left. However, this is not the case in practice. The difference between the observed and model observation forms the residual series x = t red + t white . Therefore, we are interested in both white and red noise components. The pulse residual series can be transformed into Fourier frequency space. The noise components are then described by a PSD. The PTA noise description can be found in Lentati et al. (2016); Caballero et al. (2016); Arzoumanian et al. (2016). Following the notation of the latter, the white noise is described by two parameters E k and Q k
S w = (E k W ) 2 + Q 2 k ,(2)
where W is the initial estimate of the white coming from the radiome- ter noise and template matching error, E k and Q k are the so-called EFAC and EQUAD parameters respectively. Note that S w is constant over the whole frequency range.
The red noise is described by a power law with two parameters
S r ( f ) = A 2 PTA f −γ PTA ,(3)
where A PTA is the amplitude and γ PTA is the spectral index of the power law. The total noise PSD S( f ) becomes (Vernotte & Zalamansky 1999)
S( f ) = S r ( f ) + S w .(4)
Power and Cross spectrum
In time and frequency metrology, the cross spectrum is the most often used method to measure the phase noise of oscillators. The method relies on the simultaneous measurement with two separate instruments, assuming that the background noise is statistically independent. The same hypothesis holds for pulsar observations from different radio telescopes. The telescopes are obviously independent, and sufficiently far from each other for the tropospheric/ionospheric noise to be statistically independent. We summarize the main results from our tutorial on the cross spectrum method (Rubiola & Vernotte 2010). Given a residual series x from an instrument, we can write its Fourier transform as X = ℜ(X) + i ℑ(X). The one-sided PSD can be calculated as
S x ( f ) = 2 T a XX * = 2 T a |ℜ(X)| 2 + |ℑ(X)| 2 for f > 0, (5)
where T a is the acquisition time, and the factor 2 is due to energy conservation after surpressing the negative frequencies.
Measuring a clock (pulsar) signal c simultaneously with two instruments (telescopes), we get two residual series x and y, each containing the white background noise of the measurement added to c. The cross spectrum (CS)
S yx = 2 T a Y X *(6)
converges to S c = 2 T a CC * . Notice that, after compensating for the differential path, c goes only into ℜ(S yx ), while the ℑ(S yx ) contains only the background noise of the observation. After Rubiola & Vernotte (2010), the fastest, unbiased estimator is
S c = 2 T a ℜ(S yx ) = 2 T a ℜ(X) ℜ(Y ) + ℑ(X) ℑ(Y )(7)
and can give negative values. We average over all the CSs coming from the different RT pairs to produce one simultaneous observation series for the analysis. Figure 1 shows an example of a comparison between the different spectra. The PSD can be computed on the averaged residual series (ARS) or on each residual series individually, and then averaged (PSD). As the ARS is very similar, but slightly worse than the CS, we opt to focus the analysis on the CS method, whose results should be comparable to those from the ARS. The CS is better at rejecting uncorrelated white noise and produces a lower level than the average PSD, which serves as a good reference.
We also investigated the effects of the choice of the window function on the Fourier transform by comparing the sinc function used above with a Blackman window. Although there is spectral leakage of the red noise, the amount is only noticable in the lowest frequencies and is within the uncertainty of the PSD/CS value. We found minimal effects on our analysis. In fact, as the spectral value decreases, the recovered red noise parameters provide a worse match to the injected values. We also found similar results with a prewhitening / postdarkening procedure. Therefore, we show only the results obtained with the above equations.
Variances
Different variances have been proposed to describe the amount of variation in a given residual series x with a total time span T for time steps τ. The two prominent ones are the Allan (Allan 1966) and more recently the parabolic variance (Vernotte et al. 2016), which we summarize here.
The AVAR is a simple computation comparing three observations separated by m time steps
AVAR(τ) = 1 2Mτ 2 M−1 ∑ i=0 (−x i + 2x i+m − x i+2m ) 2 ,(8)
where M = N −2m +1 is related to the total number N of data points in the residual series x. Compared to similar variances, AVAR is the most efficient at estimating for the largest τ = T /2. By contrast, the Modified Allan variance is more suitable to the analysis of shortterm fluctuations (Allan & Barnes 1981). The parabolic variance combines the original long-term computability as well as a good response to short timescales
PVAR(τ) = 72 Mm 4 τ 2 M−1 ∑ i=0 m−1 ∑ k=0 ( m − 1 − 2k 2 )(x i+k − x i+m+k ) 2 ,(9)where M = N − 2m + 2.
Similarly to the CS method, we can also use two indepedent residual series x and y to compute the ACOV
ACOV(τ) = 1 2Mτ 2 M−1 ∑ i=0 (−x i + 2x i+m − x i+2m ) (−y i + 2y i+m − y i+2m )(10)
and the PCOV
PCOV(τ) = 72 Mm 4 τ 2 M−1 ∑ i=0 m−1 ∑ k=0 ( m − 1 − 2k 2 )(x i+k − x i+m+k ) ( m − 1 − 2k 2 )(y i+k − y i+m+k ) .(11)
A comparison of the two different (co)variances can be found in figure 2. In general, the variance plot has two prominent features: (i) at lower time steps, the variance is dominated by white noise, which leads to a power law decrease with fixed spectral index, (ii) at higher time steps, red noise start to appear, which shows up as a flattening and then rising of the variance, depending on the amplitude and spectral index of the red noise. The variance curve can be parametrized as
VAR = A 2 VAR τ γ VAR + B 2 VAR τ β VAR .(12)
The relation between the PTA parameters EFAC, EQUAD, γ PTA and A PTA and the variance parameters is not trivial. As the white noise follows a fixed theoretical spectral index for a given variance β VAR = −2/ − 3 for AVAR and PVAR respectively, we can compare the overall amplitude of the variance white noise with S w from the PTA white noise in (2)
S w = 2B 2 VAR /3 for AVAR B 2 VAR /(6τ 0 ) for PVAR,(13)
where τ 0 is the sampling time.
The two red noise spectral indices can be related via
γ PTA = γ VAR + 3.(14)
The red noise amplitudes follow analytic relations for given integer spectral indices (see table 1 in Vernotte et al. 2016). Under the condition that γ PTA ∈ [1.5, 4.5] it is also possible to find an analytic relation between A VAR and A PTA for non-integer γ PTA , see also Vernotte et al. (2020),
A 2 VAR = A 2 PTA 12π 2 4π 2 yr 3+γ PTA A (γ PTA ),(15)
where A (γ PTA ) is a function that relates to the coefficients of the variance. Using the same approach as Walter (1994) we find the folowing relationships for AVAR and PVAR
A (γ PTA ) = (4 γ PTA −2 − 2 γ PTA −1 ) π γ PTA −3 Γ(1 − γ PTA ) sin(π − πγ PTA /2) for AVAR −4 γ PTA 9π γ PTA −3 (2 2−γ PTA (−1 − γ PTA ) + 2 + 3γ PTA − γ 2 PTA ) Γ(−3 − γ PTA ) sin(π − πγ PTA /2) for PVAR.(16)
SIMULATED DATASETS
Clock comparison methods require several time series of the same pulsar observed simultaneously with different telescopes over a long time span. The LEAP project uses the 5 major radio telescopes in Europe: Effelsberg in Germany, Lovell in UK, Westerbork in the Netherlands, Nançay in France and Sardinia in Italy as an interferometer with monthly simultaneous observations since 2012 (Bassa et al. 2016;Smits et al. 2017). Thus we simulate 100 realizations of 5 TOA series over a 4000day long observing campaign with a cadence of 30 days, this is of the order of the nominal LEAP observations up to today. They have been created using the libstempo package (Vallisneri 2015). For simplicity, we assume that all 5 RTs have comparable instrumental white noise, and give to all residuals the initial uncertainty W = 500 ns. As the two white noise parameters are degenerate, we will only focus on the dominant term, ie. EFAC, injecting different realizations of white noise with (EFAC = 1.05, EQUAD = 0).
We also simulate the same residual series as it would be observed by LEAP under ideal conditions. In the LEAP statistical limit (LSL) of the coherently added mode, the total S/N of the LEAP observation is the sum of the individual RTs. Since we assume that the 5 RTs are identical, the LEAP observations have a 5 times higher S/N with the TOAs having a white noise level that is 5 times lower as S/N ∝ 1/W . This also translates into an EFAC injection of 1.05/5 = 0.21, while EQUAD remains at 0.
In total, we consider three cases:
(i) white noise case: no red noise signal injected (ii) boundary case: (log 10 A PTA = −14, γ PTA = 3) (iii) red noise case: (log 10 A PTA = −13, γ PTA = 3)
For each case we simulate 100 sets of 5 residual series with different red noise realization between each set. For a given set, the red noise realization is the same amongst the 5 residual series, but the white noise realizations differ. An example of a set of 5 simulated residual series can be seen in figure 3. As we are focusing only on the noise properties, we fix the timing model at its injected values. This means that the residual series are fully determined by the injected white and red noise. In practice, there will be covariances between the timing model and the noise parameters. Here we only show the theoretically optimal results as a proof-of-concept.
METHODS
The standard PTA analysis is done via the enterprise package (Ellis et al. 2017), details of the analysis can be found in Arzoumanian et al. (2016Arzoumanian et al. ( , 2018. We use all 5 individual RT residual series for the PTA analysis comparison. For the LSL analysis we only have one residual series, but with a lower white noise level.
We compute the PSD and CS through equations (5) and (7), figure 1, and compare them directly to equation (4) to get constraints on the noise parameters. The PSD can be averaged over the 5 residual series, whilst the CS can be computed from the average of the 10 different combinations of 2 out of the 5 residual series. The Allan and parabolic Variances are calculated through equations (8) and (9), see figure 2, and matched to the parametrization in equation (12) with the use of equations (13), (14) and (15). Unlike the PSD, we use the averaged residual series to compute the variances.
We use Bayesian statistics and a nested sampling algorithm, implemented in the cpnest package (Del Pozzo & Veitch 2015), to look for the posterior constraints on the white noise properties as well as the spectral index and amplitude of the red noise parameters.
A likelihood function can be derived from a weighted least squares fitting procedure
L = − 1 2σ 2 i D − F (φ ) 2 ,(17)
where σ 2 i are the weights, D are the computed data points and F (φ ) are the corresponding parametrizations with the model parameters φ = (EFAC, log 10 EQUAD, γ PTA , log 10 A PTA ).
For the spectral analysis we work in the logarithmic space, where D corresponds to the computed log 10 PSD/CS values. From a large Monte-Carlo simulation we infer the uncertainties to be σ i = 0.2136 across all frequencies in the Gaussian approximation using only the positive estimates, see appendix A.
For the (co)variance analysis, where D are the computed (co)variances, we compute the weights σ 2 i from the effective degrees of freedom for each time step. A more detailed description can be found in appendix B.
The choice of priors follows closely the standard PTA analysis (eg. Lentati et al. (2015)) wherever possible with uniform distributions in: EFAC ∈ [0.1, 1.5], log 10 EQUAD ∈ [−10, −5], γ PTA ∈ [0, 7] and log 10 A PTA ∈ [−20, −10]. The only exception are the boundaries of γ PTA ∈ [1.5, 4.5] for the variance analysis. This is due to the limited range of validity of equation (16) (Vernotte et al. 2020).
RESULTS
To ensure that our results are statistically robust, each of the three cases is based on 100 different realization sets and doing the analysis using all the clock comparison methods and the standard PTA/LSL comparisons. All following posterior distributions and recoveries are based on the addition of all 100 individual posterior distributions, unless stated otherwise. They should thus be representative of the constraints from the analyses of a single simulated set of 5 residual series.
White noise
We first explore the performance of the clock methods in the case of pure white noise in the residual series, and compare it to the traditional PTA/LSL analysis. Figure 4 shows the PTA noise parameter marginalized 1D and 2D posterior distributions in a corner plot. As there is no red noise present, A PTA and γ PTA return very uninformative posteriors for all methods. γ PTA simply returns a nearly flat uniform prior. The quantity A PTA returns a posterior distribution that is flat for low amplitudes, but has a cutoff at an higher amplitude. This can be interpreted as a upper limit case, as red noise with a higher amplitude should have affected the residual series. Similarly, as there is no EQUAD injected, all analyses also show upper limits.
The only parameter of interest in the case of white noise is EFAC. As expected, both the PSD and PTA methods return distributions around the injected value, with the PTA distribution being more constraining than the PSD. The CS method does not seem to lower the white noise to the same level as the LSL, see top panel of figure 4. However, this is misleading, as the white noise is constrained from the positive CS values only. In fact, the CS method lowers the white noise to almost zero and could perform similarly or better to the LSL residuals by also taking the negative CS values into account. This would, however, come at the cost of a decreased performance in the recovery of the red noise parameters. In the middle panel of figure 4, both AVAR and PVAR show similar, but worse, contraints than the PTA analysis, where AVAR is closer to the injected value and PVAR has a small bias towards lower values. The bottom panel shows that the covariances reject most of the white noise, pushing the EFAC towards to lower end of the allowed values. This effect is intended as the covariances 'try' to find a correlated signal, but as there is only uncorrelated noise a upper limit on EFAC is placed.
However, we track the evolution of the performance of the different methods to recover white noise as the amount of red noise gradually increases as we move through the 3 simulated cases. The main result in figure 5 is that as the red noise increases, all methods seem to progressively perform worse in constraining the white noise. For the spectral and variance analysis, the posterior distributions progressively widen. Particularly in the strong red noise case, the AVAR and PVAR analysis are completely unable to recover the injected white noise (bottom middle panel). For the covariance analysis, the EFAC upper limit gradually increases, most notable in the strong red noise case.
A different visual aid is given in figure 6, which shows the constraints on EFAC from the individual analyses of a subset of 10 sets of realizations.
Boundary
A summary of the constraints on the PTA parameters from the standard PTA/LSL analysis and the clock comparison methods is given in table 1. The posterior distributions for all analysis methods with the recovered spectra and variances can be found in figure 7. Examples from analyses of a single simulation set are shown in figure 8. Despite no injection of a EQUAD value, we still perform the all analyses with all 4 PTA noise parameters. However, the recovered EQUAD posterior distributions are all upper limits as in the white noise case above. Thus, we will omit the EQUAD parameter from our discussion from here on, as the focus will be on the red noise parameters.
Spectral analysis
The comparison of the marginalized 1D and 2D posterior distributions of the PTA noise parameter for the boundary case from the spectral analysis can be found in the top left corner plot of figure 7. The middle and bottom left panels in figure 7 show the injected red and white noise along with the recovered PSD and CS respectively. The red noise signal is not well detectable for the PSD as only the lowest frequency bin in the left middle panel of figure 7 shows signs of deviating away from white noise. There is a small hint of the red noise. Which is sufficient to put loose constraints on the spectral index γ PTA and amplitude A PTA from the PSD method in the top left panel. The constraints on the two PTA red noise parameters are comparable to the standard PTA analysis.
On the other hand, the decrease of the white noise level through the CS improves the detection of the injected red noise and the constraints. The lowest 3 frequency bins contribute to the recovery of the red noise power law, see bottom left panel of figure 7 signal is clearly detectable and the tightest constraints can be placed in the top left panel.
Variance analysis
The middle column of figure 7 shows the comparison red noise parameter corner plot in the top panel and recovered Allan variance (middle panel) and parabolic variance (bottom panel) with the injected red noise. As equation (16) is only valid for 1.5 ≤ γ PTA ≤ 4.5, the boundaries are reduced in the top row middle panel of figure 7. With this caveat, the red noise spectral index γ PTA and amplitude A PTA can still be compared, however the cutoffs at 1.5 and 4.5 limit the comparison and can thus influence the red noise recovery. The recovered variances in the middle and bottom panels of the middle column of figure 7 illustrate the fact that the highest octave of τ with very large uncertainty is unreliable and contributes very little to the overall analysis. Since we are looking at a boundary case, the injections are close or within the uncertainties of the variance computation. Therefore, they do not necessarily coincide with the median of the recovered variance, see also table 1. Additionally, the prior choice allows for a large distribution of small amplitude red noise, producing a median variance close to white noise.
This results in a conservative estimation that there is little red noise in the boundary case with AVAR and PVAR. However, as there a small turn-up in the higher time steps, both provide some evidence of the injected red noise. This can be seen in the top row middle panel corner plot, especially in the A PTA parameter and a small improvement in constraining the spectral index. The PTA analysis is also able to pick up the red noise and performs better than the variances. With the lowest level of white noise, it is no surprise that the LSL analysis performs the best.
Covariance analysis
We can further focus on the red noise detection by sacrificing the constraints on the white noise. This is done in a similar way as with the CS by computing the covariances, such that uncorrelated white noise is rejected to allow for the correlated red noise to be more prominent. The right-hand column of figure 7 shows the posterior distributions, recovered ACOV and PCOV from top to bottom. The same limitation on the spectral index γ PTA for variances applies also to the covariances.
Despite the lower white noise from the covariances, the constraints on the red noise parameters γ PTA and A PTA are actually worse than their variance counterparts, as it can be seen in the top right panel of figure 7 and table 1. As there is no correlated white noise, the lowest time steps become upper limits with large uncertainties and do not provide any information on the red noise. This leaves only one or two time steps with information, which is shown in the right column middle and bottom panels. The overall recovery using all time steps thus favours low amplitude white and red noise. Using only time steps with small uncertainties could produce tighter, but less reliable, red noise constraints.
Red noise
The strong red noise case is of particular interest as it shows how well the different methods can in principle divide different red noise realizations in the large-signal regime. An overview on the con-straints on the PTA parameters for the red noise case is given by table 2 for the standard PTA/LSL and clock metrology analysis. The overall posterior distributions for all methods are shown in figure 9. The results of some example analyses of a single simulation set can be found in figure 10.
In general, all methods can detect the injected red noise very well. As the red noise is very large, there is very little difference in the constraints on the red noise parameters γ PTA and log 10 A PTA between the PTA and the LSL analysis. Both are centered around the injected values with tight constraints, see top panels of figure 9. So do the spectral method recoveries, although with a very small tail, which is longer for the PSD compared to the CS. The uncertainties in the spectral methods allow for small values even at the lowest frequencies. This leads to a tail of lower amplitudes being recovered, see table 2. Both variances and covariances perform very similarly to the PTA/LSL analysis in terms of the tightness of the constraints on the red noise parameters, especially PVAR and PCOV are comparable to the LSL analysis. This is due to the fact that the higher time steps are dominated by red noise and have therefore very similar VAR/COV values. Additionally, the absolute uncertainties σ i are the same for both variances and covariances. Consequently, the covariances at the lowest τ are not as well defined as the variances for the same time steps. These large relative uncertainties and the lower COV values translate into a small bias for the red noise to have steeper spectral indices γ PTA than with the variance methods. On the other hand, the variance methods are still a little bit impacted by the choice of priors, see table 2. This can be seen in the bottom middle panel of figure 9, where the median recovered PVAR is lower than the median PVAR values of the 100 realizations. The same is also true for AVAR recovery.
Table 2 also shows that both AVAR and ACOV have a longer tail towards lower red noise amplitudes log 10 A PTA when compared to PVAR and PCOV. This can be the advantage of the parabolic (co)variance transitioning from a spectral index of β VAR = −3 for white noise to γ VAR = 0 for the injected red noise. In comparison the Allan (co)variance has a transition of one order less, ie. from β VAR = −2 to γ VAR = 0.
CONCLUSIONS
We have shown that clock comparison methods, such as the cross spectrum and variances, can be applied to pulsar timing observations. Initial tests show that these methods produce slightly worse, sometimes comparable constraints on the pulsar noise parameters when compared to the standard PTA and optimal LSL analysis. Both perform very well in recovering the injected white noise, regardless of the level of red noise. The red noise recovery itself is broadly consistent with the injections, with different realizations giving similar constraints. In general, the metrology clock comparison methods perform well in the pure white noise case, slightly worse in the boundary case and comparable in the strong red noise case. The spread of recoveries between different realizations is also larger than the comparison PTA/LSL analysis.
The PSD method produces slightly broader parameter posterior distributions compared to those from the PTA analysis. The CS allows for a long tail of low values, including negative values. This means a reduction of white noise to almost zero at high frequencies, but including these negative values will skew the red noise recovery. Using Gaussian distributions also decreases the possibility for low CS values and thus give slightly more optimistic constraints for A PTA and γ PTA . Thus, the CS method performs slightly better than the PTA analysis, but still a bit worse than the LSL analysis. Since the LSL's white noise level is only a fifth of that of the PTA observations, it can put the tighest constraints on the red noise.
Similar constraints can be put on the red noise parameters for both the variances and covariances. The (co)variance methods perform particularly badly in recovering the white noise in the red noise case. By design, ACOV and PCOV should reject uncorrelated white noise and only leave the correlated red noise signal. Therefore, no useful information can be gained for the white noise, while some extra power may be pushed into the red noise. This lack of information may limit the performance of the covariances in precisely constraining the red noise and manifests as a small bias in γ PTA or a long tail of low red noise amplitudes A PTA .
The clock comparison methods complement the standard PTA analysis by giving independent constraints on the noise parameters. This can be especially valuable for certain realizations, when the different analysis methods do not converge to the same values.
We continue the study and apply the methods described to the observations from the LEAP project, where we deal with the challenges from real pulsar timing data. These include unevenly sampled data points, different levels of white noise between RTs and observations and uncertainties in the computation of the spectral densities and (co)variances.
DATA AVAILABILITY
The simulated residual series and posteriors can be obtained from the authors upon request.
APPENDIX A: ESTIMATES AND WEIGHTS OF THE CROSS SPECTRUM
For a given frequency f 0 , we consider the noise level S w from uncorrelated observational noise of the RTs and the signal level S r from the pulsar red noise, see equation (4). We perform a large Monte-Carlo simulation with 10 7 realizations at the given frequency with varying levels of noise and signal.
The large simulation allows for the evaluation of the estimates σ emp log 10 1.576 S w + 4.147S r S w + 3.833S r log 10 (2.045) = 0.3106 σ mod log 10 1.251 S w + 3.835S r S w + 3.640S r log 10 (1.635) = 0.2136 Table A1. Dependence of the position and width parameters of the CS estimates on S w and S r .
of the CS. We determine empirically the statistics and conclude that the most reliable representation can be obtained from the logarithmic CS estimates distinguishing between positive and negative estimates. The empirical distributions are described by two parameters: µ emp and σ emp are the mean and variance of the log of the CS estimates respectively. Despite the asymmetry of the probability density functions we choose to model them with Gaussians with mean µ mod and variance σ mod and find the best fit values to both positive and negative CS histograms (see inset of figure A1). By varying the noise and signal levels we obtain numerical expressions for the 4 parameters as a function of S w and S r . They can be found in table A1. The factor (S w + 5S r ) shows that we can retrieve the correct number of degrees of freedom, ie. 5, in a purely empirical way.
The important results are the following:
(i) the width parameters σ emp,mod can be considered as constants (ii) the position parameters for negative estimates µ − emp,mod only depend on S w (iii) the position parameters for positive estimates µ + emp,mod depend on S w as well as S r (iv) the shift between µ emp and µ mod is constant and can be evaluated from the 10 7 realizations to be log 10 (1.373) = 0.138. This shift between simulation and model can be explained by the long tail towards lower numbers of the empirical distributions which decrease µ emp compared to µ mod . The (almost) constancy of the width parameters confirms that the log of the CS estimates is a reliable representation. Moreover, the position parameters of the negative estimates do not depend on S r . This implies that the negative estimates do not carry any information about the signal level and should not been taken into account. The variation of the signal and noise levels makes f 0 representative for any frequency. Additionally, since the PSD is similar to the positive CS estimator, we can apply the same Gaussian model.
APPENDIX B: WEIGHTS AND UNBIASING OF THE VARIANCES
The uncertainty σ i on each time step i, whether variances or covariances, can be calculated from the average values of the variances VAR
σ 2 i = 2VAR 2 /ν,(B1)
where ν are the effective degrees of freedom, whose expressions depend on the type of noise. The expressions used in the analysis for AVAR can be found in Lesage & Audoin (1973): for white phase modulation (γ PTA = 0)
ν ≈ (N + 1)(N − 2m) 2(N − m)(B2)
and flicker frequency modulation (γ PTA = 3):
ν ≈ 5N 2 4m(N + 3m)
.
The equation for PVAR is given in Vernotte et al. (2020) as
ν ≈ 35 Am/M − B(m/M) 2 ,(B4)
where A and B are constants which depend on the type of the modulation. They are given as A = 23 for γ PTA = 0 and A ≈ 28 for γ PTA = 3, whilst B ≈ 12 for all spectral indices. As pointed out in Vernotte et al. (2020), the formula is unreliable for the highest τ. However, from the theory we know that ν = 1 at the largest time step for PVAR. We also notice a mixture of white and red noise for our realizations, especially for the boundary case. Consequently, we approximate the degrees of freedom for the boundary case using those from the white noise case as a conservative approach. The degrees of freedom used for our analysis can be found in tables B1 and B2 and are the same for the covariances. Finally, Vernotte & Lantz (2012) have shown that the largest time steps are statistically biased towards lower VAR/COV values. The factors that need to be applied for any time step τ with its degrees of freedom ν can be found in table II in Vernotte & Lantz (2012). We also show the factors used in this paper in tables B1 and B2. Table B2. Effective degrees of freedom and unbiasing factors for the variances in the white noise and boundary case.
Figure 1 .
1Average PSD (blue pluses), averaged residual spectrum (ARS) (green triangles) and CS (orange crosses/points) computed from one set of 5 different residual series in the boundary case (red dashed lines)
Figure 2 .
2Averaged data Allan variance (blue) and parabolic variance (orange), the corresponding covariances (light dotted lines) and the associated uncertainties computed from one set of 5 residual series in the boundary case (red dashed line)
Figure 3 .
3Example set of 5 different simulated residual series of a 4000-day long observing campaign with a monthly cadence in the boundary case see
Figure 4 .Figure 5 .
45Comparison of the posterior distribution corner plots for the white noise case: spectral (top), variance (middle) and covariance analyses (bottom) Evolution of the recovery of the EFAC parameter from the white noise case (top row), boundary case (middle row) to the red noise case (bottom row): spectral (left column), variance (middle column) and covariance methods (right column). The numbers in the legends represent the median values of the posterior distributions.
Figure 6 .
6Examples of the recovery of the EFAC parameter from one set of realizations of the white noise case (top row), boundary case (middle row) and the red noise case (bottom row): spectral (left column), variance (middle column) and covariance methods (right column). The numbers in the legends represent the average median values of the posterior distributions.
Figure 7 .Figure 8 .Figure 9 .Figure 10 .
78910The left, middle and right column show results from the spectral, variance and covariance analysis respectively for the boundary case. The top row shows the posterior distribution corner plots. The middle row shows the recovered PSD (left), AVAR (middle) and ACOV (right). The bottom row shows the recovered CS (left), PVAR (middle) and PCOV (right). Each recovery figure contains the values from the 100 realizations in black dots, the injection in red dotted lines and the median values and central 68 and 90% credible regions in blue bands. Examples of the recovery of the γ PTA (top row) and A PTA (bottom row) parameters to central 90% from one set of realizations of the boundary case: spectral (left column), variance (middle column) and covariance methods (right column). The numbers in the legends represent the average median values of the posterior distributions. Top and bottom rows correspond those in figure 7 respectively in the same style. However, the results are for the red noise case. Examples of the recovery of the γ PTA (top row) and A PTA (bottom row) parameters from one set of realizations of the red noise case: spectral (left column), variance (middle column) and covariance methods (right column). The numbers in the legends represent the median values of the posterior distributions.
Figure A1 .
A1Dependence on S r of the log mean (in blue) and of the Gaussian center (in red) for positive CS estimates when S w = 1. Inset: Gaussian fit of the histograms of the log of the CS estimates (S w = 3 · 10 −13 , S r = 10 −11
. Bothparameter
EFAC
γ PTA
log 10 A PTA
injection
1.05
3
−14
PTA
1.06 +0.06
−0.49
3.55 +2.32
−1.76
−14.48 +0.75
−1.24
LSL
0.23 +0.03
−0.07
3.24 +1.62
−1.10
−14.16 +0.35
−0.55
PSD
1.02 +0.09
−0.50
3.69 +2.59
−2.00
−14.75 +1.09
−1.75
CS
0.42 +0.07
−0.18
3.55 +1.44
−1.19
−14.35 +0.50
−0.71
AVAR
1.04 +0.12
−0.52
2.78 +1.33
−1.07
−15.36 +1.50
−2.91
PVAR
0.92 +0.10
−0.44
2.76 +1.48
−1.13
−15.35 +1.52
−4.03
ACOV
0.30 +0.19
−0.16
2.99 +1.26
−1.21
−15.77 +1.67
−3.27
PCOV
0.25 +0.17
−0.13
2.97 +1.35
−1.31
−16.49 +2.33
−3.16
Table 1. List of the average median posterior values and average 90% cen-
tral region bounds for the 3 constrainable PTA noise parameters from the
analyses of 100 realizations in the boundary case with different analysis
methods.
the posterior distributions in the top panel of the figure and table
1 show that the CS performs a little better with tighter constraints
than both the PSD and the standard PTA methods. In the LSL, the
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
PSD: 1.01
CS: 0.42
PTA: 1.06
LSL: 0.23
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
AVAR: 1.01
PVAR: 0.91
PTA: 1.06
LSL: 0.23
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
ACOV: 0.22
PCOV: 0.19
PTA: 1.06
LSL: 0.23
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
PSD: 1.02
CS: 0.42
PTA: 1.06
LSL: 0.23
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
AVAR: 1.04
PVAR: 0.92
PTA: 1.06
LSL: 0.23
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
ACOV: 0.30
PCOV: 0.25
PTA: 1.06
LSL: 0.23
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
PSD: 1.01
CS: 0.42
PTA: 1.07
LSL: 0.30
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
AVAR: 0.90
PVAR: 0.68
PTA: 1.07
LSL: 0.30
40
42
44
46
48
realization
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
EFAC
Injection: 1.05
ACOV: 0.57
PCOV: 0.33
PTA: 1.07
LSL: 0.30
).Parameter
Negative estimates
Positive estimates
µ emp
log 10 0.1949S w
S w + 3.876S r
S w + 4.552S r
log 10 [0.2245 (S w + 5S r )]
µ mod
log 10 0.2665S w
S w + 3.882S r
S w + 4.404S r
log 10 [0.3082 (S w + 5S r )]
Table B1. Effective degrees of freedom and unbiasing factors for the variances in the red noise case.τ
60 120 240
480
960
1920
assumed γ PTA
3
3
3
3
3
3
AVAR ν
80
38
18
7.7
3.0
1.1
PVAR ν
82
40
19
8.6
3.4
1
AVAR unbiasing factor
1
1
1
1.14 1.45
3.56
PVAR unbiasing factor
1
1
1
1.12 1.45
3.56
τ
60
120 240 480
960
1920
assumed γ PTA
0
0
0
0
0
0
AVAR ν
66
65
63
58
46
5.8
PVAR ν
100
49
23
11
4.4
1
AVAR unbiasing factor
1
1
1
1
1
1.19
PVAR unbiasing factor
1
1
1
1.1
1.31
3.56
© 2020 The Authors arXiv:2011.01912v3 [physics.data-an] 9 Mar 2021 2 Chen et al.
MNRAS 000, 1-12 (2020)
ACKNOWLEDGEMENTSThis work is funded by the ANR Programme d'Investissement d'Avenir (PIA) under the FIRST-TF network (ANR-10-LABX-48-01) project and the Oscillator IMP project (ANR-11-EQPX-0033-OSC-IMP) and the EUR EIPHI Graduate School (ANR-17-EURE-00002), and by grants from the Région Bourgogne Franche Comté intended to support the PIA.The Nançay radio Observatory is operated by the Paris Observatory, associated to the French Centre National de la Recherche Scientifique (CNRS). We acknowledge financial support from "Programme National de Cosmologie and Galaxies" (PNCG) and "Programme National Hautes Energies" (PNHE) funded by CNRS/INSU-IN2P3-INP, CEA and CNES, France.We acknowledge the support of our colleagues in the Large European Array for Pulsars and European Pulsar Timing Array collaboration.
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| In this work we show how to represent policies as programs: that is, as stochastic simulators with tunable parameters. To learn the parameters of such policies we develop connections between black box variational inference and existing policy search approaches. We then explain how such learning can be implemented in a probabilistic programming system. Using our own novel implementation of such a system we demonstrate both conciseness of policy representation and automatic policy parameter learning for a set of canonical reinforcement learning problems. | null | [
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| 5,022,334 | 1507.04635 | 0000596a026e7d5a29486e23b9e0d68f189e62a3 |
Black-Box Policy Search with Probabilistic Programs
Jan-Willem Van De Meent
Department of Engineering Science
University of Oxford
Brooks Paige
Department of Engineering Science
University of Oxford
David Tolpin
Department of Engineering Science
University of Oxford
Frank Wood
Department of Engineering Science
University of Oxford
Black-Box Policy Search with Probabilistic Programs
In this work we show how to represent policies as programs: that is, as stochastic simulators with tunable parameters. To learn the parameters of such policies we develop connections between black box variational inference and existing policy search approaches. We then explain how such learning can be implemented in a probabilistic programming system. Using our own novel implementation of such a system we demonstrate both conciseness of policy representation and automatic policy parameter learning for a set of canonical reinforcement learning problems.
Introduction
In planning under uncertainty the objective is to find a policy that selects actions, given currently available information, in a way that maximizes expected reward. In many cases an optimal policy can neither be represented compactly nor learned exactly. Online approaches to planning, such as Monte Carlo Tree Search [Kocsis and Szepesvári, 2006], are nonparametric policies that select actions based on simulations of future outcomes and rewards, also known as rollouts. While policies like this are often able to achieve near optimal performance, they are computationally intensive and do not have compact parameterizations. Policy search methods (see Deisenroth et al. [2011] for a review) learn parameterized policies offline, which then can be used without performing rollouts at test time, trading off improved test-time computation against having to choose a policy parameterization that may be insufficient to represent the optimal policy.
In this work we show how probabilistic programs can represent parametric policies in a both more general Appearing in Proceedings of the 19 th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 41. Copyright 2016 by the authors. and compact manner. We also develop automatic inference techniques for probabilistic programming systems to do model-agnostic policy search. Our proposed approach, which we call black box policy learning (BBPL), is a variant of Bayesian policy search [Wingate et al., 2011 in which policy learning is cast as stochastic gradient ascent on the marginal likelihood.
In contrast to languages that target a single domainspecific algorithm [Andre and Russell, 2002, Srivastava et al., 2014, Nitti et al., 2015, our formulation emphasizes the use of general-purpose techniques for Bayesian inference, in which learning is used for inference amortization. To this end, we adapt black-box variational inference (BBVI), a technique for approximation of the Bayesian posterior [Ranganath et al., 2014, Wingate andWeber, 2013] to perform (marginal) likelihood maximization in arbitrary programs. The resulting technique is general enough to allow implementation in a variety of probabilistic programming systems. We show that this same technique can be used to perform policy search under an appropriate planning as inference interpretation, in which a Bayesian model is weighted by the exponent of the reward. The resulting technique, BBPL is closely related to classic policy gradient methods such as REINFORCE [Williams, 1992].
We present case studies in the Canadian traveler problem, the RockSample domain, and introduce a setting inspired by Guess Who [Coster and Coster, 1979] as a benchmark for optimal diagnosis problems.
Policies as Programs
Probabilistic programming systems [Milch et al., 2007, Goodman et al., 2008, Minka et al., 2014, Pfeffer, 2009, Wood et al., 2014, Gordon et al., 2014 represent generative models as programs in a language that provides specialized syntax to instantiate random variables, as well as syntax to impose conditions on these random variables. The goal of inference in a probabilistic program is to characterize the distribution on its random variables subject to the imposed conditions, which is done using one or more generic methods provided by an inference backend. 1.)))))] ( fn policy [ u vs ] ( argmax ( zipmap vs ( map ( fn [ v ] ( Q u v )) vs )))))) Figure 1: A Canadian traveler problem (CTP) implementation in Anglican. In the CTP, an agent must travel along a graph, which represents a network of roads, to get from the start node (green) to the target node (red). Due to bad weather some roads are blocked, but the agent does not know which in advance. Upon arrival at each node the agent observes the set of open edges. The function dfs-agent walks the graph by performing depth-first search, calling a function policy to choose the next destination based on the current and unvisited locations. The function make-random-policy returns a policy function that selects destinations uniformly at random, whereas make-edge-policy constructs a policy that selects according to sampled edge preferences (Q u v). By learning a distribution on each value (Q u v) through gradient ascent on the marginal likelihood, we obtain a heuristic offline policy that follows the shortest path when all edges are open, and explores more alternate routes as more edges are closed.
In sequential decision problems we must define a stochastic simulator of an agent, which chooses actions based on current contextual information, and a stochastic simulator of the world, which may have some internal variables that are opaque to the agent, but provides new contextual information after each action.
For sufficiently simple problems, both the agent and the world simulator can be adequately described as graphical models. Here we are interested in using probabilistic programs as simulators of both the world and the agent. The trade-off made in this approach is that we can incorporate more detailed assumptions about the structure of the problem into our simulator of the agent, which decreases the size of the search space, at the expense of having to treat these simulators as black boxes from the perspective of the learning algorithm.
In Figure 1 we show an example of a program, written in the language Anglican [Wood et al., 2014], which simulates an agent in the Canadian traveler problem (CTP) domain. This agent traverses a graph using depth first search (DFS) as a base strategy, choosing edges either at random, or according to sampled preferences. Probabilistic programs can describe a family of algorithmic policies, which may make use of programming constructs such as recursion, and higher-order functions and arbitrary deterministic operations. This allows us to define structured policies that enforce basic constraints, such as the rule that you should never travel the same edge twice.
Given a base policy program, we can define different parametrizations that encode additional structure, such as the typical travel distance starting from each edge. We can then formulate a Bayesian approach to policy learning, in which we place a prior on the policy parameters and optimize its hyperparameters to maximize the reward. To do so we employ a planning as inference interpretation [Toussaint et al., 2006, Rawlik et al., 2012, Neumann, 2011, Hoffman et al., 2009a,b, Levine and Koltun, 2013] that casts policy search as stochastic gradient ascent on the marginal likelihood.
A challenge in devising methods for approximate inference in probabilistic programs is that such methods must deal gracefully with programs that may not instantiate the same set of random variables in each execution. For example, the random policy in Figure 1 will generate a different set of categorical variables in each execution, depending on the path followed through the graph. Similarly, the edge based policy samples values (Q u v) lazily, depending on the visited nodes.
In this paper we develop an approach to policy learning based on black box variational inference (BBVI) [Ranganath et al., 2014, Wingate andWeber, 2013], a technique for variational approximation of the posterior in Bayesian models. We begin by reviewing planning as inference formulations of policy search. We then show how BBVI can be adapted to perform hyperparameter optimization. In a planning as inference interpretation this method, which we call black box policy learning (BBPL), is equivalent to classic policy gradient methods. We then describe how BBPL may be implemented in the context of probabilistic programs with varying numbers of random variables, and provide a languageagnostic definition of the interface between the program and the inference back end.
Policy Search as Bayesian Inference
In sequential decision problems, an agent draws an action u t from a policy distribution π(u t | x t ), which may be deterministic, conditioned on a context x t . The agent then observes a new context x t+1 drawn from a distribution p(x t+1 | u t , x t ). In the finite horizon case, where an agent performs a fixed number of actions T , resulting in a sequence τ = (x 0 , u 0 , x 1 , u 1 , x 2 , . . . , u T −1 , x T ), which is known as a trajectory, or roll-out. Each trajectory gets a reward R(τ ). Policy search methods maximize the expected reward J θ = E p θ [R(τ )] for a family of stochastic policies π θ with parameters θ
J θ = R(τ )p θ (τ ) dτ,(1)p θ (τ ) := p(x 0 ) T −1 t=0 π(u t | x t , θ)p(x t+1 | u t , x t ).(2)
We are interested in performing upper-level policy search, a variant of the problem defined in terms of the hyperparameters λ of a distribution p λ (τ, θ) that places a prior p λ (θ) on the policy parameters
J λ = R(τ )p λ (τ, θ) dτ dθ,(3)p λ (τ, θ) := p λ (θ)p(τ | θ).(4)
Upper-level policy search can be interpreted as maximization of the normalizing constant Z λ of an unnormalized density
γ λ (τ, θ) = p λ (τ, θ) exp(βR(τ )),(5)Z λ = γ λ (τ, θ) dτ dθ (6) = E p λ [exp(βR(τ ))].(7)
The constant β > 0 has the interpretation of an 'inverse temperature' that controls how strongly the density penalizes sub-optimal actions. The normalization constant Z λ is the expected value of the exponentiated reward exp(βR(τ )), which is known as the desirability in the context of optimal control [Kappen, 2005, Todorov, 2009]. It is not a priori obvious that maximization of the expected reward J λ yields the same policy hyperparameters as maximization of the desireability Z λ , but it turns out that the two are in fact equivalent, as we will explain in section 5.
In planning as inference formulations, γ λ (τ, θ)/Z λ is often interpreted as a posterior p λ (τ, θ | r) conditioned on a pseudo observable r = 1 that is Bernoulli distributed with probability p(r = 1 | τ ) ∝ exp(βR(τ )), resulting in a joint distribution that is proportional to γ λ (τ, θ),
p(r = 1, τ, θ) ∝ p λ (τ, θ) exp(βR(τ )) = γ λ (τ, θ). (8)
Maximization of Z λ is then equivalent to the maximization of the marginal likelihood p λ (r = 1) with respect to the hyperparameters λ. In a Bayesian context this is known as empirical Bayes (EB) [Maritz and Lwin, 1989], or type II maximum likelihood estimation.
Black-box Variational Inference
Variational Bayesian methods [Wainwright and Jordan, 2008] approximate an intractable posterior with a more tractable family of distributions. For purposes of exposition we consider the case of a posterior p(z, θ | y), in which y is a set of observations, θ is a set of model parameters, and z is a set of latent variables. We write
p(z, θ | y) = γ(z, θ)/Z with γ(z, θ) = p(y | z, θ)p(z | θ)p(θ),(9)Z = γ(z, θ) dz dθ.(10)
Variational methods approximate the posterior using a parametric family of distributions q λ by maximizing a lower bound on log Z with respect to λ
L λ = E q λ [log γ(z, θ) − log q λ (z, θ)] (11) = log Z − D KL (q λ (z) || γ(z)/Z) ≤ log Z. (12)
This objective may be optimized with stochastic gradient ascent [Hoffman et al., 2013]
λ k+1 = λ k + ρ k ∇ λ L λ λ=λ k ,(13)∇ λ L λ = E q λ (z) ∇ λ log q λ (z) log γ(z, θ) q λ (z, θ)) .(14)
Here ρ k is a sequence of step sizes that satisfies the conditions
∞ k=1 ρ k = ∞ and ∞ k=1 ρ 2 k < ∞.
The calculation of the gradient ∇ λ L λ requires an integral over q λ . For certain models, specifically those where the likelihood and prior are in the conjugate exponential family [Hoffman et al., 2013], this integral can be performed analytically.
Black box variational inference targets a much broader class of models by sampling z [n] , θ [n] ∼ q λ and replacing the gradient for each component i with a sample-based estimate [Ranganath et al., 2014]
∇ λi L λ = N n=1 ∇ λi log q λ (z [n] , θ [n] )(log w [n] −b i ), (15) w [n] = γ(z [n] , θ [n] )/q λ (z [n] , θ [n] ),(16)
in whichb i is a control variate that reduces the variance of the estimator
b i = N n=1 (∇ λi log q λ (z [n] , θ [n] )) 2 w [n] N n=1 (∇ λi log q λ (z [n] , θ [n] )) 2 .(17)
Black-box Policy Search
The sample-based gradient estimator in BBVI resembles the one used in classic likelihood-ratio policy gradient methods [Deisenroth et al., 2011], such as RE-INFORCE [Williams, 1992], G(PO)MDP , and PGT [Sutton et al., 1999]. There is in fact a close connection between BBVI and these methods, as has been noted by e.g. Dayan et al. [1995], Mnih and Gregor [2014] and Ba et al. [2014].
To make this connection precise, let us consider what it would mean to perform variational inference in a planning as inference setting. In this case, we can define a lower bound L λ,λ0 on log Z λ0 in terms of a variational distribution q λ (τ, θ) with parameters λ and an unnormalized density γ λ0 (τ, θ) of the form in equation 5, with parameters λ 0
L λ,λ0 = E q λ [log γ λ0 (z, θ) − log q λ (z, θ)](18)= E q λ βR(τ ) + log p λ0 (τ, θ) q λ (τ, θ)(19)
If we now choose a variational distribution with the same form as the prior, then q λ (τ, θ) = p λ0 (τ, θ) whenever λ = λ 0 . Under this assumption, the lower bound at λ = λ 0 simplifies to
L λ,λ0 λ=λ0 = E q λ [βR(τ )] λ=λ0 = βJ λ λ=λ0 .(20)
In other words, the lower bound L λ,λ0 is proportional to the expected reward J λ when the variational posterior is equal to the prior.
The gradient of the lower bound similarly simplifies to
∇ λ L λ,λ0 λ=λ0 = E q λ ∇ λ log q λ (τ, θ) log γ λ0 (τ, θ) q λ (τ, θ) λ=λ0 = E q λ 0 ∇ λ log q λ (τ, θ) λ=λ0 βR(τ ) = dτ dθ ∇ λ q λ (τ, θ) λ=λ0 βR(τ ) = ∇ λ J λ λ=λ0 .
The implication of this identity is that we can perform gradient ascent on J λ by making a slight modification to the update equation
λ k+1 = λ k + ρ k∇λ L λ,λ k λ=λ k .(21)
The difference in these updates is that instead of calculating the gradient∇ λ L λ,λ0 estimate relative to a fixed set of prior parameters λ 0 , we update the parameters of the prior p λ k (τ, θ) after each gradient step, and calculate the gradient ∇ λ L λ,λ k . We note that the constant β is simply a scaling factor on the step sizes ρ k , and will from here on assume that β = 1.
When BBVI is performed using the update step in equation 21, and the variational family q λ is chosen to have the same form as the prior p λ , we obtain a procedure for EB estimation, which maximizes the normalizing constant Z λ with respect to the parameters λ of the prior. The difference between the EB and maximum likelihood (ML) methods is that the first calculates the gradient relative to hyperparameters λ, whereas the other calculates the gradient relative to the parameters θ. Because this difference relates only to the assumed model structure, EB estimation is sometimes referred to as Type II maximum likelihood.
As is evident from equation 20, EB estimation in the context of planning as inference formulations maximizes the expected reward J λ . In the context of a probabilistic programming system this means that we can effectively get three algorithms for the price of one: If we can provide an implementation of BBVI, then this implementation can be adapted to perform EB estimation, which in turn allows us to perform policy search by simply defining models where exponent of the reward takes the place of the likelihood terms. This results in a method that we call black box policy learning (BBPL), which is equivalent to variants of REINFORCE applied to upper-level policy search.
Learning Probabilistic Programs
An implementation of BBVI and BBPL for probabilistic program inference needs to address two domain-specific issues. The first is that probabilistic programs need not always instantiate the same set of random variables, the second is that we need to distinguish between distributions that define model parameters θ and those that define latent variables z, or variables that are part of the context x in the case of decision problems.
Let us refer back to the program in Figure 1. The function dfs-agent performs a recursive loop until a stopping criterion is met: either the target node is reached, or there are no more paths left to try. At each step dfs-agent makes a call to policy, which is created by either calling make-random-policy or make-edge-policy. A random policy samples uniformly from unexplored directions. When depth first search is performed with this policy, we are defining a model in which the number of context variables is random, since the number of steps required to reach the goal state will vary. In the case of the edge policy, we use a memoized function to sample edge preference values as needed, choosing the unexplored edge with the highest preference at each step. In this case the number of parameter variables is random, since we only instantiate preferences for edges that are (a) open, and (b) connect to the current location of the agent.
As has been noted by Wingate and Weber [2013], BBVI can deal with varying sets of random variables quite naturally. Since the gradient is computed from a sample estimate, we can compute gradients for a each random variable by simply averaging over those executions in which the variable exists. Sampling variables as needed can in fact be more statistically efficient, since irrelevant variables that never affect the trajectory of the agent will not contribute to the gradient estimate. BBVI has the additional advantage of having relatively light-weight implementation requirements; it only requires differentiation of the log proposal density, which is a product over primitive distributions of a limited number of types, for which derivatives can be computed analytically. This is in contrast to implementations based on (reverse-mode) automatic differentiation [Pearlmutter and Siskind, 2008], as is used in Stan [Kucukelbir et al., 2015], which store derivative terms for the entire computation graph.
To provide a language-agnostic definition of BBVI and BBPL, we formalize learning in probabilistic programs as the interaction between a program P and an inference back end B. The program P represents all deterministic steps in the computation and has internal state (e.g. its environment variables). The back end B performs all inference-related tasks.
A program P executes as normal, but delegates to the inference back end whenever it needs to instantiate a random variable, or evaluate a conditioning statement. The back end B then supplies a value for the random variable, or makes note of the probability associated with the conditioning statement, and then delegates back to P to continue execution. We will assume that the programming language provides some way to differentiate between latent variables z, which are simply to be sampled, and parameters θ for which a distribution is to be learned. In Anglican the syntax (sample (tag :policy d)), as used in Fig. 1, is used as a general-purpose mechanism to label distributions on random variables. An inference back end can simply ignore these labels, or implement algorithm-specific actions for labeled subsets.
In order for the learning algorithm to be well-defined in programs that instantiate varying numbers of random variables, we require that the each random variable z a is uniquely identified by an address a, which may either be generated automatically by the language runtime, or specified by the programmer. Each model parameter θ b is similarly identified by an address b.
In BBVI, the interface between a program P and the back end B can be formalized with the following rules:
• Initially B calls P with no arguments P().
• A call to P returns one of four responses to B:
1. (sample, a, f, φ): Identifies a latent random variable (not a policy parameter) z a with unique address a, distributed according to f a (· | φ a ). The back end generates a value z a ∼ f a (· | φ a ) and calls P(z a ). 2. (learn, b, f, η): For policy parameters, the address b identifies a random variable θ b in the model, distributed according to a distribution f b with parameters η b . The back end generates θ b ∼ f b (· | λ b ) conditioned on a learned variational parameter λ b and registers an im-
portance weight w b = f b (θ b | η b )/f b (θ b | λ b ).
Execution continues by calling P(θ b ). 3. (factor, c, l): Here c is a unique address for a factor with log probability l c and importance weight w c = exp(l c ). Execution continues by calling P(). 4. (return, v): Execution completes, returning a value v.
Because each call to P is deterministic, an execution history is fully characterized by the values for each random variable that are generated by B. However the set of random variables that is instantiated may vary from execution to execution. We write A, B, C for the set of addresses of each type visited in a given execution. The program P now defines an unnormalized density γ P of the form
γ P (z, θ) := p P (z, θ) c∈C exp(l c ),(22)p P (z, θ) := a∈A f a (z a | φ a ) b∈B f b (θ b | η b ) .(23)
Implicit in this notation is the fact that the distribution types f a (· | φ a ) and f b (· | η b ) are return values from calls to P, which implies that both the parameter values and the distribution type may vary from execution to execution. While f a (· | φ a ) and f b (· | η b ) are fully determined by preceding values for z and θ, we assume they are opaque to the inference algorithm, in the sense that no analysis is performed to characterize the conditional dependence of each φ a or η b on other random variables in the program.
Given the above definition of a target density γ P (z, θ), we are now in a position to define the density of a variational approximation Q λ to the program. In this density, the runtime values η b are replaced by variational parameters λ b
p Q λ (z, θ) := a∈A f a (z a | φ a ) b∈B f b (θ b | λ b ) .(24)
This density corresponds to that of a mean-field probabilistic program, where the dependency of each θ b on other random variables is ignored.
Repeated execution of P given the interface described above results in a sequence of weighted samples (w [n] , θ [n] , z [n] ), whose importance weight w [n] is defined as
w [n] := γ P (z [n] , θ [n] ) / p Q λ (z [n] , θ [n] ) = b∈B f (θ [n] b | η b ) f (θ [n] b | λ b ) c∈C exp l [n] c .(25)
With this notation in place, it is clear that we can define a lower bound L Q λ ,Q λ k analogous to that of Equation 19, and a gradient estimator analogous to that of Equation 15, in which the latent variables z take the role of the trajectory variables τ . In summary, we can describe a sequential decision problem as a probabilistic program P in which the log probabilities l c are interpreted as rewards, parameters θ b define the policy and all other latent variables z a are trajectory variables. EB inference can then be used to learn the Algorithm 1 Black-box Policy Learning generating (w [n] , θ [n] , z [n] ) according to Eqns. 24, 25
initialize parameters λ 0,b ← η b , iteration k = 0 repeat Set initial λ k+1 = {λ k,b } b∈B Run N executions of program Q λ k ,for each address b do Let N b ≤ N be the # of runs containing b Let g [n] b := ∇ λ k,b log f (θ [n] b |λ k,b ) Compute baselineb λ k,b from Eq. 17 ∇ λ k,b J λ k ← N b −1 g [n] b (log w [n] −b λ k,b ) Update λ k+1,b ← λ k,b + ρ k∇λ k,b J λ k end for k ← k + 1 until parameters λ b converge
hyperparameters λ that maximize the expected reward, as described in Algorithm 1.
An assumption that we made when deriving BBPL is that the variational distribution q λ (τ, θ) must have the same analytical form as the prior p λ0 (τ, θ). Practically this requirement means that a program P must be written in such a way that the values of the hyperparameters η b have the same constant values in every execution, since their values may not depend on those of random variables. One way to enforce this is to pass η as a parameter in the initial call P(η) by B, though we do not formalize such a requirement here.
Case Studies
We demonstrate the use of programs for policy search in three problem domains: (1) the Canadian Traveler Problem, (2) a modified version of the RockSample POMDP, and (3) an optimal diagnosis benchmark inspired by the classic children's game Guess Who.
These three domains are examples of deterministic POMDPs, in which the initial state of the world is not known, and observations may be noisy, but the state transitions are deterministic. Even for discrete variants of such problems, the number of possible information states x t = (u 0 , o 1 , . . . , u t−1 , o t ) grows exponentially with the horizon T , meaning that it is not possible to fully parameterize a distribution π(u | x, θ) in terms of a conditional probability table θ x,u . In our probabilistic program formulations for these problems, the agent is modeled as an algorithm with a number of random parameters, and we use BBPL to learn the distribution on parameters that maximizes the reward.
We implement our case studies using the probabilistic programming system Anglican [Wood et al., 2014]. We use the same experimental setup in each of the three domains. A trial begins with a learning phase, in which BBPL is used to learn the policy hyperparameters, followed by a number of testing episodes in which the agent chooses actions according to a fixed learned policy. At each gradient update step, we use 1000 samples to calculate a gradient estimate. Each testing phase consists of 1000 episodes. All shown results are based on test-phase simulations.
Stochastic gradient methods can be sensitive to the learning rate parameters. Results reported here use a RMSProp style rescaling of the gradient [Hinton et al.], which normalizes the gradient by a discounted rolling decaying average of its magnitude with decay factor 0.9. We use a step size schedule ρ k = ρ 0 /(τ + k) κ as reported in Hoffman et al. [2013], with τ = 1, κ = 0.5 in all experiments. We use a relatively conservative base learning rate ρ 0 = 0.1 in all reported experiments. For independent trials performed across a range 1, 2, 5, 10, . . . , 1000 of total gradient steps, consistent convergence was observed in all runs using over 100 gradient steps.
The source code for the case studies, as well as the BBPL implementation, is available online. 1
Canadian Traveler Problem
In the Canadian Traveler Problem (CTP) [Papadimitriou and Yannakakis, 1991], an agent must traverse a graph G = (V, E), in which edges may be missing at random. It is assumed the agent knows the distance d : E → R+ associated with each edge, as well as the probability p : E → (0, 1] that the edge is open, but has no advance knowledge of the edges that are blocked. The problem is NP-hard [Fried et al., 2013], 1 https://bitbucket.org/probprog/black-box-policy-search Edge weights indicate the frequency at which the agent moves between each pair of rocks. Starting points are in green, exit paths in red. and heuristic online and offline approaches [Eyerich et al., 2010] are used to solve problem instances.
The results in Figure 1 show that the learned policy behaves in a reasonable manner. When edges are open with high probability, the policy takes the shortest path from the start node, marked in green, to the target node, marked in red. As the fraction of closed edges increases, the policy makes more frequent use of alternate routes. Note that each edge has a fixed probability of being open in our set-up, resulting in a preference for routes that traverse fewer edges. Figure 2 shows convergence as a function of the number of gradient steps. Results are averaged over 5 domains of 20 and 50 nodes respectively. Convergence plots for each individual domain can be found in the supplementary material. We compare the learned policies against the optimistic policy, a heuristic that selects edges according to the shortest path, assuming that all unobserved edges are open. We observe that mean traveled distance for the learned policy converges to that of the optimistic policy, which is close to optimal.
RockSample POMDP
In the RockSample POMDP [Smith and Simmons, 2004], an N × N square field with M rocks is given. A rover is initially located in the middle of the left edge of the square. Each of the rocks can be either good or bad; the rover must traverse the field and collect samples of good rocks while minimizing the traveled distance. The rover can sense the quality of a rock remotely with an accuracy decreasing with the distance to the rock. We consider a finite-horizon variant of the RockSample domain, described in the supplementary material, with a structured policy in which a robot travels along rocks in a left-to-right order.
The policy plots in Figure 3 show that this simple policy results in sensible movement preferences. In particular we point out that in the 5×5 instance, the agent always visits the top-left rock when traveling to the top-middle rock, since doing so incurs no additional cost. Similarly, the agent follows an almost deterministic trajectory along the left-most 5 rocks in the 10 × 10 instance, but does not always make the detour towards the lower rocks afterwards.
Guess Who
Guess Who is a classic game in which players pick a card depicting a face, belonging to a set that is known to both players. The players then take turns asking questions until they identify the card of the other player [Coster and Coster, 1979]. We here consider a singleplayer setting where an agent asks a pre-determined number of questions, but the responses are inaccurate with some probability. This is sometimes known as a measurement selection, or optimal diagnosis problem. We make use of a feature set based on the original game, consisting of 24 individuals, characterized by 11 binary attributes and two multi-class attributes, resulting in a total of 19 possible questions. We assume a response accuracy of 0.9. By design, the structure of the domain is such that there is no clear winning opening question. However the best question at any point is highly contextual.
We assume that the agent knows the reliability of the response and has an accurate representation of the posterior belief b t (s) = p(s | x t ) for each candidate s in given questions and responses. The agent selects randomly among the highest ranked candidates after the final question. We consider 3 policy variants, two of which are parameter-free baselines. In the first baseline, questions are asked uniformly at random. In the second, questions are asked according to a myopic estimate of the value of information [Hay et al., 2012], i.e. the change in expected reward relative to the current best candidates, which is myopically optimal in this setting. Finally, we consider a policy that empirically samples questions q according to a weight v q = γ nq (Ab) q , based on the current belief b, a weight matrix A, and a discount factor γ nq based on the number of times n q a question was previously asked. Intuitively, this algorithm can be understood as learning a small set of α-vectors, one for each question, similar to those learned in point-based value iteration [Pineau et al., 2003]. The discounting effectively "shrinks" the belief-space volume associated with the α-vector of the current best question, allowing the agent to select the next-best question.
The results in Figure 4 show that the learned policy clearly outperforms both baselines, which is a surprising result given the complexity of the problem and the relatively simplistic form of this heuristic policy. While these results should not be expected to be in any way optimal, they are encouraging in that they illustrate how probabilistic programming can be used to implement and test policies that rely on transformations of the belief or information state in a straightforward manner.
Discussion
In this paper we put forward the idea that probabilistic programs can be a productive medium for describing both a problem domain and the agent in sequential decision problems. Programs can often incorporate assumptions about the structure of a problem domain to represent the space of policies in a more targeted manner, using a much smaller number of variables than would be needed in a more general formulation. By combining probabilistic programming with black-box variational inference we obtain a generalized variant of well-established policy gradient techniques that allow us to define and learn policies with arbitrary levels of algorithmic sophistication in moderately high-dimensional parameter spaces. Fundamentally, policy programs represent some form of assumptions about what contextual information is most relevant to a decision, whereas the policy parameters represent domain knowledge that generalizes across episodes. This suggests future work to explore how latent variable models may be used to represent past experiences in a manner that can be related to the current information state.
A Anglican
All case studies are implemented in Anglican, a probabilistic programming language that is closely integrated into the Clojure language. In Anglican, the macro defquery is used to define a probabilistic model. Programs may make use of user-written Clojure functions (defined with defn) as well as user-written Anglican functions (defined with defm). The difference between the two is that in Anglican functions may make use of the model special forms sample, observe, and predict, which interrupt execution and require action by the inference back end. In Clojure functions, sample is a primitive procedure that generates a random value, observe returns a log probability, and predict is not available.
Full documentation for Anglican can be found at http://www.robots.ox.ac.uk/˜fwood/anglican
The complete source code for the case studies can be found at https://bitbucket.org/probprog/black-box-policy-search
B Canadian Traveler Problem
The complete results for the Canadian traveler problem, showing the performance and convergence for the learned policies for multiple graphs of different sizes and topologies, are presented in Figures 5 and 6.
C RockSample
The RockSample problem was formulated as a benchmark for value iteration algorithms and is normally evaluated in an infinite horizon setting where the discount factor penalizes sensing and movement. In the original formulation of the problem, movement and sensing incur no cost. The agent gets a reward of 10 for each good rock, as well as for reaching the right edge, but incurs a penalty of -10 when sampling a bad rock.
Here we consider an adaptation of RockSample to a finite horizon setting. We assume sensing is free, and movement incurs a cost of -1. We structure the policy by moving along rocks in a left-to-right order. At each rock the agent sense the closest next rock and chooses to move to it, or discard it and consider the next closest rock. When the agent gets to a rock, it only samples the rock if the rock is good. The parameters describe the prior over the probability of moving to a rock conditioned on the current location and the sensor reading.
D Guess Who
In Table 1 we provide as reference the complete ontology for the Guess Who domain. At each turn, the player asks whether the unknown individual has a particular value of a single attribute. Table 1: Ontology for the Guess Who domain, consisting of 24 individuals, characterized by 11 binary attributes and two multi-class attributes.
Figure 2 :
2Convergence for CTP domains of 20 and 50 nodes. Blue lines show the mean traveled distance using the learned policy, averaged over 5 domains. Red lines show the mean traveled distance for the optimistic heuristic policy. Dash length indicates the fraction of open edges, which ranges from 1.0 to 0.6.
Figure 3 :
3Learned policies for the Rock Sample domain.
Figure 4 :
4(left) Average reward in Guess Who as a function of number of questions. (right) Convergence of rewards as function number of gradient steps. Each dot marks an independent restart.
Figure 5 :
5Canadian traveler problem: edge weights, indicating average travel frequency under the learned policy, and convergence for individual instances with 20 nodes.
Figure 6 :
6Canadian traveler problem: edge weights, indicating average travel frequency under the learned policy, and convergence for individual instances with 50 nodes. id beard ear-rings eye-color gender glasses hair-color hair-length hair-type hat moustache mouth-size nose-size red-
arXiv:1507.04635v4 [stat.ML] 4 Aug 2016( defquery ctp
" Probabilistic program representing an agent
solving the Canadian Traveler Problem "
[ graph src tgt base-prob make-policy ]
( let [ sub-graph
( sample-weather graph base-prob src tgt )
[ path dist counts ]
( dfs-agent sub-graph src tgt ( make-policy ))]
( factor (-dist ))
( predict :path path )
( predict :distance dist )
( predict :counts counts )))
( defm dfs-agent
" Run depth-first-search from start to target,
prioritizing edges according to policy "
[ graph start target policy ]
... )
( defm make-random-policy
" Policy: Select edge at random "
[]
( fn policy [ u vs ]
( sample
( categorical
( zipmap vs ( repeat ( count vs ) 1.))))))
( defm make-edge-policy
" Policy: learn priorities for each edge "
[]
( let [ Q ( mem ( fn [ u v ]
( sample [ u v ]
( tag :policy
( gamma 1.
AcknowledgementsWe would like to thank Thomas Keller for his assistance with Canadian traveler problem, and Rajesh Ranganath for helpful feedback on configuring RM-SProp for black-box variational inference. Frank Wood is supported under DARPA PPAML through the U.S. AFRL under Cooperative Agreement number FA8750-14-2-0006, Sub Award number 61160290-111668.
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| []
|
[
"A SMALL MAXIMAL SIDON SET IN",
"A SMALL MAXIMAL SIDON SET IN"
]
| [
"Z ",
"Maximus Redman ",
"Lauren Rose ",
"Raphael Walker "
]
| []
| []
| A Sidon set is a subset of an Abelian group with the property that each sum of two distinct elements is distinct. We construct a small maximal Sidon set of size O((n · 2 n ) 1/3 ) in the group Z n 2 , generalizing a result of Ruzsa concerning maximal Sidon sets in the integers. | null | [
"https://arxiv.org/pdf/2109.00292v3.pdf"
]
| 237,485,619 | 2109.00292 | c0864202d6ea16102f92dc404403f3fa154fd6ee |
A SMALL MAXIMAL SIDON SET IN
Z
Maximus Redman
Lauren Rose
Raphael Walker
A SMALL MAXIMAL SIDON SET IN
arXiv:2109.00292v3 [math.CO] 9 Apr 2022
A Sidon set is a subset of an Abelian group with the property that each sum of two distinct elements is distinct. We construct a small maximal Sidon set of size O((n · 2 n ) 1/3 ) in the group Z n 2 , generalizing a result of Ruzsa concerning maximal Sidon sets in the integers.
Introduction
Sidon sets were first studied by Simon Sidon in the 1930s [Sid32], and then by Paul Erdős in the 1940s [ET41]. At this time, they were sometimes referred to as Sidon sequences, given that they were assumed to be subsets of the integers. However, the notion of Sidon set generalizes quite naturally to other Abelian groups, and may be generalized to arbitrary groups in a number of ways. We limit our discussion here to Sidon sets contained in Abelian groups, with the following definition.
Definition 1.1. Let G be an Abelian group and S ⊆ G. We say S is Sidon if whenever a + b = x + y and a, b, x, y are in S, we have {a, b} = {x, y}.
By such a definition, if every element of G has order 2, then every Sidon set in G consists of a single element. Thus in the case G = Z n 2 , we further require that a = b and that x = y, i.e. that the sum of every pair of distinct elements be distinct.
The case G = Z n 2 is the subject of this paper. For these groups, the initial question of interest is the largest possible size of a Sidon set, which is known to be Θ(2 n /2 ) (for example, [Lin69; BR60; TW21]). However, another question of interest, which has been much less studied, is the minimal size of a maximal Sidon set, defined below.
Definition 1.2. We say a Sidon set S ⊆ G is maximal if there exists no Sidon set S ′ ⊆ G with S ⊂ S ′ .
The union of a maximal Sidon set with any additional element is not a Sidon set. That is, for any x not in S, the equation x + a = b + c is solvable for a, b, c ∈ S. The main result of this paper is the construction of maximal Sidon set in Z n 2 of size O((n 2 n ) 1 /3 ), in Theorem 3.1.
Large Sidon Sets in
Z n 2
The smallest known construction of a maximal Sidon set in the integers is due to Ruzsa [Ruz98]. Ruzsa's method for constructing a small maximal Sidon set relies on the existence of a sufficiently dense Sidon set in a smaller space than the one of ultimate interest: for Sidon sets in the integers, Singer's theorem [Sin38] on the existence of a Sidon set of size p + 1 in Z 1+p+p 2 sufficed to provide this smaller set.
A construction in the case of Z n 2 comes from the field of coding theory, in particular the Bose-Chaudhuri-Hocquenghem (BCH) codes [BR60], which are a type of error-correction system making use of polynomials in finite fields. When constructed over F 2 , BCH codes of length n and distance 5 are in fact Sidon sets. The Sidon sets resulting from such constructions are suitable for the proof of Theorem 3.1, our generalization of Ruzsa's construction.
Proposition 2.1. Let S n ⊂ F 2 n/2 × F 2 n/2 be defined by
{(x, x 3 ) | x ∈ F 2 n/2 }.
Then S n is a Sidon set for all even n ≥ 1.
Proof. Apply Lemma 2 in [BR60,p. 74] with m = n/2, l = 4, and t = 2.
As F 2 n/2 × F 2 n/2 is additively isomorphic to Z n 2 , the image of S n under any such isomorphism is a Sidon set in Z n 2 . By abuse of notation, we will henceforth consider S n ⊂ Z n 2 to be some such image.
If S is a Sidon set in G, for some points x of G \ S, S ∪ {x} is a Sidon set, and for other points it is not: that is, we have x + a = b + c, for some a, b, c ∈ S. Recall that for a maximal Sidon set, S ∪ {x} is never a Sidon set. Yet the number of "collisions" -that is, the number of solutions to x + a = b + c -varies depending on the choice of x. We use the following definition to measure the number of such solutions.
Definition 2.2. Let S ⊆ Z n 2 be a Sidon set. We say a point x ∈ Z n 2 \ S is covered k times by S if there exist k distinct unordered solutions {a, b, c} to a + b + c = x, for a, b, c ∈ S.
It follows immediately that if a point x is covered k times by S, then the k triples {a, b, c} satisfying a + b + c = x are disjoint by the Sidon property of S. The maximal Sidon sets are exactly those that cover every point at least once, and maximal Sidon sets that cover each point many times are relatively "denser" than those which cover each point fewer times in the same space. Thus to search for large Sidon sets, we try to maximize how many times each point is covered, while to search for small Sidon sets, we want to minimize the same quantity.
We show that the BCH Code construction of S 2n ⊂ Z 2n 2 (above) covers every point of Z 2n 2 \ S 2n relatively many times. Note that notationally, we say f (n) = Ω(g(n)) if there exists c > 0 and n 0 such that for all n > n 0 , f (n) ≥ cg(n). Compare to the more common big-O notation:
f (n) = O(g(n)) if f (n) ≤ cg(n).
Theorem 2.3. The set S 2n given by Theorem 2.1 covers every element of Z 2n 2 \ S 2n at least Ω(2 n ) times.
Proof. As in Theorem 2.1, let
S = {(x, x 3 ) | x ∈ F 2 n }.
Let (x, y) ∈ F 2 2 n \ S. That is, let x, y ∈ F 2 n , where y = x 3 . To determine how many times S covers (x, y), we count the number of solutions (a, b, c) over F 2 n to
a + b + c = x (1) a 3 + b 3 + c 3 = y.(2)
If Eq. (1) holds, then c = x + a + b, and it suffices to solve the equation
0 = y + a 3 + b 3 + (x + a + b) 3 .
We can homogenize this to get a projective curve C of degree 3, defined by
F (a, b, p) = a 2 b + b 2 a + a 2 px + b 2 px + ap 2 x 2 + bp 2 x 2 + p 3 x 3 + p 3 y = 0.
The polynomial F is absolutely irreducible: that is, irreducible over F 2 n [a, b, p], where F 2 n is the algebraic closure of F 2 n . To see this, specialize to a polynomial in F 2 n [a, b] by setting p = 1. Supposing that F (a, b, 1) is the product of two non-constant polynomials, we have without loss of generality
F (a, b, 1) = (s 1 a 2 + s 2 b 2 + s 3 ab + s 4 a + s 5 b + s 6 )(t 1 a + t 2 b + t 3 ).
Expanding the product and setting the coefficient of each resulting term equal to the coefficient of the corresponding term in F (a, b, 1) gives the following system of equations, which we will solve by hand:
s 1 t 1 = 0 (3) s 2 t 2 = 0 (4) s 3 t 1 + s 1 t 2 = 1 (5) s 2 t 1 + s 3 t 2 = 1 (6) s 4 t 1 + s 1 t 3 = x (7) s 5 t 2 + s 2 t 3 = x (8) s 5 t 1 + s 4 t 2 + s 3 t 3 = 0 (9) s 6 t 1 + s 4 t 3 = x 2 (10) s 6 t 2 + s 5 t 3 = x 2 (11) s 6 t 3 = x 3 + y.(12)
By (3), the possibilities for (s 1 , t 1 ) are (1, 0), (0, 1), or (0, 0) without loss of generality. (5) rules out the possibility that s 1 = t 1 = 0, so it remains to check the two cases s 1 = 0, t 1 = 1 and s 1 = 1, t 1 = 0.
Suppose s 1 = 1, t 1 = 0. By (5) we have t 2 = 1 and hence s 2 = 0. By (6) we have s 3 = 1; by (7), t 3 = x; by (8), s 5 = x; by (9), s 4 = x; by (11), s 6 = 0; and hence by (12) that x 3 = y, which contradicts the assumption that (x, y) ∈ S.
On the other hand, suppose s 1 = 0, t 1 = 1. Then we have by (5) that s 3 = 1 and by (6) that s 2 + t 2 = 1. Combining this with (4), we have that either s 2 = 1 and t 2 = 0 or s 2 = 0 and t 2 = 1. Since the polynomial F (a, b, 1) is symmetric in a and b, (i.e, F (a, b, 1) = F (b, a, 1)), the case s 2 = 1 and t 2 = 0 reduces to the case above, which we have seen to be inconsistent. So without loss of generality we have s 2 = 0, t 2 = 1. Then by (7) we have s 4 = x; by (8), s 5 = x; by (9), t 3 = 0; and hence by (12) that x 3 = y which is as before a contradiction.
Thus the system of equations for the factorization of F (a, b, 1) into two non-constant polynomials is inconsistent when x 3 = y, and so F (a, b, 1) is absolutely irreducible. As the specialization in F 2 n [a, b] is irreducible, so is the original polynomial in F 2 n [a, b, p].
Since F is absolutely irreducible, by the Hasse-Weil theorem (see, for instance, [Hur03, p. 6]),
2 n + 1 − 2g √ 2 n ≤ #C ≤ 2 n + 1 + 2g √ 2 n ,
where g is the geometric genus of C and #C is the number of projective points on C (i.e., the number of solutions up to scalar multiplication of (a, b, p)). By the Riemann-Hurwitz formula (see, for instance, [Sil09, p. 37]), the genus of C can be computed from the degree d and the number of singular points s of C. In particular,
g = (d − 1)(d − 2) 2 − s = 1 − s ≤ 1,
and hence
2 n + 1 − 2 √ 2 n ≤ #C ≤ 2 n + 1 + 2 √ 2 n .
The solutions of interest are the affine solutions, given by
F (a, b, 1) = 0.
Let A be the number of affine solutions as above, and P the number of projective solutions, given by F (a, b, 0) = 0. We have #C = A + P .
P is the number of solutions to F (a, b, 0) = a 2 b + b 2 a = ab(a + b) = 0, which is satisfied by the pairs (0, 1), (1, 0), and (1, 1), and so P = 3. Then we have
A = #C − P ≥ 2 n − 2 √ 2 n − 2.
Each of the A affine points (a, b, 1) on C corresponds to an ordered triple of distinct elements (a, b, a + b + x) such that (a, a 3 ) y). From our assumption that (x, y) ∈ S (i.e., y = x 3 ), it follows that for each pair, a, b, and a + b + x are distinct. Thus the point (x, y) is covered A/6 ≥ 2 n −2 √ 2 n −2 6
+ (b, b 3 ) + (a + b + x, (a + b + x) 3 ) = (x,
= Ω(2 n ) times by S.
For n = 3, 5, 7, 9, we have observed by direct computation that the Sidon set given by the method above covers each point exactly 2 n −2 6 times, and we conjecture that the pattern continues for larger odd n. Nonetheless, the fact that each point is covered Ω(2 n ) times is sufficiently strong for the requirements of the main theorem of this paper, in which we construct a small maximal Sidon set in Z n 2 by "projecting" this Sidon set S from a subspace.
An Analogue of Ruzsa's Construction
In this section we discuss maximal Sidon sets and their sizes. While rather tight bounds on the largest possible size of a Sidon set are known for many groups (the size of the largest Sidon set in Z q is around √ q; see, for example, [OBr04]), very little is known about the minimal sizes of maximal Sidon sets.
In the case of G = Z n 2 , any maximal Sidon set S satisfies |S| 3 + |S| ≥ 2 n , as S is maximal if and only if every point of G \ S is covered at least once by S, and hence |S| = Ω(2 n /3 ). We suspect that this bound is not sharp in general. On the other hand, there has not been until now an upper bound on the size of the smallest maximal Sidon set in Z n 2 except for the upper bound on the size of any Sidon set: |S| = O(2 n /2 ). In the case of the integers, Ruzsa [Ruz98] showed that there exists a maximal Sidon set in [1, N ] with size O ((N log N ) 1 /3 ). The main theorem of this paper is a similar bound on the size of maximal Sidon sets in Z n 2 . Theorem 3.1. There exists a maximal Sidon set S ⊆ Z n 2 such that |S| = O (n · 2 n ) 1 /3 .
In this section, we adopt the technique from [Ruz98] in order to construct a small maximal Sidon set in Z n 2 , using a method that generalizes easily to arbitrary Abelian groups, provided a sufficiently dense Sidon set can be found in a quotient of the desired group. In the future, we hope to use this method to construct small maximal Sidon sets in groups of the form Z n p , for p > 2. Proof of Theorem 3.1. Let T > 0 be a fixed constant, such that for each t, the Sidon set S 2t ⊂ Z 2t 2 given by Theorem 2.3 covers every point of Z 2t 2 \ S 2t at least 2 t T times (i.e., Ω(2 t )). Let m be the least even integer satisfying m > 2 3 log 2 (T ln(2) n 2 n ), and let Q < Z n 2 be a subgroup isomorphic to Z n−m 2 . Observe that the quotient group Z n 2 /Q is isomorphic to Z m 2 , which is the additive group of F m 2 . For an element x of Z n 2 , let x be the coset x + Q. Any such set B can be extended to a maximal Sidon set S. We can add an element x to B and still have a Sidon set if any only if x is not covered by B: that is, if there is no solution to
x = a + b + c for a, b, c ∈ B.
We show that it is possible to choose B such that every x satisfying x ∈ A is covered by B. That is, if A covers x, then B covers x. Since we chose each b i randomly, we determine the probability that such an element (i.e., with x ∈ A) is covered by B.
Since A covers every element of Z n 2 /Q \ A at least 2 m/2 /T times, let (a u j , a v j , a w j ) for 1 ≤ j ≤ J (where J ≥ 2 m/2 /T ) be a sequence of disjoint triples of elements of A, such that for each j, a u j + a v j + a w j = x, and hence b u j + b v j + b w j ∈ x. Since each b i was chosen with a uniform distribution from the coset a i , for each j we have
P (b u j + b v j + b w j = x) =
# of triples of elements in x whose sum is x # of triples of elements in x = |Q| 2 |Q| 3 = 2 m−n Each pair of triples of indices (u j , v j , w j ) and (u k , v k , w k ) is disjoint if j = k, and so if we apply the computed probability for the J independent events, we have
P (b u j + b v j + b w j = x, for all 1 ≤ j ≤ J) = (1 − 2 m−n ) J ≤ e −J 2 m−n ≤ e − 2 (3/2)m−n T ,
and due to the initial choice of m > 2 3 log 2 (T ln(2) n 2 n ),
P (b u j + b v j + b w j = x, for all 1 ≤ j ≤ J) ≤ e − 2 (3/2)m−n T < 2 −n .
As the probability of the union of events is at most the sum of the probabilities of each event, we have P (∃x such that x ∈ A and x is not covered by B) ≤ (2 n − |Q||A|) 2 −n < 1, and hence P (B covers each x satisfying x ∈ A) > 0. Since this probability is positive, there exists a choice of B which covers each x satisfying x ∈ A. Let B 0 be such a choice. The set B 0 is not necessarily maximal, but we may bound the number of elements required to extend it to a maximal Sidon set. So let S = B 0 ∪ X be a maximal Sidon set. For each element s i of X, s i ∈ A, so let a t i = s i . Then we have b t i = s i + q i for some q i ∈ Q. But then q i = s i + b t i , and since S is Sidon, q i = q j for i = j. Thus by the pigeonhole principle, there can be at most |Q| elements in X, we have |X| ≤ |Q| = 2 n−m = O(2 n /3 ).
Finally, as |S| = |B 0 | + |X| and |B 0 | = |A|, we achieve |S| ≤ |A| + |Q| ≤ 2 m/2 + 2 n−m ≤ O((n · 2 n ) 1 /3 ) + O(2 n /3 ) = O((n · 2 n ) 1 /3 ).
This result provides an upper bound on the smallest maximal Sidon set, and thus if S is the smallest maximal Sidon set in Z n 2 , we have Ω((2 n ) 1 /3 ) ≤ |S| ≤ O((n · 2 n ) 1 /3 ).
This pair of bounds parallels the best-known bounds on the minimal size of maximal Sidon sets in the integers [1, N ]. Probabilistic estimates by P. Bennett and T. Bohman [BB16] on the size of greedilyconstructed maximal sets in regular hypergraphs predict that in the case of Z n 2 , the righthand side of this interval is the best-possible upper bound, as a randomly constructed maximal Sidon set in Z n 2 has size Ω((n · 2 n ) 1 /3 ) with high probability. Thus the minimal size of maximal Sidon sets may be easier to compute here than in the integers, and hence we anticipate that this question will be resolved definitively in the future.
By Theorem 2.3, there exists a Sidon set A ⊂ Z n 2 /Q (i.e., A = S m ) such that |A| = 2 m /2 and A covers every p ∈ Z n 2 /Q \ A at minimum 2 m/2 T times. For each a i in A, pick a random representative b i of the coset a i , and let B = {b i | 1 ≤ i ≤ 2 m /2 }. Choose each representative b i from a uniform distribution on a i , independently of the random choices for each other representative, such that each of the (2 n−m ) m 2 possible choices for B has the same probability. Regardless of which representatives are chosen, B is a Sidon set in Z n 2 .
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Ein Satz über trigonomietrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen. Simon Sidon, 10.1007/BF01455900Mathematische Annalen. 106Simon Sidon. "Ein Satz über trigonomietrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen". Mathematische Annalen 106 (1932), pp. 536- 539. doi: 10.1007/BF01455900.
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"PLASMA DIAGNOSTICS OF CORONAL DIMMING EVENTS",
"PLASMA DIAGNOSTICS OF CORONAL DIMMING EVENTS"
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| [
"Kamalam Vanninathan \nInstitute of Physics\nUniversity of Graz\n8010GrazAustria\n",
"Astrid M Veronig [email protected] \nInstitute of Physics\nUniversity of Graz\n8010GrazAustria\n\nKanzelhöhe Observatory for Solar and Environmental Research\nUniversity of Graz\n9521TreffenAustria\n",
"Karin Dissauer \nInstitute of Physics\nUniversity of Graz\n8010GrazAustria\n",
"Manuela Temmer \nInstitute of Physics\nUniversity of Graz\n8010GrazAustria\n",
"Astrid M Veronig "
]
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"Institute of Physics\nUniversity of Graz\n8010GrazAustria",
"Institute of Physics\nUniversity of Graz\n8010GrazAustria",
"Kanzelhöhe Observatory for Solar and Environmental Research\nUniversity of Graz\n9521TreffenAustria",
"Institute of Physics\nUniversity of Graz\n8010GrazAustria",
"Institute of Physics\nUniversity of Graz\n8010GrazAustria"
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| []
| Coronal mass ejections (CMEs) are often associated with coronal dimmings, i.e. transient dark regions that are most distinctly observed in Extreme Ultra-violet (EUV) wavelengths. Using Atmospheric Imaging Assembly (AIA) data, we apply Differential Emission Measure (DEM) diagnostics to study the plasma characteristics of six coronal dimming events. In the core dimming region, we find a steep and impulsive decrease of density with values up to 50-70%. Five of the events also reveal an associated drop in temperature of 5-25%. The secondary dimming regions also show a distinct decrease in density, but less strong, decreasing by 10-45%. In both the core and the secondary dimming the density changes are much larger than the temperature changes, confirming that the dimming regions are mainly caused by plasma evacuation. In the core dimming, the plasma density reduces rapidly within the first 20-30 min after the flare start, and does not recover for at least 10 hrs later, whereas the secondary dimming tends to be more gradual and starts to replenish after 1-2 hrs. The pre-event temperatures are higher in the core dimming (1.7-2.6 MK) than in the secondary dimming regions (1.6-2.0 MK). Both core and secondary dimmings are best observed in the AIA 211Å and 193Å filters. These findings suggest that the core dimming corresponds to the footpoints of the erupting flux rope rooted in the AR, while the secondary dimming represents plasma from overlying coronal structures that expand during the CME eruption. | 10.3847/1538-4357/aab09a | [
"https://arxiv.org/pdf/1802.06152v1.pdf"
]
| 118,864,203 | 1802.06152 | f3d028992f5c5187f9a525641f9bab168057137a |
PLASMA DIAGNOSTICS OF CORONAL DIMMING EVENTS
July 26, 2018 16 Feb 2018
Kamalam Vanninathan
Institute of Physics
University of Graz
8010GrazAustria
Astrid M Veronig [email protected]
Institute of Physics
University of Graz
8010GrazAustria
Kanzelhöhe Observatory for Solar and Environmental Research
University of Graz
9521TreffenAustria
Karin Dissauer
Institute of Physics
University of Graz
8010GrazAustria
Manuela Temmer
Institute of Physics
University of Graz
8010GrazAustria
Astrid M Veronig
PLASMA DIAGNOSTICS OF CORONAL DIMMING EVENTS
July 26, 2018 16 Feb 2018(Received July 26, 2018; Revised xxxxxxxx xx, xxxx; Accepted xxxxxxxx xx, xxxx) Submitted to ApJDraft version Typeset using L A T E X default style in AASTeX61 Corresponding author: 2 Vanninathan et al.sun: atmospheresun: coronasun: coronal mass ejections (CMEs)sun: flares
Coronal mass ejections (CMEs) are often associated with coronal dimmings, i.e. transient dark regions that are most distinctly observed in Extreme Ultra-violet (EUV) wavelengths. Using Atmospheric Imaging Assembly (AIA) data, we apply Differential Emission Measure (DEM) diagnostics to study the plasma characteristics of six coronal dimming events. In the core dimming region, we find a steep and impulsive decrease of density with values up to 50-70%. Five of the events also reveal an associated drop in temperature of 5-25%. The secondary dimming regions also show a distinct decrease in density, but less strong, decreasing by 10-45%. In both the core and the secondary dimming the density changes are much larger than the temperature changes, confirming that the dimming regions are mainly caused by plasma evacuation. In the core dimming, the plasma density reduces rapidly within the first 20-30 min after the flare start, and does not recover for at least 10 hrs later, whereas the secondary dimming tends to be more gradual and starts to replenish after 1-2 hrs. The pre-event temperatures are higher in the core dimming (1.7-2.6 MK) than in the secondary dimming regions (1.6-2.0 MK). Both core and secondary dimmings are best observed in the AIA 211Å and 193Å filters. These findings suggest that the core dimming corresponds to the footpoints of the erupting flux rope rooted in the AR, while the secondary dimming represents plasma from overlying coronal structures that expand during the CME eruption.
INTRODUCTION
Coronal Mass Ejections (CMEs) are large-scale eruptions on the Sun often associated with flares (Harrison 1995;Zhou et al. 2003;Compagnino et al. 2017). Substantial amounts of energy, plasma and magnetic field are released into the interplanetary medium during such ejections (Vourlidas et al. 2000;Webb & Howard 2012). Being the main drivers of severe space weather disturbances (Schwenn et al. 2005;Lugaz et al. 2016;Riley et al. 2018), CMEs are of high relevance for solar as well as space weather research and predictions (e.g. Schwenn 2006;Gopalswamy 2009;Howard 2014;Green et al. 2018).
CMEs originating from regions located centrally on the solar disk are most likely to be directed towards Earth (St. Cyr et al. 2000). However, they are difficult to observe in the Thomson-scattered white light by coronagraphs on-board spacecraft in the Sun-Earth line, since their propagation direction is far away from the plane of sky, and the measurements may be strongly affected by projection effects (e.g. Gopalswamy et al. 2000). For these reasons, there is also a need for indirect means of studying the early evolution of Earth-directed CMEs. Hansen et al. (1974) were the first to report intensity 'dimmings' in coronameter images from Mauna Loa occurring simultaneously with CMEs. They have an appearance similar to coronal holes and are thus also referred to as 'transient coronal holes' (Rust 1983;Hewish et al. 1985). A great advance in observing coronal dimmings associated with CMEs has been made with the advent of the Yohkoh/Soft X-Ray Telescope and Solar and Heliospheric Observatory/Extreme Ultraviolet Imaging Telescope (SoHO/EIT) (Hudson et al. 1996;Sterling & Hudson 1997;Zarro et al. 1999; Thompson et al. 2000). Since then, many case studies and statistical studies were performed using imaging at EUV and SXR wavelengths, in order to better understand the physics of coronal dimmings and how they relate to the CME evolution (e.g. Zhukov & Auchère 2004;Reinard & Biesecker 2008;Bewsher et al. 2008).
Coronal dimmings occur in association with the early eruption phase of CMEs. It is believed that such dimmings are a consequence of density decrease in the low corona due to mass carried out by the CME (Hudson et al. 1998;Mason et al. 2014). Coronal dimming regions can be used as indirect means to estimate the source location, width and mass of CMEs (Harrison et al. 2003;Aschwanden 2016). While observations of CMEs originating from regions close to the center of the solar disk are subject to substantial projection effects, the coronal dimming region is most clearly discernible here making them a good proxy for studying Earth-directed CMEs.
Coronal dimmings can be differentiated into two types: core and secondary dimmings. The core dimmings are localized regions that occur in pairs on opposite sides of the source active region, and are rooted in opposite magnetic polarities. They are interpreted as a signature of the footpoints of the erupting flux rope (Sterling & Hudson 1997;Webb et al. 2000). In contrast, secondary dimmings appear more shallow, diffuse and widespread. They are assumed to be caused by the expansion of the overall CME structure formed, and thus to correspond to the spatial extent of the CME observed in coronagraph data (Mandrini et al. 2007).
Several spectroscopic results support the presence of outflowing plasma in the dimming regions (Harra & Sterling 2001;Harra et al. 2007;Miklenic et al. 2011;Tian et al. 2012) with velocities decreasing with time (Jin et al. 2009) and prevalent thermal line broadening (McIntosh 2009;Chen et al. 2010). Density decrease of 35-40% within the dimming regions (Cheng et al. 2012) and plasma flows to replenish it (Landi et al. 2012) have been identified using Differential Emission Measure (DEM) analysis.
In this paper, we study the plasma characteristics of coronal dimmings using DEM analysis. For this purpose, we selected six dimming events associated with CME/flare events of different speeds and flare classes for detailed case studies. The plasma evolution in the core and the secondary dimming regions are studied in high cadence over a period of 12 hrs. The paper is organized as follows, in Section 2 we describe our data and methods of analysis. The results are presented in Section 3 and discussed in Section 4. Finally, in Section 5 we give the summary and conclusions of this work.
OBSERVATIONS AND DATA ANALYSIS
The six EUV filters of the Atmospheric Imaging Assembly (AIA, Lemen et al. 2012) instrument on-board the Solar Dynamics Observatory (SDO, Pesnell et al. 2012) are sensitive to plasma over a wide temperature range from ≈10 5 to above 10 7 K which facilitates the use of their data for the purpose of DEM reconstruction. Many DEM codes have been developed in the past years to use with AIA data (Aschwanden et al. 2013;Hannah & Kontar 2012Plowman et al. 2013;Cheung et al. 2015). For this work the code developed by Hannah & Kontar (2013) has been used, henceforth referred to as HK code. The HK code uses regularized inversion to reconstruct a DEM from broad/narrow band filter observations using the equation
I filter = R filter (T ) φ(T ) dT,(1)
where I filter is the intensity measured in a particular filter, R filter (T ) is the temperature response function of the instrument, and φ(T ) is the required DEM. For our analysis, we first binned the AIA images to a size of 512 pixels × 512 pixels (from the original images of 4096 × 4096) while conserving flux and then cut to a sub-map around the AR where the dimming could be clearly seen. The binning increases the signal to noise ratio, which is especially relevant in the AIA channels sensitive to hotter plasma where the counts are relatively low (94 and 131Å), thus improving the DEMs derived. From the DEM maps, we then calculated for each rebinned pixel the emission measure (EM), the DEM weighted average temperature (T ) and plasma density (n) using the following expressions:
EM = φ(T ) dT,(2)T = φ(T ) T dT φ(T ) dT ,(3)n = φ(T ) dT h ,(4)
where h is the distance along the line-of-sight (LOS). For this study we have used h=60 Mm corresponding to the coronal scale height for a temperature of about 1 MK (see also Vanninathan et al. 2015). Note that each of our rebinned pixels corresponds to 8 × 8 original AIA pixels, i.e. covers a field of about 4 × 4 on the solar disk. For the current work we have selected six coronal dimming events, which were observed against the solar disk, for detailed case study. We made sure that the list represented events associated with flares of different classes, ranging from C3 to X2 (and located within 30 • from the central meridian), and CMEs of different speeds, ranging from ≈400-1100 km s −1 . Four of the events are associated with an EUV wave (Nitta et al. 2013), indicating the presence of large amplitude magnetosonic waves initiated by the impulsive lateral CME expansion (e.g. Kienreich et al. 2009;Patsourakos & Vourlidas 2009;Veronig et al. 2010;Long et al. 2017). The events selected along with information on the associated flare, CME and EUV wave are summarized in Table 1. The CME speed given is the linear speed derived in the LASCO field-of-view, as listed in the LASCO/SOHO CME catalogue (https://cdaw.gsfc.nasa.gov/CME_list/), the EUV wave speeds are from the EUV wave catalogue compiled in Nitta et al. (2013).
RESULTS
Using the HK code we have constructed DEM maps for all the selected events. EM, density and temperature maps were derived from the DEMs as given in Equations 2-4. In addition, we derived base-ratio maps for each of the parameters using a pre-event image (30 min before the flare) as the base image. These maps were then used to characterize the plasma properties in the dimming region and their changes during the event. The differences between the core and secondary dimmings were studied and compared for all the events. Below we demonstrate our analysis in detail on two sample events: the 06 September 2011 event which was associated with the strongest flare in our list and a CME and EUV wave with high speed, and the event on 14 March 2012 which was associated with a slow CME. Thereafter, we summarize the main results obtained for the other four events, and add the related figures in the Appendix.
Event on 06 September 2011
The coronal dimming event of 06 September 2011 was associated with a X2.1 class flare in NOAA Active Region 11283, peaking at 22:12 UT, and a CME with a linear speed of 575 km s −1 in LASCO data. 1 There was also a strong EUV wave with a speed of 1250 km s −1 associated with this event, studied in detail in Dissauer et al. (2016).
To identify the core and secondary dimming regions, we use the aglorithm described in Dissauer et al. (2018). The dimming detection is based on thresholding and region growing techniques applied to sequences of logarithmic baseratio images. To identify locations of potential core dimmings within the overall dimming detections, the pixels with the smallest intensities reached over the whole event duration are extracted from so-called minimum intensity maps. The results from the dimming detection are shown in Figure 1. Here, the time evolution of the dimming region in the AIA 211Å channel (left panels), the corresponding logarithmically scaled base-ratio images (middle panels) as well as the dimming regions detected (right panels) are shown. Red (yellow) colors indicate the secondary (core) dimming regions identified. The first row in the image represents the state of the corona before the start of the event. Hence there is no dimming present yet. The second row shows the time step soon after the peak in the associated flare emission, where the dimming has already started to form. The third row shows an evolved dimming, with the number of dimming pixels increasing. This is close to the time step when the dimming area was at its maximum. In the last row, which is about 4 hrs after the start of the associated flare, we see that the number of pixels detected as dimming has reduced and the area of the dimming region has already shrunk, indicating that the corona has started to be replenished.
We also compared the detected dimming region with photospheric magnetic field maps as shown in Figure 2. These are LOS magnetic field data from the Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012) onboard SDO. The core dimming regions (left panel) are localized, and do not show much expansion through the progress of the event. They occur in pairs of opposite polarity within the active region where the magnetic field density is high. This scenario conforms to the interpretation that these regions mark the footpoints of the erupting flux rope (Sterling & Hudson 1997;Mandrini et al. 2005). The secondary dimming region (right panel), on the other hand, first expands, reaches a maximum and thereafter shrinks with time, and is located predominantly in regions of weak magnetic fields.
Using the HK code we constructed DEM maps from which we were able to derive EM, density and temperature maps. These maps are shown in Figure 3 together with the corresponding images from the AIA 211Å filter. The first panel under EM is replaced with magnetic field data which can give us some information about the magnetic structure within the dimming region. We can see that as the dimming evolves, there is a significant decrease in EM and density. There is also a decrease observed in the temperature maps but this is not as substantial as the other quantities.
To better see the changes in the derived quantities, we use base-ratio maps for enhancement. These images are shown in Figure 4. Each panel in Figure 4 corresponds to the same panel in Figure 3. From the dimming evolution in Figure 4 we clearly see a decrease in EM and density within the dimming region. The changes in these quantities closely reflect the changes seen in the AIA 211Å channel. On the other hand, the temperature maps show a decrease only in certain small regions. In many pixels, especially those around the region of the strong X2.1 flare, we see an increase in temperature. This effect in the temperature reconstruction from the DEMs most probably arises from strong scattered light in these pixels, caused by the bright flaring regions.
For detailed analysis we selected subregions within the core and secondary dimming and checked the shape of the DEM curves. The evolution of these DEM profiles are shown in Figure 5. Before the dimming region is formed, the DEMs in both the regions are quite high. The core dimming shows a double peak at log T [K] = 6.20 and log T [K] = 6.45, suggesting that plasma at quiet coronal temperatures as well as plasma at active region temperatures is included in the LOS of the selected core dimming region. The secondary dimming shows one broad peak at log T [K] = 6.25, indicative that most plasma along the LOS is at typical quiet coronal temperatures. As the dimming evolves, we see that the overall DEM curve diminishes drastically in both the regions. This indicates strong reduction of emission along the LOS due to depletion of plasma. In the core dimming region, it is noticeable, that the peak at log T [K] = 6.45 fully vanishes, indicating that active regions loops are ejected. In the secondary dimming region one can clearly see that not only the overall DEM distribution strongly reduces, but also that the peak moves to smaller temperatures, indicating plasma cooling associated with the density decrease (as is expected from adiabatic expansion).
To further study the evolution in the core and secondary dimming regions we derive lightcurves from the six EUV channels of AIA in the selected subregions. We study the evolution of the dimming region using the median of the lightcurves of pixels within the subregion. The selected regions of interest for core dimming (green) and secondary dimming (red) are marked in Figure 4, left. The lightcurves, normalized to the pre-event value for each filter, are plotted in Figure 6 along with the absolute deviation. They give direct evidence on the percentage change of the emission with respect to the state before the dimming. We can see from Figure 6 that in the core dimming region, all AIA channels show a strong reduction in intensity (up to about 75% decrease) immediately after the flare. We see the steepest changes within the first 10 min after the flare start. It then takes 20 more minutes to reach a minimum. The lightcurves do not recover from the decrease for at least the next 10 hrs covered by our analysis.
Within the secondary dimming region, the strongest decrease of ≈60-70% is seen in the 193Å, 211Å and 335Å channels. The other channels show a lesser decrease of ≈30-50%. The secondary dimming region is not as impulsive as the core dimming region. The lightcurves are seen to decrease and reach a minimum after about 30 min. However, unlike the core dimming region which stays at strongly reduced levels for the following 10 hrs, the intensity in the secondary dimming begins to increase again within 10-30 min after reaching its minimum.
We also studied the changes in the derived plasma parameters in a similar way and the results are shown in Figure 7. The EM and density in both the regions behave similar to the EUV lightcurves i.e. they drop abruptly after the flare in the dimming pixels. In the core dimming region there is a distinct density drop of 58% and temperature drop of 19%. The secondary dimming region shows lesser decrease than the core dimming region, same as we saw in the AIA lightcurves. There is a 40% decrease in density and a 8% decrease in temperature. The larger decrease in density in both the core and secondary dimming regions as compared to the decrease in temperature suggests that the dimming regions are mainly formed by plasma evacuation rather than by temperature changes.
Event on 14 March 2012
The dimming event of 14 March 2012 was associated with an M2.8 class flare that occurred in NOAA Active Region 11432 (GOES peak at 15:08 UT) and a CME with a speed of speed 411 km s −1 , which is the slowest CME in our selection. It also had a moderate EUV wave of speed 485 km s −1 associated with it.
In Figure 8 we show the time evolution of the dimming in AIA 211Å together with the EM, density and temperature maps. We see a clear coronal dimming formed, which is distinguishable from EM and density images but in the temperature images this region is partly obscured by the emission from the flare.
The DEM curves for this event are shown in Figure 9. We see that the curves peak at around log T [K] = 6.2 in both the secondary and core dimming regions. As the dimming evolves, the DEM curves show a distinct decrease. In particular in the core dimming region, the overall DEM distribution drastically diminishes, indicating that a large portion of the emitting plasma along the LOS is removed. The pre-event peak values of the DEM curves (T = 1.7 MK) indicate that most of the plasma that is removed is at quiet coronal temperatures.
The AIA lightcurves in Figure 10 show a similar trend as in the event on 06 September 2011. In these lightcurves, we first see a peak due to the flare. In the core dimming region, we see immediately after this peak a decrease of more than 80% in the lightcurves from all the AIA channels; the largest drop (90%) is observed in the AIA 193Å filter. This drastic decrease occurs within the first 30 min after the flare start, and the lightcurves do not recover for at least the next 10 hrs. The changes in the secondary dimming region evolve much slower than the core dimming region. The lightcurves reduce gradually, and take more than 2 hrs to reach a minimum. Thereafter, they start to slowly rise again and return to background levels. For this region, the maximum intensity drop of 54% is observed in the AIA 193Å channel.
The time evolution of the plasma parameters is shown in Figure 11. In the core dimming region, we observe a steep gradient in the density and temperature evolution for about 20 min, where the density decreases by 64% and the temperature by 9%. There is very little variation in these values for the next 10 hrs. Within the secondary dimming region, the plasma parameters take more than 2 hrs to reach their minimum. The density decreases by 22%, and after the minimum is reached eventually starts to climb to reach background levels. The temperature curve does not show any significant decrease from the background levels.
All other events
Here we briefly discuss also all the other events we have studied (cf. Table 1 ). The corresponding figures for these events are given in the Appendix.
The dimming event on 01 August 2010 was associated to a small C3.2 class flare peaking at 07:24 UT from NOAA Active Region 11092 and a CME with speed of 850 km s −1 . An EUV wave was also associated with it, with a speed of 312 km s −1 . Figure 12 shows images from the AIA 211Å filter and the corresponding maps of the plasma parameters derived, illustrating the evolution of the dimming region. In Figure 13, we show the AIA lightcurves, and in Figure 14 the time evolution of the plasma parameters derived. The strongest density decrease occurs within 10 min after the flare start, followed by a more gradual further decrease, reaching a maximum drop of 68% about 1 hr after flare start. The temperature curve shows some fluctuation during the flare time, which we attribute to uncertainties in the DEM reconstruction. Eventually it shows a steady decrease for about 2 hrs, attaining a maximum drop of 25%. In the secondary dimming region, the lightcurves evolve slowly and take about 40-60 min to reach a minimum. The percentage decrease of density and temperature in the secondary dimming region is 31% and 12%, respectively.
The dimming event on 21 June 2011 was associated with a C7.7 class flare from NOAA Active Region 11236 (peak time 01:22 UT) and a CME of speed 719 km s −1 . No EUV wave was associated with the event. Figure 15 shows the evolution of the dimming region. For this event the dimming region is initially obscured by loops along the LOS. This is characterized by the prolonged brightness in the AIA lightcurves and plasma parameters at the beginning of the event (see Figures 16 and 17). The loops finally open up to reveal a well formed dimming region. The changes in the core dimming region in this event are more gradual compared to the other events, which is due to the presence of loops. The density in the core dimming region reaches a minimum of 68%, whereas the temperature shows only a 6% decrease. In the secondary dimming region, the AIA lightcurves and the plasma parameters show many fluctuations that we attribute to the presence of loops along the LOS. The temperature curve does not show a clear decrease while the density curve reaches a minimum of 18%.
The dimming event on 19 January 2012 was associated with an M3.2 class flare peaking at 13:44 UT from NOAA Active Region 11402. This event was associated with the fastest CME within our event selection, with a speed 1120 km s −1 but no EUV wave was associated. The evolution of the dimming region is shown in Figure 18. From the AIA lightcurves and time evolution of the plasma parameters, shown in Figures 19 and 20 respectively, we see a pronounced dimming in both the core and secondary dimming regions. In the core dimming region, the fastest decrease occurs in the first 30 min followed by a more gradual decrease thereafter, with the density dropping by 51%. The secondary dimming reached a maximum drop in density by 45% in about 60 min. In the secondary dimming region, the reconstructed temperature shows a more or less steady decrease, whereas in the core dimming region there are some fluctuations in the beginning.
The dimming event on 09 March 2012 was associated with an M6.3 class flare peaking at 03:22 UT from NOAA Active Region 11429. This event was associated with a CME of speed 950 km s −1 and an EUV wave of speed 690 km s −1 . Figure 21 shows maps of the AIA 211Å filter and of the plasma parameters derived, illustrating the evolution of the dimming region. Lightcurves in the AIA channels and in the plasma parameters are shown in Figures 22 and 23, respectively. In this event, the plasma parameters show a rather gradual change. The density decreased by 52%, reached after about 4 hrs after the flare, while the temperature curve shows almost no decrease. The evolution of the secondary dimming region is also slow and takes several hours to reach a minimum. The density decreases by 10% but there is no clear decrease in the temperature curve.
In Tables 2 and 3, we summarize all the results obtained for the plasma parameters as well as for the changes in the AIA lightcurves for all six events under study. Table 2. The percentage decrease in the plasma parameters (EM, density, temperature) within the core and secondary dimming regions for all the events. Also given are the minimum values attained in density and temperature during the dimming evolution.
DISCUSSION
We have reconstructed DEM maps from SDO/AIA filtergrams for six CME-associated coronal dimming events to study the plasma characteristics and evolution in both the core and the secondary dimming regions. The most distinct difference we find is that in the core dimming region the plasma parameters show a much faster and deeper decrease than in the secondary dimming region, and stay at these low levels for the overall duration of our study. In most cases, the core dimming region shows the steepest changes within 20-30 min after the flare start, while the secondary dimming evolution tends to be more gradual and takes longer to reach its minimum (30-90 min). After reaching a minimum, within the core dimming region, there was not much change in the time evolution of the plasma parameters. They remained low and showed (almost) no signs of increase for the entire duration of this study ( 10 hrs after the flare). This is consistent with the findings that CMEs may keep their connection to the Sun over days, as is suggested from measuring bidirectional electron streams at 1AU (e.g., Bothmer et al. 1996). On the other hand, in the secondary dimming region we observe a gradual increase about 1-2 hrs after the minimum was reached, indicative of replenishment of these coronal regions with plasma after the CME eruption. These findings suggest that the core dimming region corresponds to the footpoints of the erupting flux rope where the magnetic field lines open to interplanetary space enable continuous outflow of plasma, preventing the refill of this region, while the secondary dimming corresponds to overlying fields higher up in the corona that expand due to the CME eruption.
In Table 2 we have summarized the results of the plasma parameters for all the events. We have calculated the percentage decrease of the minimum value attained by the plasma parameters as compared to the pre-flare background for all the events. We have listed the percentage decrease in EM, density and temperature separately for the core dimming and the secondary dimming regions for all the events. In addition we also list the absolute minimum values attained by density and temperature for both these regions. We found that within the core dimming region the decrease in EM is 80-90%, density decrease is 50-70% and temperature decrease is 5-25%. While within the secondary dimming we found the decrease in EM to be around 20-70%, density decrease around 10-45%; not all cases showed a decrease in temperature.
From the results summarized in Table 2 we see that for a given event the percentage decrease of each plasma parameter is significantly higher within the core dimming region than in the secondary dimming region. In some cases the changes within the core dimming region are almost twice more than the changes in the secondary dimming region. We also notice that for each event, the percentage decrease in temperature is much lower than the decrease in EM and density, indicating that the main cause of the dimming is plasma evacuation rather than temperature changes. Comparing the maximum density and temperature drops observed, we find that in all cases the temperature decreases are smaller than would be predicted from adiabatic expansion, T /T 0 = (n/n 0 ) γ−1 with γ = 5/3 for a fully ionized plasma (e.g. Aschwanden 2004).
However, at this stage, it is important to discuss the different limitations in the reconstructions of EM, density and temperature from the DEM analysis. In our analysis we have seen that some events do not reveal a clear temperature decrease or show strong fluctuations in the temperature evolution curves, even in cases where the reconstructed EM and density evolution showed a clear and smooth evolution. The estimates of EM and density (see Equations 2, 4) are determined from the integral over the whole DEM curve reconstructed, whereas the determination of the mean plasma temperature (see Equation 3) is strongly dependent on the shape of the DEM curve (Cheng et al. 2012). This explains why the reconstructed EM and density maps and curves are much more robust than the derived temperatures with regard to problems in the reconstructed DEM curves. Such problems may, e.g., arise due to uncertainties in the instrument response, due to saturation, blooming effects or scattered light from the bright flaring regions affecting the selected subegions in some filters at some time steps. Or, they may also arise, when very different structures (e.g. flare and dimming) are included in the same selected pixel (note that we have binned our maps by 8 × 8 AIA pixels for better signal-to-noise ratio). In the selection of our subregions in the core and secondary dimmings for detailed analysis, we tried to avoid areas that are affected by such effects, but we cannot assure this holds for all the AIA channels during all the time steps. Thus, we caution on those curves that do not show a smooth evolution but strong fluctuations.
The effects discussed above are much less severe in the pre-event state, i.e. before the dimming and the associated strong flare emission take place. Comparing the pre-event densities and temperatures in the core and secondary dimming regions, which characterize the plasma that is later expanding or ejected from this region, we find the following. In the core dimming, the pre-event densities are in the range 6 × 10 8 to 1.3 × 10 9 cm −3 and the pre-event temperatures 1.7-2.6 MK, whereas the corresponding values in the secondary dimming regions are 4 × 10 8 cm −3 to Table 3. The percentage decrease in emission in each of the AIA channels within the core and secondary dimming region for all the events. The largest and second largest values of decrease for each type of dimming are highlighted. Green corresponds to core and red to secondary dimming regions.
7×10 8 cm −3 and 1.6-2.0 MK, respectively. Systematically, in each event, both the pre-event densities and temperatures are smaller in the core than in the secondary dimming region. We also note that the event with the highest density and temperature (06 September 2011) showed a double component in the pre-event DEMs at log T [K] = 6.20 and log T [K] = 6.45 (cf. Figure 5). The higher temperature component fully disappeared during the dimming, and the lower temperature component also strongly reduced. This together with the high density derived for the core dimming region, 1.1 × 10 9 cm −3 , implies that in this event also hot and dense active region loops were ejected from the selected region in the core dimming region.
In order see in which of the AIA filters we can observe the dimming most clearly, we also analyzed the time evolution of the dimming regions in the different AIA filters, and identified which one reveals the highest percentage decrease compared to the pre-event intensities. The results for all the events are shown in Table 3. We found that within the core dimming region the highest decrease occurs in the 211Å and 193Å channels, with values of about 80 to 90%. For the secondary dimming regions we found that again the 193Å and 211Å filter reveal the strongest decrease, with values of about 40 to 75%, but for several events also the 171Å filter sensitive to cooler plasma shows strong decreases up to 50%. Other authors have also calculated percentage decrease in dimmings using the EIT 195Å and 171Å filters (Chertok & Grechnev 2005;McIntosh et al. 2007). They showed 40-50% decrease in the 195Å and up to 80% in the 171Å filter.
SUMMARY AND CONCLUSIONS
In this paper, we use SDO/AIA DEM analysis to study the plasma characteristics and evolution of coronal dimmings caused by CMEs. The six events under study are associated with CMEs of speeds ranging from 400-1100 km s −1 and flare classes from C3 to X2. We have constructed DEM maps and derived EM, density and temperature maps to study the time evolution of each dimming event over a duration of 12 hrs. Particular focus was drawn on the differences between the core and the secondary dimming regions. Our main results and related conclusions are the following:
1. For each event, the decrease in density is much larger in the core dimming than in the secondary dimming region. In both types of dimmings, the decrease in density is consistently more pronounced than the decrease in temperature. In the core dimming region, we observe reductions in density of 50-70% while the temperature drops by 5-25%, which is smaller than is expected from adiabatic expansion. In the secondary dimming region we observe density decreases of 10-45%, but not all events reveal a measurable temperature decrease.
2. In the core dimming region, the main changes in the plasma characteristics occur impulsively over a period of ≈20-30 min after the start of the associated flare. The core dimmings occur in opposite polarity regions, localized in the active region, and remain evacuated for extended periods of time ( 10 hrs). The secondary dimmings are more dispersed and start to recover already after 1-2 hrs indicating replenishment of the corona after the eruption.
3. In each event, the pre-event densities and temperatures are higher in the core than in the secondary dimming regions. The pre-event densities in the core dimming are in the range n = 6 × 10 8 to 1.3 × 10 9 cm −3 and the pre-event temperatures T = 1.7 − 2.6 MK, whereas for the secondary dimming regions we obtain n = 4 × 10 8 to 7 × 10 8 cm −3 and T = 1.6 − 2.0 MK. These values and systematic differences indicate that the plasma in the core dimming region stems from active regions, whereas the consistently lower densities and temperatures in the secondary dimming regions are indicative of plasma from overlying regions higher in the corona.
4. Both core and secondary dimming regions are best observed in the AIA 211Å and 193Å filters, with percentage decreases up to 80-90% in the core dimming and 40-75% in the secondary dimming regions with respect to the pre-event intensities.
Our results confirm that coronal dimmings associated with CMEs are formed due to plasma evacuation rather than temperature changes, and thus coronal dimmings provide indeed a valuable alternative means to estimate the mass ejected by CMEs (e.g. Harrison et al. 2003;Mason et al. 2014;Aschwanden 2016). In addition, we provide further support of the interpretation that the core dimming regions are formed at the location of the footpoints of the erupting flux rope. Our findings that the core dimming regions remain evacuated for 10 hrs is indicative of magnetic field lines open to interplanetary space, which enable continuous outflow of plasma and preventing the refill of this region -as it is expected for an erupting flux rope still connected to the Sun (e.g., Temmer et al. 2017). The secondary dimmings correspond to plasma from higher lying coronal regions that expand during the CME eruption, and start to be replenished already 1-2 hrs after the start of the event. Finally, we note that the gradual evolution and replenishment derived for the secondary dimming regions may also bear general implications and characteristic time scales for the formation of the corona.
Figure 1 .Figure 2 .
12Time evolution of the coronal dimming in SDO/AIA 211Å direct images (left) and logarithmic base-ratio images scaled from −0.4 to +0.4 (middle) for the event on 06 September 2011. In the right panels, we show the base ratio images overplotted with the instantaneously detected dimming regions (red pixels). The total extent of the core dimming region, as derived from the overall time series, is shown in the third panel (yellow pixels). SDO/HMI LOS magnetic field data (scaled to ±100 G) for the event on 06 September 2011 at a timestep close to the start of the flare. Contours of the core (left) and secondary (right) dimming regions identified at different timesteps (color coded, see legend) are superimposed. The timesteps are the same as inFigure 1.
Figure 3 .
3Time evolution of SDO/AIA 211Å images (first column), EM (second column), density (third column) and temperature (last column) for the event on 06 September 2011. The first panel of the second column shows the LOS magnetic field at a time close to the start of the flare scaled to ±100 G.
Figure 4 .
4Sequence of base-ratio maps visualizing the relative changes in the SDO/AIA 211Å emission (first column), EM (second column), density (third column) and temperature (last column) for the event on 06 September 2011. The base-ratio maps plotted here correspond to the direct maps shown inFigure 3. The green and red boxes indicate a location within the core and secondary dimming region, selected for detailed study.
Figure 5 .
5DEM curves, for the event on 06 September 2011, derived from a subregion in the core dimming region (left) and the secondary dimming region (right), as indicated by green and red boxes inFigure 4, respectively.
Figure 6 .
6Normalized lightcurves from the six SDO/AIA channels of the core dimming (left) and secondary dimming (right) for the event on 06 September 2011. Note that in the left panels the lightcurves are cut on the positive y-axis (dominated by the strong flare peak), to make the decrease due to the coronal dimming better visible. The region used for these lightcurves are indicated by green and red boxes inFigure 4, respectively.
Figure 7 .
7Time evolution of the plasma parameters within the core dimming (left) and secondary dimming (right) regions for the event on 06 September 2011. The left axes give the relative changes with respect to the pre-event, the right axes give the absolute values.
Figure 8 .
8Same as Figure 4 but for the event on 14 March 2012.
Figure 9 .
9Same as Figure 5 but for the event on 14 March 2012.
Figure 10 .
10Same asFigure 6but for the event on 14 March 2012. We note there is a short data gap around 19 UT.
Figure 11 .
11Same as Figure 7 but for the event on 14 March 2012.
This work is supported by the Austrian Space Applications Program of the Austrian Research Promotion Agency FFG (ASAP-11 4900217) and the Austrian Science Fund (FWF): P24092-N16.
Figure 13 .
13Same as Figure 6 but for the event on 01 August 2010.
Figure 14 .Figure 15 .
1415Same as Figure 7 but for the event on 01 August 2010. Same as Figure 4 but for the event on 21 June 2011.
Figure 16 .
16Same as Figure 6 but for the event on 21 June 2011.
Figure 17 .Figure 18 .
1718Same as Figure 7 but for the event on 21 June 2011. Same as Figure 4 but for the event on 19 January 2012.
Figure 19 .
19Same asFigure 6but for the event on 19 January 2012.
Figure 20 .Figure 21 .
2021Same as Figure 7 but for the event on 19 January 2012. Same as Figure 4 but for the event on 09 March 2012.
Figure 22 .
22Same as Figure 6 but for the event on 09 March 2012.
Figure 23 .
23Same as Figure 7 but for the event on 09 March 2012.
Table 1. List of coronal dimming events under study. ‡ CME speed from SOHO/LASCO catalog. † EUV wave speed fromNitta et al. (2013).Date
GOES flare class Flare location CME speed ‡
EUV wave †
(km s −1 )
speed (km s −1 )
01 August 2010
C3.2
N13 E21
850
312
21 June 2011
C7.7
N17 W21
719
-
06 September 2011
X2.1
N14 W18
575
1246
19 January 2012
M3.2
N28 E13
1120
-
09 March 2012
M6.3
N17 W13
950
689
14 March 2012
M2.8
N14 E01
411
485
We note that in this event the linear speed of the CME as observed by STEREO-A (EUVI-COR1-COR2) on the limb (v = 990 km s −1 ) is much larger than the LASCO plane-of-sky speed (v = 575 km s −1 ), indicative of strong projection effects(Dissauer et al. 2016).
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| []
|
[
"Enhancing correlation times for edge spins through dissipation",
"Enhancing correlation times for edge spins through dissipation"
]
| [
"Loredana M Vasiloiu \nSchool of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems\nUniversity of Nottingham\nNG7 2RDNottinghamUK\n",
"Federico Carollo \nSchool of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems\nUniversity of Nottingham\nNG7 2RDNottinghamUK\n",
"Juan P Garrahan \nSchool of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems\nUniversity of Nottingham\nNG7 2RDNottinghamUK\n"
]
| [
"School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems\nUniversity of Nottingham\nNG7 2RDNottinghamUK",
"School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems\nUniversity of Nottingham\nNG7 2RDNottinghamUK",
"School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems\nUniversity of Nottingham\nNG7 2RDNottinghamUK"
]
| []
| Spin chains with open boundaries, such as the transverse field Ising model, can display coherence times for edge spins that diverge with the system size as a consequence of almost conserved operators, the so-called strong zero modes. Here, we discuss the fate of these coherence times when the system is perturbed in two different ways. First, we consider the effects of a unitary coupling connecting the ends of the chain; when the coupling is weak and non-interacting, we observe stable long-lived harmonic oscillations between the strong zero modes. Second, and more interestingly, we consider the case when dynamics becomes dissipative. While in general dissipation induces decoherence and loss of information, here we show that particularly simple environments can actually enhance correlation times beyond those of the purely unitary case. This allows us to generalise the notion of strong zero modes to irreversible Markovian time-evolutions, thus defining conditions for dissipative strong zero maps. Our results show how dissipation could, in principle, play a useful role in protocols for storing information in quantum devices. | 10.1103/physrevb.98.094308 | [
"https://arxiv.org/pdf/1805.10060v1.pdf"
]
| 51,694,484 | 1805.10060 | c6b817b6f30054c4a9e64e72061a77c20355cfbe |
Enhancing correlation times for edge spins through dissipation
Loredana M Vasiloiu
School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems
University of Nottingham
NG7 2RDNottinghamUK
Federico Carollo
School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems
University of Nottingham
NG7 2RDNottinghamUK
Juan P Garrahan
School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems
University of Nottingham
NG7 2RDNottinghamUK
Enhancing correlation times for edge spins through dissipation
(Dated: May 28, 2018)
Spin chains with open boundaries, such as the transverse field Ising model, can display coherence times for edge spins that diverge with the system size as a consequence of almost conserved operators, the so-called strong zero modes. Here, we discuss the fate of these coherence times when the system is perturbed in two different ways. First, we consider the effects of a unitary coupling connecting the ends of the chain; when the coupling is weak and non-interacting, we observe stable long-lived harmonic oscillations between the strong zero modes. Second, and more interestingly, we consider the case when dynamics becomes dissipative. While in general dissipation induces decoherence and loss of information, here we show that particularly simple environments can actually enhance correlation times beyond those of the purely unitary case. This allows us to generalise the notion of strong zero modes to irreversible Markovian time-evolutions, thus defining conditions for dissipative strong zero maps. Our results show how dissipation could, in principle, play a useful role in protocols for storing information in quantum devices.
I. INTRODUCTION
Recent results have demonstrated the possibility of observing in many-body quantum chains with open boundary conditions coherence times for edge spins that diverge exponentially with the size of the system [1][2][3][4][5][6]. One of the interests in this phenomenon stems from the possibility of storing and protecting the information encoded in quantum states for very long times, with possible applications in future quantum technologies [7,8]. From this perspective, for the applicability of long coherence times for boundary degrees of freedom -so far only established in isolated (unitary) quantum systems [1,2] -the effect of the inevitable interaction of the system of interest with the surrounding environment must be considered. This is the problem we address in this paper.
Acting as a source of dissipation and noise, the interaction of a system with an environment usually leads to suppression of truly quantum features in the system and to the emergence of classical-like behavior. However, in some instances it has been shown that engineered systemenvironment couplings can actually enhance or even generate quantum correlations, such as entanglement [9][10][11][12][13][14][15][16][17][18]. While robustness of coherence times for edge spins has been extensively studied in the presence of interactions or of integrability-breaking terms and also in the presence of disorder in the Hamiltonian [1,2,19], less is known about their behavior under irreversible open quantum dynamics [20][21][22][23][24]. In particular, it is not clear whether under Markovian (memory-less) dynamics these long edge time-correlations can be observed and general conditions for their existence are not known. Understanding the behavior of these time-correlations in open quantum systems would allow for the possibility of exploiting the protected information encoded in edge spins for applications in quantum devices also in realistic non-equilibrium settings.
Long coherence times of edge spins are due to the pres- , localized at the two edges, perturbed either by weakly closing the chain (blue arrow) or by dephasing one edge (red box). (b) Behavior in time of the infinite temperature time-correlations of the z-magnetization of the first spin C∞(t), given by Eq. (7), when the edges are joined through a non-interacting Hamiltonian perturbation (λ = 0). Coherent oscillations between the two strong zero modes take place on faster time-scales than those of the free boundary case (λ = 0). (c) In the presence of dephasing (Γ = 0) acting only on one edge and λ = 0 oscillations between the strong zero modes are suppressed; however the correlation time for the spin at the edge is doubled in order of magnitude. In the plot time is represented on a log-scale.
ence of almost conserved operators -the so-called strong zero modes (SZM) [1][2][3][4][5][6]. The simplest example is that of the transverse field Ising model (TFIM) with open boundaries where the SZMs operators can be written down explicitly in compact form [1,3]. For finite sized systems, the SZMs are coupled by the dynamics but their evolution happens on very slow time-scales, so that initial information stored in the edge spins is preserved for arXiv:1805.10060v1 [cond-mat.stat-mech] 25 May 2018 times which are exponential in the system size [1].
Here we consider the effect of perturbations away from the optimal conditions for the dynamical protection of the SZMs. One can think of two possible scenarios. The first one, illustrated in Fig. 1(a), consists of connecting the two ends of an open chain via a unitary perturbation. Figure 1(b) illustrates our main result in this case: when the Hamiltonian perturbation is non-interacting (in a way we specify below) we observe macroscopic coherent oscillations between the two SZMs, occurring on faster time-scales than in the disconnected case [cf. Fig. 1(c)] as the two ends are, in this case, directly connected.
More interestingly, we then address the case in which the system undergoes a dissipative Markovian dynamics induced by the presence of an external environment weakly interacting with it. Not only we show that Markovian quantum dynamics can sustain finite yet long correlation times, but also that suitably engineered -but still relatively simple -environments can increase coherence times by orders of magnitude with respect to the equilibrium closed system scenario. Figure 1(c) illustrates this result. In fact, these correlation times diverge with system size with a faster exponential rate in the dissipative case than in the purely coherent case.
Indeed, as we will show, specific type of dissipations can induce fast decoherence effects affecting solely one of the two localized operators: coherent oscillations are suppressed but the information stored in one of the two edges is protected for much longer time-scales than the unitary ones. In particular, even a small perturbation, such as dephasing on the last site of the chain, can double in order of magnitude the correlation time for the first spin [see Fig. 1(c)]. While for finite systems these correlations will eventually decay, in the thermodynamic limit they can persist for infinitely long times also in the presence of dissipation.
We mainly focus on the TFIM [25,26] allowing for numerical diagonalization of large system size and amenable analytical considerations; nonetheless, we expect our findings to be very general and we show how they apply also for a dissipative interacting spin chain featuring almost conserved operators -the XYZ chain [2,27].
Finally, we introduce the conditions for the existence of strong zero maps, which generalize the notion of SZMs to irreversible Markovian quantum dynamics. These are obtained exploiting the possible presence of symmetry sectors in the dynamical generator of the open quantum dynamics [28,29].
II. STRONG ZERO MODES IN THE TFIM
The simplest many-body system where the emergence of long correlation times for the edge spins can be observed is the TFIM with free boundary conditions [25,26]. It consists of a chain of two-level systems de-scribed by the Hamiltonian
H = −J L−1 i=1 σ z i σ z i+1 − h L i=1 σ x i ,(1)
where σ β i , with β = x, y, z, represents the β Pauli matrix corresponding to the i-th site. The parameter J embodies the strength of magnetic interactions between neighboring sites, while h is a transverse magnetic field in the x direction. This Hamiltonian describes non-interacting fermions. Indeed, by introducing the Majorana operators
γ A i = σ z i i−1 k=1 σ x k , γ B i = σ y i i−1 k=1 σ x k ,(2)such that {γ X i , γ Y j } = 2δ X,Y δ i,j ,
the Hamiltonian can be written as
H = −iJ L−1 i=1 γ B i γ A i+1 − ih L i=1 γ A i γ B i .(3)
Introducing the parity operator
P = L i=1 σ x i ,(4)
it is immediate to check that [H, P ] = 0, implying that the eigenvectors of the Hamiltonian, as well as its eigenvalues, can be divided into even and odd sectors. Interestingly, in the magnetically ordered phase |h| < |J|, the even and odd parts of the Hamiltonian spectrum become identical exponentially fast in the system size L. This can be shown by introducing the so-called SZMs [1][2][3][4][5][6]
φ 1 = C L i=1 h J i−1 γ A i , φ L = C L i=1 h J L−i γ B i(5)
where the constant
C 2 = [1 − (h/J) 2 ]/[1 − (h/J) 2L ]
normalizes the operators in such a way that φ 2 1/L = 1. First of all, one notices that φ 1 and φ L anticommute with the parity operator P , and almost commute with the Hamiltonian H,
[H, φ 1/L ] = ±2iCJ (h/J) L γ B/A L/1 ,(6)
i.e., these commutators decay exponentially with system size. Therefore, with |ψ i being an even (odd) eigenvector of the Hamiltonian associated to the eigenvalue i , one has that φ 1 |ψ i is a vector which is odd (even) under parity transformation. Then, because [H, φ 1 ] ≈ 0 for large L, one has
Hφ 1 |ψ i ≈ i φ 1 |ψ i ,
showing that i becomes, in the thermodynamic limit, an eigenvalue associated to both an even and an odd eigenvector. These two vectors are mapped one onto the other by the action of the SZM. These properties of the Hamiltonian have remarkable practical consequences: since the operators are localized at the two ends of the chain when |h| |J|, the information in the boundary sites can be stored for times which are exponentially long in the system size [1]. Even more interestingly, this can actually be observed at hightemperatures as witnessed by the (infinite temperature) time-correlations
C ∞ (t) := 1 2 L Tr e −iHt σ z 1 e iHt σ z 1 .(7)
III. WEAK BOND IN THE RING GEOMETRY FOR THE TFIM
While the nice properties discussed above are completely lost when considering the same model in the ring geometry (i.e. the system is made translationally invariant through periodic boundary conditions), a question one may ask is what occurs in intermediate regimes that interpolate between the free boundary case with its long coherence times, and the ring geometry where they are absent. To this aim we introduce a Hamiltonian perturbation H b to the TFIM Hamiltonian of Eq. (1), consisting of a term directly coupling the ends of the chain, as illustrated in Fig.1(a); the unitary time-evolution is then
governed byĤ = H + λH b .
First, we consider a non-interacting perturbation H b = −iJγ B L γ A 1 : in this situation, for small λ, as we display in Fig.1(b), the time-correlations C ∞ (t) manifest stable macroscopic oscillations. This feature emerges from the fact that the Hamiltonian perturbation connects harmonically the modes φ 1/L , now with a finite (i.e. not decreasing with the system size) frequency that is isolated from the region of the spectrum that becomes dense in the large L limit. Thus, while correlations of all other modes are rapidly washed out by destructive interference, the two SZMs display stable long-lived oscillations with frequency ω λ = 2λJ. When λ ≈ 1, ω λ enters in the continuous region of the spectrum and the stable oscillations fade away. Completely different is the scenario in which boundary sites of the Ising chain are connected through the Hamiltonian H b = −Jσ z L σ z 1 ; in this case, the firstorder perturbation correction vanishes and thus the modification to the evolution comes as a second-order map determining a decay of the correlation function C ∞ (t) [see Fig. 2].
IV. DISSIPATIVE DYNAMICS IN THE OPEN CHAIN TFIM AND ENHANCEMENT OF CORRELATION TIMES
The results of the previous section concern instances of unitary quantum dynamics, thus describing a system which is perfectly isolated from its thermal surrounding. In this case, we see that C∞(t) decays instead of undergoing macroscopic oscillations as shown in Fig. 1(b). For increasing system sizes the characteristic time of this decay increases, however it does not show an exponential dependence on L.
This situation is an idealized one and, in order to account for more realistic settings, one needs to consider open quantum evolutions [20][21][22][23][24]. For simplicity, we will consider the case of an environment weakly interacting with the system. Usually, such interaction leads to loss of quantum coherence and of quantum correlations. However, as we now discuss, certain open quantum dynamics can still feature long correlation times for edge spins as a consequence of the existence of SZMs.
In the Markovian regime, the irreversible evolution of system observables is generated byẊ t = L * [X t ] where L * is the dual of the Lindbladian map L [20,21,23], which generates the dynamics of the state of the system, ρ, throughρ = L[ρ]. The dual map has the following structure
L * [X] := i[H, X]+ k L † k XL k − 1 2 L † k L k , X .(8)
In the generator above, the first term represents the coherent part of the dynamics while the terms in the sum encode noisy effects due to the environment. We now assume that the presence of the environment induces decoherence in the spin chain, which can be described by Lindblad operators
L i = Γ i σ z i ,(9)
with Γ i being the dephasing rate at site i. As one would expect, when dephasing acts uniformly on the whole system, Γ i = Γ, the spin-chain cannot sustain correlation times that grow with system size [see Fig.3(a)]. Nevertheless, with Γ small, the presence of SZMs of the coherent case, leads to correlations times for the boundary spins which, while finite, are longer than those where SZMs are not present (for instance when h = J). When Γ 1 the physics changes, and timecorrelations increase irrespective of the presence or the In the presence of dissipation only on a small portion of the chain, α < 1, we observe that C∞(t) can decay later than in the unitary case (data shown for the same parameters J, h and L as in (a) with Γ = 1 and different α.) absence of SZMs in the unitary case. This corresponds to a quantum Zeno regime, with characteristic decay time given by τ z ≈ Γ/(2h 2 ), which can be obtained straightforwardly from second-order perturbation theory.
A very different scenario emerges when one considers dissipation only on a fraction α of the chain,
Γ i = 0 i < L(1 − α) Γ i = Γ i ≥ L(1 − α)(10)
as illustrated in Fig.3(b). Also in this dissipative case we see that characteristic decay times for C ∞ (t) are exponentially large with the size of the system. This is a surprising result suggesting that SZMs and infinitely long time-correlations can be observed in open quantum dynamics. However, there is a much more interesting effect that we can observe: not only, as we already pointed out, SZMs can exist in dissipative settings, but, strikingly, the characteristic decay time of time-correlations associated to their existence can be enhanced by the presence of an external environment. This is clearly displayed in Fig. 3(b): decreasing the portion of the chain affected by dephasing -which we denote by α ∈ [0, 1], cf. (10)we can see that the time-correlations C ∞ (t) stay almost invariant for larger times and, interestingly, for certain values of α these can be much greater than the unitary characteristic time τ u , where τ u is the first instance when C ∞ (t) = 1/e. In order to understand this feature we consider the extreme case in which only the last site of the chain is subject to dephasing. Notice that this represents, in the large L limit, a perturbation to the unitary timeevolution which is infinitely far apart from the SZM φ 1 which is instead localized around the first edge. In this scenario, considering a small transverse field h, the effective "slow" dynamics of the SZMs is given, at the largest order in h, by the following system of differential equations d dt
φ 1 φ L = 0 −2J h J L 2J h J L −2Γ φ 1 φ L .(11)
Computing the eigenvalues of the matrix in the above
equation one finds ∆ ± = −Γ ± Γ 2 − 4J 2 (h/J) 2L ; for
increasing L one always reaches a regime where Γ (h/J) L and thus we can expand the square root in order to obtain a prediction for the characteristic decay time
log τ ≈ log Γ 2J 2 + 2L log J h .(12)
Noticing that for the unitary case one expects log τ u ∝ L log(J/h) [1], we immediately see how Eq.(12) predicts, for the dissipative case under investigation, a characteristic time for the edge correlation function C ∞ (t) which is doubled in order of magnitude with respect to the unitary evolution. In Fig. 4 we show how the prediction in Eq.(12) is confirmed by numerical results. While it is remarkable that a small modification of the dynamics affecting a site far apart from the first edge can have such a strong effect on C ∞ (t), the same scaling of the correlation times can also be found for a dissipative dynamics affecting the whole chain and described by jump operators given by σ z k σ y k+1 for k = 1, 2, . . . L − 1.
V. DISSIPATIVE STRONG ZERO MAPS
In this section we now proceed to a formal definition of SZMs in dissipative contexts. This is achieved by promoting the notion of SZM from an operator acting on vectors to that of a map acting on operators. Let us consider a Lindblad generator L as in equation (8) and assume that it commutes with a map implementing a discrete symmetry transformation on the system operators. Given the generator S of the symmetry, the map can be written as π s [X] = SXS; thus if the operator X is even under this transformation one has π s [X] = X, while if it is odd one has π s [X] = −X. The fact that the Lindblad operator commutes with the transformation L * •π s = π s •L * means that one can divide eigenmatrices and eigenvalues of L into even and odd sectors. If there exists a map Ψ commuting, up to exponentially small corrections in system size, with the Lindblad generator and anti-commuting with the parity transformation
L * • Ψ[X] ≈ Ψ • L * [X] , ∀X ,(13)Ψ • π s = −π s • Ψ ,(14)
then we can show that the even part of the spectrum of the Lindblad operator L is exponentially (in system size) close to the odd part. Indeed, considering X + to be an even eigenmatrix of L, namely π s (X + ) = X + ,
L * [X + ] = + X + ,(15)
one has that Ψ[X + ] is an odd operator under the transformation, π s • Ψ[X + ] = −Ψ[X + ], and, because of the almost commutation of Ψ with L, also
L * Ψ[X + ] ≈ Ψ L * [X + ] = + Ψ[X + ] .
This shows that Ψ[X + ] is, in the thermodynamic limit, an odd eigenmatrix of L with eigenvalue + , meaning that the odd and even parts of the spectrum of the Lindblad operator are indeed paired. In order for the above derivation to be meaningful it is important that Ψ[X] = X , ∀X or, at least, that the norm Ψ[X] is neither zero nor divergent for all operators X. As a consequence of this pairing, with dissipative quantum dynamics (and not just coherent dynamics), correlation times of system observables which diverge exponentially with system size can be observed at infinite temperature.
For In the examples of the previous section, the strong zero map is given by Ψ[X] = φ 1/L X or equivalently by Ψ[X] = Xφ 1/L , and S = P .
VI. STRONG ZERO MAP IN AN INTERACTING SYSTEM
The construction of the dissipative strong zero maps (DSZMs) of the previous section is generic and does not depend on a specific model. So far, however, we have only discussed the case of the TFIM which is in effect a system of non-interacting fermions. To show the more general applicability, we now study the case of an XYZ chain with open boundaries which, in contrast to the TFIM, corresponds to an interacting system.
The Hamiltonian we consider is
H XY Z = L−1 k=1 β=x,y,z J β σ β k σ β k+1 ,(17)
with couplings J β which in principle can be different for the three components β = x, y, z. Furthermore, we consider dissipative dynamics, Eq. (8), with dephasing acting on the last site of the lattice, that is, with jump operators (9) with Γ i = Γδ i,L . Figure 5 shows the comparison between the boundary correlation function C ∞ (t) for the case where dynamics is unitary generated by Eq. (17) and the case where there is dephasing on the last site. From Ref. [2] we know that in the coherent case, the existence of a SZM in the XYZ model gives rise to long coherence time (see black curve in Fig. 5), for analogous reasons to what occurs in the TFIM case. The corresponding correlator in the dissipative case is shown as the red curve in Fig. 5. Due to the existence of a DSZM, as in the TFIM case (cf. Fig. 3), correlation times are significantly enhanced due to dissipation at the opposite edge. The inset to Fig. 5 show that indeed the (complex) spectrum of the Lindbladian is paired (approximately for the finite system shown, but the splitting will vanish exponentially in system size) between even and odd eigenstates of L.
VII. CONCLUSIONS
We have considered the stability of strong zero modes in quantum chains with open boundaries. Our central result is that local dissipation can significantly enhance the correlation time of boundary spins. On the one hand this is surprising, as dissipation often acts to suppress memory of initial states as it tends to open up relaxation channels beyond those existing in its absence. We find that for both the non-interacting transverse field Ising model and for the interacting XYZ model, dephasing at one end of an open chain increases the correlation time of spins at the opposite end. This observation led us to define disspative strong zero maps, corresponding to superoperators that play the role in the case of dissipative (and Markovian) dynamics of the strong zero mode operators present in the unitary case. It would be interesting to probe our results by means of chains of Rydberg atom or trapped ion systems [30][31][32][33][34].
FIG. 1 .
1(a) Pictorial representation of a quantum spin chain featuring almost conserved operators, (φ1, φL)
FIG. 2 .
2Behavior of the time correlation function C∞(t) in the unitary case (J = 1 and h = 0.1) when joining the edges of the chain through the Hamiltonian H b = −Jσ z L σ z 1 with a small perturbative parameter λ = 0.1.
FIG. 3 .
3Transverse field Ising model subject to dephasing on a fraction α of the chain starting from the last site. (a) For α = 1 the infinite temperature time-correlation C∞ (t) decays much faster than in the unitary case (the plot is for J = 1, h = 0.2, L = 8 and different Γ). When Γ 1 the time-scales of the decay of C∞(t) are dominated by the Zeno effect and the presence or not of a strong zero mode is irrelevant. (b)
FIG. 4 .
4Scaling of characteristic time of edge correlations in the unitary caseBehavior of the logarithm of the characteristic time of the correlation functions C∞(t) as a function of x = L log(J/h). In the unitary case the scaling is characterized by slope equal to one (log τ ≈ x). Remarkably, the presence of dephasing affecting the last site of the chain, enhances the scaling of log τ which clearly shows, in this case, a behavior log τ ≈ 2x, meaning that correlation time is doubled in order of magnitude with respect to the unitary case. In the plot we fix J = 1 and explore different values of L, h and Γ.
FIG. 5 .
5Time correlation function C∞(t) for the XYZ chain (L = 8) with Jx = 0.2, Jy = 0.3, Jz = 1 in the unitary case as well as with dephasing Γ = 1 on the right boundary. Also for this interacting Hamiltonian we see that the presence of dephasing enhances the characteristic correlation time for the first spin. In the inset, we display a portion of the spectrum for the XYZ chain (L = 6) with Jx = Jy = 0.1 and Jz = 1: crosses represent eigenvalues from the odd sector while squares are from the even one. We found the largest distance between corresponding eigenvalues in the two sectors to be of order 10 −4 .
example, considering the operator Ψ[1] where 1 is the identity, one has 1 2 L Tr e tL * Ψ[1] † Ψ[1] ≈ 1 2 L Tr Ψ e tL * [1] †Ψ[1]
=
1
2 L Tr Ψ[1]
†
Ψ[1] .
ACKNOWLEDGMENTSThe research leading to these results has received
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| []
|
[
"Comment on \"Photon energy and carrier density dependence of spin dynamics in bulk CdTe crystal at room temperature\"",
"Comment on \"Photon energy and carrier density dependence of spin dynamics in bulk CdTe crystal at room temperature\""
]
| [
"J H Jiang \nHefei National Laboratory for Physical Sciences at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n",
"M W Wu \nHefei National Laboratory for Physical Sciences at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n"
]
| [
"Hefei National Laboratory for Physical Sciences at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Hefei National Laboratory for Physical Sciences at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina"
]
| []
| We comment on the conclusion by Ma et al. [Appl. Phys. Lett. 94, 241112 (2009)] that the Elliott-Yafet mechanism is more important than the D'yakonov-Perel' mechanism at high carrier density in intrinsic bulk CdTe at room temperature. We point out that the spin relaxation is solely from the D'yakonov-Perel' mechanism. The observed peak in the density dependence of spin relaxation time is exactly what we predicted in a recent work [Phys. Rev. B 79, 125206 (2009)]. | 10.1063/1.3371817 | [
"https://arxiv.org/pdf/0910.1506v1.pdf"
]
| 103,432,249 | 0910.1506 | c6f19a0e358cba798e8e51ae3e73de7efd65961e |
Comment on "Photon energy and carrier density dependence of spin dynamics in bulk CdTe crystal at room temperature"
8 Oct 2009 (Dated: October 8, 2009)
J H Jiang
Hefei National Laboratory for Physical Sciences at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiChina
M W Wu
Hefei National Laboratory for Physical Sciences at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiChina
Comment on "Photon energy and carrier density dependence of spin dynamics in bulk CdTe crystal at room temperature"
8 Oct 2009 (Dated: October 8, 2009)
We comment on the conclusion by Ma et al. [Appl. Phys. Lett. 94, 241112 (2009)] that the Elliott-Yafet mechanism is more important than the D'yakonov-Perel' mechanism at high carrier density in intrinsic bulk CdTe at room temperature. We point out that the spin relaxation is solely from the D'yakonov-Perel' mechanism. The observed peak in the density dependence of spin relaxation time is exactly what we predicted in a recent work [Phys. Rev. B 79, 125206 (2009)].
In a recent Letter, 1 Ma et al. measured the density dependence of electron spin relaxation time in intrinsic bulk CdTe at room temperature. They found that the electron spin lifetime first increases then decreases with increasing excitation density. They attributed the increase of spin lifetime at low excitation density to the D'yakonov-Perel' mechanism whereas the decrease at high excitation density to the Elliott-Yafet mechanism. They concluded that the Elliott-Yafet mechanism dominates spin relaxation at high excitation density in CdTe at room temperature.
Their conclusion can not be correct. As shown in our recent work, the Elliott-Yafet mechanism is unimportant even in narrow band gap semiconductors such as InAs and InSb for n-type and intrinsic samples in metallic regime. 2 For CdTe, which has a large band gap of E g = 1.45 eV, the Elliott-Yafet mechanism can not be important for intrinsic samples, especially at such high temperature of 300 K.
Below, through a fully microscopic calculation, we show that the Elliott-Yafet mechanism is totally irrelevant to spin relaxation under the experimental condition. The calculation is based on the fully microscopic kinetic spin Bloch equation approach with all relevant scatterings, such as electron-impurity, electron-phonon, electron-electron and electron-hole scatterings, explicitly included. 2 The spin-flip process due to the Elliott-Yafet mechanism is fully incorprated in all these scatterings. The calculation based on kinetic spin Bloch equation approach has achieved good agreements with different ex-
periments [e.g., see Appendix A of Ref. 2].
The calculation of the spin relaxation due to the Elliott-Yafet mechanism is based on the following spinflip scattering,
Γ s (k) = 2 k ′ 1 τ p (k → k ′ ) |Λ ↑↓ k,k ′ | 2 (1) where 1 τp(k→k ′ ) is the momentum scattering rate from state k to state k ′ and Λ ↑↓ k,k ′ = −iλ c (k × k ′ ) · σ ↑↓ . 3 Here λ c = 2 η(1−η/2) 3mcEg(1−η/3) with η =∆SO
∆SO+Eg . 3 m c is the conduction band effective mass. E g and ∆ SO are the band-gap and the spin-orbit splitting of the valence band, respectively. The momentum scattering rate is determined by all relevant scatterings, such as the electron-impurity, electron-phonon, electron-electron and electron-hole scatterings: 1 τp = 1 τei + 1 τep + 1 τee + 1 τ eh . The corresponding scatterings are given in detail in Ref.
2.
The spin relaxation time τ s is then obtained by average over Γ s (k), 1/τ s = Γ s (k) . It is noted that there is no fitting parameter in the calculation. The material parameters of CdTe are taken from the standard handbooks of From the parameter-free fully microscopic calculation, we obtain the spin lifetime limited by the Elliott-Yafet mechanism, τ s 800 ps in the excitation density range of 10 14 to 10 17 cm −3 . This is at least two-orders of magnitude larger than the one observed in the experiment by Ma et al. 1 Hence the Elliott-Yafet mechanism is totally irrelevant under the experimental condition. The spin relaxation is then solely determined by the D'yakonov-Perel' mechanism.
As we have pointed in a recent work, 2 the density dependence of spin relaxation time due to the D'yakonov-Perel' mechanism τ s ∼ 1/[ Ω(k) 2 τ p ] is nonmononotic in intrinsic bulk III-V semiconductors: spin relaxation time increases with increasing density in non-degenerate regime due to decrease of momentum scattering time τ p but decreases in degenerate regime due to the enhancement of inhomoegeneous broadening Ω(k) 2 . There is a peak in the crossover regime. For II-VI semiconductors with zinc-blende structure, the spin-orbit coupling and the band structure is similar to III-V semiconductors. Hence the same behavior is also expected. Actually, the band and material parameters of CdTe are very similar to GaAs. In intrinsic GaAs at room temperature, the peak density is 9 × 10 16 cm −3 . 2 In the experiment by Ma et al., the peak density is 3 × 10 11 cm −2 . As the authors did not determine the penetration depth of the laser, a rough estimation gives the peak density of 6 × 10 16 cm −3 which is close to the one in GaAs. This indicates that the observed peak in density dependence of spin lifetime should be samilar to what we have predicted in III-V semiconductors. 2 However, due to the uncertainty in the penetration depth and the possible effect of hot-electron effect of the photo-excited carriers (as also indicated by the experimental results in the photon en-
F.Meier and B. P. Zakharchenya, Optical Orientation (North-Holland, Amsterdam, 1984). 4 Semiconductors,Landolt-Börnstein, New Series, Vol. 17b, ed. by O. Madelung (Springer-Verlag, Berlin, 1987).
This work was supported by the Natural Science Foundation of China under Grant No. it is premature to give a quantitative comparisondependence), it is premature to give a quantitative comparison. This work was supported by the Natural Science Foun- dation of China under Grant No. 10725417.
Author to whom correspondence should be addressed. Electronic address: [email protected]* Author to whom correspondence should be addressed; Elec- tronic address: [email protected].
. H Ma, Z Jin, G Ma, W Liu, S H Tang, Appl. Phys. Lett. 94241112H. Ma, Z. Jin, G. Ma, W. Liu, and S. H. Tang, Appl. Phys. Lett. 94, 241112 (2009).
. J H Jiang, M W Wu, Phys. Rev. B. 79125206J. H. Jiang and M. W. Wu, Phys. Rev. B 79, 125206 (2009).
| []
|
[
"The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion",
"The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion"
]
| [
"Hakima Bessaih ",
"Benedetta Ferrario "
]
| []
| []
| In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized through a smoothing kernel of order α in the nonlinear term and with a β-fractional Laplacian; we are in the critical case α + β = 5 4 . The temperature equation is a pure transport equation. We prove regularity results when the initial velocity is in H r and the initial temperature is in H r−β for r > max 5 2 − 2α, β + 1 with β ≥ 1 2 and α ≥ 0. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions with respect to the initial conditions. | 10.1016/j.jde.2016.10.032 | [
"https://arxiv.org/pdf/1504.05067v2.pdf"
]
| 119,313,022 | 1504.05067 | 1b7811b4f3a422105fa933100e4f87649c68f0c6 |
The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion
20 Apr 2015 April 21, 2015
Hakima Bessaih
Benedetta Ferrario
The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion
20 Apr 2015 April 21, 2015arXiv:1504.05067v1 [math.AP]Boussinesq equationsLeray-alpha modelsTransport equationCommutators Mathematics Subject Classification 2010: Primary 35Q3576D03; Sec- ondary 35Q86
In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized through a smoothing kernel of order α in the nonlinear term and with a β-fractional Laplacian; we are in the critical case α + β = 5 4 . The temperature equation is a pure transport equation. We prove regularity results when the initial velocity is in H r and the initial temperature is in H r−β for r > max 5 2 − 2α, β + 1 with β ≥ 1 2 and α ≥ 0. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions with respect to the initial conditions.
Introduction
We consider the Boussinesq system in a d-dimensional space:
(1.1) ∂ t v + (v · ∇)v − ν∆v + ∇p = θe d ∂ t θ + v · ∇θ = 0 ∇ · v = 0
where v = v(t, x) denotes the velocity vector field, p = p(t, x) the scalar pressure and θ = θ(t, x) a scalar quantity, which can represent either the temperature of the fluid or the concentration of a chemical component; e d is the unit vector (0, . . . , 0, 1), the viscosity ν is a positive constant. Suitable initial conditions v 0 , θ 0 and boundary conditions (if needed) are given.
For d = 2, the well posedness of system (1.1) in the plane has been studied by several authors under different assumptions on the initial data (see [12,7,1,11,8,9]). For d = 3, very little is known; it has been proven that there exists a local smooth solution. Some regularity criterions to get a global (in time) solution have been obtained in [20,10]. Otherwise, in the particular case of axisymmetric initial data, [2] shows the global well posedness for the Boussinesq system in the whole space.
To overcome the difficulties of the three-dimensional case, different models have been proposed. For instance, one can regularize the equation for the velocity by putting a fractional power of the Laplacian; this hyper-dissipative Boussinesq system takes the form
(1.2) ∂ t v + (v · ∇)v + ν(−∆) β v + ∇p = θe 3 ∂ t θ + v · ∇θ = 0 ∇ · v = 0
For β > 5 4 , [26] proved the global well posedness. This result has been improved by Ye [25], allowing β = 5 4 . Notice that for zero initial temperature θ 0 , the Boussinesq system reduces to the Navier-Stokes equations. It is well known that the three-dimensional Navier-Stokes equations have either a unique local smooth solution or a global weak solution. The questions related to the local smooth solution being global or the global weak solution being unique are very challenging problems that are still open since the seminal work of Leray. For this reason, modifications of different types have been considered for the threedimensional Navier-Stokes equations. On one side there is the hyper-viscous model, i.e. (1.2) with zero initial temperature; when β ≥ 5 4 , uniqueness of the weak solutions has been proved in [16] (see Remark 6.11 of Chapter 1) and [17] . On the other hand, Olson and Titi in [19] suggested to regularize the equations by modifying two terms. For a particular model of fluid dynamics, they replaced the dissipative term by a fractional power of the Laplacian and they regularized the bilinear term of vorticity stretchingà la Leray. The well posedness of those equations is obtained by asking a balance between the modification of the nonlinearity and of the viscous dissipation; at least one of them has to be strong enough, while the other might be weak. Similarly, Barbato, Morandin and Romito in [4] considered the Leray-α Navier-Stokes equations with fractional dissipation
(1.3) ∂ t v + (u · ∇)v + ν(−∆) β v + ∇p = 0 v = u + (−∆) α u ∇ · u = ∇ · v = 0
and proved that this system is well posed when α + β ≥ 5 4 (with α, β ≥ 0); even some logarithmic corrections can be included, but we do not specify this detail, since it is not related to our analysis. It is worth mentioning the result of the current authors with Barbato in [3], where a stochastic version of the associated inviscid system to (1.3) (when ν = 0) has been studied. In fact, by choosing an appropriate stochastic perturbation to the system to be formally conservative, they were able to prove global existence and uniqueness of solutions in law for α > 3 4 . This is a very strong result although the uniqueness is to be understood in law. Similar regularization have been used for the MHD models, see for eg. [24] and the references therein. Since these models are quite different from the ones considered in the current paper, we don't state their results and we refer interested readers to the literature related to these models.
Inspired by [4], in this paper we consider the modified Boussinesq system for d = 3, where the equation for the velocity has fractional dissipation whereas the temperature equation has no dissipation term; a Lerayregularization for the velocity appears in the quadratic terms. This system is
(1.4) ∂ t v + (u · ∇)v + ν(−∆) β v + ∇p = θe 3 ∂ t θ + u · ∇θ = 0 v = u + (−∆) α u ∇ · u = ∇ · v = 0
Our goal is to generalize the results of Ye [25] and Barbato, Morandin, Romito [4], by proving well posedness of system (1.4) for α + β ≥ 5 4 when v 0 , θ 0 are regular enough. So the interesting case is for β < 5 4 with α = 5 4 − β > 0; indeed, the result of Ye corresponds to α = 0 and β ≥ 5 4 and that of Barbato, Morandin, Romito does not include the temperature equation, i.e. corresponds to our system (1.4) with θ 0 = 0. We have to point out that the temperature satisfies a pure transport equation, without thermal diffusivity; hence, the uniqueness result for the unknown θ requires v to be smooth enough. This imposes β to be not too small.
We can summarize our result in the following
Theorem 1.1 Assume α ≥ 0, β ≥ 1 2 with α + β ≥ 5 4 .
Then, system (1.4) has a unique global smooth solution for every smooth initial conditions v 0 , θ 0 .
Our proofs rely on the commutator estimates introduced in [14], also used in [25]. However in contrast to [25], we first prove global existence (for any α ≥ 0 and β > 0) and then uniqueness of these solutions; moreover we consider different order of space regularity for v and θ (H r -regularity for v and H r−β -regularity for θ), whereas in [25] the same order of regularity for both v and θ is considered. We point out that the requirement on the regularity on the initial data is needed only to guarantee uniqueness.
The paper is organized as follows. Section 2 is devoted to the mathematical framework. Our main functional spaces, the regularization operator Λ s with its properties given in Lemma 2.6 are defined. The bilinear operator of the Navier-Stokes equations, the transport operator and the commutator operator are defined and their properties are stated in Lemma 2.1, Remark 2.3 and Remark 2.4 and Lemma 2.5. The main system is then written in its abstract (operator) form and the definition of weak solutions is given. At the end of this section, we recall the Gagliardo-Nirenberg and Brézis-Gallouet-Wainger inequalities and some continuity results. In Section 3, we prove global existence of weak solutions with their uniform estimates. Slightly better estimates are performed. However, they are not enough to prove the uniqueness of solutions. The main result of the paper is stated in Section 4, Theorem 4.1, where we prove global existence of regular solutions; this regularity is enough to prove uniqueness of solutions and their continuous dependence with respect to the initial conditions, see Theorem 4.3 and Theorem 4.4. The main tool used is the commutator estimate. Let us point out that the results of Section 4 Theorem 1.1, i.e. every smooth initial data gives rise to a unique smooth solution. Section 5 is devoted to showing in more details the crucial estimates used in Section 4.
Mathematical framework
We consider the evolution for positive times and the spatial variable belongs to a bounded domain of R 3 ; for simplicity and because of the lack of natural boundary conditions, we work on the torus, i.e. the spatial variable x ∈ T = [0, 2π] 3 and periodic boundary conditions are assumed. We set L p = L p (T).
As usual in the periodic setting, we can restrict ourselves to deal with initial data with vanishing spatial averages; then the solutions will enjoy the same property at any fixed time t > 0.
Therefore we can represent any T-periodic function f :
R 3 → R as f (x) = k∈Z 3 0 f k e ik·x , with f k ∈ C, f −k = f k ∀k.
where Z 3 0 = Z 3 \ 0. For s ∈ R we define the spaces
H s = {f = k∈Z 3 0 f k e ik·x : f −k = f k and k∈Z 3 0 |f k | 2 |k| 2s < ∞}.
They are a Hilbert spaces with scalar product
f, g Vs = k∈Z 3 0 f k g −k |k| 2s .
We simply denote by f, g the scalar product in H 0 and also the dual pairing
of H s − H −s , i.e. f, g = k f k g −k .
The space H s+ǫ is compactly embedded in H s for any ǫ > 0. Moreover, if 0 ≤ s < 3 2 and 1 p = 1 2 − s 3 , then H s ⊂ L p and there exists a constant C (depending on s and p) such that
(2.1) f Lp ≤ C f H s .
Similarly, we define the spaces for the divergence free velocity vectors, which are periodic and have zero spatial average. For w : R 3 → R 3 we write formally
w(x) = k∈Z 3 0 w k e ik·x , with w k ∈ C 3 , w −k = w k , w k · k = 0 ∀k and for s ∈ R define V s = {w = k∈Z 3 0 w k e ik·x : w −k = w k , w k · k = 0 and k∈Z 3 0 |w k | 2 |k| 2s < ∞}.
This is a Hilbert space with scalar product v, w Vs =
k∈Z 3 0 v k · w −k |k| 2s .
We define the linear operator Λ = (−∆) 1/2 , i.e.
f = k∈Z 3 0 f k e ik·x =⇒ Λf = k∈Z 3 0 |k|f k e ik·x
and its powers Λ s :
Λ s f = k∈Z 3 0 |k| s f k e ik·x ; hence Λ 2 = −∆. Note, in particular that Λ s maps H r onto H r−s .
For simplicity, we shall use the same notation for Λ in the scalar spaces H s and in the vector spaces V s .
Let us denote by Π the Leray-Helmotz projection from L 2 onto V 0 . The operators Π and Λ s commute.
Finally we define the bilinear operator B :
V 1 × V 1 → V −1 by B(u, v), w = T (u · ∇)v · w dx i.e. B(u, v) = Π (u · ∇)v for smooth vectors u, v.
We summarize the properties of the nonlinear terms; these are classical results, see e.g. [23].
Lemma 2.1 For any u, v, w ∈ V 1 and θ, η ∈ H 1 we have (2.2) B(u, v), w = − B(u, w), v , B(u, v), v = 0, (2.3) v · ∇θ, η = − v · ∇η, θ , v · ∇θ, θ = 0
and more generally for any u, v, w giving a meaning to the trilinear forms above, as stated precisely in the following:
(2.4) B(u, v), w ≤ C u V m 1 v V 1+m 2 w V m 3
for any m i ≥ 0 with at least one of the three parameters positive and such that
m 1 + m 2 + m 3 ≥ 3 2 .
Hereafter, we denote by the same symbol C different constants. Now, we are ready to give the abstract formulation of problem (1.4); we apply the projection operator Π to the first equation in order to get rid of the pressure. In addition due to the periodic setting, we regularize u in a different, but equivalent way. Therefore, our system in abstract form is
(2.5) ∂ t v + B(u, v) + νΛ 2β v = Π(θe 3 ) ∂ t θ + u · ∇θ = 0 v = Λ 2α u
We focus our analysis on the unknowns v and θ. The pressure p will be recovered by taking the curl of the equation for the velocity in (1.4), i.e. p solves the equation
∆p = −∇ · [(u · ∇)v − θe 3 ] = −∇ · [((Λ −2α v) · ∇)v − θe 3 ].
Therefore we give the following definition in terms of v and θ only. The finite time interval [0, T ] is fixed throughout the paper.
Definition 2.2 Let α ≥ 0 and β > 0. We are given v 0 ∈ V 0 , θ 0 ∈ H 0 . We say that the couple (v, θ) is a weak solution to system (2.5) over the time interval [0, T ] if v ∈ L ∞ (0, T ; V 0 ) ∩ L 2 (0, T ; V β ) ∩ C w ([0, T ]; V 0 ) θ ∈ L ∞ (0, T ; H 0 ) ∩ C w ([0, T ]; H 0 ) and, given any ψ ∈ V 5 2 , φ ∈ H 5 2 , they satisfy v(t), ψ − t 0 B(u(s), ψ), v(s) ds + ν t 0 Λ β v(s), Λ β ψ ds = v 0 , ψ + t 0 θ(s)e 3 , ψ ds (2.6) (2.7) θ(t), φ − t 0 u(s) · ∇φ, θ(s) ds = θ 0 , φ for every t ∈ [0, T ].
Remark 2.3
In the weak formulations above, the trilinear terms are well
defined; indeed, if 2α + β ≤ 3 2 | B(u, ψ), v | ≤ C u V 2α ψ V 5 2 −2α−β v V β by (2.4) ≤ C v V 0 v V β ψ V 5 2 (2.8) and if 2α + β > 3 2 | B(u, ψ), v | ≤ C u L∞ ∇ψ L 2 v L 2 by Hölder inequality ≤ C u V 2α+β v V 0 ψ V 1 since V 2α+β ⊂ L ∞ ≤ C v V β v V 0 ψ V 1 . (2.9)
Similarly for the temperature:
if 2α + β < 3 2 | u · ∇φ, θ | ≤ u Lp 1 ∇φ Lp 2 θ L 2 ≤ C u V 2α+β ∇φ H 3 2 −2α−β θ H 0 ≤ C v V β φ H 5 2 θ H 0 (2.10)
where we used first the Hölder inequality with 1
p 1 = 1 2 − 2α+β 3 , 1 p 2 = 1 2 − 1 p 1 and then the embedding theorems; if 2α + β ≥ 3 2 | u · ∇φ, θ | ≤ u L 4 ∇φ L 4 θ L 2 ≤ C u V 2α+β ∇φ H 3 4 θ H 0 ≤ C v V β φ H 7 4 θ H 0 ≤ C v V β θ H 0 φ H 5 2 (2.11)
where we used first the Hölder inequality and then the embedding theorems V 2α+β ⊂ L q for any finite q, H
5 2 ⊂ H 7 4 , H 3 4 ⊂ L 4 .
For more regular solutions, the trilinear term B(u, ψ), v is equal to − B(u, v), ψ and we recover the term appearing in the equation for the velocity. The same holds for the temperature.
Remark 2. 4 We point out that the estimates by means of Sobolev embeddings need some restriction for the parameters; but, for bigger values of the parameters they are easier to prove and the details will be skipped. This means for instance that
(2.8) with (2.2) gives B(u, v) V − 5 2 ≤ C v V 0 v V β assuming 2α + β ≤ 3
2 , whereas for 2α + β > 3 2 we get something stronger in (2.9):
B(u, v) V −1 ≤ C v V 0 v V β
which is proven in another way. But for sure, from the proof of (2.8) one can
say that B(u, v) V − 5 2 ≤ C v V 0 v V β also for 2α + β > 3 2 without proving it.
In this last part of the section, we summarize the technical tools to be used later on.
To estimate an L ∞ -norm we use either the embedding theorem H r ⊂ L ∞ with r > 3 2 or the Brézis-Gallouet-Wainger inequality (see [5,6]): for any r > 3 2 there exists a constant C such that
(2.12) g L∞ ≤ C g H 3 2 1 + ln(1 + g H r g H 3 2
) .
Actually, we shall use the stronger form of this inequality, as given for instance in [25]: for any r > 3 2 there exists a constant C such that
(2.13) g L∞ ≤ C 1 + g H 3 2 + g H 3 2 ln(e + g H r ) .
Gagliardo-Niremberg inequality (see [18]) Let 1 ≤ q, r ≤ ∞, 0 < s < m, s m ≤ a < 1 and
1 p = s 3 + 1 q − m 3 a + 1 − a r
then there exists a constant C such that
(2.14) Λ s g Lp ≤ C g 1−a Lr Λ m g a Lq .
Define the commutator
[Λ s , f ]g = Λ s (f g) − f Λ s g.
From [14], [15] we have Lemma 2.5 (Commutator lemma) Let s > 0, 1 < p < ∞ and p 2 , p 3 ∈ (1, ∞) be such that
1 p ≥ 1 p 1 + 1 p 2 , 1 p ≥ 1 p 3 + 1 p 4 .
Then
[Λ s , f ]g Lp ≤ C ∇f Lp 1 Λ s−1 g Lp 2 + Λ s f Lp 3 g Lp 4 .
and Lemma 2.6 Let s > 0, 1 < p < ∞ and p 2 , p 3 ∈ (1, ∞) be such that
1 p ≥ 1 p 1 + 1 p 2 , 1 p ≥ 1 p 3 + 1 p 4 .
Then
Λ s (f g) Lp ≤ C f Lp 1 Λ s g Lp 2 + Λ s f Lp 3 g Lp 4 .
We shall use the commutator acting also on vectors; in particular for
u, v ∈ R 3 , θ ∈ R [Λ s , u] · ∇θ = Λ s (u · ∇θ) − u · ∇Λ s θ and [Λ s , u] · ∇v = Λ s (u · ∇)v − (u · ∇)Λ s v. Therefore (2.15) Λ s (u · ∇θ), Λ s θ = [Λ s , u] · ∇θ, Λ s θ + u · ∇Λ s θ, Λ s θ =0 by (2.3) and (2.16) Λ s (u · ∇)v , Λ s v = [Λ s , u] · ∇v, Λ s v + (u · ∇)Λ s v, Λ s v =0 by (2.2)
About the continuity in time, we have the strong continuity result (see [21] or Lemma 1.4, Chap III in [22])
Lemma 2.7 Let s ∈ R and h > 0. If v ∈ L 2 (0, T ; V s+h ) and dv dt ∈ L 2 (0, T ; V s−h ), then v ∈ C([0, T ]; V s ) and d dt v(t) 2 V s = 2 Λ −h dv dt (t), Λ h v(t) Vs
and the weak continuity result (see [21]).
Lemma 2.8 Let X and Y be Banach spaces, X reflexive, X a dense subset of Y and the inclusion map of X into Y continuous. Then
L ∞ (0, T ; X) ∩ C w (0, T ; Y ) = C w (0, T ; X).
Existence of weak solutions
Existence of a global weak solution of system (2.5) can be obtained easily; the technique is very similar to that for the classical Boussinesq system. The equation for θ is a pure transport equation; then the L q -norm of θ is conserved in time (for any q ≤ +∞). On the other hand, it is enough to have some regularization in the velocity equation (i.e. β > 0) in order to get a weak solution as in Definition 2.2; moreover, this solution satisfies an energy inequality. Of course, the bigger are the parameters α, β, the more regular is the velocity v. for any q ≥ 2 (including q = ∞).
Proof. We define the finite dimensional projector operator Π n in V 0 as Π n v = 0<|k|≤n v k e ik·x for v = k∈Z 3 0 v k e ik·x ; similarly for the scalar case, i.e. Π n in H 0 . We set B n (u, v) = Π n B(u, v).
We consider the finite dimensional approximation of system (2.5) in the unknowns v n = Π n v, u n = Π n u and θ n = Π n θ. This is the Galerkin approximation for n = 1, 2, . . .
(3.1) ∂ t v n + B n (u n , v n ) + νΛ 2β v n = Π(θ n e 3 ) ∂ t θ n + Π n (u n · ∇θ n ) = 0 v n = Λ 2α u n
We take the L 2 -scalar product of the equation for the velocity v n with v n itself; bearing in mind (2.2) we get
1 2 d dt v n (t) 2 V 0 + ν v n (t) 2 V β = − B n (u n (t), v n (t)), v n (t) + Π(θ n (t)e 3 ), v n (t) = − B(u n (t), v n (t)), v n (t) + θ n (t)e 3 , v n (t) ≤ 1 2 θ n (t) 2 H 0 + 1 2 v n (t) 2 V 0
and similarly for the second equation d dt θ n (t) 2 H 0 = − Π n (u n (t) · ∇θ n (t)), θ n (t) = − u n (t) · ∇θ n (t), θ n (t) = 0.
In both cases the trilinear forms vanish according to (2.2), (2.3). Adding these estimates, by means of Gronwall's lemma we get the basic L 2 -energy estimate: there exists a constant K 1 independent of n such that
sup 0≤t≤T ( v n (t) 2 V 0 + θ n (t) 2 H 0 ) + ν T 0 v n (t) 2 V β dt ≤ K 1
for any n. From the equation for the velocity v n , one has that dvn dt is expressed as the sum of three terms involving v n , u n and θ n . In particular, the dissipative term Λ 2β v n ∈ L 2 (0, T ; V −β ); by (2.8), (2.9) we have B n (u n , v n ) ∈ L 2 (0, T ; V −s ) for some finite s ≥ 1. Therefore there exist constants γ > 0 and K 2 independent of n, such that dv n dt This means that v n is bounded in L 2 (0, T ; V β ) ∩ W 1,2 (0, T ; V −γ ) (with β > 0 and γ > 0), which is compactly embedded in L 2 (0, T ; V 0 ) (see Lemma 2.2. in [22]). Hence we can extract a subsequence, still denoted by {v n } and
{θ n }, such that v n −→ v weakly in L 2 (0, T ; V β ) v n −→ v weakly * in L ∞ (0, T ; V 0 ) v n −→ v strongly in L 2 (0, T ; V 0 ) θ n −→ θ weakly * in L ∞ (0, T ; H 0 ).
Using these convergences, it is a classical result to pass to the limit in the variational formulation (2.6) and (2.7) and prove that (v, θ) is solution of (2.5) and inherits all the regularity from (
v n , θ n ), i.e. v ∈ L ∞ (0, T ; V 0 ) ∩ L 2 (0, T ; V β ), θ ∈ L ∞ (0, T ; H 0 ).
Moreover, it is a classical result (see [25]) that
(3.2) sup 0≤t≤T θ n (t) Lq ≤ θ 0 Lq for any q ≤ ∞.
Hence, the sequence {θ n } n is uniformly bounded in L ∞ (0, T ; L q ) which implies (up to a subsequence still denoted θ n ) that
θ n −→ θ weakly * in L ∞ (0, T ; L q ) and (3.3) sup 0≤t≤T θ(t) Lq ≤ θ 0 Lq .
Now, let us prove that v ∈ C w (0, T ; V 0 ) and θ ∈ C w (0, T ; L q ). We integrate in time the equation for v:
v(t) = v 0 + t 0 [−B(u(s), v(s)) − νΛ 2β v(s) + Πθ(s)e 3 ]ds.
Bearing in mind (2.2) and the estimates of Remark 2.3, we get that B(u, v) ∈ L 2 (0, T ; V − 5 2 ); therefore v ∈ C(0, T ; V −m ) for some positive m. By Lemma 2.8 we get that v ∈ C w (0, T ; V 0 ). Now we look for the weak continuity of θ. Assume that φ ∈ C ∞ # (T) which is the space of infinitely continuously differentiable functions on T that are periodic. Then for t, s ∈ [0, T ], we have that
| θ(t) − θ(s), φ | = | t s u(r) · ∇φ, θ(r) dr| ≤ t s ∇φ L∞ u(r) L 2 θ(r) L 2 dr ≤ ∇φ L∞ θ L ∞ (0,T ;H 0 ) t s u(r) V 0 dr. Using the density of C ∞ # (T) in L q ′ (with 1 q + 1 q ′ ≤ 1), we deduce that lim t→s θ(t) − θ(s), φ = 0 ∀φ ∈ L q ′
which means that θ ∈ C w (0, T ; L q ). A similar argument can be used for q = ∞ and this completes the proof. ✷ Remark 3.2 Take α ≥ 0 and β > 0 such that
2α + β ≤ 3 2 , α + β ≥ 5 4 .
For this to hold it is necessary that α is not too big (α ≤ 1 4 ) and β not too small (1 ≤ β ≤ 3 2 ). Then, from the first estimate in (2.8) we get B(u, v) ∈ L 2 (0, T ; V −β ). Hence, going back to the proof of the previous theorem we get that dv dt ∈ L 2 (0, T ; V −β ); by Lemma 2.7 this implies that v ∈ C([0, T ]; V 0 ), which is stronger than the weak continuity result of Theorem 3.1.
In addition, for more regular initial data we have
α + β ≥ 5 4 . Let s ≥ 0 with 1 − β ≤ s ≤ β.
Then, given v 0 ∈ V s , θ 0 ∈ H 0 , any weak solution of (2.5) obtained in Theorem 3.1 is more regular; indeed, the velocity is more regular
v ∈ C([0, T ]; V s ) ∩ L 2 (0, T ; V s+β ).
Proof. We look for a priori estimates for v. We proceed as before, but for more regular norms. We have
1 2 d dt v(t) 2 V s + ν v(t) 2 V s+β = − B(u(t), v(t)), Λ 2s v(t) + Πθ(t)e 3 , Λ 2s v(t) ≤ | Λ s (u(t) · ∇)v(t) , Λ s v(t) | + Λ s−β θ(t) L 2 Λ s+β v(t) L 2 ≤ [Λ s , u(t)] · ∇v(t) L 2 Λ s v(t) L 2 + C θ(t) H 0 v(t) V s+β
where we used (2.16) and that H 0 ⊂ H s−β . We use the Commutator Lemma 2.5
[Λ s , u] · ∇v L 2 ≤ C Λu Lp 1 Λ s v Lp 2 + Λ s u Lp 3 ∇v Lp 4 = C Λ 1−2α v Lp 1 Λ s v Lp 2 + Λ s−2α v Lp 3 ∇v Lp 4 (3.4) with 1 p 1 = 1 2 − s+β−(1−2α) 3 , 1 p 2 = 1 2 − β−s 3 , 1 p 3 = 1 2 − β−(s−2α) 3 , 1 p 4 = 1 2 − β+s−1 3 .
Notice that by our assumptions we get s 5 4 . The choice of the p i 's allows to use the Sobolev embedding theorem; we have
+ β ≥ 1 − 2α, β ≥ s, β ≥ s − 2α, s + β ≥ 1. Both conditions 1 p 1 + 1 p 2 ≤ 1 2 and 1 p 3 + 1 p 4 ≤ 1 2 are equivalent to α + β ≥Λ 1−2α v Lp 1 ≤ C v V s+β Λ s v Lp 2 ≤ C v V β and Λ s−2α v Lp 3 ≤ C v V β ∇v Lp 4 ≤ C v V s+β
Hence, we conclude that
1 2 d dt v(t) 2 V s + ν v(t) 2 V s+β ≤ C v(t) V β v(t) V s+β v(t) V s + C θ(t) H 0 v(t) V s+β ≤ ν 2 v(t) 2 V s+β + C ν v(t) 2 V β v(t) 2 V s + C ν θ(t) 2 H 0 (3.5)
by Young inequality. In particular,
d dt v(t) 2 V s ≤ C ν v 2 V β v 2 V s + C ν θ 2 H 0 .
Since v ∈ L 2 (0, T ; V β ) and θ ∈ L ∞ (0, T ; H 0 ) from the previous theorem, we can proceed by means of Gronwall lemma to get the estimate for the L ∞ (0, T ; V s )-norm:
sup 0≤t≤T v(t) 2 V s ≤ v 0 2 V s e Cν T 0 v(s) 2 V β ds + C ν T 0 e Cν T r v(s) 2 V β ds θ(r) 2 H 0 dr.
Integrating in time (3.5), we also get
(3.6) ν 2 T 0 v(t) 2 V s+β dt ≤ 1 2 v 0 2 V s + C ν v 2 L ∞ (0,T ;V s ) T 0 v(t) 2 V β dt + C ν T 0 θ(t) 2 H 0 dt.
Summing up, we get that v ∈ L ∞ (0, T ; V s ) ∩ L 2 (0, T ; V s+β ). Now, we study the time regularity. We recall property (2.4) for the nonlinear term B(u, v) with m 1 = 5 2 − 2β, m 2 = s + β − 1 ≥ 0, m 3 = β − s ≥ 0; this can be used if β < 5 4 . But for bigger values of β the estimate we are looking for is even easier to prove (see Remark 2.4). We have
dv dt (t) V s−β = sup ψ V β−s ≤1 dv dt (t), ψ = sup ψ V β−s ≤1 − B(u(t), v(t)), ψ − ν Λ 2β v(t), ψ + θ(t)e 3 , ψ ≤ sup ψ V β−s ≤1 C u(t) V 5 2 −2β v(t) V s+β + ν Λ β+s v(t) V 0 + Λ s−β θ(t) H 0 ψ V β−s ≤ C v(t) V 5 2 −2β−2α v(t) V s+β + ν v(t) V s+β + C θ(t) H 0 since s − β ≤ 0 ≤ C v(t) V 0 v(t) V s+β + ν v(t) V s+β + C θ(t) H 0 since V 0 ⊆ V 5 2 −2β−2α for α + β ≥ 5
4 . Hence, using the regularity of v, θ we get that dv dt ∈ L 2 (0, T ; V s−β ). Now using Lemma 2.7, we deduce that v ∈ C([0, T ]; V s ). ✷ In particular we have Then given v 0 ∈ V β , θ 0 ∈ H 0 , any weak solution of (2.5) obtained in Theorem 3.1 is more regular; indeed, the velocity is more regular v ∈ C([0, T ]; V β ) ∩ L 2 (0, T ; V 2β ).
Regular solutions: global existence, uniqueness and continuous dependence on the initial data
The regularity of solutions from the previous section is not enough to prove uniqueness. To this end, we seek classical solutions. These are solutions for which the derivatives in the equations of (2.5) exist. Indeed, we shall get Remark 4.2). The crucial point is to show that these regular solutions are defined on any given time interval [0, T ]; their local existence is easy to prove.
that v ∈ C([0, T ]; V r ) ∩ L 2 (0, T ; V r+β ) and θ ∈ C([0, T ]; H r−β ) with r ≥ 3 2 , r ≥ 2β, r − β ≥ 1 (see
Unlike the previous section, here we will consider H s -regularity for θ(t) (with s > 0). This will help prove the uniqueness of solutions.
Theorem 4.1 We are given α ≥ 0, β ≥ 1 2 with (4.1) α + β ≥ 5 4 . Let r > max 5 2 − 2α, β + 1 .
Then, for any v 0 ∈ V r , θ 0 ∈ H r−β , there exists a solution (v, θ) to (2.5) such
that v ∈ C([0, T ]; V r ) ∩ L 2 (0, T ; V r+β ), θ ∈ C([0, T ]; H r−β ).
Proof. We proceed as before. We take the L 2 -scalar product of the first equation of (2.5) with Λ 2r v; then
1 2 d dt v(t) 2 V r + ν v(t) 2 V r+β = − B(u(t), v(t)), Λ 2r v(t) + θ(t)e 3 , Λ 2r v(t) = − B(Λ −2α v(t), v(t)), Λ 2r v(t) + Λ r−β θ(t)e 3 , Λ r+β v(t) ≤ C v(t) V 2β v(t) V r+β v(t) V r + C θ(t) H r−β v(t) V r+β ≤ ν 4 v(t) 2 V r+β + C ν v(t) 2 V 2β v(t) 2 V r + C ν θ(t) 2 H r−β (4.2)
where we used first Lemma 5.1 and then Young inequality. Now for θ, we take the L 2 -scalar product of the second equation of (2.5) with Λ 2r−2β θ(t); then
1 2 d dt θ(t) 2 H r−β = − u(t) · ∇θ(t), Λ 2r−2β θ(t)
We estimate the r.h.s.
u · ∇θ, Λ 2r−2β θ = Λ r−β (u · ∇θ), Λ r−β θ = [Λ r−β , u] · ∇θ, Λ r−β θ by (2.15) ≤ [Λ r−β , Λ −2α v] · ∇θ L 2 Λ r−β θ L 2
and the Commutator Lemma 2.5 gives
≤ C Λ 1−2α v L∞ Λ r−β θ L 2 + Λ r−β−2α v Lq 3 Λθ Lq 4 Λ r−β θ L 2 with 1 q 3 + 1 q 4 ≤ 1 2 ;
we continue by means of the Brézis-Gallouet-Wainger estimate (2.13) (with g = Λ 1−2α v) and Lemma 5.4
≤ C 1 + Λ 5 2 −2α v L 2 + Λ 5 2 −2α v L 2 ln(e + v V m+1−2α ) θ 2 H r−β + C v a V 2β v 1−a V r+β θ 1−a Lq θ 1+a
H r−β for any m > 3 2 and for suitable q > 2, a ∈ (0, 1); m will be chosen later on. Finally we use that V 2β ⊂ V 5 2 −2α since α + β ≥ 5 4 :
≤ C (1 + v V 2β + v V 2β ln(e + v V m+1−2α )) θ 2 H r−β + C v a V 2β v 1−a V r+β θ 1−a Lq θ 1+a H r−β . Now, we use Young inequality: v a V 2β v 1−a V r+β θ 1−a Lq θ 1+a H r−β ≤ ν 4 v 2 V r+β + C ν v 2a 1+a V 2β θ 2(1−a) 1+a Lq θ 2 H r−β . Set φ := v 2a 1+a V 2β θ 2(1−a) 1+a
Lq ; then φ ∈ L 1 (0, T ) according to Theorem 3.4 and (3.3). Thus
(4.3) 1 2 d dt θ(t) 2 H r−β ≤ C 1+ v(t) V 2β + v(t) V 2β ln(e+ v(t) V m+1−2α ) θ(t) 2 H r−β + ν 4 v(t) 2 V r+β + C ν φ(t) θ(t) 2 H r−β .
Adding the estimates (4.2) for v and (4.3) for θ, we get
(4.4) d dt ( v(t) 2 V r + θ(t) 2 H r−β ) + ν v(t) 2 V r+β ≤ C v(t) 2 V 2β v(t) 2 V r + C 1 + v(t) V 2β + v(t) V 2β ln(e + v V m+1−2α ) + φ(t) θ(t) 2 H r−β .
Recall that r > 5 2 − 2α by assumption; then there exists m > 3 2 such that V r ⊂ V m+1−2α . Thus, we get
(4.5) d dt ( v(t) 2 V r + θ(t) 2 H r−β ) + ν v(t) 2 V r+β ≤ C v(t) 2 V 2β v(t) 2 V r + C 1 + v(t) V 2β + v(t) V 2β ln(e + v(t) V r ) + φ(t) θ(t) 2 H r−β Set X(t) = v(t) 2 V r + θ(t) 2 H r−β . Then, from (4.5) we easily get dX dt (t) ≤ C 1 + v(t) V 2β ln(e + 1 + X(t)) + v(t) 2 V 2β + φ(t) X(t) ≤ C 1 + v(t) V 2β ln(e + 1 + X(t)) + v(t) 2 V 2β + φ(t) (e + 1 + X(t)).
This implies that Y (t) = ln(e + 1 + X(t)) satisfies
Y ′ (t) ≤ C 1 + v(t) V 2β Y (t) + v(t) 2 V 2β + φ(t) .
By Gronwall lemma we get
sup 0≤t≤T Y (t) ≤ Y (0)e C T 0 v(s) V 2β ds +C T 0 e C T s v(r) V 2β dr 1+ v(s) 2 V 2β +φ(s) ds.
Since v ∈ L 2 (0, T ; V 2β ) by Theorem 3.4 and φ ∈ L 1 (0, T ), we get that sup 0≤t≤T Y (t) ≤ K 3 and therefore going back to the unknown X sup 0≤t≤T X(t) ≤ K 4 ; from (4.5), after integration on [0, T ] we get also
T 0 v(t) 2 V r+β dt ≤ K 5 .
Therefore we have proved that
v ∈ L ∞ (0, T ; V r ) ∩ L 2 (0, T ; V r+β ), θ ∈ L ∞ (0, T ; H r−β ).
Now we consider the continuity in time. Lemma 2.6 (with p = p 2 = 2,
p 1 = ∞) gives B(u, v) V r−β ≤ C u L∞ v V r−β+1 + Λ r−β u Lp 3 Λv Lp 4 . Take 1 p 3 = 1 2 − β+2α 3 , 1 p 4 = 1 2 − 3 2 −β−2α 3
; by Sobolev embeddings we get
u L∞ ≤ C Λ −2α v L∞ ≤ C v V 5 2 −2α ≤ C v V r since r > 5 2 − 2α v V r−β+1 ≤ C v V r+β since β ≥ 1 2 Λ r−β u Lp 3 ≤ C Λ r−β−2α v Lp 3 ≤ C v V r Λv Lp 4 ≤ C Λv V 3 2 −β−2α = C v V 5 2 −β−2α .
Using that V r+β ⊂ V 5 2 −β−2α when α + β ≥ 5 4 and r ≥ 0, we obtain that
B(u, v) V r−β ≤ C v V r v V r+β .
This implies
dv dt = −B(u, v) − νΛ 2β v + Πθe 3 ∈ L 2 (0, T ; V r−β )
By Lemma 2.7 we deduce that v ∈ C([0, T ]; V r ).
As far as the continuity in time for θ is concerned, we have that θ satisfies a transport equation
∂θ + u · ∇θ = 0
where the velocity is given and in particular u ∈ C([0, T ]; V r+2α ). [13] considers this equation in R 2 ; but a straightforward modification of Lemma 4.4 of [13] allows to prove in the three dimensional case that given u ∈ C([0, T ]; V ρ ) with ρ > 5 2 and θ 0 ∈ H k with 0 ≤ k < [ρ], then there exists a unique solution θ ∈ C([0, T ]; H k ). Taking ρ = r + 2α and k = r − β, we get the continuity result for θ. ✷ Remark 4.2 Let β ≥ 1 2 . As far as the range of values of r is concerned, we have that when α + β = 5
4 max 5 2 − 2α, β + 1 ≡ max 2β, β + 1 = β + 1 for 1 2 ≤ β ≤ 1 2β
for β ≥ 1
Therefore we have r > 3 2 and r − β > 1 at least. In addition, r > 2β.
This regularity is enough to get uniqueness.
α + β ≥ 5 4 . Let r > max 5 2 − 2α, β + 1 .
Then, the solutions given in Theorem 4.1 are unique.
Proof. Let (v 1 , θ 1 ) and (v 2 , θ 2 ) be two solutions given by Theorem 4.1. We
define V = v 1 − v 2 , U = u 1 − u 2 and Φ = θ 1 − θ 2 .
Using the bilinearity we have that they satisfy
∂ t V + νΛ 2β V + B(u 1 , V ) + B(U, v 2 ) = ΠΦe 3 ∂ t Φ + U · ∇θ 1 + u 2 · ∇Φ = 0
As before, using (2.2) we get
1 2 d dt V (t) 2 V 0 + ν V (t) 2 V β = − B(u 1 (t), V (t)), V (t) − B(U(t), v 2 (t)), V (t) + Φ(t)e 3 , V (t) ≤ − B(U(t), v 2 (t)), V (t) + Φ(t) H 0 V (t) V 0 .
And similarly, using (2.3)
1 2 d dt Φ(t) 2 H 0 = − (U(t) · ∇θ 1 (t)), Φ(t) − u 2 (t) · ∇Φ(t), Φ(t) = − (U(t) · ∇θ 1 (t)), Φ(t) .
Let us estimate the terms on the right hand side of each of the relationships above. For the velocity equation, we proceed as usual by means of the Sobolev embeddings:
| B(U, v 2 ), V | ≤ (U · ∇)v 2 L 2 V L 2 ≤ U Lp 1 ∇v 2 Lp 2 V V 0 if 1 p 1 + 1 p 2 ≤ 1 2 ≤ C U V β+2α v 2 V 2β V V 0 if 1 p 1 = 1 2 − β+2α 3 , 1 p 2 = 1 2 − 2β−1 3 ≤ C V V β v 2 V 2β V V 0 ≤ ν 4 V 2 V β + C ν v 2 2 V 2β V 2 V 0
and the condition 1 p 1 + 1 p 2 ≤ 1 2 is equivalent to 3 2 β + α ≥ 5 4 , which is trivially stafisfied.
Similarly, for the temperature equation:
| U · ∇θ 1 , Φ | ≤ U · ∇θ 1 L 2 Φ L 2 ≤ U Lp 3 ∇θ 1 Lp 4 Φ H 0 if 1 p 3 + 1 p 4 ≤ 1 2 ≤ C U V β+2α θ 1 H r−β Φ H 0 if 1 p 3 = 1 2 − β+2α 3 , 1 p 4 = 1 2 − r−β−1 3 ≤ C V V β θ 1 H r−β Φ H 0 ≤ ν 4 V 2 V β + C ν θ 1 2 H r−β Φ 2 H 0 .
Summing up, we have obtained
d dt V (t) 2 V 0 + ν V (t) 2 V β + d dt Φ(t) 2 H 0 ≤ C v 2 (t) 2 V 2β V (t) 2 V 0 + θ 1 (t) 2 H r−β Φ(t) 2 H 0 + Φ(t) 2 H 0 + V (t) 2 V 0 . If we define Z(t) = V (t) 2 V 0 + Φ(t) 2 H 0 , we have Z(0) = 0 and Z ′ (t) ≤ C( v 2 (t) 2 V 2β + θ 1 (t) 2 H r−β + 1)Z(t).
By Gronwall lemma we get Z(t) = 0 for all t, and this completes the proof. ✷ Theorem 4.4 (Continuous dependence on the initial data) We are given α ≥ 0 and β ≥ 1 2 with α + β ≥ 5 4 .
Let r ≥ β + 2.
Then, given any initial conditions v 1,0 , v 2,0 ∈ V r and θ 1,0 , θ 2,0 ∈ H r−β we have
(4.6) v 1 − v 2 L ∞ (0,T ;V r−1 ) + v 1 − v 2 L 2 (0,T ;V r−1+β ) + θ 1 − θ 2 L ∞ (0,T ;H r−β−1 ) ≤ C ( v 1,0 − v 2,0 V r−1 + θ 1,0 − θ 2,0 H r−β−1 )
where the constant C depends on T , θ 1 L ∞ (0,T ;H r−β ) , v i L 2 (0,T ;V r+β−1 ) and v i L ∞ (0,T ;V r ) .
Proof. We begin by pointing out that, under the assumptions on α and β, if r ≥ β + 2 then the conditions on r given in Theorem 4.1 (r > 5 2 − 2α and r > β + 1) hold true.
Using the same setting as in the proof of Theorem 4.3, we get
1 2 d dt V (t) 2 V r−1 + ν V (t) 2 V r−1+β = − B(Λ −2α v 1 (t), V (t)), Λ 2r−2 V (t) − B(Λ −2α V (t)
, v 2 (t)), Λ 2r−2 V (t) + Λ r−β−1 Φ(t)e 3 , Λ r−1+β V (t) .
We estimate the first two terms of r.h.s. by means of Lemma 5.2
| B(Λ −2α v 1 (t), V (t)), Λ 2r−2 V (t) | + | B(Λ −2α V (t), v 2 (t)), Λ 2r−2 V (t) | ≤ C v 1 V r V V r−1 + v 1 V r+β−1 V V r+β−1 V V r−1 + C V V r−1 v 2 V r+β−1 V V r+β−1 .
Using Young inequality, we get
(4.7) 1 2 d dt V (t) 2 V r−1 + ν V (t) 2 V r−1+β ≤ ν 2 V (t) 2 V r+β−1 + C ν Φ(t) 2 H r−β−1 + C ν ( v 1 (t) V r + v 1 (t) 2 V r+β−1 + v 2 (t) 2 V r+β−1 ) V (t) 2 V r−1 .
Similarly, for the temperature difference; we use Lemma 5.3 and Young inequality
1 2 d dt Φ(t) 2 H r−β−1 = − U(t) · ∇θ 1 (t), Λ 2r−2β−2 Φ(t) − u 2 (t) · ∇Φ(t), Λ 2r−2β−2 Φ(t) ≤ C V (t) V r−1 θ 1 (t) H r−β Φ(t) H r−β−1 + C v 2 (t) V r Φ(t) 2 H r−β−1 ≤ C V (t) 2 V r−1 + C θ 1 (t) 2 H r−β + v 2 (t) V r Φ(t) 2 H r−β−1 .
Finally, we consider the sum V (t) 2 V r−1 + Φ(t) 2 H r−β−1 := W (t) and define a(t) = 1 + θ 1 (t) 2 H r−β + v 1 (t) 2 V r+β−1 + v 2 (t) 2 V r+β−1 + v 1 (t) V r + v 2 (t) V r ; we have a ∈ L 1 (0, T ) and (4.8) W ′ (t) + ν V (t) 2 V r+β−1 ≤ Ca(t)W (t).
Gronwall lemma applied to
W ′ (t) ≤ Ca(t)W (t)
gives sup 0≤t≤T W (t) ≤ W (0)e C T 0 a(t) dt .
Integrating in time (4.8) and using the latter result we get the estimate for T 0 V (t) 2 H r+β−1 dt. This concludes the proof. ✷
Theorem 3. 1
1Let α ≥ 0 and β > 0. For any v 0 ∈ V 0 , θ 0 ∈ H 0 , there exists a weak solution (v, θ) of (2.5) on the time interval [0, T ]. Moreover θ ∈ C w (0, T ; L q )
Theorem 3. 3 (
3More regularity) We are given α ≥ 0 and β ≥ 1 2 with
Theorem 3.4 (s = β)We are given α ≥ 0 and β ≥
Theorem 4. 3 (
3Uniqueness) We are given α ≥ 0 and β ≥ 1 2 with
L 2 (0,T ;V −γ ) ≤ K 2 .
Acknowledgements The research of Hakima Bessaih was supported by the NSF grants DMS-1416689 and DMS-1418838. Part of this research started while Hakima Bessaih was visiting the Department of Mathematics of the University of Pavia and was partially supported by the GNAMPA-INDAM project "Regolarità e dissipazione in fluidodinamica"; she would like to thank the hospitality of the Department.In this section we prove the lemma used in the proofs of the previous section.Lemma 5.1 Let α ≥ 0, β ≥ 1 2 with α + β ≥5 4. Then for any r > 0 there esists a constant C > 0 such thatThen, we use the Commutator Lemma 2.5 with p = 2:For the latter estimate we have used the embeddings (5.1)The condition 1 p 1 + 1 p 2 ≤ 1 2 is equivalent to α + 3 2 β ≥ 5 4 , which holds when α + β ≥5 4. Moreover α + β ≥ 5 4 implies that the condition 2β ≥ 1 − 2α, needed for the embedding, holds true.Similarly for the other two terms:The condition 1Proof. First, notice that we also have r ≥ 1 and r ≥ 2 − β.To prove the first inequality, we use the Commutator Lemma 2.5 with p =; the condition 1, which is fulfilled if α + β ≥ 5 4 and r ≥ 1. Then we use the Sobolev embeddings:For the second inequality, we use Lemma 2.6 with p = 2, 1; the conditions 1 p 1 + 1 p 2 ≤ 1 2 and 1 p 3 + 1 p 4 ≤ 1 2 are again equivalent to α + β + r 2 ≥ 7 4 . Then we use the Sobolev embeddings:then there exists a constant C > 0 such thatProof. To prove the first inequality, we use Lemma 2.6 with p = p 2 = 2,; the condition 1 p 3 + 1 p 4 ≤ 1 2 is equivalent to r ≥ 5 2 − 2α. Then the Sobolev embeddings:For the second inequality, we use the Commutator Lemma 2.5 with p = p 2 = 2, p 1 = ∞, 1; the condition 1is equivalent again to r ≥ 5 2 − 2α. Then we use the Sobolev embeddings as before:Let α ≥ 0, β > 0 with α + β ≥ 5 4 and r > β + 1. Then, there exist q 3 , q 4 > 2 with 1 q 3 + 1 q 4 ≤ 1 2 and q > 2, a ∈ (0, 1), C > 0 such thatProof. We use Sobolev embedding theorem, interpolation theorem and the Gagliardo-Nirenberg inequality; then for some a ∈ (0, 1) and q ≥ 2 to be defined later on we look forr−β ≤ a < 1 equivalent to (since r > β by assumption)The second equation is satisfied for some q (big enough) when 1 3 + a 2 < 2 3 (α + β); therefore we need to find a ∈ (0, 1) such that (5.4) 1 r − β ≤ a < min 2 α + β r − β , 2 2(α + β) − 1 3 .Assuming α + β ≥ 5 4 , we have 2(α + β) ≥ 5 2 > 1 and 2 2(α+β)−1 3 ≥ 1; this implies that, under the conditions r − β > 1 there exists a ∈ (0, 1) satisfying(5.4).✷
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| []
|
[
"Investigation of alpha particle induced reactions on natural silver in the 40-50 MeV energy range",
"Investigation of alpha particle induced reactions on natural silver in the 40-50 MeV energy range"
]
| [
"F Ditrói \nInstitute for Nuclear Research\nHungarian Academy of Sciences\nDebrecenHungary\n",
"S Takács \nInstitute for Nuclear Research\nHungarian Academy of Sciences\nDebrecenHungary\n",
"H Haba \nNishina Center for Accelerator-Based Science\nRIKEN\nWakoJapan\n",
"Y Komori \nNishina Center for Accelerator-Based Science\nRIKEN\nWakoJapan\n",
"M Aikawa \nFaculty of Science\nHokkaido University\nSapporoJapan\n",
"M Saito \nGraduate School of Biomediacal Science and Engineering\nHokkaido University\nSapporoJapan\n",
"T Murata \nSchool of Science\nHokkaido University\nSapporoJapan\n"
]
| [
"Institute for Nuclear Research\nHungarian Academy of Sciences\nDebrecenHungary",
"Institute for Nuclear Research\nHungarian Academy of Sciences\nDebrecenHungary",
"Nishina Center for Accelerator-Based Science\nRIKEN\nWakoJapan",
"Nishina Center for Accelerator-Based Science\nRIKEN\nWakoJapan",
"Faculty of Science\nHokkaido University\nSapporoJapan",
"Graduate School of Biomediacal Science and Engineering\nHokkaido University\nSapporoJapan",
"School of Science\nHokkaido University\nSapporoJapan"
]
| []
| Natural silver targets have been irradiated by using a 50 MeV alpha-particle beam in order to measure the activation cross sections of radioisotopes in the 40-50 MeV energy range. Among the radio-products there are medically important isotopes such as 110m In and 111 In. For optimizing the production of these radioisotopes and regarding their purity and specific activity the cross section data for every produced radioisotope are important. New data are measured in this energy range and the results of some previous measurements have been confirmed. Physical yield curves have been calculated by using the new cross section data completed with the results from the literature. | 10.1016/j.nimb.2018.09.018 | [
"https://arxiv.org/pdf/2103.10783v1.pdf"
]
| 106,117,542 | 2103.10783 | ef076c3b2857158b2762ac8cbf21ea17c8a6fe96 |
Investigation of alpha particle induced reactions on natural silver in the 40-50 MeV energy range
F Ditrói
Institute for Nuclear Research
Hungarian Academy of Sciences
DebrecenHungary
S Takács
Institute for Nuclear Research
Hungarian Academy of Sciences
DebrecenHungary
H Haba
Nishina Center for Accelerator-Based Science
RIKEN
WakoJapan
Y Komori
Nishina Center for Accelerator-Based Science
RIKEN
WakoJapan
M Aikawa
Faculty of Science
Hokkaido University
SapporoJapan
M Saito
Graduate School of Biomediacal Science and Engineering
Hokkaido University
SapporoJapan
T Murata
School of Science
Hokkaido University
SapporoJapan
Investigation of alpha particle induced reactions on natural silver in the 40-50 MeV energy range
1 * Corresponding author: [email protected] 2Silver, cadmium and indium radioisotopesalpha-particle irradiationnatural silver targetcross sectionmodel calculationphysical yieldTLA
Natural silver targets have been irradiated by using a 50 MeV alpha-particle beam in order to measure the activation cross sections of radioisotopes in the 40-50 MeV energy range. Among the radio-products there are medically important isotopes such as 110m In and 111 In. For optimizing the production of these radioisotopes and regarding their purity and specific activity the cross section data for every produced radioisotope are important. New data are measured in this energy range and the results of some previous measurements have been confirmed. Physical yield curves have been calculated by using the new cross section data completed with the results from the literature.
Introduction
As silver is a basic material for producing radioisotopes for medical and for industrial applications it is of fundamental importance to know the corresponding nuclear data of the produced radioisotopes (cross section and yield) with reasonable accuracy. Among the produced radioisotopes e.g. 111 In plays an important role in nuclear medicine as diagnostic isotope [1], but its relatively long half-life allows also industrial and research applications [2]. 109 Cd is an interesting radioisotope from the point of view of medical applications as a parent of 109m Ag [3][4][5]. Nuclear data for alpha particle induced reactions on silver have already been measured by several authors [6][7][8][9][10][11][12]. In this work in the frame of a higher energy alpha experiment series by using stacked foil technique we covered the energy range 40-50 MeV using high purity silver foils as targets in order to complement the failing data in this energy range and resolve the contradictions between the existing data.
Experimental
The irradiations have been performed on a dedicated beam line of the K70-MeV AVF cyclotron of the RIKEN RI Beam Factory by using an Eα = 50.73±0.3 MeV beam. The exact beam energy was determined by using the time of flight (TOF) setup [13]. The well-established stacked foil technique was used for assessing the excitation function of the different nuclear reactions. A combined stack was constructed, it contained foils for two experiments, i.e. Ag+α and Ni+α cross section measurements. In this work we discuss only the results on silver. The used target foils were high purity (at least 99.99%) Goodfellow © foils with the following thicknesses: Ag: 30 and 8.25 µm. The actual thickness of the foils, which was in general different from the nominal one, was determined by weighting the whole metal sheet purchased, and measuring the exact lateral size, from which an average thickness was determined, assuming that the whole sheet was homogeneous and had even thickness all over the sheet. The nat Ti(α,x) 51 Cr reaction on titanium foils (Ti: 10.9 µm) was used as a monitor reaction to check and correct the beam intensity and energy degradation through the whole stack. The foils are ordered in groups in such a way that we could compensate or avoid the activity loss or excess activity because of recoil effect of the radioisotope in question. Only those foils were involved in the final evaluation, where the recoil from the particular target foil was compensated by the recoil from the preceding foil (same material), or it was measured together with the following foil (different material) if the preceding foil was not the same material. The Ag foils were arranged in one block, i.e. one after each other, because for silver we were interested in the high energy part. (50 -40 MeV range). The first 10 foils of the stack were silver. The first Ag foil was thick (30 µm), because in this case the recoil was proportionally small, all the others were thinner (8.25 µm). In the case of the thin Ag foils the recoil effect was compensated, because foil from the same material, Ag was before each target foil. The irradiation lasted for one hour at 200 nA beam current. After a short cooling time the stack was disassembled into single foils or pairs of foils and the gamma-ray measurements begun.
A HPGe semiconductor detector based spectrometer was used for the gamma-ray measurements. Three series of measurements have been performed on silver targets with different cooling times from 20 hours to 10 days. The spectra were evaluated later by using the automatic software [14] and in special cases (overlapping multiple and/or weak peaks) manual evaluation by using a home-developed evaluation software [15] was also performed. The activity of the Ti monitor foils, which were also inserted into the stack in pairs, in order to compensate the recoil effect of the lower energy foil of the pair, were also measured later in order to fit the beam energy and intensity. The whole excitation function of the nat Ti(α,x) 51 Cr monitor reaction was re-measured and the results were compared with the recommended values of the IAEA monitor reaction database [16]. The final beam energy and intensity were adjusted according to the best fit with the recommended values [17]. Based on these results the initial parameters for all calculations were set. The used nuclear data are listed in Table 1 for all measured radioisotopes.
The uncertainties of the single cross section values were estimated by calculating square root of the sum in quadrature of all single contributions [18]: beam current (5%), target thicknesses (3%), detector efficiency (5%), nuclear data (3%), peak area and counting statistics (1-20%), the overall uncertainty in the results (cross sections) was 7-20 %.
Theoretical model calculations
We wanted to test the prediction capability of the theoretical nuclear reaction model codes.
Calculations for the measured cross sections were made by using the modified TALYS 1.8 code [21] presented in the TENDL-2017 on-line library [22]. Production cross sections with the latest version of the EMPIRE code [23], (EMPIRE 3.2 (Malta) [24]), which contained the latest reference input parameters library RIPL-3 [25], were also given for comparison. The code run with the default input parameters by considering all possible reactions involved, including emission of complex particles above the reaction thresholds at the given bombarding energies.
Results and discussion
The experimental cross sections deduced for the nat Ti(α,x) 51 Cr monitor reaction were compared with the recommended cross section from the IAEA monitor reaction database [16]. After very small adjustment of the beam intensity and using the actual foil thicknesses, an excellent agreement was found between the recommended and the measured data ( Fig. 1). Tables 2-3.
nat Ag(α,x) 111 In nuclear reaction
This reaction is possible only on the 109 Ag target isotope by emission of two neutrons. The 107 Ag target isotope does not play a role in this case. The ground state has a long enough half-life to measure the activity of the foils in all the three spectrum series. The short-lived (7.6 min) metastable state decays completely to the ground state by internal conversion. The results given in
nat Ag(α,x) 110g In nuclear reaction
This radioisotope is produced with 1 and 3 neutron emissions from the 107 Ag and 109 Ag target isotopes, respectively. It has a relatively short half-life (4.9 h), so only the earliest measured spectra could be used for the final evaluation. The decay of 110g In is followed by emission of several strong gamma-lines, but because of interferences the 937.48 keV peak was used for the final calculations. While both the ground-state and the meta-stable state decay independently to
nat Ag(α,x) 110m In nuclear reaction
nat Ag(α,x) 108g In nuclear reaction
The 108g In radionuclide is produced by direct reactions from the 107 Ag and 109 Ag target isotopes with 3 and 5 neutrons emission, respectively. It has a relatively short half-life (58 min) so the results for this reaction were deduced only from the first spectrum of the first series, which means that we could only deduce a single experimental point (see Fig. 6). The even shorter-lived isomeric state 108m In (38,6 min, which could not be assessed from this experiment) decays completely into the stable 108 Cd, so cumulative effect and/or interfering radiation do not take place. The results are presented in Fig. 6. Fig. 6 it is seen that the newly measured single data point gives a good continuation to the previous data of Takács et al. [10]. The two theoretical model codes give similar trends with different values, and the results from TENDL-2017 seem to be a better approximation. It is also seen from the results of the model calculations that mainly the nuclear reactions on 107 Ag dominate for the production of the 108 In radioisotope.
nat Ag(α,x) 111g Ag nuclear reaction
The radioisotope 111 Ag has a very short-lived isomeric state (65 s) and a relatively longer-lived (7.45 d) ground-state. The isomeric state decays completely into the ground-state and because of its very short half-life all measurements series gave cumulative cross sections for the 111 Ag 13 radioisotope. 111 Ag is produced exclusively from the 109 Ag target isotope with two protons emission. Taking into account the longer half-life the best (but not very good) statistics was resulted in from the measurement series with longer cooling and measuring times. The 342.13 keV independent gamma-line was used for the evaluation. The results are presented in Fig. 7. gives closer values to our new experimental results. 14 The 110 Ag has a long-lived (249.76 d) isomeric state and a short-lived ground state (24.6 s). The ground state in this experiment could not be measured. The results were deduced from the measurement series with the longest cooling times and longest measuring times. The results are presented in Fig. 8.
nat Ag(α,x) 110m Ag nuclear reaction
nat Ag(α,x) 106m Ag nuclear reaction
The radioisotope 106 Ag can be produced from both stable target isotopes 107 Ag and 109 Ag by emissions of 2 protons, and 3 and 5 neutrons, respectively. The ground state has a too short halflife (23.96 min) to be included in our evaluation and anyway has no proper gamma-line. The excited state (8.28 d) has several relatively strong gamma-lines, so those having good statistics were involved into the evaluation. The results are shown in Fig. 9. The new data in Fig. 9 show a good continuation to the previous results of Takács et al. [10], lower than the data of Patel et al. [8] and show a good agreement with the data of Wasilewsky et al [12], in spite of their energy shift. Both TENDL-2017 and EMPIRE 3.2 give completely different trends and values. Up to 50 MeV the TENDL-2017 estimation seems to be better with a small overestimation.
nat Ag(α,x) 105g Ag nuclear reaction
The 105 Ag has two isomeric states, a short-lived (7.23 min) metastable state, which could not be assessed in this work and a longer-lived (41.29 d) ground state. The higher energy isomer state decays 100% into the ground-state. Fig. 10 it is seen that the newly deduced data give a good continuation to the previous results of Takács et al. [10]. Data points of Patel et al. [8] are scattered. The previous results of Wasilewsky [12] show partial agreement, but the energy shift is still visible. Both the theoretical model code predictions strongly underestimate the experiments above 35 MeV, though they give the same trend and values up to 65 MeV.
nat Ag(α,x) 109 Cd nuclear reaction
109 Cd can be produced from both the stable isotopes of Ag target elements with the emission of one proton and several neutrons. The 109 Cd radionuclide has a long half-life (461.9 d), and the coproduced 109 In decays into it, so reliable results could be deduced from the long measurements with long cooling time. Only its low-intensity gamma peak at 88.03 keV could be measured in this work. The cumulative results are presented in Fig. 11. Fig. 11 it is seen that our new data are in excellent agreement with the previous results of
Haasbroek [6] and also with the prediction of the TENDL-2017 on-line data library. The results of Porges [9] are not comparable, because they are in the energy range below 35 MeV. Both the theoretical model codes give similar trends describing the contributions on the two target isotopes. EMPIRE seems to be energy-shifted and the predicted amplitudes of TENDL-2017 are better at least in the 38-55 MeV energy range.
Physical yield
Integral physical yields have been calculated from the measured excitation functions for the investigated nuclear reactions. The experimental literature data were used to complete the low energy part of the excitation function, if they were available, in order to cover the whole energy region in the yield calculation. Searching the literature only one previous work was found [26] with experimental thick target yield measurements on natural silver. Fig. 12 it is seen that the yields cover 10 orders of magnitudes in different energy ranges.
The highest production yields are found for the two reactions to produce the medically interesting 109g In and 110g In isotopes. Both isotopes can be produced in considerable amount (above 1GBq/C) using proper energy range (above 20 MeV) cyclotrons. Considering the deduced thick target yield for three radioisotopes ( 110g In, 109g In and 111 In), good agreements were found with the previously published results by Abe et al. [26].
Industrial applications
Beside the medical applications some of the produced radioisotopes can also be used in the industry as radioactive tracers. The selection criteria are the proper (enough long) half-life, proper (medium energy and high intensity) gamma-lines, and good production parameters. The first two criteria limit the selection for the Ag radioisotopes, except 111 Ag which has only weak gamma lines.
The 106m Ag has the highest production yield and the production energy window still can be reached by medium energy accelerators. Its half-life allows tracing of relatively quick processes within a month application time. On the other hand, the shorter half-life ensures low environmental load.
The most interesting application of radioisotopes in tracer technology is TLA (Thin Layer Activation) [27,28]. With 106m Ag it is possible to perform TLA investigations to measure wear, corrosion or erosion rate of parts containing silver, within a period of about one month. As an example four activity distributions are presented in Fig. 13, with 4 different parameter sets. All the four activitydepth distribution provide almost a constant activity for a certain depth measured from the surface down (see Table 4). Table 4.
From Fig. 13 and Table 4 it is seen that already by 1 hour irradiation reasonable specific activity can be produced in the surface layer for wear measurements, not only at the end of the irradiation (EOB) but even after 10 days cooling time. The bombarding energy (37.6 MeV) was chosen in such a way that in the first several µm-s from the irradiated surface the activity is almost homogeneous (and the specific activity is almost constant) for a certain depth. This type of activity depth distribution can be reached for those nuclear reactions, which has an excitation function with a local maximum in the available energy range. By changing the irradiation angle the depth profile of the produced activity (and the thickness of the constant specific activity layer) can be controlled together with the specific activity. The total and the specific activity can be multiplied by increasing the beam current. A longer irradiation also provides higher activity but the irradiation time vs.
activity function is not a linear one. The produced total activity should be below the FHL (Free Handling Limit =1000 kBq for 106m Ag) [28] level, if the industrial laboratory or site has no license to handle and store radioactive materials, which is the case according to Table 4. For a licensed laboratory higher activity can be produced (with additional transport problems).
Conclusions
Excitation functions for the nat Ag(α,x) Among the possible industrial applications, the TLA method was demonstrated by using the 106m Ag the best radioisotope for this purpose. It has been proved that by using 106m Ag as tracer, the wear measurement with actual parameters can be performed. Table 4.
Fig. 1 6 From
16Re-measured cross section data of the nat Ti(α,x) 51 Cr monitor reaction compared with the recommended dataFig. 1it is seen that the first monitor foil in the stack is placed behind the last silver target foil. Since we have a good agreement among the measured and the recommended values for the monitor reaction in the low energy segment of the stack, the energy degradation calculation within the high energy segment of the stack, the silver foils of the stack, was correct. The new results for all measured radioisotopes are presented in Figs.2-11 and in
Fig 2 .
2are deduced from the intense 171.28 keV gamma-line, which is free from interferences. All the three series gave consistent results. The other intense, 245.35 keV gamma-line was excluded from the evaluation because of interferences with other produced radioisotopes.
Fig. 2
2Measured excitation function of the nat Ag(α,x) 111 In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations From Fig 2. it is seen that our new data are in excellent agreement with the previous results of Haasbroek et al. [6] and Takács et al. [10] in the overlapping energy range and also with the prediction of the EMPIRE 3.2 calculations. EMPIRE also follow the results of the previous experiments in the lower and upper energy regions. TENDL-2017 does not predict the maximum around 28 MeV correctly even produces two local maxima. The values given by TENDL-2017 are acceptable only above 65 MeV.
Fig. 3
3Measured excitation function of the nat Ag(α,x) 110g In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 110g In has several strong gamma-lines, but utmost care should have been taken because most of them interfere with the excited state of 110 In and/or with other produced radioisotopes. From Fig. 3 it is seen that our new data give a good continuation to the previous results of Takács et al. [10] at the high energy part of the local maximum on the excitation function. The data of Wasilewsky [12] was recalculated for absolute cross section. The trend is similar except the large energy shift and lower values below 50 MeV. Both TENDL-2017 and EMPIRE 3.2 give similar trend, EMPIRE describes the low energy maximum well both in position and in amplitude. The higher energy local maximum is a little bit shifted by TENDL-2017, while the position prediction of EMPIRE 3.2 is better. Both model codes underestimate the experimental values around the higher energy maximum.
Fig. 4
4Measured excitation function of the nat Ag(α,x) 110m In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 4.4 nat Ag(α,x) 109g In nuclear reaction The 109g In radionuclide is produced by direct reactions from the 107 Ag and 109 Ag target isotopes with 2 and 4 neutrons emission, respectively. Because of its relatively short half-life (4.159 h) reasonable results could only be assessed from the first measurement series. Due to its shortlived (1.34 min) meta-stable state, which completely decays into the ground-state, the measured cross section is cumulative. The 203.3 MeV gamma-line, which was used for the evaluation, is independent. The results are presented in Fig. 5.
Fig. 5
5Measured excitation function of the nat Ag(α,x) 109g In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations FromFig. 5it is seen that our new results rather support the previous data of Takács et al.[10] giving a good continuation to their previous data. All the other data show large discrepancies, only the data of Wasilewsky agree well with our new results in spite of their large energy shift in the lower energy region. The agreement with the TENDL-2017 prediction is also very good. The both theoretical model codes give similar trend and describe both local maxima correctly, but EMPIRE 3.2 underestimates the cross sections in the energy region above 40 MeV.
12 Fig. 6
126Measured excitation function of the nat Ag(α,x) 108g In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculationsFrom
Fig. 7
7Measured excitation function of the nat Ag(α,x) 111g Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations From Fig. 7 it is seen that our new results are completely different from the previously measured data. Both the theoretical model calculations cannot reproduce our new results, though EMPIRE
Fig. 8
8Measured excitation function of the nat Ag(α,x) 110m Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations From Fig. 8 it is seen that our new data give a good continuation to the previous results of Takács et al. [10]. The EMPIRE 3.2 prediction is in reasonably good agreement with our new data and the previous results. The results from the TENDL-2017 on-line library give a bit higher values up to 45 MeV and a different trend above this energy.
Fig. 9
9Measured excitation function of the nat Ag(α,x) 106m Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations
Fig. 10
10Measured excitation function of the nat Ag(α,x) 105g Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculationsFrom
Fig. 11
11Measured excitation function of the nat Ag(α,x) 109 Cd nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculationsFrom
Fig 12
12Calculated physical yields from selected α-particle induced nuclear reactions on natural silver compared with the literature dataFrom
Fig. 13
13Activity distribution of the 106m Ag radioisotope irradiated and measured according to the parameters in
Measured excitation function of the nat Ag(α,x) 111 In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 3. Measured excitation function of the nat Ag(α,x) 110g In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 4. Measured excitation function of the nat Ag(α,x) 110m In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 5. Measured excitation function of the nat Ag(α,x) 109g In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 6. Measured excitation function of the nat Ag(α,x) 108g In nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 7. Measured excitation function of the nat Ag(α,x) 111g Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 8. Measured excitation function of the nat Ag(α,x) 110m Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 9. Measured excitation function of the nat Ag(α,x) 106m Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 10. Measured excitation function of the nat Ag(α,x) 105g Ag nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 11. Measured excitation function of the nat Ag(α,x) 109 Cd nuclear reaction compared with the previous results from the literature and with the results of theoretical model code calculations 12. Calculated physical yields from selected α-particle induced nuclear reactions on natural silver compared with the literature data 13. Activity distribution of the 106m Ag radioisotope irradiated and measured according to the parameters in
Table 1
1Nuclear data for the radioisotopes produced[19,20] Isotope
spin
level energy(keV)
Half-life
Decay mode
E
(keV)
I
(%)
Contributing reactions
Q-value
(MeV)
111g In
9/2 +
2.8047 d
ε: 100%
171.28
90.7
109 Ag(α,2n)
-14.05
245.35
94.7
110g In
7 +
4.9 h
ε: 100%
657.75
98
107 Ag(α,n)
-7.58
884.68
93
109 Ag(α,3n)
-24.04
937.48
68.4
110m In
2 +
62.084
69.1 min
ε: 100%
β + : 61.3%
657.75
97.74
107 Ag(α,n)
-7.58
109 Ag(α,3n)
-24.04
109g In
9/2 +
4.159 h
ε: 100%
β + : 4.64%
203.3
74.2
107 Ag(α,2n)
-15.63
109 Ag(α,4n)
-32.09
108g In
7 +
58 min
ε: 100%
β + : 24.8%
632.9
100
107 Ag(α,3n)
-26.75
875.4
100
109 Ag(α,5n)
-44.09
111g Ag
½ -
7.45 d
β -: 100%
245.4
1.24
109 Ag(α,2p)
-12.66
110m Ag
6 +
117.595
249.76 d
ΙΤ: 1.33%
β -: 100.4%
657.76
95.61
109 Ag(α,2pn)
-21.49
763.94
22.6
884.68
75
937.49
35
106m Ag
6 +
89.667
8.28 d
ε: 100%
450.98
28.2
107 Ag(α,2p3n)
-37.83
717.34
28.9
109 Ag(α,2p5n)
-54.29
1045.83
29.6
1527.65
16.3
105g Ag
1/2 -
41.29 d
ε: 100%
280.44
30.2
107 Ag(α,2p4n)
-45.77
344.52
41.4
109 Ag(α,2p6n)
-62.23
443.37
10.5
109 Cd
5/2 +
461.9 d
ε: 100%
88.03
3.64
107 Ag(α,2p5n)
-55.80
109 Ag(α,2p7n)
-72.26
Increase the Q-values if compound particles are emitted by: np-d, +2.2 MeV; 2np-t, +8.48 MeV; n2p-3 He,
+7.72 MeV; 2n2p-α, +28.30 MeV.
Decrease Q-values for isomeric states with level energy of the isomer
Abundances: 107 Ag (51.83%), 109 Ag (48.17%)
The110 In radionuclide has a meta-stable state with an only 69.1 min half-life. It means that only data from the earliest measured spectra could be used for evaluation. The earliest measurements began about 10 hours after the end of bombardment and only the last Ag foil (lowest energy) could be measured. In the time of the next Ag measurement there was no measurable 110m In activity in the samples. The only strong and measurable gamma-line is the 657.65 keV line, but unfortunately the ground-state and 110m Ag have this same line. The contribution of the 110m In meta-stable state could be easily separated by using the independent gamma-lines of its groundstate. In the case of the 110m Ag isotope the situation was a bit more complicated, because it had no peaks with reasonable statistics in the earliest measured series, due to its long half-life and the short measuring time. The result is presented inFig. 4. Our new point is closer to the results of Patel et al.[8]. The theoretical model codes estimate the first two maxima almost correctly, but give different values above 25 MeV.
Table 2
2Experimental cross sections of the nat Ag(α,x) 111,110g,110m,109g,108g In nuclear reactions
Table 3 Experimental
3cross sections of the nat Ag(α,x) 111g,110m,106m,105g Ag, 109 Cd nuclear reactions
Alpha energy
Cross section (mb)
ΜeV
111
Ag
11om
Ag
106m
Ag
105g Ag
109
Cd
E
∆E
σ
∆σ
σ
∆σ
σ
∆σ
σ
∆σ
σ
∆σ
49.49
0.20
0.08
0.03
2.06
0.24
26.76
2.90
64.83
7.01
445.62
48.21
47.89
0.24
0.48
0.10
2.29
0.34
21.80
2.37
74.36
8.05
455.86
49.41
47.18
0.25
0.65
0.11
1.94
0.31
20.44
2.22
72.30
7.83
435.09
47.16
46.47
0.27
0.60
0.10
1.91
0.28
19.51
2.12
71.39
7.73
411.08
44.54
45.75
0.29
0.43
0.05
1.34
0.22
18.23
1.99
65.63
7.11
373.44
40.47
45.02
0.30
0.62
0.10
1.04
0.23
18.20
1.98
63.77
6.91
348.02
37.69
44.28
0.32
0.36
0.07
1.01
0.19
18.52
2.01
59.91
6.48
317.39
34.35
43.53
0.34
0.33
0.08
1.25
0.27
19.58
2.13
55.32
5.99
290.29
31.54
42.77
0.35
0.16
0.08
20.53
2.23
46.44
5.03
268.70
29.25
42.01
0.37
0.16
0.06
0.52
0.18
21.40
2.33
42.96
4.66
250.41
27.18
Alpha energy
Cross section (mb)
ΜeV
111
In
110g In
110m
In
109g In
108g In
E
∆E
σ
∆σ
σ
∆σ
σ
∆σ
σ
∆σ
σ
∆σ
49.49
0.20
24.06
2.60
195.52
21.16
357.83
38.71
47.89
0.24
32.91
3.56
270.67
29.37
323.90
35.07
47.18
0.25
34.28
3.71
311.59
33.77
306.29
33.18
46.47
0.27
37.28
4.03
342.95
37.16
289.45
31.36
45.75
0.29
38.10
4.12
372.00
40.28
262.03
28.38
45.02
0.30
41.48
4.49
394.68
42.74
237.73
25.75
44.28
0.32
44.74
4.84
425.05
46.03
213.25
23.11
43.53
0.34
49.41
5.34
451.76
48.90
189.55
20.53
42.77
0.35
55.07
5.96
477.04
51.64
174.56
18.92
42.01
0.37
58.52
6.33
445.35
48.21
311.15
65.00
156.57
16.94
289.35
32.68
Table 4
4Parameters of activity distributions in Fig. 13. (EOB=End of Bombardment)Bombarding energy (MeV)
37.6
37.6
37.6
37.6
Irradiation angle
90 o
90 o
15 o
15 o
Cooling time (day)
0(EOB)
10
0(EOB)
10
Irradiation time (h)
1
1
1
1
Beam current (µA)
2
2
2
2
Total activity (kBq)
238.24
103.15
238.24
103.15
Activity of the constant layer (kBq)
58.35
25.26
58.35
25.26
Specific activity at the surface (kBq/µm)
3.33
1.44
12.88
5.58
Thickness of the quasi-constant layer (µm)
17.50
17.50
4.53
4.53
111,110m,110g,109g,108g In, 111,110m,106m,105g Ag, 109 Cd were measured in the energy range of 40-50 MeV. The newly determined cross section data help to clarify the problems between the previous literature results. Cross section deduced for production of 110g In, 109g In, 108g In, 111 Ag and 110g Ag in most cases show a good continuation of the eventually existing literature data in a lower energy region. In the case of 111 In and 109 Cd the agreement with the previous literature data is excellent. The results of the theoretical nuclear reaction model codes are not systematic and give only partly good estimations for several reactions. There are reactions, for which both (EMPIRE and TENDL) completely fail. From the measured cross sections thick target physical yield curves were calculated. The excitation functions for these calculations were constructed by using our new results combined with data from the literature. The literature values agree well with our results.
Cd the cross sections of both states could be determined independently.
AcknowledgementsThis work was performed at the RI Beam Factory operated by the RIKEN Nishina Center and CNS, University of Tokyo. The authors express their gratitude to the crew of the AVF cyclotron for their invaluable assistance in the course of the experiment. The work of Masayuki Aikawa was supported by JSPS KAKENHI Grant Number 17K07004.
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Charged particle cross-section database for medical radioisotope production: diagnostic radioisotopes and monitor reactions. F Tárkányi, S Takács, K Gul, A Hermanne, M G Mustafa, M Nortier, P Oblozinsky, S M Qaim, B Scholten, Y N Shubin, Beam monitor reactions. 49TECDOC 1211, IAEAF. Tárkányi, S. Takács, K. Gul, A. Hermanne, M.G. Mustafa, M. Nortier, P. Oblozinsky, S.M. Qaim, B. Scholten, Y.N. Shubin, Z. Youxiang, Beam monitor reactions (Chapter 4). Charged particle cross-section database for medical radioisotope production: diagnostic radioisotopes and monitor reactions. , in: TECDOC 1211, IAEA, 2001, pp. 49.
Determination of effective bombarding energies and fluxes using improved stacked-foil technique. F Tárkányi, F Szelecsényi, S Takács, Acta Radiol., Suppl. 37672F. Tárkányi, F. Szelecsényi, S. Takács, Determination of effective bombarding energies and fluxes using improved stacked-foil technique, Acta Radiol., Suppl., 376 (1991) 72.
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| []
|
[
"Formation and field-driven dynamics of nematic spheroids",
"Formation and field-driven dynamics of nematic spheroids"
]
| [
"Fred Fu ",
"Nasser Mohieddin Abukhdeir [email protected] ",
"\nof Chemical Engineering\n‡Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n",
"\n¶Waterloo Institute for Nanotechnology\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n",
"\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n"
]
| [
"of Chemical Engineering\n‡Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada",
"¶Waterloo Institute for Nanotechnology\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada",
"University of Waterloo\nN2L 3G1WaterlooOntarioCanada"
]
| []
| Emerging technologies based on liquid crystal (LC) materials increasingly leverage the presence of nanoscale defects, unlike the canonical application of LCs -LC displays. The inherent nanoscale characteristics of LC defects present both significant opportunities and barriers for the application of this fascinating class of materials.Simulation-based approaches to the study of the effects of confinement and interface anchoring conditions on LC domains has resulted in significant progress over the past decade, where simulations are now able to access experimentally-relevant micron-scales while simultaneously capturing nanoscale defect structures. In this work, continuum simulations were performed in order to study the dynamics of micron-scale nematic LC droplets of varying spheroidal geometry. Nematic droplets are one of the simplest inherently defect-containing LC structures and are also relevant to polymer-dispersed 1 arXiv:1702.05404v1 [cond-mat.soft] 17 Feb 2017 LC-based "smart" window technology. Simulation results include nematic phase formation and external field-switching dynamics of droplets ranging in shape from oblate to prolate. Results include both qualitative and quantitative insight into the complex coupling of nanoscale defect dynamics and structure transitions to micron-scale reorientation. Dynamic mechanisms are presented and related to structural transitions in LC defects present in the droplet. Droplet-scale metrics including order parameters and response times are determined for a range of experimentally-accessible electric field strengths. These results have both fundamental and technological relevance, in that increased understanding of LC dynamics in the presence of defects is a key barrier to continued advancement in the field. Keywords liquid crystals, nematic phase, defect dynamics, polymer-dispersed liquid crystals, simulation Liquid crystals (LCs) are materials which exhibit properties characteristic of both disordered liquids and crystalline solids. Their anisotropic nature imparts unique optical properties and makes them susceptible to external fields. These properties have resulted in a wide array of electro-optical applications, such as liquid crystal displays (LCDs). However, unlike LCDs, which are designed using uniform defect-free domains, next-generation LCbased technologies are increasingly leveraging the presence of nanoscale topological defects.These emerging technologies include tunable photonics based on blue LC phases, 1 molecular self-assembly, 2 and bistable optical devices. 3 Consequently, understanding and predicting defect-enabled LC phenomena is a key barrier to continued advancements, both fundamental and technological. Theoretical and computational research is necessary to overcome this barrier due to the nanoscale lengths and times associated with LC structure and dynamics, which are currently inaccessible via experimental methods.One of the simplest inherently defect-containing structures is an LC droplet. When LC material is confined in this way, a frustrated domain with significant spatial variation in LC | 10.1039/c7sm00484b | [
"https://arxiv.org/pdf/1702.05404v1.pdf"
]
| 34,971,150 | 1702.05404 | 82fcbf3352b32079358bf95c2e974051d7db302d |
Formation and field-driven dynamics of nematic spheroids
17 Feb 2017
Fred Fu
Nasser Mohieddin Abukhdeir [email protected]
of Chemical Engineering
‡Department of Physics and Astronomy
University of Waterloo
N2L 3G1WaterlooOntarioCanada
¶Waterloo Institute for Nanotechnology
University of Waterloo
N2L 3G1WaterlooOntarioCanada
University of Waterloo
N2L 3G1WaterlooOntarioCanada
Formation and field-driven dynamics of nematic spheroids
17 Feb 2017
Emerging technologies based on liquid crystal (LC) materials increasingly leverage the presence of nanoscale defects, unlike the canonical application of LCs -LC displays. The inherent nanoscale characteristics of LC defects present both significant opportunities and barriers for the application of this fascinating class of materials.Simulation-based approaches to the study of the effects of confinement and interface anchoring conditions on LC domains has resulted in significant progress over the past decade, where simulations are now able to access experimentally-relevant micron-scales while simultaneously capturing nanoscale defect structures. In this work, continuum simulations were performed in order to study the dynamics of micron-scale nematic LC droplets of varying spheroidal geometry. Nematic droplets are one of the simplest inherently defect-containing LC structures and are also relevant to polymer-dispersed 1 arXiv:1702.05404v1 [cond-mat.soft] 17 Feb 2017 LC-based "smart" window technology. Simulation results include nematic phase formation and external field-switching dynamics of droplets ranging in shape from oblate to prolate. Results include both qualitative and quantitative insight into the complex coupling of nanoscale defect dynamics and structure transitions to micron-scale reorientation. Dynamic mechanisms are presented and related to structural transitions in LC defects present in the droplet. Droplet-scale metrics including order parameters and response times are determined for a range of experimentally-accessible electric field strengths. These results have both fundamental and technological relevance, in that increased understanding of LC dynamics in the presence of defects is a key barrier to continued advancement in the field. Keywords liquid crystals, nematic phase, defect dynamics, polymer-dispersed liquid crystals, simulation Liquid crystals (LCs) are materials which exhibit properties characteristic of both disordered liquids and crystalline solids. Their anisotropic nature imparts unique optical properties and makes them susceptible to external fields. These properties have resulted in a wide array of electro-optical applications, such as liquid crystal displays (LCDs). However, unlike LCDs, which are designed using uniform defect-free domains, next-generation LCbased technologies are increasingly leveraging the presence of nanoscale topological defects.These emerging technologies include tunable photonics based on blue LC phases, 1 molecular self-assembly, 2 and bistable optical devices. 3 Consequently, understanding and predicting defect-enabled LC phenomena is a key barrier to continued advancements, both fundamental and technological. Theoretical and computational research is necessary to overcome this barrier due to the nanoscale lengths and times associated with LC structure and dynamics, which are currently inaccessible via experimental methods.One of the simplest inherently defect-containing structures is an LC droplet. When LC material is confined in this way, a frustrated domain with significant spatial variation in LC
order can emerge. This so-called LC "texture" can differ depending on LC/solid anchoring conditions, domain shape, and LC material properties. 4 LC droplets play a major role in polymer-dispersed liquid crystal (PDLC) films, which are typically fabricated through a "bottom-up" process which results in nano-to-microscale LC domains dispersed in a polymer matrix. PDLCs are optical functional materials which exhibit an optical response when subjected to thermal or external field actuation (Figure 1a), introducing complex dynamics and constraints on response and relaxation times between equilibrium states. PDLC films have traditionally been used for optical light shutter technology, 5 in which LC domains are micron-sized. More recently however, PDLCs incorporating nano-sized domains have been incorporated into novel applications such as holographic PDLC (H-PDLC) lasers and tunable microlens arrays. 1,6 PDLC performance is governed by a variety of material and operating parameters, including LC defect-mediated structure and dynamics due to the topological constraints on LC order resulting from spheroidal confinement. It has been more than two decades since Drzaic 5 found that domain shape, specifically anisometry, strongly affects device performance. 7 Since then, it has been demonstrated that this anisometry can be directly controlled through various means, 8,9 the simplest of which is by uniaxial mechanical stretching of the PDLC film to produce highly prolate spheroidal domains ( Figure 1b). 10,11 While a significant body of past mesoscale simulation work exists for cylindrical nematic capillaries 12,13 and spherical droplets, [14][15][16] elliptic or ellipsoidal domains have been far less studied. [17][18][19][20] Furthermore, of this work, most use theoretical models which are unable to accurately capture nematic defects and phase transition. Only recently have simulations been performed which capture nematic dynamics, 21 as opposed to just determining equilibrium states. As a result, while past research has provided some insight into the nanoscale defect structure present in these domains, as of yet there have been no simulations of the dynamics of nematic spheroids on relevant length and timescales. Thus our aim is to predict the dynamic mechanisms involved in the formation, field switching, and relaxation of nematic spheroids, such as those present in PDLC-based devices. This objective has both fundamental and technological relevance in that these dynamic mechanisms are both poorly understood and necessary for advancement of this technology. From a fundamental perspective, PDLCs provide ideal templates for the study of nanoscale defect behaviour in confined LC domains. From a technological perspective, significant improvement in the performance of PDLCs as electro-and thermo-optical functional materials is required for their broader commercialization.
E=0 E≠0
(a) (b) Figure 1: (a) Schematic of the operation of a PDLC-based "smart" window where light is scattered by (left) randomly oriented nematic droplets in the absence of an electric field (translucent mode), which when exposed to an external field (right) are aligned in the direction normal to film (transparent mode). (b) SEM images of an (left) unstretched and (right) uniaxially stretched PDLC film where the resulting droplet shape is anisometric. Reproduced with permission from ref. 10.
NEMATIC PROPERTIES AND DYNAMIC MODEL
LCs include a wide variety of phases, referred to as mesophases, with the simplest mesophase being the nematic phase. Nematics exhibit not only translational disorder like a traditional liquid but also long-range orientational order, as shown by their tendency to self-align at the molecular scale. Technological applications of nematic LCs, such as LCDs, mainly involve domains that are at or close to hydrostatic equilibrium which is likely due to the significant complexity of accounting for LC hydrodynamics. 22,23 Dynamics within this regime are referred to as reorientation dynamics, in which the orientation of individual LC molecules, or mesogens, evolve in response to thermodynamic or external stimuli. This LC orientation can be described using the continuum Landau-de Gennes model of the nematic phase, 24 which introduces a symmetric traceless tensor order parameter called the alignment tensor, 25
Q ij = S(n i n j − 1 3 δ ij ) + P (m i m j − l i l j )(1)
which approximates the local orientational distribution function of the mesogens at each point in space. The alignment tensor Q may be decomposed into its eigenvalues and eigenvectors, which describe the local orientational axis or nematic director n, the uniaxial scalar order parameter S, and the biaxial scalar order parameter P (and its associated axes, given by m and l). For a nematic domain, S = P = 0 corresponds to the isotropic phase (a traditional disordered liquid), while 0 < S < 1 and P = 0 corresponds to the (uniaxial) nematic phase where a higher value of S corresponds to greater alignment. Biaxial orientational ordering occurs when both S and P are non-zero.
The majority of past simulation-based research on nematic LCs neglects variations in S, which results in a simplified model involving only the nematic director n, 26,27 f (n, ∇n)
= f 0 + 1 2 k 11 (∇ · n) 2 + 1 2 k 22 (n · ∇ × n) 2 + 1 2 k 33 (n × ∇ × n) 2 − 1 2 k 24 ∇ · (n(∇ · n) + n × ∇ × n) 2 ) (2)
which includes elastic energy terms that quantify the nematic response to orientational deformations of splay k 11 , twist k 22 , bend k 33 , and saddle-splay k 24 . Many past simulation studies of elliptic nematic capillaries and ellipsoidal droplets 14,17,19 use this simplified model despite its inability to accurately capture nanoscale defects in nematic order, called disclinations. 25 Disclinations correspond to singularities in the nematic director n, and are therefore also regions of high biaxial nematic order (S, P > 0), as opposed to isotropic regions of disorder (S = P = 0). Figure 2 shows schematics of the two main types of disclinations relevant to nematic droplets: +1 line and + 1 2 loop disclinations.
(a) (b) Figure 2: Schematics of (a) +1 and (b) + 1 2 disclination lines using a combination of hyperstreamlines to indicate nematic orientation and an isosurface indicating the nanoscale defect "core" region.
In contrast, the Landau-de Gennes model (see Methods section) is able to accurately capture both the presence of disclinations in nematic domains as well as their dynamics. A review of recent studies using this model to simulate nematic dynamics may be found in ref. 21. However, there are two major shortcomings of past simulation studies of nematic LC droplets. The first is the widely-used single elastic constant approximation, where it is assumed that k 11 = k 22 = k 33 and k 24 = 0, despite the fact that these constants can widely differ, even for commonly studied LCs 28 and may significantly affect simulation outcomes. 29 The second shortcoming is the sparsity of dynamic simulations, which can offer greater insight than simply solving for equilibrium nematic textures. In this study, material parameters are used that correspond to the 4'-pentyl-4-cyanobiphenyl (5CB), a well-characterized nematic LC. The domain is assumed to be isothermal, at hydrostatic equilibrium (v = 0), and fluctuations in nematic order are neglected. These assumptions are consistent with past simulations 30,31 except that simplifications of nematic elasticity are not made in this work.
Finally, in addition to nematic elasticity, interfacial surface anchoring effects arising from factors such as PDLC composition 32 must be considered. Surface anchoring may result in a preferred nematic director n at the droplet interface and also the enhancement of nematic ordering S > S 0 , where S 0 is the value of S at thermodynamic equilibrium. In this study, the case of homeotropic anchoring is investigated, in which n k is energetically preferred, where k is the unit normal vector to the LC droplet surface. Several experimental studies of PDLC dynamics have been performed under these conditions. [33][34][35][36] In order to study the formation and field-driven dynamics of spheroidal nematic domains relevant to electro-optical applications of PDLCs, simulations were performed of nematic spheroids with fixed volume corresponding to an initial "unstretched" sphere of diameter 500 nm. To emulate stretching of the droplets, the initial sphere was consistently elongated or contracted along a single direction. Droplet aspect ratio R is defined as the length ratio between the axis of elongation/contraction and the remaining (equivalent) axes of the spheroid, resulting in oblate droplets for R < 1 and prolate droplets for R > 1. Various aspect ratio R domains were simulated in the interval [0.5, 2] based upon experimental evidence regarding the expected variation in droplet shape. 5,8,37,38 These simulations were performed in three stages: (i) formation of the nematic phase from an initially disordered (high temperature) phase, (ii) application of an electric field corresponding to the "on" (transparent) state of a PDLC film, and (iii) relaxation resulting from release of the electric field, corresponding to the "off" (translucent) state of a PDLC film. For the formation dynamics simulations, heterogeneous nucleation of the nematic phase at the solid/LC interface was assumed based on recent experimental observations. 39 For the field dynamics simulations, a range of experimentally accessible electric field strengths up to 14 V µm −1 were applied. Further details of the nematic dynamics model, numerical methods, and auxiliary conditions used in these simulations may be found in the Methods section.
FORMATION FROM DISORDERED PHASE
Formation dynamics simulations were initially performed for oblate spheroids of aspect ratio R ∈ [0.5, 1). This geometry can be considered a rotational extrusion of a two-dimensional ellipse about its minor axis. It is therefore comparable to previous simulations of nematic elliptic capillaries, 20 in which a sequence of three different growth regimes were identified during droplet formation: (i) free growth, (ii) defect formation, and (iii) bulk relaxation. The free growth regime consists of the stable nematic "shell" growing into an unstable isotropic phase, with the bulk nematic orientation being commensurate with the homeotropic surface anchoring conditions. Next, the defect formation regime involves the impingement of the nematic-isotropic interface on itself. This resultes in the simultaneous formation of a pair of + 1 2 disclination lines along the major axis of the elliptic cross-section of the capillary. Finally, during bulk relaxation, the domain as a whole relaxes to its equilibrium state through simultaneous disclination motion and bulk reorientation.
The simulation results of the formation process for a R = 0.5 oblate droplet are shown in Figure 3. The same set of growth regimes can be identified, starting with the initial free growth of the stable nematic boundary layer into the central unstable isotropic region (Figures 3a-b). As free growth proceeds, the curvature of the isotropic/nematic interface increases in the focal regions of the spheroid and simultaneously the interface velocity decreases. This critical slowing down of the nematic/isotropic interface 20 may be explained by an approximation of the interface velocity v, 29
βv = C − ∆F(3)
where β is an effective viscosity term, ∆F is the difference in energy between the nematic and isotropic phases, and C the capillary force. For an isothermal domain, ∆F is constant and, in the absence of curvature of the interface (C = 0), the model predicts constant interface velocity v. While the isotropic/nematic interfaces in the equatorial region of the droplet are able to grow inwards with minimal increase in interface curvature, this is not the case for focal regions of the interface. As the radii of curvature of the interfaces in this region approach the nematic coherence length λ n ≈ 10 nm, 24 the capillary force C approaches the difference in free energy resulting from the transition ∆F and v → 0. At this point, the free growth regime transitions to the defect formation regime.
Figures 3c-d show the defect formation regime dynamics. Simultaneously, a + 1 2 disclination loop forms in the focal region through a interface-driven defect "shedding" mechanism 40 and the isotropic/nematic fronts in the equatorial region impinge. This is followed by the bulk relaxation regime where the droplet texture relaxes through bulk reorientation and the disclination loop expands towards the focal boundaries. As expected, the formation process of oblate nematic droplets is analogous to that of nematic elliptic capillaries due to their geometric similarities. Figure 3: Simulation visualizations of the formation process of an oblate (R = 0.5) nematic droplet from an initially isotropic (disordered) state. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orientation (alignment tensor) and isosurfaces indicate nanoscale defect "core" regions.
However, prolate nematic droplets behave differently. While prolate spheroids can also be generated by extruding an ellipse, the homeotropic surface anchoring conditions distort the symmetric nature of the system. Figure 4 shows the formation dynamics of a prolate nematic droplet, which is found to exhibit the same general regimes as the oblate droplet: free growth (Figure 4a Figure 4: Simulation visualizations of the formation process of a prolate (R = 2) nematic droplet from an initially isotropic (disordered) state. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orientation (alignment tensor) and isosurfaces indicate nanoscale defect "core" regions.
axial, in agreement with past theoretical predictions. 46 Figure 5c shows the initial distorted
EXTERNAL FIELD-DRIVEN REORIENTATION AND RELAXATION
Electric-field driven reorientation of nematic droplets is a key process in the operation of PDLC-based technology. Past experimental research has shown that droplet shape has a significant effect on the electro-optical switching process and can result in shorter switching times. 7,9 Subsequently, simulations were performed for both oblate and prolate spheroids using the equilibrium states resulting from the formation process (Figures 3f and 4f, respectively). Electric fields ranging from 0 −14 V µm −1 were applied in the direction parallel to the major axis of the droplets (x-axis), which, for the case of film stretching, corresponds to the direction orthogonal to the optical axis of the droplet at equilibrium. Since 5CB is a positive dielectric anisotropy LC, nematic orientation parallel to the electric field is energetically favored. Thus, imposition of the electric field orthogonal to the optical axis results in the maximum amount of field-induced reorientation, leading to more complex and inter-esting dynamics. This corresponds to an in-plane switching mode that has been explored experimentally for PDLC-based devices. 47,48 Two different field-switching regimes were observed depending on the magnitude of the electric field, corresponding to a Fredericks-like transition. For electric fields strengths E below a critical value E c , the nematic texture changes only slightly without undergoing bulk reorientation in the field direction. In contrast, for E above E c , a complex reorientation process occurs with defect dynamics and a reorientation of the nematic texture to a fieldaligned state. Overall, a general sequence of three dynamic regimes, consistent with nematic capillaries, 20 can be identified during this process, consisting of:
Regime I. Bulk growth and recession, involving growth of the field-aligned focal regions and recession of the misaligned central region;
Regime II-A. Disclination and bulk rotation, involving rotation of the disclination loop orthogonal to the field direction; and
Regime II-B. Bulk relaxation, involving expansion of the disclination loop until the force from the applied field equilibrates with the elastic and anchoring forces in the system. In all cases it was observed that upon release of the electric field, the nematic texture was restored to the initial equilibrium texture resulting from the earlier formation process. Figure 6: (a-f) Simulation visualizations of the electric field-switching process for E = 14 V µm −1 > E c applied along the x-axis of an oblate (R = 0.5) nematic droplet starting from (a) the equilibrium texture (following formation) and proceeding to the (f) the field-driven equilibrium texture. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orientation (alignment tensor) and isosurfaces indicate nanoscale defect "core" regions. Figure 7 shows the field-driven switching dynamics of a R = 2 prolate droplet where E > E c . The dynamic regimes observed here are more similar to those for nematic capillaries than for oblate spheroids. In particular, the transition between dynamic regime II-A and II-B is more distinct. This result can be attributed to the difference in disclination loop structure between the two cases, which is imposed by their geometries. For the prolate droplet, as the size of the disclination loop decreases following application of the electric field, the loop becomes circular and its size is nanoscale. In contrast, the oblate droplet disclination loop transitions from circular to elliptic after application of the field and the major axis of the elliptic loop maintains the micron-scale size of the overall droplet. Next, dynamic regime II-A proceeds (Figures 7c-d) Figure 7: (a-f) Simulation visualizations of the electric field-switching process for E = 14 V µm −1 > E c applied along the x-axis of a prolate (R = 2) nematic droplet starting from (a) the equilibrium texture (following formation) and proceeding to the (f) the field-driven equilibrium texture. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orientation (alignment tensor) and isosurfaces indicate nanoscale defect "core" regions.
Upon release of the external field, the nematic texture at equilibrium while the field was applied is now a high-energy state. Relaxation of the texture back to the original equilibrium state is due to a so-called "restoring" force 5 which arises from a combination of confinement geometry and surface anchoring conditions. Figures 8-9 show simulation results of these dynamic mechanisms for R = 0.5 oblate and R = 2 prolate droplets, respectively. Here, the relaxation process is, qualitatively, the reverse of the field-on process. One significant difference was observed for the oblate droplet, in which the shape of the disclination loop during the relaxation process is different than that for the field-on case.
Focusing on the oblate droplet, the disclination loop shape during field-on conditions ( Figure 6c) is elliptic while during release conditions (Figure 8b) it is circular. For the fieldon case, the elliptic disclination loop has a minor axis parallel to the field direction. This elliptic shape is initially driven by the growth of the field-aligned regions and recession of the unaligned central region within the droplet. As dynamic regime II-A proceeds, the elliptic character of the disclination loop is enhanced due to its proximity to the droplet's elliptic cross-section. For the release case, the disclination loop is circular at the beginning of the rotation regime, and continues to maintain this shape throughout the rotation process. As the disclination loop recedes from the elliptic part of the nematic/solid interface, it continuously transitions toward a state of minimum mean curvature which results in a circular shape. As the loop rotates, this circular character of disclination loop is enhanced due to its proximity to a circular cross-section of the nematic/solid interface. Additionally, unlike in the field-driven case there is a distinct transition from dynamic regime II-A (Figures 8b-d) to the dynamic regime II-B (Figures 8e-f). Figure 8: (a-f) Simulation visualizations of the field-off relaxation process after applying a field E = 14 V µm −1 > E c along the x-axis of an oblate (R = 0.5) nematic droplet starting from (a) the field-on equilibrium texture and proceeding to the (f) the field-off equilibrium texture. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orientation (alignment tensor) and isosurfaces indicate nanoscale defect "core" regions. Figure 7d, the disclination loop is slightly elliptic with minor axis parallel to the field direction. As it rotates the disclination loop transitions to a circular shape resulting from its proximity to a circular cross-section of the nematic/solid interface, shown in Figure 7f. Upon relaxation of the field (Figure 9c), the disclination loop adopts a circular shape throughout the rotation process which is followed by transition to an elliptic shape due to its proximity to the elliptic cross-section of the droplet. Figure 9: (a-f) Simulation visualizations of the field-off relaxation process after applying a field E = 14 V µm −1 > E c along the x-axis of a prolate (R = 2) nematic droplet starting from (a) the field-on equilibrium texture and proceeding to the (f) the field-off equilibrium texture. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orientation (alignment tensor) and isosurfaces indicate nanoscale defect "core" regions.
DROPLET-SCALE DYNAMICS
In order to analyze the external field-switching and relaxation dynamics quantitatively, a volume-averaged droplet uniaxial scalar order parameter S d and director n d can be determined 49 through eigendecomposition of the volume-averaged alignment tensor Q d :
Q d,ij = V −1 V Q ij dV(4)
where V is the volume of the domain. cases, however, the evolution of S d for spherical and prolate droplets is found to involve three dynamic regimes, while the oblate droplet involves only two. These quantitative findings support the qualitative observations from the previous section, where for oblate droplets dynamic regimes II-A and II-B occur simultaneously, whereas for the prolate droplets they are distinct. Furthermore, the spherical droplet field-switching dynamics are found to be comparable to that of the prolate droplet, except for that the field-alignment of the droplet director n d occurs very early in the field-on process for the spherical case.
These trends also indicate that dynamic regime I for spherical and prolate droplets occurs in two stages, unlike for the oblate droplet case. In Figures 10a and 10c (spherical and prolate droplets), S d initially decreases during regime I, followed by a rotation of the droplet director n d and a simultaneous increase in S d . This is more pronounced for the prolate droplet than the spherical droplet. In Figure 10b, S d does not exhibit this nonmonotonic evolution during dynamic regime I.
The difference in field-on dynamics between prolate/spherical and oblate droplets may be explained through qualitative comparison of the disclination dynamics of the prolate and oblate droplets during the initial bulk growth/recession regime. Focusing on the prolate droplet case, during the first stage of this dynamic regime, the field-aligned regions of the droplet grow while simultaneously the disclination loop diameter decreases. The decrease is disclination loop diameter does not initially result in interaction of adjacent regions of the loop, which would result in a high-energy elastic interaction of the nanoscale defect "core" regions. 43 During the second stage of this regime, the droplet scalar order parameter evolution decreases resulting from an overall slowing of the reorientation dynamics. This is due to a slowing down of the macroscale field-alignment in the bulk domain as the disclination loop diameter approaches a critical value where adjacent defect core regions interact. 20 Following this, the domain transitions to dynamic regime II-A which occurs rapidly followed by a long timescale regime II-B. For the oblate droplet case, the dynamic regime I is not observed to have two stages, implying different dynamics of the disclination loop during this regime.
Referring back to Figure 6, as the disclination loop reduces in size, it forms an elliptic shape which results in the focal segments of the loop having high curvature. These high-energy regions preclude the possibility of adjacent disclination cores approach each other, and thus dynamic regime I for the oblate droplet does not involve interaction of adjacent defect core regions of the loop, unlike in the prolate case.
As mentioned in the previous section, the field-off/release dynamics, also shown in Figure 10, are inherently different from the field-on dynamics due to the absence of an external field.
The restoring force resulting from the frustration of the field-on nematic texture with respect to the combination of the geometry, surface anchoring conditions, and nematic elasticity is substantially different for the spherical droplet case compared to both the oblate and prolate droplets in that there is only a very weakly imposed droplet director due to the geometry being essentially isometric. Thus the release dynamics for this droplet involve only a bulk relaxation of the nematic texture. As was described in the previous section, oblate and prolate droplets exhibit dynamics qualitatively similar to the field-on case, except in reverse. (Figures 6 and 8), and (c) R = 2 prolate (Figures 7 and 9) nematic droplets resulting from application (left column) and release (right column) of electric fields with strengths ranging from E = 2 − 14V µm −1 . Curves represent the droplet scalar order parameter S d with solid/dotted lines corresponding to the droplet director n d orthogonal/parallel to the electric field direction. Vertical bars with labels indicate the simulation time at which the corresponding simulation snapshots were taken for the oblate (Figures 6 and 8) and prolate (Figures 7 and 9) switching dynamics.
Analysis of the droplet order parameter evolution for the field-off case shown in Figure 10 indicates that the dynamics are qualitatively similar, but both prolate and oblate droplets exhibit only two dynamic regimes with dynamic regimes II-A and II-B combined.
Equilibrium droplet scalar order parameter values and response times for a range of electric field strengths were also determined from simulations, which are of interest for PDLCbased devices and other technological applications. Figure 11 shows simulation results of droplet order parameter S d at equilibrium, field-on response times τ on , and field-off response times τ off for oblate, spherical, and prolate droplets for a range of electric field strengths.
Measurements for τ on and τ off were estimated based on the time for S d to reach steady-state in order to be more comparable to experimental measurements, which are based on changes in optical film transmission. 38 As shown in Figure 11, equilibrium S d values varied significantly depending on both droplet shape and field strength. Spherical droplets, which exhibit the lowest E c , lack a strongly preferred droplet director, meaning that even relatively weak electric fields are effective for field-aligning the nematic texture. Furthermore, the droplet order parameter S d disclination line structure into a + 1 2 disclination loop is predicted to result in an "unraveling" of the nanoscale loop structure, similar to the nematic elastica behavior observed in nematic capillaries.
Simulations of electric field-driven reorientation and relaxation dynamics reveal the mechanisms of the reorientation process, which are highly dependent on domain shape and external field strength. Both oblate and prolate spheroidal droplets are found to have qualitatively similar dynamic reorientation mechanisms, with the critical (reorientation) electric field strength E c being significantly higher than for spherical droplets. For electric fields E < E c , the nematic texture of anisometric droplets becomes increasingly frustrated between the orientation imposed by the external field and that preferred by the geometry and anchoring conditions. This corresponds to an optical state that is increasingly light scattering. For electric fields E > E c , the nematic texture transitions to a field-aligned state through a series of complex and distinct dynamic mechanisms involving both micron-scale reorientation and nanoscale defect dynamics.
In summary, the presented results provide both qualitative and quantitative insight into the dynamics of nematic spheroids with resolution of the nanoscale length and timescales inherent to LC domains which include defects. These simulations include the dynamic regimes relevant to PDLC-based devices and thus could be used to guide the design and optimization of their performance as optical functional materials. Additionally, these results provide fundamental insight into the effects of nanoscale defect dynamics on confined LC domains.
METHODS
Nematic Reorientation Dynamics Model. Simulations are performed using the Landaude Gennes continuum model for the nematic phase, 24,51 which uses an alignment tensor, 25 or Q-tensor, order parameter to quantify nematic order:
Q ij = S(n i n j − 1 3 δ ij ) + P (m i m j − l i l j )(5)
where S and P are uniaxial and biaxial nematic scalar order parameters, n i is the nematic director, and m i , l i are the biaxial orientation vectors. The Helmholtz free energy density of the domain is: 24,51 f
b − f iso = 1 2 a(Q ij Q ji ) + 1 3 b(Q ij Q jk Q ki ) + 1 4 c(Q ij Q ji ) 2 + 1 2 L 1 (∂ i Q jk ∂ i Q kj ) + 1 2 L 2 (∂ i Q ij ∂ k Q kj ) + 1 2 L 3 (Q ij ∂ i Q kl ∂ j Q kl ) + 1 2 L 24 (∂ k Q ij ∂ j Q ik ) − • 8π + 2 ⊥ 3 δ ij + ( − ⊥ )Q ij E j E i (6)
where f iso is the free energy of the isotropic phase, which is assumed to be constant. All three second-order terms in Q ij are used, while the third-order L 3 term is used in order to resolve splay-bend anisotropy, and L 24 is used to quantify saddle-splay elasticity. The L 24
term is also referred to as L 3 or L 4 , depending on the reference source. 31,51,52 Additionally, a contribution to the free energy from the solid/nematic interface corresponding to homeotropic surface anchoring is used: 53
f s = αk i Q ij k j(7)
where k i is the surface unit normal and α is the surface anchoring strength. A value of α = −1.0 × 10 −4 J/m 2 was used, which is corresponds moderately strong surface anchoring with a surface extrapolation length ξ s = L 1 α ≈ 100 nm). 30,53 The total free energy of the domain includes both bulk and surface contributions:
F [Q ij ] = V f b dV + S f s dS.(8)
Nematic reorientation dynamics are modelled using the time-dependent Ginzburg-Landau
model: 54 ∂Q ij ∂t = −Γ δF δQ ij ST(9)
where Γ = µ −1 r where µ r is the rotational viscosity of the nematic phase, and [] ST indicates the symmetric-traceless component.
Numerical solution of the resulting system of nonlinear partial differential equations was performed using the finite element method with the software package FEniCS 55 on meshes of spheroid geometries or "droplets" with aspect ratio R = c a , where c and a correspond to the lengths of the major and minor axes of the spheroid. Droplet volume was maintained constant for each geometry and set to be equivalent to the volume of a perfectly spherical droplet with diameter 500 nm with mesh spacing less than the nematic coherence length in order to accurately resolve the defect structure.
The governing equations were nondimensionalized before solving, which gives rise to a time scale t s :t = t t s , t s = µ r a 0 T ni (10) An estimate for t s can be calculated using the parameters given in Table 1.
Simulation Method Conditions. The model parameters used approximate the liquid crystal 4-cyano-4'-pentylbiphenyl (5CB) and are given in Table 1. Values were chosen according to experimental data for a temperature of T = 307 K. 28,40,56 were determined using known empirical models. 28,56,58 The saddle-splay constant k 24 , which has been difficult for researchers to measure consistently for 5CB, [58][59][60] was chosen such that the elastic energy penalty term L 24 remained positive (k 24 = 0.25k 22 ). However, this is not a strict condition and negative L 24 is possible as long as the Frank elastic constants satisfy Ericksen's inequalities. 31 The simulation for determining field-off equilibrium droplet textures was initialized using a uniaxial boundary layer with scalar order parameter S 0 = S eq . The boundary is aligned perpendicular to the surface in accordance with the surface boundary conditions ref. 20.
Simulations of electric field switching were conducted using these equilibrium textures as initial conditions for each field strength studied.
Visualization. Three-dimensional visualizations of the droplets were generated using hyperstreamline seeding of the Q-tensor field. 61 Hyperstreamlines are used to represent the orientational order tensor Q ij (x, t). 61,62 These structures are an extension of streamlines and orient along the director field n i (x, t), with varying width in order to visualize the additional degrees of freedom associated with tensorial data. Hyperstreamlines are colored according to the scalar order parameter S. Disinclination lines are indicated the blue contour surfaces which were computed for a fixed biaxial scalar order parameter P > 0.
order parameter S are used to visualize nematic orientation (alignment tensor) and
isosurfaces indicate nanoscale defect "core" regions (refer to Methods section). Time is given as a dimensionless quantity (see eqn. 10).
• oblate_formation.mpg: Video of formation dynamics for a R = 0.5 oblate droplet simulation, corresponding to Figure 3.
• prolate_formation.mpg: Video of formation dynamics for a R = 2 prolate droplet simulation, corresponding to Figure 4.
• sphere_fieldon_14Vum.mpg: Video of field-switching dynamics of a R ≈ 1 spherical droplet for E = 14 V µm −1 > E c applied along the x-axis.
• oblate_fieldon_14Vum.mpg: Video of field-switching dynamics of a R = 0.5 oblate droplet for E = 14 V µm −1 > E c applied along the x-axis, corresponding to Figure 6.
• prolate_fieldon_14Vum.mpg: Video of field-switching dynamics of a R = 2 prolate droplet for E = 14 V µm −1 > E c applied along the x-axis, corresponding to Figure 7.
• sphere_fieldrelease_from_14Vum.mpg: Video of field-off relaxation dynamics for a R ≈ 1 spherical droplet simulation.
• oblate_fieldrelease_from_14Vum.mpg: Video of field-off relaxation dynamics for a R = 0.5 oblate droplet simulation, corresponding to Figure 8.
• prolate_fieldrelease_from_14Vum.mpg: Video of field-off relaxation dynamics for a R = 2 prolate droplet simulation, corresponding to Figure 9.
), defect formation (Figures 4b-d), and bulk relaxation (Figures 4e,f).Despite being topologically equivalent to the oblate droplet, the defect formation mechanism for a prolate droplet is substantially more complex. First, a pair of +1 point defectlike structures form as the high-curvature focal regions impinge(Figure 4b). These structures are not true point defects in that the nematic phase within the droplet is not fullyformed. The defect formation mechanism proceeds through the continued impingement of the isotropic/nematic interface along the droplet equator. This results in the point-like defects growing into the center of the droplet where they impinge to form a high-energy +1 disclination line. As expected based on past two-dimensional simulation results 41,42 and defect energy scaling analysis, 43 this defect line then splits into a + 1 2 disclination loop for the prolate and spherical droplet (not shown) cases. Notably, the simulations predict the dynamic mechanism through which this transition occurs.Figures 4c,dshow that there is a degeneracy in the direction in which the +1 disclination line splits, which results in this splitting direction varying along its length. The resulting + 1 2 disclination loop then undergoes an elastic relaxation process driven by defect line tension, bending, and torsion.
Figure 5
5shows the uniaxial S and biaxial P nematic order parameters in the vicinity of the central region of the prolate droplet during the disclination splitting process. This process is similar to disclination line-loop dynamics observed by Shams and Rey, 44,45 for which they developed a nematic elastica model for defect dynamics which captures line tension and bending of disclinations. In the presently observed defect splitting process, line torsion, in addition to tension and bending, would need to be accounted for which could be accomplished through incorporating higher order terms in the nematic elastica model.
Figures
5a-b show the formation of an unstable +1 disclination line originating from the joining of a pair of +1 disclination lines growing into the unstable isotropic center of the prolate droplet.
Figure 5b
5bshows that the central region of the fully-formed +1 disclination line is uni-
loop immediately following the splitting process. The loop has significant bending and torsion resulting from the degeneracy in the splitting process. It eventually relaxes into a loop with no torsion (Figure 5d) where the central region of the droplet is well-aligned with little distortion of the nematic director.Finally, following the complex defect formation regime, the bulk relaxation regime is observed where the fully-formed nematic texture of the droplet relaxes through bulk reorientation and expansion of the disclination loop. Comparing the equilibrium nematic textures of the oblate(Figure 3f) and prolate droplets(Figure 4f), the oblate droplet exhibits a relatively uniform nematic texture due to the surface exhibiting commensurate anchoring conditions with the bulk elasticity. The prolate droplet exhibits a more non-uniform texture, and is more similar to the radial-like textures often observed in spherical nematic droplets.
Figure 5 :
5Plot of uniaxial S and biaxial P nematic order parameters versus position along the major axis of the R = 2 droplet (illustrated in red) showing the progression of +1 disclination formation and splitting process.
Figure 6
6shows simulation results of the field-driven switching dynamics of a R = 0.5 oblate nematic droplet where E > E c . Initially, the disclination loop contracts along the x-axis (Figures 6b-c, dynamic regime I) as the field-aligned regions grow. This process ends once the defect loop is "compressed" sufficiently into an elliptic shape such that the elastic energy penalty resulting further shape dynamics of the defect loop approaches that of the applied field. For oblate droplet simulations where E < E c (not shown), dynamic regime I was the only dynamic regime observed. Figures 6d-e show the disclination/bulk rotation regime that follows. Unlike the dynamic mechanism for field-switching of nematic capillaries, 20 the rotation of the loop is accompanied by both expansion of the loop and bulk rotation of the nematic director throughout the droplet. This corresponds to a combination of dynamic regimes II-A and II-B.
with a minimal increase in the defect loop diameter, unlike in the oblate case. Following this, dynamic regime II-B is observed which the disclination loop diameter transitions from nanoscale to micron-scale, corresponding to the length scale imposed by the droplet geometry.
Figure 9
9shows simulation results of the relaxation of a R = 2 prolate droplet. For this case, the difference in the disclination loop shape between the field-on (Figure 7d) and release (Figure 9c) is more subtle, but similar to the oblate case. For the field-on case, shown in
Figure 10 shows
10the evolution of S d and n d for the field-switching and relaxation simulations of oblate (Figures 6 and 8), prolate (Figures 7 and 9), and (not shown) spherical nematic droplets which were presented in the previous section. For the field-on dynamics, evolution of S d for E < E c exhibits a single bulk growth/recession regime. For the E > E c
Figure 10 :
10Droplet-scale order evolution plots for (a) R ≈ 1 spherical (not shown), (b) R = 0.5 oblate
Figure 11 :
11increases monotonically with increasing field strength. In contrast, for both oblate and prolate droplets, S d is nonmonotonic with respect to electric field strength, initially decreasing for E < E c and then increasing as E > E c . For the cases where E < E c , oblate and prolate droplet responses do not involve reorientation of the droplet director. Instead, S d decreases corresponding to decreased nematic alignment about the intrinsic droplet director resulting from geometry and anchoring conditions. For the cases where E > E c , full droplet director reorientation occurs in both prolate and oblate droplets, but to differing degrees. The critical field strength for the oblate droplet reorientation is relatively high (10-12 V µm −1 ), due to the large portion of the nematic/solid interface promoting alignment along the intrinsic droplet director. In contrast, the critical field strength for the prolate droplet rorientation is relatively low (6-8 V µm −1 ) for the opposite reason. Once reorientation occurs, the droplet scalar order parameter increases linearly with E as the electric field influence overcomes surface anchoring forces. (top) Equilibrium droplet scalar order parameter S d versus electric field strength.(middle) Response times to reach field-driven equilibrium τ on versus electric field strength.(bottom) Response times to reach field-release equilibrium τ off versus electric field strength. Unfilled points correspond to droplet textures that are not field-aligned, while filled points correspond to those which are.The results for field-on response times for both prolate and spherical droplets are comparable to experimental results for spherical droplets under similar conditions (≈ 1 ms). 34 Both field-on and field-off response times for oblate droplets are significantly lower, on the order of 0.1 ms, which is due to their negligible change in texture in response to an applied field, as indicated by very little change in the droplet order parameter between field-on and field-off states. However, simulation results for field-off response times for both prolate and spherical droplets are somewhat lower than experimental results, 34 ≈ 10 ms versus ≈ 30 ms, respectively. This can be attributed to the significantly larger length scale of nematic droplets studied experimentally, 1 − 7µm, which results in a decreased ratio of restoring to viscous forces, slowing down droplet dynamics.CONCLUSIONSIn this work continuum simulations were performed in order to predict the dynamic mechanisms involved in the formation, field switching, and relaxation of nematic LC droplets with varying spheroidal geometry. The presented simulation results have both fundamental and technological relevance in that formation and field-switching dynamic mechanisms were previously poorly understood and of significant relevance to the performance of PDLC-based optical functional materials. The key feature of these nematic domains is the presence of nanoscale defect structures which contribute to the dynamics of the micron-scale domain in complex ways.Simulations of formation dynamics from an initially unstable isotropic phase predict intrinsically different defect formation mechanisms in anisometric droplets (oblate and prolate) compared to spherical ones. Defect loop structures, which are topologically imposed by domain geometry and anchoring conditions, are observed to form through the combination of defect shedding and splitting dynamic mechanisms. A degeneracy in the splitting of a +1
The values of the elastic constants L 1 to L 24 were derived 31,57 from the Frank elastic constants k 11 = 2.5 × 10 −12 J/m, k 22 = 1.7 × 10 −12 J/m, and k 33 = 3.0 × 10 −12 J/m, which
The droplet scalar order parameter S d is analogous to nematic scalar order parameter S in Equation 5, where S d → 0 corresponds to a nematic droplet with no preferred alignment and S d → 1 corresponds to uniform aligned along n d .The case where S d → 0 may correspond to two possible states of the nematic droplet: a
fully isotropic (disordered) state or a symmetrically radial nematic texture. In this work, all
analysis is performed for fully-formed nematic droplets and thus S d → 0 corresponds to the
latter state. From a general optical applications perspective, lower values of S d correspond
to nematic droplets which scatter light, while higher values of S d correspond to nematic
droplets with improved optical transparency. 50
Table 1 :
1Material parameters for 5CB.
AcknowledgementThis work was supported by the Natural Sciences and Engineering Research Council of Canada and Compute Canada.Supporting Information AvailableThe following files are available free of charge.• sphere_formation.mpg: Video of formation dynamics for a R ≈ 1 spherical droplet simulation. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar
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| []
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[
"Towards Graphene Nanoribbon-based Electronics",
"Towards Graphene Nanoribbon-based Electronics"
]
| [
"Bing Huang \nDepartment of Physics\nTsinghua University\n100084BeijingPeople's Republic of China\n",
"Qimin Yan \nDepartment of Physics\nTsinghua University\n100084BeijingPeople's Republic of China\n",
"Zuanyi Li \nDepartment of Physics\nTsinghua University\n100084BeijingPeople's Republic of China\n",
"Wenhui Duan \nDepartment of Physics\nTsinghua University\n100084BeijingPeople's Republic of China\n"
]
| [
"Department of Physics\nTsinghua University\n100084BeijingPeople's Republic of China",
"Department of Physics\nTsinghua University\n100084BeijingPeople's Republic of China",
"Department of Physics\nTsinghua University\n100084BeijingPeople's Republic of China",
"Department of Physics\nTsinghua University\n100084BeijingPeople's Republic of China"
]
| []
| The successful fabrication of single layer graphene has greatly stimulated the progress of the research on graphene. In this article, focusing on the basic electronic and transport properties of graphene nanoribbons (GNRs), we review the recent progress of experimental fabrication of GNRs, and the theoretical and experimental investigations of physical properties and device applications of GNRs. We also briefly discuss the research efforts on the spin polarization of GNRs in relation to the edge states. | 10.1007/s11467-009-0029-3 | [
"https://arxiv.org/pdf/1002.4461v1.pdf"
]
| 118,683,596 | 1002.4461 | 6258082154ae31d6ae83271abba1f319c8cdf272 |
Towards Graphene Nanoribbon-based Electronics
24 Feb 2010 (Dated: February 24, 2010)
Bing Huang
Department of Physics
Tsinghua University
100084BeijingPeople's Republic of China
Qimin Yan
Department of Physics
Tsinghua University
100084BeijingPeople's Republic of China
Zuanyi Li
Department of Physics
Tsinghua University
100084BeijingPeople's Republic of China
Wenhui Duan
Department of Physics
Tsinghua University
100084BeijingPeople's Republic of China
Towards Graphene Nanoribbon-based Electronics
24 Feb 2010 (Dated: February 24, 2010)
The successful fabrication of single layer graphene has greatly stimulated the progress of the research on graphene. In this article, focusing on the basic electronic and transport properties of graphene nanoribbons (GNRs), we review the recent progress of experimental fabrication of GNRs, and the theoretical and experimental investigations of physical properties and device applications of GNRs. We also briefly discuss the research efforts on the spin polarization of GNRs in relation to the edge states.
I. INTRODUCTION
Graphene, one monolayer of carbon atoms tightly packed into a two-dimensional honeycomb lattice, is actively being pursued as a material for next-generation electronics due to its promising electronic properties, such as high carrier mobility [1,2], long phase coherence lengths [3]. On the other side, the unique twodimensional atomic structure of graphene implies unique confinement on electron system and offers a perfect platform to explore the amazing physics phenomenons, such as quantum Hall effect [4][5][6][7] and massless Dirac fermions [7][8][9][10][11][12].
The first task for experimentalists to study graphene electronics is to fabricate high quality single layer graphene. Until now, several different experimental methods have been proposed and realized to prepare single layer (or few layers) graphene, including mechanical exfoliation of highly oriented pyrolytic graphite [1], patterned epitaxially grown graphene on silicon carbide or transition metal (e.g. Ru, Ni) substrates [2,[13][14][15], liquid-phase exfoliation of graphite [16][17][18], substratefree gas-phase synthesis [19], and chemical vapor deposition [20,21]. The success in fabricating single layer graphene has stimulated the extensive research efforts (both theoretical and experimental) in graphene related research area.
The ultimate goal of the use of graphene in nextgeneration electronics is to realize all-graphene circuit with functional devices built from graphene layers or graphene nanoribbons (GNRs) [22,23]. As the basic building blocks of such circuit, the concept of electronic devices based on graphene have been proposed theoretically and realized by experiments recently, such as field effect transistors [23][24][25][26], p-n junctions [27][28][29][30], gas molecule sensor [31][32][33], and so on.
In this article, we will focus our discussion on the basic electronic and transport properties of GNRs and their application to electronic devices. In particular, the theoretical investigations of GNRs physics and the technical aspects of GNR based electronic devices will be reviewed * E-mail address: [email protected] in detail. For other topics on the recent experimental and theoretical research efforts on graphene, please refer to the reviews by Katsnelson [34], Geim et al. [35], Beenakker [36], and Castro Neto et al. [37],
II. EXPERIMENTAL FABRICATION OF GRAPHENE NANORIBBONS
The realization of graphene electronics relies on the ability to modify the electronic properties of finite-size graphenes (for example, from semiconducting to metallic) by varying their size, shape, and edge orientation. Such unique property compared to traditional semiconductor materials, such as silicon, would ultimately enable the design and miniaturization of future electronic circuit by patterned graphene. One of the most important issues in patterned graphene fabrication is the control of the nanoribbon width. In order to take advantage of quantum confinement effects in graphene, the ribbon width should go down to nanometer scale. To realize the patterning of graphene with nano-scale width, several different techniques have been proposed including standard e-beam lithography (Fig. 1a) [38,39], microscope lithography ( Fig. 1b) [40][41][42], chemical method ( Fig. 1c) [43], metallic nanoparticle etching [44], and e-beam irradiation of ultrathin poly(methylmethacrylate) (PMMA) [45]. As shown in Fig. 1a, the scanning electron microscopy (SEM) image reveals the graphene can be patterned by traditional e-beam lithography technique into nanoribbons with various widths ranging from 20 to 500 nm [39]. Figure 1b shows 10-nm-wide nanoribbon etched via scanning tunnelling microscope (STM) lithography. By setting the optimal lithographic parameters, it is possible to cut GNRs with suitably regular edges, which constitutes a great advance towards the reproducibility of GNR-based devices [40]. Figure 1c shows atomic force microscopy images of chemically derived GNRs with various widths ranging from 50 nm to sub-10 nm. These GNRs have atomic-scale ultrasmooth edges [43].
The electronic properties of GNRs exhibit a strong dependence on the orientation of their edges. As two typical types, armchair GNRs (AGNRs) and zigzag GNRs (ZGNRs) can be obtained by lithography technology along the specific orientation on graphene (Fig. 2b) [38,40]. Actually, the detailed edge structures (both armchair and zigzag) have already been clearly observed in recent experiments [46][47][48]. One of the most serious obstacle to graphene electronic application is the reliable control of the edge structure of GNRs. Theoretical studies predict that edge states (in a manner similar to the well-known concept of surface states of a 3D crystal) in graphene are strongly dependent on the edge termination and affect the physical properties of GNRs [23,[49][50][51][52][53][54]. However, until now there is no reliable experimental method which is able to exactly control the edge structures and reduce their roughness. An interesting experimental observation is that the band gaps of GNRs show little orientation dependence [38] and all fabricated GNRs show semiconducting behavior [55], which seems inconsistent with theoretical results [49][50][51]. One of the reason for such inconsistency comes from the roughness of GNR edges and our explanation is also given in the following part of the article. Another issue related to GNR edges is the edge passivation. Since the dangling bonds from the edge carbon atoms have relatively high chemical activity, there is the possibility that other chemical elements present in the material fabrication process (such as C, O, N, H and other chemical groups formed by these atoms) would interact with the edge atoms and modify the electronic properties of GNRs. To the best of our knowledge, this issue has not been properly solved experimentally.
III. ELEMENTARY ELECTRONIC AND TRANSPORT PROPERTIES OF GRAPHENE NANORIBBONS
Next we will review some basic electronic and transport properties of GNRs from the theoretical viewpoint. Figs. 2a and 2b show two typical models of armchair and zigzag GNRs in first-principles or other atomic-level electronic structure calculations, noting as 11-AGNR and 6-ZGNR, respectively. Here the numbers 11 and 6 are defined as the width index, N . In order to remove the effect of dangling bonds, the edges of GNRs are saturated by hydrogen atoms. As geometrically terminated graphene, the electronic structure of GNRs can be modelled by imposing appropriate boundary conditions on Schrödinger's equation with simple tight-binding (TB) approximations based on π-states of carbon [49,50]. Another way to get the band structure is to solve two-dimensional Dirac's equation of massless free particles with an effective speed of light to model GNR system [56]. Within these models, it is predicted that GNRs with armchair-shaped edges can be either metallic or semiconducting depending on their widths, as shown in Fig. 3a. On the other side, the GNRs with zigzag-shaped edges are metallic with peculiar edge states on both sides of ribbons regardless of their widths, as shown in Fig. 4a. [49,50,57] Further detailed ab initio and GW quasiparticle calculations show that all of the AGNRs exhibit semiconducting behavior and the energy gaps decrease as a function of increasing ribbon widths. The variation in energy gaps can be separated into three distinct family behaviors [23,52,53,58], as shown in Fig. 3b. As mentioned above, such dependence of band gap on the geometrical structure of GNR offers unique possibility to modify the electronic properties of GNRs simply by controlling the width and edge orientation in order to realize allgraphene functional devices.
Upon inclusion of the spin degrees of freedom within density functional theory (DFT) calculations, ZGNRs are predicted to have a magnetic insulating ground state with ferromagnetic ordering at each zigzag edge and antiparallel spin orientation between the two edges [52,57], as shown in Fig. 4b. The spin polarization originates from the edge states that introduce a high density of state (DOS) at the Fermi energy. It can be qualitatively understood in terms of the stoner magnetism of sp electrons (in analogy to conventional d electrons), which occupy a very narrow edge band and render instability of spin-band splitting [59]. What is more interesting, the zigzag GNRs show half-metallic behavior when external transverse electric field is applied across the ZGNRs along the lateral direction [57], as shown in Fig. 4c. However, such spin related half-metallic phenomenon becomes weak with increasing ribbon width (since the total energy difference per edge atom between spin-unpolarized and spin-polarized edge states is only about 20 meV in their simulation system and decreases with increasing width) and is not energetically stable if the width of GNR is significantly larger than the decay length of the spin-polarized edge states [60,61]. On the other hand, it is predicted that the half-metallicity can be also achieved in edge-modified or doped ZGNRs [62][63][64][65] Another important issue regarding the basic electronic structures of GNRs relies on the edge states. Due to the presence of the edge states, the π and π * subbands of metallic ZGNRs (in the spin-unpolarized state) do not cross with each other at the Fermi level to span the whole energy range like metallic armchair carbon nanotubes (CNTs) (the left panel of Fig. 5a). This leads to the fact that the transport property of ZGNRs under a low bias voltage (or a small potential step) is only determined by the transmission between π and π * subbands (as shown in Fig. 5a). With the presence of such unique band structures, ZGNRs exhibit two distinct transport behaviors depending on the existence of σ mirror symmetry with respect to the midplane between two edges [66][67][68][69], although all the ZGNRs have similar metallic energy band structures. Since the π and π * subbands of symmetric ZGNRs (i.e., width index N is an even number) have opposite definite σ parities, the transmission between them is forbidden (the left panel of Fig. 5b). For asymmetric ZGNRs (i.e., width index N is an odd number), however, their π and π * subbands do not have definite σ parities, so the coupling between them can contribute to about one conductance quantum (the right panel of Fig. 5b). This transport difference can be clearly reflected in the current-bias-voltage (I-V bias ) characteristics of ZGNRs by using the first-principle transport simulation, as shown in Fig. 5c. Although metallic armchair carbon nanotubes also have π and π * subbands with definite parities, such symmetry-depending (or band-selective) I-V bias characteristics cannot be observed in them because of the crossover of their subbands, i.e., the absence of edge states. Recently, theoretical work predicts a very large magnetoresistance in a graphene nanoribbon device due to the existence of edge states [70].
Besides the fabrication and theoretical study of monolayer graphene and GNRs, recent experimental [71][72][73] and theoretical [74][75][76][77] studies are also carried out on bilayer graphene and GNRs. Theoretically, it is shown that the bilayer GNRs and monolayer GNRs have some similar electronic properties such as edge states localized at the zigzag edges and semiconducting behavior of armchair bilayer GNRs [75][76][77]. Experimentally, it is found that the bilayer graphene has unique features such as anomalous integer quantum Hall effects [71], which is absent in single layer graphene. And the size of energy gap of such bilayer structures can be controlled by adjusting carrier concentration [72] as well as by an external electric field [73]. These unique properties open an opportunity to implement bilayer graphene or GNRs in various electronic applications.
IV. EDGE DISORDER IN GRAPHENE NANORIBBONS
As mentioned above, current experimental techniques (such as lithography) are not able to realize exact control of the edge structures of GNRs and the edges are always very rough due to the limitation of the fabrication technology [38,39]. There are theoretical evidences that such edge disorders can significantly change the electronic properties of GNRs [78][79][80][81][82][83], and lead to some unexpected physics effect, such as the Anderson localization [84,85] and Coulomb blockade effect [86]. These effects have already been observed in lithographically obtained graphene nanoribbons [29,38,[87][88][89].
The edge roughness is also crucial for the spin polarized properties of GNRs. As we know, the magnetic properties of GNRs depend on the highly degenerate edge states. In principle a perfect edge structure is necessary for stabilizing magnetic properties of GNRs as theoretically predicted. An important question is, how robust the spin-polarized state is in the presence of edge defects and impurities? The answer to this question is not only scientifically interesting to better understand the physical mechanism of spin polarization in GNRs but also has important technological implications in the reliability of GNRs as a new class of spintronic materials. First-principles theoretical studies reveals the effect of edge vacancies and substitutionally doped boron atoms, as typical examples of structural edge defects and impurities, on the spin-polarization of ZGNRs [59]. The calculated energy difference between the magnetic state [both antiferromagnetic (AF) and ferromagnetic (FM)] and the nonmagnetic state is found to rapidly decrease with increasing defect concentration and eventually decrease to zero (nonmagnetic), as shown in Fig. 6. The critical defect (impurity) concentration is found to be ∼ 0.10/Å when the ribbon width is larger than 2 nm.
Evidently, the magnetism in GNRs depends on a high density of state (DOS) around the Fermi energy coming from the highly degenerate edge states in a perfect ribbon edge (E F ) that renders instability of spin polarization [59]. The presence of edge vacancies and impurities would decrease the DOS at E F since they do not contribute to the same edge state. From Stoner model, such decrease of DOS will suppress the spin polarization of GNR systems. Therefore, the practical realization of the spin polarization in GNRs for spintronics applications could be rather challenging [59]. Recently, an interesting theoretical work systematically studied the spin current in rough GNRs and predicted that only GNRs with imperfect edges exhibit a nonzero spin conductance while there is no spin current in perfect GNRs [90]. It confirms that the edge effect is of great importance to spin related properties of GNRs.
Furthermore, the problem of edge passivation has not yet clearly resolved by experiment until now. From the theoretical viewpoint, the edge passivation can be well modeled by the modifications of the hopping energies in the tight-binding approach [91] or via additional phases in the boundary conditions [92]. Recent theoretical modeling and calculations have indicated that the edge passivation has a strong effect on the electronic and spinpolarized properties of GNRs [62][63][64]. The possible passivation species include hydrogen, carbon, oxygen, nitrogen, and other chemical groups. Further experimental works are needed to explore the realistic edge structures of GNRs at atomic scale and determine which types of edge passivation are favorable.
V. TRANSISTORS BASED ON GRAPHENE NANORIBBONS
The interesting and unique electronic properties of GNRs, such as orientation and width dependence of transport behavior, offer great possibilities for their electronic device applications. Compared with other electronic materials, one of the most promising advantage of GNRs is that GNR-based devices and even integrated circuits can be fabricated by a single process of patterning a graphene sheet [23]. Figures 7a-7c illustrate three basic device building blocks: (i) metal-semiconductor junction, (ii) p-n junction, and (iii) hetero-junction, which can be, respectively, made by patterned GNRs (i) along different direction, (ii) with different edge doping, and (iii) with different widths. It was proposed that a variety of devices can be constructed from these building blocks. For example, a field effect transistor (FET) can be made simply by two metal-semiconductor junctions, as shown in Fig. 7d. There are some potential key advantages in designing and constructing device architectures based on GNRs. The first advantage is the perfect atomic interface, a feature that is difficult to achieve for the interconnection between nanotubes of different diameter and chirality. Second, it is generally difficult to find a robust method to make contact with the molecular device unit, because there exists usually a large contact resistance between the metal electrodes and molecules (e.g., single-walled CNT) due to a very small contact area. This difficulty may be circumvented by using GNRs, because the GNR-based devices can be connected to the outside circuits exclusively via metallic GNRs (or graphene), as illustrated in Fig. 7d, which serve as extensions of metal electrodes to make contact with the semiconducting GNRs so that an atomically smooth metal-semiconductor interface is maintained with minimum contact resistance. Last but not the least, the edges of GNRs may serve as effective sites for doping. In principle, by introducing different types of dopants at different sections of GNR edges, one can realize a p-n junction by selective doping, as shown in Fig. 7b.
One of the most important electronic applications based on GNRs is field effect transistors. Recently, ex- perimental studies [38,39,43,55] have indicated the possibility of fabricating GNR-based transistors. The advantage of GNRs as an alternative material for transistors is that it could bypass the chirality challenge of CNTs while retaining the excellent electronic properties of graphene sheets, such as the high I on /I off ratio and excellent electron/hole mobilities. The performance of one sub-10-nm GNR-FET in the latest work from Dai's group is shown in Figs. 8a and 8b (the transfer and output characteristics, respectively, for the GNR device with the width of ∼ 2 ± 0.5 nm and the channel length of ∼ 236 nm) [55]. This device delivered I on ∼ 4 µA at V ds = 1 V, I on /I off ratio > 10 6 at V ds = 0.5 V, subthreshold slope S=dV gate /dlogI ∼ 210 mV/decade and transconductance ∼ 1.8 µS (∼900 µS/µm). The device performance is comparable with the best CNT-based transistors. However, the Dirac point was not observed around zero gate bias in this measurement, indicating p-doping effects at the edges or by physisorbed species during the chemical treatment steps.
Together with experimental progress on GNR-based transistors, theoretical studies using semiclassical and quantum transport models show that GNR-based FETs could have a similar performance as CNT-based FETs and might outperform traditional Si-based FETs [23,[93][94][95]. Figure 9 shows a first-principles study on the performance of a typical GNR-based FET made with a 5.91 nm long intrinsic semiconducting 10-AGNR channel connected to two metallic 7-ZGNR leads (source and drain) [23]. In Fig. 9a, the near-symmetric I − V gate curve shows an excellent ambipolar transistor with ON/OFF ratio I on /I off ∼ 2000 and subthreshold swing of S ∼ 60 mV/decade, which are comparable to those of high performance CNT-FETs. Such the field effect can be clearly reflected in the change of I − V bias characteristics under different gate voltages (Fig. 9b). Figure 9c shows the I − V gate curves of the GNR-FETs made from the same 10-AGNR channel with its length ranging from 1.69 to 6.76 nm, from which the values of S are derived as a function of L as shown in Figure 9d. Clearly, S decreases with increasing L, and gradually approaches ∼60 mV/decade when L becomes longer than 6 nm. Meanwhile, the ON-current stays the same, independent of L, but the OFF-state leakage current increases rapidly with decreasing L, which gives rise to a large S. The performance of the ambipolar GNR-FETs made of intrinsic semiconductor channels can be understood in terms of metal-semiconductor tunneling junctionh within the semiclassical band-bending model. Compared with the basic ambipolar FETs, it is well known that n-type (or p-type) FETs serve as critical transistor devices for digital electronics applications [96,97]. To realize such device design based on GNRs, a method was proposed using N (or B) atoms as selective dopants at the channel region of perfect GNR-FETs (the positions of B or N are indicated by arrows in Fig. 7d). Figure 9e (9f) shows the calculated I − V gate curves under V bias = 20 mV, exhibiting the typical behavior of a n-type (p-type) FET [23]. It is suggested that all of the functional transistor devices that work in traditional Sibased circuits could be realized by GNRs and GNR-based junctions in principle.
Noting the current experimental difficulty to get an accurate Z-shape junction (i.e., FET shown in Fig. 7d) due to the limitation of lithography technique, a new type of field effect transistor has also been proposed taking advantage of the metal-semiconductor transition in ZGNRs induced by substitutional doping of nitrogen or boron atoms at their edges [98], as shown in Fig. 10a. Besides simplifying the fabrication process, such a linear configuration can also increase the device density in electronic circuits. Figure 10b shows a typical I −V gate curve for the N-doped GNR-FET (with the channel length of 8.54 nm) under the bias voltage V bias = 0.01 V. Clearly, the doped FET exhibits ambipolar characteristics, similar to the Zshape FETs. The relationship between the device performance and the channel length is demonstrated by calculating I−V gate curve of N-doped GNR-FETs as a function of the doped channel length while keeping the bias volt-age V bias at 0.01 V. As shown in Fig. 10c, the subthreshold swing S of these doped GNR-FETs decreases and the ON/OFF current ratio increases exponentially. It can be seen that for good device performance with small S value (e.g., below 100 mV/decade) and high ON/OFF current ratio (e.g., above 100), the doped channel length should be longer than 5 nm. The minimum leakage current of those FETs with the doped channels shorter than this critical length will be greatly enhanced by direct tunneling, which lowers the device performance.
Besides ideal case, some more practice issues concerning GNR-based FETs are discussed in recent theoretical works. For example, the effects of the various contact types and shapes on the performance of Schottkybarrier-type GNR-FETs have been investigated theoretically [99], which indicates that the semi-infinite normal metal can potentially provide promising performance. In addition, the effect of edge roughness and carrier scattering on GNR-FETs have been studied [100][101][102]. The presence of edge disorder significantly reduces ON-state currents and increases OFF-state currents (the ON/OFF ratio decreases), and introduces wide variability across devices. These effects become weaker for GNRs with larger width and smoother edges. However, the band gap decreases with increasing width, thereby increasing the band-to-band tunneling mediated subthreshold leakage current even with perfect GNRs. Obviously, without atomically precise edge control during fabrication, it is hard to get reliable and stable performance of GNR-FETs.
Due to their unusual basic properties, GNRs as well as graphene are promising for a large number of applications [35,83], from spin filters [63,90,103], valley filters [104], to chemical sensors [31,33,105]. GNRs can be chemically and/or structurally modified in order to change its functionality and hence its potential applications.
VI. SUMMARY
In summary, we review the basic electronic and transport properties of graphene nanoribbons, and discuss recent theoretical and experimental progress on GNRbased field effect transistors from the viewpoint of device application. Due to the interesting electronic and magnetic properties, GNRs have been demonstrated as a promising candidate material for future post-silicon electronics such as transport materials, field effect transistors, and spin injection or filter. More experimental efforts will focus on fabricating high quality nanoribbon samples with accurate control of the edge structures.
FIG. 2 :
2(Color online) The structures of H-passivated 11-AGNR (a) and 6-ZGNR (b), where big green balls and small blue balls represent carbon atoms and hydrogen atoms, respectively. Integer N is their width index.
FIG. 3 :
3(Color online) The variation of band gaps of Na-AGNRs as a function of width (wa) obtained (a) from TB calculations and (b) from first-principles calculations (symbols). (c) First-principles band structures of Na-AGNRs with Na= 12, 13, and 14, respectively. Reprinted with permission from Ref. [52], Y. -W. Son et al., Phys. Rev. Lett. 97, 216803 (2006). c 2006, American Physical Society.
FIG. 4 :
4(Color online) Electronic structures of graphene nanoribbons. In all figures, the Fermi energy (EF ) is set to zero. a, The spin-unpolarized band structure of a 16-ZGNR. b, The spatial distribution of the charge difference between α-spin and β-spin for the ground state when there is no external field. The magnetization per edge atom for each spin on each sublattice is 0.43µB with opposite orientation, where µB is the Bohr magneton. The graph is the electron density integrated in the z direction, and the scale bar is in units of 10 −2 eÅ −2 . c, From left to right, the spin-resolved band structures of a 16-ZGNR with the external field of 0.0, 0.05 and 0.1VÅ, respectively. The red and blue lines denote bands of α-spin and β-spin states, respectively. Reprinted with permission from Ref. [57], Y. -W. Son et al., Nature 444, 347 (2006). c 2006, Nature Publishing Group.
FIG. 5 :
5(Color online) (a) Schematic band structure around the Fermi level of a ZGNR under a positive (Vg+) and a negative (Vg−) potential step. (b) Conductance of 8-ZGNR and 7-ZGNR under two potential steps shown in (a). (c) I-V bias curves of the two-probe system (see the inset) made of ZGNRs with different widths N . Reprinted with permission from Ref. [66], Z. Li et al., Phys. Rev. Lett 100, 206802 (2008). c 2008, American Physical Society.
FIG. 6 :
6(Color online) The energy difference per edge atom between the magnetic (AF or FM) and paramagnetic (PA) state as a function of vacancy concentration in the edge. Two different ribbon widths of N =6 and N =4 are shown. The inset shows the critical concentration as a function of the ribbon width up to 5 nm. Reprinted with permission from Ref. [59], B. Huang et al., Phys. Rev. B 77, 153411 (2008). c 2008, American Physical Society.
FIG. 7 :
7(Color online) Schematics of three device building blocks: (a) a metal-semiconductor junction between a zigzag and an armchair GNR, (b) a p-n junction between two armchair GNRs with different edge doping, and (c) a heterojunction between two armchair GNRs of different widths (band gaps). (d) Schematics of a GNR-FET, made from one semiconducting 10-AGNR channel and two metallic 7-ZGNR leads connected to two external metal electrodes. Reprinted with permission from Ref. [23], Q. Yan et al., Nano Lett. 7, 1469 (2007). c 2007, American Chemical Society.
FIG. 8 :
8(Color online) (a) and (b) Transistor performance of GNR-FETs with width of ∼ 2 nm and channel length of ∼ 236 nm [(c) and (d), width of ∼ 60 nm, and channel length of ∼ 190 nm ]. (a) Transfer characteristics (current vs gate voltage I ds -Vgs) under various V ds . Ion/I off ratio of > 10 6 is achieved at room temperature. (b) Output characteristics (I ds -V ds ) under various Vgs. On current density is ∼ 2000 µA/µm in this device. (c) Transfer and (d) output characteristics of the 60 nm width GNR-FET device. Reprinted with permission from Ref. [55], X. Wang et al., Phys. Rev. Lett. 100, 206803 (2008). c 2008, American Physical Society.
FIG. 9 :
9(Color online) (a) I −Vgate curve for a 5.91 nm long intrinsic 10-AGNR channel (V bias =20 mV). (b) I − V bias curves under different gate voltage (Vgate). (c) I −Vgate curves for different channel lengths (V bias =20 mV). (d) The subthreshold swing (S) as a function of channel length L. (e),(f) I − Vgate curves for a 5.91 nm long 10-AGNR channel with selective N and B doping, respectively (V bias =20 mV). Reprinted with permission from Ref. [23], Q. Yan et al., Nano Lett. 7, 1469 (2007). c 2007, American Chemical Society.
FIG. 10 :
10(Color online) (a) The schematic structure of the field effect transistor (FET) made from a single 5-ZGNR. The semiconducting channel is obtained by edge doping of N in a finite-length region (the center region). (b) Simulated I-Vgate curves of N-doped GNR-FETs under V bias = 0.01 V. The channel length is 8.54 nm and the linear doping concentration is 0.1365Å −1 . (c) The dependence of the subthreshold swing S (blue line) and the ON/OFF current ratio (red line) on the channel length L. Reprinted with permission from Ref. [98], B. Huang et al., Appl. Phys. Lett. 91, 253122 (2007). c 2007, American Institute of Physics.
FIG. 1: (Color online) various GNRs got from different experimental methods: (a) The SEM image of GNRs patterned by e-beam lithography. Reprinted with permission from Ref. [39], Z. Chen et al., Physica E 40, 228 (2007). c 2007, Elsevier. (b) An 8-nm-wide 30 • GNR bent junction connecting an armchair and a zigzag nanoribbon etched by STM lithography. Reprinted with permission from Ref. [40], L. Tapasztó et al., Nat. Nanotechnol. 3, 397 (2008). c 2008, Nature Publishing Group. (c) GNRs are got by using simple chemical methods. Reprinted with permission from Ref. [43], X. Li et al., Science 319, 1229 (2008). c 2008, American Association for the Advancement of Science.
AcknowledgmentsThis work was supported by the Ministry of Science and Technology of China (Grant Nos. 2006CB605105
and the National Natural Science Foundation of China. 2006CB0L0601), and the National Natural Science Foundation of China.
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| []
|
[
"IRREDUCIBLE COMPLETELY POINTED MODULES OF QUANTUM GROUPS OF TYPE A",
"IRREDUCIBLE COMPLETELY POINTED MODULES OF QUANTUM GROUPS OF TYPE A"
]
| [
"Vyacheslav Futorny ",
"Jonas Hartwig ",
"ANDEvan Wilson "
]
| []
| []
| We give a classification of all irreducible completely pointed U q (sl n+1 ) modules over a characteristic zero field in which q is not a root of unity. This generalizes the classification result of Benkart, Britten and Lemire in the non quantum case. We also show that any infinite-dimensional irreducible completely pointed U q (sl n+1 ) can be obtained from some irreducible completely pointed module over the quantized Weyl algebra A q n+1 .p. The first main theorem of the paper is as follows, where V + is the u-invariant subset of V :Theorem I. Let V be an irreducible, infinite-dimensional, completely-pointed U q (sl n+1 )-module and let v + ∈ V + be given. Then the action of U q (sl n+1 ) on V can be extended to a U q (gl n+1 ) action such that the following relations hold:; 0] · v λ for any weight vector v λ ∈ V . | 10.1016/j.jalgebra.2015.03.006 | [
"https://arxiv.org/pdf/1404.0305v2.pdf"
]
| 119,630,243 | 1404.0305 | fdacc804c312ba78b99aefbd521ceffe559a65c7 |
IRREDUCIBLE COMPLETELY POINTED MODULES OF QUANTUM GROUPS OF TYPE A
23 Jun 2014
Vyacheslav Futorny
Jonas Hartwig
ANDEvan Wilson
IRREDUCIBLE COMPLETELY POINTED MODULES OF QUANTUM GROUPS OF TYPE A
23 Jun 2014quantum groupsrepresentation theoryweight modules of bounded multiplicity
We give a classification of all irreducible completely pointed U q (sl n+1 ) modules over a characteristic zero field in which q is not a root of unity. This generalizes the classification result of Benkart, Britten and Lemire in the non quantum case. We also show that any infinite-dimensional irreducible completely pointed U q (sl n+1 ) can be obtained from some irreducible completely pointed module over the quantized Weyl algebra A q n+1 .p. The first main theorem of the paper is as follows, where V + is the u-invariant subset of V :Theorem I. Let V be an irreducible, infinite-dimensional, completely-pointed U q (sl n+1 )-module and let v + ∈ V + be given. Then the action of U q (sl n+1 ) on V can be extended to a U q (gl n+1 ) action such that the following relations hold:; 0] · v λ for any weight vector v λ ∈ V .
Introduction
Let U q (g) be the quantum group of finite-dimensional semisimple Lie algebra over a characteristic zero field in which q is not a root of unity. A U q (g) weight module is called completely pointed if all of its weight spaces are one dimensional. This paper is a generalization of the classification given by Benkart, Britten and Lemire ( [1]) of infinite dimensional completely pointed modules of semisimple Lie algebras. In the Lie algebra case such modules can only exist if every ideal of g is of type A or C. In the current paper we consider the case of U q (sl n+1 ), i.e. the quantum group of type A.
Throughout the paper, we will make extensive use of certain generators E α of U q (g) introduced by Lusztig ( [10]) where α ranges over the roots of g, which are analogues of the root vectors of g. These generators are not unique, but depend on a choice of reduced decomposition of the longest Weyl group element, w 0 . For a fixed reduced decomposition of w 0 , and a irreducible weight module V we see that each E α acts either locally nilpotently or injectively on V . This set is very much like root vectors parabolic subalgebra p of sl n+1 though sl n+1 ⊂ U q (sl n+1 ) as a Lie algebra so this correspondence is not precise. Nevertheless, we call the U q (sl n+1 )-subalgebra generated by locally nilpotent root vectors U q (p) and there exists another U q (sl n+1 )-submodule U q (u) where u is analogous to the nilradical of Using this theorem one sees, for example, that the action of the cyclic subalgebra C(U q (sl n+1 )) is completely determined by the action of theK i (see Lemma 3.3), hence this gives a classification of irreducible completely pointed U q (sl n+1 ) modules (with all finite dimensional ones given in Proposition 2.2).
In our next two main results, we construct the infinite-dimensional completely pointed U q (sl n+1 )-modules. Let A q n+1 be the rank n + 1 quantum Weyl algebra and π be the homomorphism from U q (gl n+1 ) to A q n+1 (see [6]) which restricts to U q (sl n+1 ). Then we have the following:
Theorem II A. Let W be an irreducible completely pointed A q n+1 -module. Let π * W be the U q (gl n+1 )-module, given as the π-pullback of W . Then π * W is completely reducible, and each irreducible submodule is completely pointed, and occurs with multiplicity one.
This gives a construction of irreducible infinite dimensional, completely pointed U q (sl n+1 )-modules. An application of Theorem I then gives the following, which completes our classification:
Theorem II B. Any infinite-dimensional irreducible completely pointed U q (sl n+1 ) is isomorphic to a direct summand of π * W for some irreducible completely pointed A q n+1 -module W .
Preliminaries
Let F be a field of characteristic 0 closed under quadratic extensions and suppose q ∈ F is nonzero and not a root of unity. For us, U q (gl n+1 ) is the associative unital F-algebra with generators E i , F i ,K ±1 j , i ∈ {1, . . . , n}, j ∈ {1, . . . , n + 1} and defining relations (1)
K j E iK −1 j = q δ ij −δ j,i+1 E i ,K j F iK −1 j = q −(δ ij −δ j,i+1 ) F i , i ∈ {1, .
. . , n}, j ∈ {1, . . . , n + 1}
(2) [E i , F j ] = δ ijK iK −1 i+1 −K −1 iK i+1 q−q −1
, i, j ∈ {1, . . . , n + 1}
(3) [E ± i , E ± j ] = 0 for |i − j| > 1,
(4) (E ± i ) 2 E ± j − [2] q E ± j E ± i E ± j + E ± j (E ± i ) 2 = 0, for |i − j| = 1, where E + i := E i , E − i := F i , and [k] q = q k −q −k q−q −1 , k ∈ Z ≥0
. Then, U q (sl n+1 ) is the subalgebra of U q (gl n+1 ) generated by E i , F i , and K i :=K iK −1 i+1 . For g = sl n+1 recall the following automorphisms T i : U q (g) → U q (g), 1 ≤ i ≤ n as given by Lusztig [10]:
T i (E j ) = −F i K i , if i = j q −1 E j E i − E i E j , if |i − j| = 1 E j ,
otherwise,
T i (F j ) = −K −1 i E i , if i = j −F j F i + qF i F j , if |i − j| = 1 F j , otherwise, T i (K j ) = K −1 j , if i = j, K i K j , if |i − j| = 1, K j , otherwise.
We also recall the braid relations satisfied by the T i :
T i T i+1 T i = T i+1 T i T i+1 , T i T j = T j T i , if |i − j| > 1.
To each root α we assign a corresponding root vector E α in using following method. Let w 0 = s i 1 s i 2 · · · s ir be a reduced decomposition of the longest Weyl group element. Then every positive root occurs exactly once in the following sequence:
β 1 = α i 1 , β 2 = s i 1 (α i 2 )
, . . . , β r = s i 1 s i 2 · · · s i r−1 (α ir ). The positive root vector E β k is defined to be T i 1 T i 2 · · · T i k−1 (E i k ), and the negative root vector E −β k is defined by the same sequence of T i 's acting on F i k . We choose w 0 = s 1 s 2 · · · s n s 1 s 2 · · · s n−1 · · · s 1 s 2 s 1 as our reduced expression of the longest Weyl group element of U q (sl n+1 ), which gives the following sequence of positive roots:
ε 1 − ε 2 , ε 1 − ε 3 , ε 1 − ε 4 , . . . , ε 1 − ε n+1 , ε 2 − ε 3 , ε 2 − ε 4 , . . . , ε 2 − ε n+1 , · · · ε n−1 − ε n , ε n − ε n+1 , ε n − ε n+1 .
We recall also from [3, Example 8.1.5] the following identity:
T i T i+1 (E i ) = E i+1 .
Using this fact, and the braid relations, one obtains the following simplified form for the root vectors:
E ε 1 −ε 2 = E 1 , E ε 1 −ε 3 = T 1 (E 2 ), E ε 1 −ε 4 = T 1 T 2 (E 3 ), . . . , E ε 1 −ε n−1 = T 1 T 2 · · · T n−1 (E n ), E ε 2 −ε 3 = E 2 , E ε 2 −ε 4 = T 2 (E 3 ), . . . , E ε 2 −ε n+1 = T 2 T 3 · · · T n−1 (E n ), . . . E ε n−1 −εn = E n−1 , E ε n−1 −ε n+1 = T n−1 (E n ), E εn−ε n+1 = E n
and similarly for the negative root vectors. Let 1 ≤ i < j < k ≤ n. Double induction on i and j gives the following, where [x, y] v = xy − vyx:
E ε i −ε k = −[E ε i −ε j , E ε j −ε k ] q −1 , (1.1) E −ε i +ε k = −[E −ε j +ε k , E −ε i +ε j ] q . (1.2)
Similarly we have: (3) and (4) of the definition of U q (sl n+1 ) lead to the following:
[E ε j −ε k , E −ε i +ε k ] = −qK −1 jk E −ε i +ε j , [E ε i −ε k , E −ε j +ε k ] = −K jk E ε i −ε j , (1.3) [E ε i −ε j , E −ε i +ε k ] = K ij E −ε j +ε k , [E ε i −ε k , E −ε i +ε j ] = q −1 K −1 ij E ε j −ε k , (1.4) where K ij = j−1 k=i K k . Also, relationsE ε j −ε k E ε i −ε k = q −1 E ε i −ε k E ε j −ε k , E ε i −ε k E ε i −ε j = q −1 E ε i −ε j E ε i −ε k , (1.5) E −ε j +ε k E −ε i +ε k = q −1 E −ε i +ε k E −ε j +ε k , E −ε i +ε k E −ε i +ε j = q −1 E −ε i +ε j E −ε i +ε k . (1.6)
Finally, if 1 ≤ i < j < k < l ≤ n then:
[E ε i −ε j , E −ε k +ε l ] = [E ε i −ε j , E ε k −ε l ] = [E ε i −ε l , E ε j −ε k ] = 0 (1.7) [E ε i −ε k , E ε j −ε l ] = (q − q −1 )E ε i −ε l E ε j −ε k (1.8) [E −ε i +ε k , E −ε j +ε l ] = (q − q −1 )E −ε i +ε l E −ε j +ε k (1.9) [E ε i −ε k , E −ε j +ε l ] = −(q − q −1 )K jk E −ε i +ε j E ε k −ε l (1.10) [E ε j −ε l , E −ε i +ε k ] = (q − q −1 )K −1 jk E ε k −ε l E −ε i +ε j . (1.11)
Finally, since the T i are U q (g) automorphisms, we have:
[E ε i −ε j , E −ε i +ε j ] = [K ij ; 0] (1.12) where [K; j] = q j K−q −j K −1 q−q −1 for invertible K ∈ F[K ±1 1 , K ±1 2 , . . . , K ±1 n ] and j ∈ Z. Let V be a U q (sl n+1 )-module. For λ ∈ (F × ) n , the weight space V λ is defined to be the subspace {v ∈ V |K i · v = λ i v}.
It is easy to show that the sum of weight spaces in V over all λ ∈ (F × ) n is direct. Moreover, if V is finite-dimensional then it is the sum of its weight spaces (see [3]) though the same is not necessarily true if V is infinite-dimensional. A U q (sl n+1 )-module that is the direct sum of its weight spaces is called a U q (sl n+1 ) weight module. Throughout this paper, we will consider only irreducible modules in the category of U q (sl n+1 ) weight modules. 4
Classification of irreducible completely pointed modules
Let g = sl n+1 and Φ = Φ(g) be the root system of g. Let V be an irreducible U q (g) weight module and α ∈ Φ. On a weight module, the only possible eigenvalue of E α is 0, hence E α either acts nilpotently or injectively on a given weight vector. The subset of vectors on which E α acts nilpotently (resp. injectively) is a submodule of V . Since V is irreducible, we see that E α acts nilpotently on all of V or else it acts injectively. In the first case, E α is called locally nilpotent and in the second it is called torsion free. Highest weight modules are a special case where every positive root vector E α is locally nilpotent. The other extreme is where every root vector is torsion free. Finally, there are cases where a certain subset of positive root vectors are locally nilpotent but not necessarily all of them. We discuss each case below.
2.1. Highest weight modules. The irreducible highest weight U q (g)-module with highest weight λ ∈ (F × ) n is denoted L(λ). Note that for us, unlike the finite-dimensional case for example, K i can have arbitrary eigenvalues in F\{0}, not just powers of q. Also this is done so that we can have examples of torsion-free modules (see below).
Lemma 2.1. Assume V is a completely pointed U q (g)-module and v ∈ V is a weight vector. For θ in the positive root lattice, suppose x 1 , x 2 ∈ U q (g) θ and y 1 ,
y 2 ∈ U q (g) −θ . Then y i x j · v = γ i,j v for some γ i,j ∈ F and i, j ∈ {1, 2}, and the 2 × 2 matrix (γ ij ) is singular.
Proof. Same as the proof in [1, Lemma 3.2], with F in place of C. Proof. Let v + be a highest weight vector of L(λ), where λ = (λ 1 , λ 2 , . . . , λ n ) ∈ (F × ) n . All irreducible highest weight U q (sl 2 )-modules are completely pointed, and equal to L(c) for some c which proves the n = 1 case.
Suppose n > 1. Let
x 1 = E −ε i +ε i+2 , x 2 = E −ε i +ε i+1 E −ε i+1 +ε i+2 ∈ U q (g) −ε i +ε i+2 , and y 1 = E ε i −ε i+2 , y 2 = E ε i −ε i+1 E ε i+1 −ε i+2 ∈ U q (g) ε i −ε i+2 for i ∈ {1, 2, . . . , n − 1}.
We denote λ ij = j−1 k=i λ k and compute:
y 1 x 1 · v λ = E ε i −ε i+2 E −ε i +ε i+2 · v λ = E −ε i +ε i+2 E ε i −ε i+2 · v λ + [K i,i+2 ; 0] · v λ = [λ i,i+2 ; 0]v λ , y 1 x 2 · v λ = E ε i −ε i+2 E −ε i +ε i+1 E −ε i+1 +ε i+2 · v λ = q −1 K −1 i E ε i+1 −ε i+2 E −ε i+1 +ε i+2 · v λ 5 = q −1 K −1 i [K i+1 ; 0] · v λ = q −1 λ −1 i [λ i+1 ; 0]v λ , y 2 x 1 · v λ = E ε i −ε i+1 E ε i+1 −ε i+2 E −ε i +ε i+2 · v λ = −E ε i −ε i+1 qK −1 i+1 E −ε i +ε i+1 · v λ = −K −1 i+1 [K i ; 0] · v λ = −λ −1 i+1 [λ i ; 0]v λ , y 2 x 2 · v λ = E ε i −ε i+1 E ε i+1 −ε i+2 E −ε i +ε i+1 E −ε i+1 +ε i+2 · v λ = [λ i ; 0][λ i+1 ; 0]v λ .
Therefore, Lemma 2.1 gives:
[λ i ; 0][λ i+1 ; 0]([λ i,i+2 ; 0] + q −1 λ −1 i,i+2 ) = 0, (2.1) from which we see λ i = ±1, λ i+1 = ±1, or λ i λ i+1 = ±q −1 .
This finishes the proof for n < 3, so assume n ≥ 3.
For 1 ≤ i < j < k < l ≤ n + 1, let x 1 = E −ε i +ε l , x 2 = E −ε i +ε k E −ε k +ε l ∈ U q (g) −ε i +ε l and y 1 = E ε i −ε l , y 2 = E ε i −ε j E ε j +ε l ∈ U q (g) ε i −ε l .
We compute:
y 1 x 1 · v λ = E ε i −ε l E −ε i +ε l · v λ = [λ il ; 0]v λ , y 1 x 2 · v λ = E ε i −ε l E −ε i +ε k E −ε k +ε l · v λ = q −1 K −1 ik E ε k −ε l E −ε k +ε l · v λ = q −1 λ −1 ik [λ kl ; 0]v λ , y 2 x 1 · v λ = E ε i −ε j E ε j −ε l E −ε i +ε l · v λ = −E ε i −ε j qK −1 jl E −ε i +ε j · v λ = −q 2 λ −1 jl [λ ij ; 0]v λ , y 2 x 2 · v λ = E ε i −ε j E ε j −ε l E −ε i +ε k E −ε k +ε l · v λ = E ε i −ε j E −ε i +ε k E ε j −ε l E −ε k +ε l · v λ + E ε i −ε j (q − q −1 )K −1 jk E −ε i +ε j E ε k −ε l E −ε k +ε l · v λ = (q 2 − 1)K −1 jk E ε i −ε j E −ε i +ε j E ε k −ε l E −ε k +ε l · v λ = (q 2 − 1)K −1 jk [K ij ; 0][K kl ; 0] · v λ = (q 2 − 1)λ −1 jk [λ ij ; 0][λ kl ; 0] · v λ . It follows from Lemma 2.1 that ((q 2 − 1)λ −1 jk [λ il ; 0] + qλ −1 jl λ −1 ik )[λ ij ; 0][λ kl ; 0] = qλ ij λ kl [λ ij ; 0][λ kl ; 0] = 0 (2.2)
Therefore λ ij = ±1 or λ kl = ±1, for 1 ≤ i < j < k < l ≤ n + 1. Let i be minimal such that λ i = ±1. Then we have λ j = ±1 for all j > i + 1. Since i was chosen to 6 be minimal such that
λ i = ±1 the other index j such that λ j = 0 is j = i + 1. If λ i+1 = ±1 the previous paragraph implies that λ i λ i+1 = ±q −1 .
This leaves the case such that only λ i = ±1. Let λ i = c ∈ F\{0} and suppose 1 < i < n + 1 (if i is not in that range, then c is not fixed in the statement of the theorem). Let
x 1 = E −ε i−1 +ε i+1 E −ε i +ε i+2 , x 2 = E −ε i +ε i+1 E −ε i−1 +ε i+2 , y 1 = E ε i−1 −ε i+1 E ε i −ε i+2 , y 2 = E ε i −ε i+1 E ε i−1 −ε i+2 .
We compute:
y 1 x 1 · v λ = E ε i−1 −ε i+1 E ε i −ε i+2 E −ε i−1 +ε i+1 E −ε i +ε i+2 · v λ = [K i−1,i+1 ; 0][K i,i+2 ; 0] · v λ + E ε i−1 −ε i+1 K −1 i (q − q −1 )E −ε i−1 +ε i E ε i+1 −ε i+2 E −ε i +ε i+2 · v λ = [K i−1,i+1 ; 0][K i,i+2 ; 0] · v λ − (q 2 − 1)K −1 i E ε i−1 −ε i+1 E −ε i−1 +ε i qK −1 i+1 E −ε i +ε i+1 · v λ = [K i−1,i+1 ; 0][K i,i+2 ; 0] · v λ − (q − q −1 )K −1 i−1,i+2 [K i ; 0] · v λ = λ i−1 λ i+1 ([c; 0] 2 − 1 + c −2 ) · v λ y 1 x 2 · v λ = E ε i−1 −ε i+1 E ε i −ε i+2 E −ε i +ε i+1 E −ε i−1 +ε i+2 · v λ = E ε i−1 −ε i+1 E −ε i +ε i+1 E ε i −ε i+2 E −ε i−1 +ε i+2 · v λ + E ε i−1 −ε i+1 q −1 K −1 i E ε i+1 −ε i+2 E −ε i−1 +ε i+2 · v λ (after several steps, using the fact that λ i−1 = ±1) = −K −1 i,i+2 E ε i−1 −ε i+1 E −ε i−1 +ε i+1 · v λ = −K −1 i,i+2 [K i−1,i+1 ; 0] · v λ = −λ i−1 λ −1 i+1 c −1 [c; 0]v λ y 2 x 1 · v λ = E ε i −ε i+1 E ε i−1 −ε i+2 E −ε i−1 +ε i+1 E −ε i +ε i+2 · v λ = E ε i −ε i+1 q −1 K −1 i−1,i+1 E ε i+1 −ε i+2 E −ε i +ε i+2 ) · v λ = −E ε i −ε i+1 K −1 i−1,i+2 E −ε i +ε i+1 · v λ = −K −1 i−1,i+2 E ε i −ε i+1 E −ε i +ε i+1 · v λ = −λ −1 i−1 λ −1 i+1 c −1 [c; 0]v λ y 2 x 2 · v λ = E ε i −ε i+1 E ε i−1 −ε i+2 E −ε i +ε i+1 E −ε i−1 +ε i+2 · v λ = E ε i −ε i+1 E −ε i +ε i+1 E ε i−1 −ε i+2 E −ε i−1 +ε i+2 · v λ = [K i ; 0][K i−1,i+2 ; 0] · v λ = λ i−1 λ i+1 [c; 0] 2 v λ .
Therefore, by Lemma 2.1 we conclude that [c; 0] 2 ([c; 0] 2 − 1) = 0. From this we see that c = ±1, or c = ±q ±1 , which finishes the proof.
Example: Let V be the natural representation of U q (sl n+1 ). The representations S r q (V ), r ∈ Z ≥0 and Λ i q (V ), i ∈ {1, 2, . . . , n} of highest weight (q r , 1, 1, . . . , 1) and (1, 1, . . . , 1, q, 1, . . . , 1) with q in the ith slot, are completely pointed (see [6] for 7 an explicit construction). They are, up to isomorphism and tensoring with onedimensional modules, the only finite dimensional representations in our classification (recalling that L(q r , 1, 1, . . . , 1) ∼ = L(1, 1, 1, . . . , q r ) from the Dynkin diagram symmetry). Example: In this example we take
F = Q(q). Let L(q + 1) be the U q (sl 2 )- module isomorphic to U q (sl 2 )/J where J is the left U q (sl 2 ) ideal generated by the set {E, K −(q + 1) · 1}. This is a highest weight U q (sl 2 )-module with highest weight vector v 0 = 1+J. As a Q(q) vector space, L(q+1) has basis {v k = F (k) ·v 0 |m ∈ Z ≥0 }, where F (k) = F k /[k] q !.
The weight of v k is given by the following computation:
K · v k = K · (F (k) · v 0 ) = q −2k (1 + q)v k .
We see that each v k spans a one-dimensional weight space of weight q −2k (1 + q). Also, L(q + 1) is irreducible, as we show by the following standard argument. Suppose that L(q +1) had a proper submodule V ′ . Then V ′ would have a maximal vector of weight q −2k (q + 1) for some k > 0. This maximal vector would have to be proportional to v k for k > 0. But we have the following:
E · v k = E · (F (k) · v 0 ) = EF (k) · v 0 = F (k−1) Kq −k+1 − K −1 q k−1 q − q −1 − F (k) e · v 0 = (1 + q)q −k+1 − (1 + q) −1 q k−1 q − q −1 F (k−1) · v 0 = [1 + q; −k + 1]v k−1 which is non-zero when k > 0. We now consider the A-form of L(q+1), where A = Q[q, q −1 ].
Recall that U A (sl 2 ) is defined to be the A-subalgebra of U q (sl 2 ) generated by the elements E, F, K, and
[K; 0] = (K − K −1 )/(q − q −1 ). The A-form of L(q + 1) is now the U A (sl 2 )-module L A (q + 1) = U A (sl 2 ) · v 0 . We have: v (k) 0 = [K; 0] k · v 0 = K − K −1 q − q −1 k · v 0 = (q + 1) − (q + 1) −1 q − q −1 k v 0
are elements of L A (q + 1) for all k > 0 that are not in the A-submodule generated by v 0 . However, these elements satisfy the following relations over A:
(q − q −1 )(q + 1)v (k+1) 0 = ((q + 1) 2 − 1)v (k)
0 . Notice in this above example that the problem was not that the A-form in question did not exist-indeed one can consider U A (g) acting on any U q (g)-module. The problem was that passing to the q = 1 limit provided no information about the U q (g)-module we wanted to study. 8
Put differently, if we define the classical limit of a highest weight U q (g)-module L(λ) as the U(g)-module L A (λ)/(q−1)L A (λ), then the above equation (with k = 0) shows that v (0) 0 ∈ (q − 1)L A (q + 1). Consequently the classical limit of the U q (sl 2 )module L(q + 1) is trivial. The conclusion is that the class of completely pointed U q (g)-modules is richer than in the classical case, since it consists not only of q-deformations of completely pointed U(g)-modules.
2.2.
Torsion free modules. Recall that U q (sl 2 ) is embedded in U q (gl 2 ) by adding the invertible elementK 2 satisfying the relationsK 2 EK −1
2 = q −1 E,K 2 FK −1 2 = qF , and K ±K ± 2 =K ± 2 K ± and definingK ± 1 = (KK −1 2 ) ± . Lemma 2.3.
Let V be an irreducible, torsion free, completely pointed U q (sl 2 )module and v λ a weight vector in V . The action of U q (sl 2 ) on V can be extended to a U q (gl 2 ) action such that the following relations hold:
F E · v λ = [K 1 ; 1][K 2 ; 0] · v λ .
Proof. Let c be the Casimir element of U q (sl 2 ):
c = F E + qK + q −1 K −1 (q − q −1 ) 2
Since V is completely pointed and irreducible, c acts as a scalar τ ∈ F. Therefore, on a weight vector v λ we have:
F E · v λ = τ − qλ + (qλ) −1 (q − q −1 ) 2 v λ .
Since F is closed under quadratic extensions there are µ 1 and µ 2 satisfying the two equations τ = qµ 1 µ 2 +(qµ 1 µ 2 ) −1 (q−q −1 ) 2 and λ = µ 1 µ −1 2 . This gives:
F E · v λ = qµ 1 − (qµ 1 ) −1 q − q −1 µ 2 − µ −1 2 q − q −1 v λ .
We can make V into a U q (gl 2 )-module by lettingK 2 act as µ 2 on v λ and demanding that the additional relations of U q (gl 2 ) be satisfied, i.e.
K 2 EK −1 2 · v λ = q −1 Ev λ andK 2 FK −1 2 · v λ = qF · v λ .
ThenK 1 acts as µ 1 which gives the desired result. As before, U q (sl n+1 ) is embedded in U q (gl n+1 ) by adding the elementK 2 and defining inductivelyK 1
= K 1K2 andK i+1 = K −1 i+1K i . Theorem 2.4. Let V be a irreducible, torsion free, completely pointed U q (sl n+1 )- module and v λ a weight vector in V .
The action of U q (sl n+1 ) on V can be extended to a U q (gl n+1 ) action such that the following relations hold:
E −ε i +ε j E ε i −ε j · v λ = [K i ; 1][K j ; 0] · v λ .
Proof. If n = 1 then the result is the previous lemma. So assume n > 1.
Since V is completely pointed and torsion free we have,
for 1 ≤ i < j < k ≤ n+1, E ε i −ε j E ε j −ε k · v λ = κ ijk E ε i −ε k · v λ , for some κ ijk ∈ F. Let z ij ∈ F, 1 ≤ i < j ≤ n + 1 be the scalars by which E −ε i +ε j E ε i −ε j act on v λ . Recall that K ij = j−1 k=i K k and λ ij = j−1 k=i λ k . We compute: 0 = E −ε i +ε j E ε i −ε j (E ε i −ε j E ε j −ε k − κ ijk E ε i −ε k ) · v λ = E −ε i +ε j E ε i −ε j (q −1 E ε j −ε k E ε i −ε j − (κ ijk + 1)E ε i −ε k ) · v λ = E −ε i +ε j (q −2 E ε j −ε k E ε i −ε j − (q(κ ijk + 1) + q −1 )E ε i −ε k )E ε i −ε j · v λ = (−q −1 [K ij ; 0] + K −1 ij (κ ijk + 1))E ε j −ε k E ε i −ε j · v λ + (q −2 E ε j −ε k E ε i −ε j − (q(κ ijk + 1) + q −1 )E ε i −ε k )z ij · v λ = (κ ijk + 1)(−[λ ij ; 1] + (κ ijk + 1)λ −1 ij )E ε i −ε k · v λ − ((q − q −1 )κ ijk + q)z ij E ε i −ε k · v λ = φ ijk − q −1 q − q −1 λ −1 ij φ ijk − qλ ij q − q −1 E ε i −ε k · v λ − φ ijk z ij E ε i −ε k · v λ (*) and 0 = E −ε j +ε k E ε j −ε k (E ε i −ε j E ε j −ε k − κ ijk E ε i −ε k ) · v λ = E −ε j +ε k (qE ε i −ε j E ε j −ε k + (q − κ ijk q −1 )E ε i −ε k )E ε j −ε k · v λ = (−[K jk ; 0] − κ ijk q −1 K jk )E ε i −ε j E ε j −ε k · v λ + (qE ε i −ε j E ε j −ε k + (q − κ ijk q −1 )E ε i −ε k )E −ε j +ε k E ε j −ε k · v λ = κ ijk (−[λ jk ; 1] − κ ijk λ jk )E ε i −ε k · v λ + ((q − q −1 )κ ijk + q)z jk E ε i −ε k · v λ = φ ijk − q q − q −1 q −1 λ −1 jk − φ ijk λ jk q − q −1 E ε i −ε k · v λ + φ ijk z jk E ε i −ε k · v λ . (**) where φ i 1 ,i 2 ,i 3 = q + (q − q −1 )κ i i ,i 2 ,i 3 for arbitrary i 1 , i 2 , i 3 .
If φ ijk = 0 then (*) gives the following:
0 = λ ij (q − q −1 ) 2
which is a contradiction (since every K i acts as an invertible scalar). Therefore, φ ijk = 0. In addition, we have:
z ij z jk v λ = E ε j −ε i E ε i −ε j E ε k −ε j E ε j −ε k · v λ = E ε j −ε i E ε k −ε j E ε i −ε j E ε j −ε k · v λ = κ ijk E ε j −ε i E ε k −ε j E ε i −ε k · v λ = (q −1 κ ijk E ε k −ε j E ε j −ε i E ε i −ε k + q −1 κ ijk E ε k −ε i E ε i −ε k ) · v λ = q −1 κ ijk (κ ijk z ik − (q − q −1 )z ij z jk )v λ + q −1 κ ijk z ik v λ whence κ ijk (κ ijk + 1)z ik = φ ijk z ij z jk .
(***) From (*), (**), and (***) we deduce:
z ij = φ −1 ijk φ ijk − q −1 q − q −1 λ −1 ij φ ijk − qλ ij q − q −1 (2.3) z jk = −φ −1 ijk φ ijk − q q − q −1 q −1 λ −1 jk − φ ijk λ jk q − q −1 (2.4) z ik = −φ −1 ijk λ −1 ij φ ijk − qλ ij q − q −1 q −1 λ −1 jk − φ ijk λ jk q − q −1 . (2.5)
This covers the case where n = 2, so suppose n > 2. If 1 ≤ i < j < k < l ≤ n + 1, then (2.4) and (2.5) give:
− φ −1 ijl φ ijl − q q − q −1 q −1 λ −1 jl − φ ijl λ jl q − q −1 = −φ −1 jkl λ −1 jk φ jkl − qλ jk q − q −1 q −1 λ −1 kl − φ jkl λ kl q − q −1 (2.6)
which yields:
(φ jkl φ ijl − λ −2 kl )(λ 2 jk φ ijl − φ jkl ) = 0.
(2.7) A similar argument using (2.3) and (2.5) gives:
(φ ikl φ ijk − λ 2 ij )(λ 2 jk φ ijk − φ ikl ) = 0. (2.8)
We compute:
κ jkl E ε j −ε l · v λ = E ε j −ε k E ε k −ε l · v λ = (−E ε j −ε l + q −1 E ε k −ε l E ε j −ε k ) · v λ hence q(κ jkl + 1)E ε j −ε l · v λ = E ε k −ε l E ε j −ε k · v λ .
Therefore we have:
qκ ijl (κ jkl + 1)E ε i −ε l · v λ = q(κ jkl + 1)E ε i −ε j E ε j −ε l · v λ = E ε k −ε l E ε i −ε j E ε j −ε k · v λ = κ ijk E ε k −ε l E ε i −ε k · v λ = qκ ijk (κ ikl + 1)E ε i −ε l · v λ .
Therefore, since E ε i −ε l · v λ = 0, we have:
κ ijl (κ jkl + 1) = κ ijk (κ ikl + 1) (2.9)
and analogously: κ jil (κ ikl + 1) = κ jik (κ jkl + 1).
(2.10) 11
Now we derive more relations for κ i 1 ,i 2 ,i 3 :
E ε i −ε j E ε j −ε l E ε j −ε k · v λ = (−E ε i −ε l E ε j −ε k + q −1 E ε j −ε l E ε i −ε j E ε j −ε k ) · v λ = (−E ε i −ε l E ε j −ε k + q −1 κ ijk E ε j −ε l E ε i −ε k ) · v λ = (−q −1 (q − q −1 )κ ijk − 1)E ε i −ε l E ε j −ε k · v λ + q −1 κ ijk E ε i −ε k E ε j −ε l · v λ and E ε i −ε j E ε j −ε l E ε j −ε k · v λ = q −1 E ε i −ε j E ε j −ε k E ε j −ε l · v λ = (−q −1 E ε i −ε k E ε j −ε l + q −2 E ε j −ε k E ε i −ε j E ε j −ε l ) · v λ = (−q −1 E ε i −ε k E ε j −ε l + q −2 κ ijl E ε j −ε k E ε i −ε l ) · v λ .
Therefore:
0 = κ jkl (q −1 (q − q −1 )κ ijk + 1 + q −2 κ ijl )E ε i −ε l E ε j −ε k · v λ − q −1 κ jkl (κ ijk + 1)E ε i −ε k E ε j −ε l · v λ = κ jkl (q −1 (q − q −1 )κ ijk + 1 + q −2 κ ijl )E ε i −ε l E ε j −ε k · v λ − q −1 (κ ijk + 1)E ε i −ε k E ε j −ε k E ε k −ε l · v λ = κ jkl (q −1 (q − q −1 )κ ijk + 1 + q −2 κ ijl )E ε i −ε l E ε j −ε k · v λ − κ ikl (κ ijk + 1)E ε i −ε l E ε j −ε k · v λ which gives: κ jkl (q −1 (q − q −1 )κ ijk + 1 + q −2 κ ijl ) = κ ikl (κ ijk + 1).
(2.11) Subtracting (2.9) from the above we see:
κ jkl (q −1 (q − q −1 )κ ijk + 1 − q −1 (q − q −1 )κ ijl ) − κ ijl = κ ikl − κ ijk (2.12) (q −1 φ jkl − 1)(φ ijk − φ ijl ) + φ jkl − φ ijl = φ ikl − φ ijk (2.13) q −1 φ jkl (φ ijk − φ ijl ) + φ jkl − φ ikl = 0.
(2.14)
We repeat the same argument with the indices i and j transposed to obtain:
E ε j −ε i E ε i −ε l E ε i −ε k · v λ = (−q −1 K −1 ij E ε j −ε l E ε i −ε k + E ε i −ε l E ε j −ε i E ε i −ε k ) · v λ = (−q −1 λ −1 ij E ε j −ε l E ε i −ε k + κ jik E ε i −ε l E ε j −ε k ) · v λ = −q −1 λ −1 ij E ε i −ε k E ε j −ε l · v λ + (κ jik + q −1 (q − q −1 )λ −1 ij )E ε i −ε l E ε j −ε k · v λ and E ε j −ε i E ε i −ε l E ε i −ε k · v λ = q −1 E ε j −ε i E ε i −ε k E ε i −ε l · v λ = (−q −2 K −1 ij E ε j −ε k E ε i −ε l + q −1 E ε i −ε k E ε j −ε i E ε i −ε l ) · v λ = (−q −2 λ −1 ij E ε i −ε l E ε j −ε k + q −1 κ jil E ε i −ε k E ε j −ε l ) · v λ . 12
Hence:
0 = κ ikl (κ jik + λ −1 ij )E ε i −ε l E ε j −ε k · v λ − q −1 κ ikl (κ jil + λ −1 ij )E ε i −ε k E ε j −ε l · v λ = (κ jik + λ −1 ij )E ε j −ε k E ε i −ε k E ε k −ε l · v λ − q −1 κ ikl (κ jil + λ −1 ij )E ε i −ε k E ε j −ε l · v λ = q −1 κ jkl (κ jik + λ −1 ij )E ε i −ε k E ε j −ε l · v λ − q −1 κ ikl (κ jil + λ −1 ij )E ε i −ε k E ε j −ε l · v λ which gives: κ jkl (κ jik + λ −1 ij ) = κ ikl (κ jil + λ −1 ij ). (2.15)
Subtracting (2.10) from the above gives:
λ −1 ij κ jkl − κ jik = λ −1 ij κ ikl − κ jil (2.16) λ −1 ij (κ jkl − κ ikl ) = κ jik − κ jil . (2.17)
We compute:
0 = (E ε k −ε j E ε j −ε i E ε i −ε k − κ jik E ε k −ε j E ε j −ε k ) · v λ = (κ ijk z ik − (q − q −1 )z ij z jk − κ jik z jk )v λ = ((κ ijk + 1) −1 φ ijk − (q − q −1 ))z ij z jk v λ − κ jik z jk v λ .
Hence, using (2.3)-(2.5) we see:
κ jik = q −1 λ −1 ij − λ ij φ −1 ijk q − q −1 (2.18)
and similarly,
κ jil = q −1 λ −1 ij − λ ij φ −1 ijl q − q −1 . (2.19)
Using the above in (2.17) gives:
λ −1 ij φ ijk φ ijl (φ jkl − φ ikl ) = λ ij (φ ijk − φ ijl ). (2.20)
Equations (2.7), (2.8), (2.14), and (2.20) have a unique simultaneous solution for φ ijl , φ ikl , φ jkl in terms of φ ijk , namely: We remark that analogues of equations (2.14) and (2.20) were found in [2] and in the q = 1 limit they imply the first two identities in (2.21), but in our case all four equations are needed. 13 2.3. Modules with torsion. We now set out to prove an extension of the Theorem 2.4 to the torsion case. Before we do so, let us introduce some notation. Let V be an irreducible U q (sl n+1 )-module.
φ ijl = φ ijk , φ ikl = φ jkl = λ 2 jk φ ijk .Let N = {β ∈ Φ|∀v ∈ V, ∃k > 0 such that E k β · v = 0}, T = {β ∈ Φ|∀v ∈ V, E β · v = 0}, N s = N ∩ (−N), T s = T ∩ (−T ), N a = N\N s and T a = T \T s . Finally, define V + = {v ∈ V |∀β ∈ N a ∪ N + s , E β · v = 0} where N + s = Φ + ∩ N s .
Using that q is not a root of unity it is easy to show that Φ = N ∪ T . The following is an analogue of (4.6) and (4.12) in [4], but in our proof for completely-pointed modules we avoid working with the center of U q (g).
Proposition 2.5. Let V be an irreducible completely-pointed U q (sl n+1 ) weight module. Then N and T are closed subsets of Φ.
Proof. Let α, β ∈ N be such that α+β ∈ Φ. Then there exist 0 = v + ∈ V such that E α ·v + = 0 and s ∈ Z >0 such that E s β ·v + = 0 and E s−1 β ·v + = 0. Note that equations
(1.1)-(1.4) imply KE α+β = ±(q j E α E β − q k E β E α ) for some j, k ∈ {−1, 0, 1}, and invertible K ∈ F[K ±1 1 , K ±1 2 , . . . , K ±1 n ].
Using this, we compute:
0 = E α E s β · v + = ± s−1 i=0 q k i E i β KE α+β E s−i−1 β · v + = ± s−1 i=0 q k i +r i KE α+β · (E s−1 β · v + ) (2.22) where k i , r i ∈ Z, i ∈ {1, 2, .
. . , n} are increasing or decreasing sequences. Therefore α + β ∈ N, which gives that N is closed. Now let α, β ∈ T be such that α +β ∈ Φ. Since α, β ∈ T we have E k α ·v λ = 0 and E r β ·v λ = 0 for all k, r ∈ Z ≥0 and all weights λ ∈ (F × ) n+1 , hence q Z ≥0 (α+β) supp(V ) ⊆ supp(V ). If E α+β ∈ N then, given λ ∈ wt(V ), we have an infinite sequence of vectors v i ∈ V q m i (α+β) λ where m i ∈ Z ≥0 is an increasing sequence such that E α+β · v i = 0. We compute:
0 = E α+β · v i = cE α+β (E −α−β ) m i+1 −m i · v i+1 (for some c ∈ F\{0} since V is completely pointed) = c m i+1 −m i −1 i=0 (E −α−β ) i [K α+β ; 0](E −α−β ) m i+1 −m i −i · v i+1 = c[m i+1 − m i ] q [K α+β ; m i+1 − m i − 1](E −α−β ) m i+1 −m i −1 · v i+1 . Since c(E −α−β ) m i+1 −m i −1 · v i+1 = 0 and m i+1 > m i we see that [K α+β ; m i+1 − m i − 1]
· v i+1 = 0. Therefore, letting λ α+β ∈ F\{0} be the eigenvalue of K α+β on v i+1 , we see that λ α+β = ±q −(m i+1 −m i −1) -a non-positive integral power of q times ±1. However, since K α+β · v i+j = q 2(m j −m i+1 ) λ α+β v i+j and m j is an increasing sequence of integers, for some j * > i we must have K α+β acting as a non-negative power of q times ±1 on v j * . This j * is therefore the highest index in the sequence of m j , contrary to it being an infinite sequence. Therefore α + β / ∈ N and must be in T since Φ = N ∪ T .
As a corollary, we have that N s and T s are root subsystems of Φ.
Lemma 2.6. If g is simply-laced, then there exists a base B of Φ(g) such that N a ⊆ Φ + B , and every α ∈ B\N a is a positive root (with respect to the usual base of Φ).
Proof. Lemma 4.7 (i ) of [1] proves the existence of a base B of Φ = Φ(g) such that N + a ⊆ Φ + B . We may apply their result since the proof only uses results on root subsystems that satisfy the same hypotheses as in our case. We show how to choose a new base B n satisfying the same condition, but with every α ∈ B\N a positive with respect to the usual base, B u . By the previous proposition we see that N s ant T s are a root subsystems of Φ. Let W N and W T be the Weyl group of the root subsystems N s and T s respectively. We may choose a base B p of N s ∪ T s such that (N s ∪ T s ) ∩ B p is contained in Φ + Bu . It is a well known fact of finite root systems (see [8,Section 10.1]) that the Weyl group permutes bases. Let w ∈ W T × W N be the Weyl group element taking B ∩ (N s ∪ T s ) to B p . We want to show that w preserves N a . Let α ∈ N a and β ∈ B ∩ (N s ∪ T s ) be given. We have r β (α) = α − β, α β ∈ Φ. Since g is simply laced, β, α = 0 or −1. In the first case, r β (α) = α ∈ N a . Otherwise, r β (α) = α + β. By (ii ) of the above lemma, we have α + β ∈ N a . Hence, w(α) ∈ N a , and w(B) is the desired basis.
We let U q (g Ns ) (resp. U q (n Ns )) denote the subalgebra of U q (g) generated by {E α |α ∈ N s } and (resp. {E α |α ∈ N + s }). Proposition 2.7. Suppose V is an irreducible completely-pointed U q (g)-module. Then the following hold: Proof. Proof of (1). Let v * ∈ V \{0} and {β 1 , β 2 , . . . , β l } = N ∩ B where B is the base of Φ given in the previous lemma. Then there exist r j ∈ Z >0 , j ∈ {1, 2, . . . , l} such that E
(1) if N = ∅ then the set V + is non-zero, i.e. there exists v + ∈ V \{0} such that E α · v + = 0 for α ∈ N + s ∪ N a ,(2)r j β j l k=j+1 E r k −1 β k · v * = 0 and r j is minimal that this occurs. Let v + = l k=1 E r k −1 β k · v * . Then v + = 0 and E β j · v + = 0, j ∈ {1, 2, . . . , l}. Since V is completely pointed, we have E α E β · v λ = cE β E α · v λ for some c ∈ F whenever v λ is a weight vector such that E β · v λ = 0, hence E β k · v + = 0 for all k ∈ {1, 2, . . . , l}. Now, let γ ∈ N a ∪ N +
s . Then γ ∈ Φ + B hence it can be written as a positive integral combination of elements of B. At least one of these simple roots must be in N ∩ B since otherwise γ ∈ T which is a contradiction. Let α, β ∈ B such that E β · v + = 0 15 and α + β ∈ Φ. We compute
E α+β · v + = ±K(q j E β E α − q k E α E β ) · v + = ±K(q j − cq k )E β E α · v + = 0 for some j, k ∈ {−1, 0, 1}, K ∈ F[K ±1 1 , K ±1 2 , . . . , K ±1 n ]
and c ∈ F. Let β i ∈ N ∩ B be one of the simple roots in the decomposition of γ. Then there exists a sequence of roots γ 0 = β i , γ 1 , . . . , γ r = γ with γ i+1 − γ i ∈ B. Since E β i · v + = 0 we may use induction on the sequence of γ k to see that E γ · v + = 0. Therefore v + is the vector we are looking for.
The proof of (2) is similar to Lemma 4.9 of [1] and we leave it to the reader.
We are now ready to prove:
Theorem I. Let V be an irreducible, infinite-dimensional, completely-pointed U q (sl n+1 )-module and let v + ∈ V + be given. Then the action of U q (sl n+1 ) on V can be extended to a U q (gl n+1 ) action such that the following relations hold:
E −ε i +ε j E ε i −ε j · v + = [K i ; 1][K j ; 0] · v + . Also, we have F i E i · v λ = [K i ; 1][K i+1 ; 0] · v λ for any weight vector v λ ∈ V .
Proof. There are three cases: the extreme cases T s = ∅, T s = Φ and the intermediate case ∅ T s Φ. Case i : T s = ∅. In this case, V is a highest weight module, hence, by Proposition 2.2, is isomorphic to L(λ) where λ = (c, ±1, ±1, · · · , ±1), (±1, ±1, · · · , ±1, c), with c not a positive integer power of q, or (±1, ±1, · · · , c, c −1 q −1 , · · · , ±1). As before, let z ij ∈ F, 1 ≤ i < j ≤ n + 1 be the scalars by which E −ε i +ε j E ε i −ε j act on v λ . We have z ij = 0 for i < j ∈ {1, 2, . . . , n + 1}. In the first case, we may choose µ 1 = c, and µ i = ±1, 1 < i ≤ n + 1. In the second case µ n+1 = q −1 c −1 and µ i = ±q −1 , 1 ≤ i < n + 1 gives the desired result. In the third case, if λ k = c then we choose µ i = ±q −1 , 1 ≤ i < k, µ k = q −1 c −1 , and µ i = ±1, k < i ≤ n + 1 satisfies the conditions of the theorem.
Case ii: T s = Φ. In this case, V is a torsion free module. By Theorem 2.4 the conclusion holds for some weight vector v λ . As in the last part of the next case one shows that then it holds for any weight vector.
Case iii: ∅ T s Φ. Let β 1 , β 2 , . . . , β n be a base of Φ such that N a ⊆ Φ + B with β i ∈ T s for i ∈ {k, k + 1, . . . , l}. Let v + be an invariant vector of weight λ for E α , α ∈ N a ∪ N + s . Using Theorem 2.4 we can choose µ s i , i ∈ {k, k + 1, . . . , l + 1} such that z s i ,s j = [µ s i ; 1][µ s j ; 0] and λ s i ,s j = µ s i µ −1 s j for i, j ∈ {k, k + 1, . . . , l + 1}. For β ∈ N + s ∪ N a we have E −β E β · v λ = 0. We need to show that this choice 16
of µ s i induces a choice µ ′ i , i ∈ {1, 2, . . . , n + 1} satisfying z ij = [µ ′ i ; 1][µ ′ j ; 0] and λ ij = µ ′ i µ ′−1 j for i = j. Let β = ε t −ε u ∈ N a , β ′ = ε u −ε v ∈ T + s . We have z ut v + = E β E −β v + = [λ tu ; 0]v + . Since −β, −β ′ ∈ T we can define κ vut similarly to Theorem 2.4 by E −β E −β ′ · v + = κ vut E −β−β ′ · v + .
By similar computations to those used to prove (2.3)-(2.5) we obtain the following for 1 ≤ i < j < k ≤ n + 1:
(q −1 − (q − q −1 )κ kji )z kj = (κ kji + 1)(λ −1 jk κ kji + q −1 [λ jk ; 0]) (2.23) (q −1 − (q − q −1 )κ kji )z ji = κ kji (λ ij κ kji − [λ ij ; −1]) (2.24) (q 2 λ ij + (q − q −1 )κ kij )z ki = (κ kij + qλ ij )(q −1 λ jk κ kij + q[λ ik ; 0]) (2.25) (q 2 λ ij + (q − q −1 )κ kij )z ij = κ kij (q −1 κ kij + [λ ij ; 1]) (2.26) (q −2 λ jk − (q − q −1 )κ ikj )z ik = (κ ikj + q −1 λ jk )(qλ ij κ ikj − q −1 [λ ik ; 0]) (2.27) (q −2 λ jk − (q − q −1 )κ ikj )z kj = κ ikj (qκ ikj − [λ jk ; −1]) (2.28) (qλ −1 jk + (q − q −1 )κ jki )z jk = (κ jki + λ −1 jk )(κ jki − q[λ jk ; 0]) (2.29) (qλ −1 jk + (q − q −1 )κ jki )z ki = κ jki (λ −1 ij κ jki − [λ ik ; −1]) (2.30) (q −1 λ −1 ij − (q − q −1 )κ jik )z ji = (κ jik + λ −1 ij )(κ jik + q −1 [λ ij ; 0]) (2.31) (q −1 λ −1 ij − (q − q −1 )κ jik )z ik = κ jik (λ −1 jk κ jik + [λ ik ; 1]) (2.32)
The proof now depends on the order of t, u, v. We consider the case t < u < v, with similar arguments giving the other cases. Putting i = t, j = u, k = v, equations (2.23) and (2.24) hold. Since z ji = [λ ij ; 0] this gives:
(q −1 − (q − q −1 )κ kji )[λ ij ; 0] = κ kji (λ ij κ kji − [λ ij ; −1])
which may be solved to give κ kji = −1 or q −1 λ −1 ij [λ ij ; 0]. But, if κ kji = −1 then (2.23) gives z kj = 0 contradicting ε j − ε k ∈ T s . Hence κ kji = q −1 λ −1 ij [λ ij ; 0]. Inserting this in (2.23) gives:
q −1 λ −2 ij z kj = −q −1 λ −2 ij + q q − q −1 −q −1 λ −1 jk λ −2 ij + q −1 λ jk q − q −1 z kj = qλ ij − q −1 λ −1 ij q − q −1 λ ij λ jk − (λ ij λ jk ) −1 q − q −1 .
In light of Theorem 2.4, for the root subsystem T s we have z kj = [µ k ; 1][µ j ; 0], λ jk = µ j µ −1 k for some µ k , µ j = 0 in F (using that F is closed under quadratic extensions). Therefore λ ij = ±µ k or ±q −1 µ −1 j . Similarly, for any order of t, u, v we have
λ tu = ±µ v or ±q −1 µ −1 u . Defining β = ε t − ε u ∈ T s , β ′ = ε u − ε v ∈ N a we similarly see λ uv = ±µ u or ±q −1 µ −1 t for some µ t , µ u = 0 such that z tu = [µ t ; 1][µ u ; 0], λ tu = µ t µ −1 u .
For ε t − ε u and ε u − ε v ∈ N + s ∪ N a , by computations similar to (2.1), λ tu λ uv = ±q −1 , ±λ uv or ±λ tu . Since ε s k − ε s k+1 ∈ T + s and ε s k−2 − ε s k ∈ N a , we have λ s k−2 ,s k−1 λ s k−1 ,s k = ±q −1 µ −1 s k or ±µ s k+1 . If λ s k−2 ,s k−1 λ s k−1 ,s k = ±q −1 , then µ s k = ±1 or µ s k+1 ± q −1 , each of which imply z s k+1 ,s k = 0 contradicting ε s k − ε s k+1 ∈ T s . Therefore λ s k−2 ,s k−1 = ±1. In the following, we let I 1 = {1, 2, . . . , k − 1}, I 2 = {k, k + 1, . . . , l + 1}, I 3 = {l + 2, l + 3, . . . , n + 1}. By (2.2), we have λ s i ,s j = ±1 for i < j ∈ I 1 . Similarly, λ s i ,s j = ±1 for i < j ∈ I 3 . Now, let i ∈ I 2 , j ∈ I 3 . By construction ε s i − ε s l+1 ∈ T + s and ε s l+1 − ε s j ∈ N a , and we have λ s l+1 ,s j = ±µ s l+1 or ±q −1 µ −1 s i . If the first case holds for some i ∈ I 2 then we keep µ ′ s i = µ s i , i ∈ I 2 . Otherwise, it is the case that λ s l+1 ,s j = ±q −1 µ −1 sr for all r ∈ I, which implies that µ sr = c for r ∈ I. Now, having ε s i − ε s l ∈ T + s and ε s l − ε s j ∈ N a gives λ s l ,s l+1 λ s l+1 ,s j = ±c or ±q −1 c −1 , hence µ s l+1 = ±q −1 c −1 or ±c.
In the first case, we we keep µ ′ s i = µ s i , i ∈ I 2 . In the second case, we make a change of variables µ ′
s i = q −1 µ −1 s i , k ≤ i ≤ l +1 giving [µ s i ; 1][µ s i+1 ; 0] = [µ ′ s i+1 ; 1][µ ′ s i ; 0] and µ s i µ −1 s i+1 = µ ′ s i+1 µ ′−1 s i ,
hence these relations are preserved. In each case above, with our choice of µ ′ s i , we have λ s l+1 ,s j = ±µ ′ s l+1 . Therefore λ s i ,s j = λ s i ,s l+1 λ s l+1 ,s j = ±µ ′ i for i ∈ I 2 and j ∈ I 3 . Finally, for i ∈ I 1 and j ∈ I 2 , since λ s i ,s j = ±1 and λ s j ,s l+2 = ±1, we must have λ s i ,s j λ s j ,s l+2 = ±q −1 . Therefore λ s i ,s j = ±q −1 µ ′−1 j . Summarizing, we have:
λ s i ,s j = ±q −1 µ ′−1 j for i ∈ I 1 , j ∈ I 2 ±µ ′ i for i ∈ I 2 , j ∈ I 3 ±µ ′ i µ ′−1 j for i < j ∈ I 2 , ±q −1 for i ∈ I 1 , j ∈ I 3 ±1
otherwise.
Therefore, we can put µ ′ i = ±q −1 for i ∈ I 1 and µ ′ j = ±1 for j ∈ I 3 . Hence, for i < j ∈ {1, 2, . . . , n + 1} we have λ s i ,
s j = µ ′ s i µ ′−1 s j and z s i ,s j = [µ ′ s i ; 1][µ ′ s j ; 0]. It follows that for 1 ≤ i ≤ n + 1 we have F i E i · v λ = [µ i ; 1][µ i+1 ; 0] · v λ = [K i ; 1][K i+1
; 0] · v λ for some weight vector v λ ∈ V . It remains to show that the same holds for any weight vector in V . Let ε j − ε k ∈ T ∪ N − s be given, and suppose j < k (the case k < j is symmetric). If i = j −1, j, k −1 or k then E ε j −ε k commutes with F i E i ,K i andK i+1 , hence we have
F i E i E ε j −ε k · v + = [K i ; 1][K i+1 ; 0]E ε j −ε k · v + .
First, suppose i = j − 1. We compute:
F i E i E ε i+1 −ε k · v + = −F i E ε i −ε k · v + + q −1 F i E ε i+1 −ε k E i · v + = −κ i+1,i,k E ε i+1 −ε k · v + + q −1 E ε i+1 −ε k F i E i · v + = µ i+1 [µ i ; 1]E ε i+1 −ε k · v + + q −1 [µ i ; 1][µ i+1 ; 0]E ε i+1 −ε k · v + = [µ i ; 1][µ i+1 ; 1]E ε i+1 −ε k · v + = [K i ; 1][K i+1 ; 0]E ε i+1 −ε k · v + , 18
and similarly if i = k. If i = k − 1 and i > j then we compute:
F i E i E ε j −ε i+1 · v + = qF i E ε j −ε i+1 E i · v + = q 2 K −1 j,i+1 E −ε i +ε j E i · v + + qE ε j −ε i+1 F i E i · v + = µ −1 j µ i+1 κ j,i,i+1 E ε j −ε i+1 · v + + q[µ i ; 1][µ i+1 ; 0]E ε j −ε i+1 · v + = −µ i+1 [µ i ; 1]E ε j −ε i+1 · v + + q[µ i ; 1][µ i+1 ; 0]E ε j −ε i+1 · v + = [µ i ; 1][µ i+1 ; −1]E ε j −ε i+1 · v + = [K i ; 1][K i+1 ; 0]E ε j −ε i+1 · v + ,
and similarly if i = j and i < k. If i = j and i = k − 1 it is easy to see that
F i E i E i · v + = [K i ; 1][K i+1 ; 0]E i · v + . Since V is irreducible, it is generated by v + , hence it is equal to U q (p) · v + .
The result follows by induction on the degree of monomials in U q (g).
Construction of irreducible completely pointed modules
In this section we find a quantum version of the construction in [1] of irreducible completely pointed weight gl n+1 -modules. Then we show that any irreducible completely pointed U q (gl n+1 )-module occurs in this way.
As in [6, Theorem 3.2], one checks that there is an F-algebra homomorphism
π : U q (gl n+1 ) −→ A q n+1 E i −→ x i y i+1 , F i −→ x i+1 y i , K i −→ ω i . (3.1)
where A q n+1 is the quantized Weyl algebra, defined as the associative unital Falgebra with generators ω i , ω −1 i , x i , y i , i ∈ {1, 2, . . . , n + 1} and defining relations
ω i ω j = ω j ω i , (3.2a) ω i ω −1 i = ω −1 i ω i = 1, (3.2b) ω i x j ω −1 i = q δ ij x j , (3.2c) ω i y j ω −1 i = q −δ ij y j , (3.2d) y i x j = x j y i , i = j, (3.2e) y i x i − q −1 x i y i = ω i , (3.2f) y i x i − qx i y i = ω −1 i (3.2g)
where i, j ∈ {1, 2, . . . , n + 1}. The last two relations are equivalent to the two relations
y i x i = qω i − (qω i ) −1 q − q −1 , x i y i = ω i − ω −1 i q − q −1 (3.2h)
Thus, A q n+1 is isomorphic to the rank n generalized Weyl algebra R(σ, t) where
R = F[ω ±1 1 , . . . , ω ±1 n+1 ], σ j (ω i ) = q −δ ij ω i , t i = qω i −(qω i ) −1 q−q −1
for i ∈ {1, 2, . . . , n + 1}. The central element I n+1 =K 1 · · ·K n+1 of U q (gl n+1 ) is mapped by π to the element E q := ω 1 ω 2 · · · ω n+1 . E q should be thought of as q i x i ∂i : a q-analogue of the Euler operator.
Lemma 3.1. The following identities hold.
E q x i E −1 q = qx i , E q y i E −1 q = q −1 y i , E q ω i E −1 q = ω i , i ∈ {1, 2, . . . , n + 1}, (3.3) A q n+1 = m∈Z A q n+1 [m], A q n+1 [m] = a ∈ A q n+1 | E q aE −1 q = q m a ,(3.
4)
A q n+1 [m 1 ] · A q n+1 [m 2 ] ⊆ A q n+1 [m 1 + m 2 ], (3.5) π U q (gl n+1 ) = A q n+1 [0] = C A q n+1 (E q ). (3.6) where C A q n+1 (E q ) denotes the centralizer of E q in A q n+1 .
Proof. The identities (3.3) follow directly from the commutation relations (3.2) in A q n+1 . Identities (3.4) and (3.5) follow from (3.3) and that A q n+1 is generated by x i , y i and ω i . The second equality of (3.6) is trivial. By definition,
(3.1), of π it is clear that π(E i ), π(F i ), π(K i ) ∈ A q n+1 [0] for all i ∈ {1, . . . , n}. Since U q (gl n+1 ) is generated by the set {E i , F i ,K i } n+1 i=1
, it follows that π U q (gl n ) ⊆ A q n [0]. It remains to prove that A q n+1 [0] ⊆ π U q (gl n+1 ) . First observe that A q n+1 [0] is invariant under left multiplication by elements from R = F[ω ±1 i | i = 1, . . . , n + 1]. Since A q n+1 is a generalized Weyl algebra, it follows that A q n+1 [0] is generated as a left R-module by all monomials a = x k 1 1 x k 2 2 · · · x k n+1 n+1 y l 1 1 y l 2 2 · · · y l n+1 n+1
where k, l ∈ (Z ≥0 ) n+1 are such that i k i = i l i and k i · l i = 0 for all i ∈ {1, 2, . . . , n + 1}. Since any such monomial a is a product of elements of the form x i y j , where i = j, it suffices to show that x i y j lies in the image of π for any i = j. We prove by induction on j that x i y j ∈ π U q (gl n+1 ) whenever i < j. If j = i + 1, then x i y i+1 = π(E i ). If j > i + 1, note that by (3.2f),
x i y j = ω −1 j−1 [x i y j−1 , π(E j−1 )] q (3.7)
(recalling that [a, b] u := ab − uba), which by the induction hypothesis lies in the image of π. Similarly one can use π(F i ) to prove that x i y j ∈ π U q (gl n+1 ) if i > j. This finishes the proof of (3.6).
Lemma 3.2. Let V be an irreducible A q n+1 weight module and m ∈ Specm(R) with V m = 0. Then dim R/m V m = 1. If in addition dim F R/m = 1, then V is completely pointed.
Proof. Let A = A q n+1 . Since A is a generalized Weyl algebra and V is an irreducible weight A-module, each weight space V m is an irreducible C(m)-module, where C(m) = g∈Stab Z n+1 (m) A g is the cyclic subalgebra of A with respect to m, (see e.g. [11,Prop. 7.1]). Since q is not a root of unity, the action of Z n+1 on Specm(R) is faithful, and therefore C(m) = R, which is commutative. This implies that dim R/m V m ≤ 1. The second claim follows the fact that the support of an indecomposable weight module over a generalized Weyl algebra is invariant under the automorphisms σ 1 , . . . , σ n and that dim F R/m = dim F R/τ (m) for any F-algebra automorphism τ of R.
Let Specm 1 (R) denote the set of all maximal ideals m of R such that R/m is onedimensional over F. Thus m = (ω 1 − µ 1 , . . . , ω n+1 − µ n+1 ), where (µ 1 , . . . , µ n+1 ) ∈ (F × ) n+1 .
Theorem II A. Let W be an irreducible completely pointed A q n -module. Let π * W be the U q (gl n+1 )-module, given as the π-pullback of W , where π is the map (3.1). Then π * W is completely reducible, and each irreducible submodule is completely pointed, and occurs with multiplicity one.
Proof. Since E q ∈ R, the A q n+1 -module W decomposes in particular into eigenspaces with respect to E q . Due to the commutation relations in A q n+1 , the ratio of any two eigenvalues is a power of q. That is, there exists a non-zero ξ ∈ F such that is an U q (gl n+1 )-submodule of W . Since π(K i ) = ω i for each i ∈ {1, . . . , n + 1}, and W is completely pointed as an A q n+1 -module, it follows that each W [m], m ∈ Z is a completely pointed U q (gl n+1 )module. It remains to prove that for each m ∈ Z, the U q (gl n+1 )-module W [m] is either zero or irreducible. By (3.6), proving that W [m] is irreducible as an U q (gl n+1 )-module is the same thing as proving that W [m] is irreducible as an A q n+1 [0]-submodule of W . Suppose W [m] = {0} and let w 0 and w 1 be any two non-zero weight vectors of W [m] of weights m 0 and m 1 respectively. A n+1 is generated as a left R-module by monomials of the form a = x k 1 1 x k 2 2 · · · x k n+1 n+1 · y l 1 1 y l 2 2 · · · y l n+1 n+1 , where k, l ∈ (Z ≥0 ) n+1 and k i l i = 0 for each i. Moreover, there is at most one such monomial a such that (aW m 0 ) ∩ W m 1 = {0}. Since W is irreducible as an 21
Proposition 2 . 2 (
22Analogous to[1, Proposition 3.2]). The irreducible highest weight U q (sl n+1 )-module L(λ) is completely pointed only if λ = ±1, λ i = ±q, λ 1 = c, λ n = c, λ i λ i+1 = ±q −1 for some i = 1, 2, . . . , n − 1, where c ∈ F × is arbitrary and all unspecified entries are ±1.
relations and equations (2.3)-(2.5) we see that all the z ij are determined by φ 123 . Using that F is closed under quadratic extensions, we choose µ 2 = ±(qφ 123 ) −1/2 and obtain the desired result: z ij = [µ i ; 1][µ j ; 0].
either B ∩ T is empty or it corresponds to a connected part of the Dynkin diagram of Φ B where B is the basis of the previous lemma.
W =
Wm∈Z W [m], W [m] = w ∈ W | E q w = ξq m w (3.8) Each W [m] is a direct sum of certain R-weight spaces of W . More precisely, for each m ∈ F we have W [m] = m∈Specm 1 (R) Eq−q m ξ∈m W m . By Lemma 3.1, each subspace W [m]
A n+1 -module, there exists r ∈ R and a single monomial a such that raw 0 = w 1 . Since w 0 , w 1 ∈ W [m], this forces i k i = i l i , which implies that a ∈ A q n+1 [0]. This proves that W [m] is irreducible as an A q n+1 [0]-module.The cyclic algebra of U q (g)-C(U q (g))-is defined to be the subalgebra of all elements commuting with K ±1 i , i ∈ {1, 2, . . . , n + 1}. Lemma 3.3. Let V be an irreducible, infinite dimensional, completely pointed U q (sl n+1 )-module. Then ker π ⊆ Ann Uq(sl n+1 ) V , where π is the map (3.1)., and the sequences i = (i 1 , i 2 , . . . , i l ), j = (j 1 , j 2 , . . . , j l ) are such that i is a permutation of j. We have:where s r (resp. s ′ r ) denotes the number of times the element i r (resp. i r − 1) appears in the sequence (j 1 , j 2 , . . . , j l )\(i r+1 , i r+2 , . . . , i l ) and t r (resp. t ′ r ) denotes the number of times j r + 1 (resp. j r ) appears in the sequence (j r+1 , j r+2 , . . . , j l ). We prove this by induction on l. If l = 1 then we have s 1 = 1, s ′ 1 = 0, t 1 = 0, t ′ 1 = 0 and compute:For l > 1 observe that x i l +1 y i l commutes with x j k y j k +1 if and only if j k = i l . Let k ∈ {1, 2, . . . l} be the minimum such that i l = j k (we know such a k exists since j is a permutation of i). We have:l r=1x ir+1 y ir l r=1x jr y jr+1and apply induction to obtain the desired result. Now, let V be an infinite-dimensional irreducible completely-pointed U q (sl n+1 )module. Then, using Theorem I, we extend the U q (sl n+1 ) action on V to a U q (gl n+1 ) action that satisfiesBy a similar computation we see:where s r , s ′ r , t r , and t ′ r are as before. Therefore, π(x) is a Laurent polynomial in the ω i and x acts on v by the same Laurent polynomial evaluated at ω i = µ i . If x ∈ ker(π) then this Laurent polynomial must be identically 0, henceNext we prove ker(π) ⊆ Ann Uq(sl n+1 ) (V ). Let x ∈ ker(π). Without loss of generality we can assume x is homogeneous with respect to the root lattice grading:x = x β for some β ∈ Q and K i xK −1 i = q β,α i x for i = 1, 2, . . . , n. Assume for the sake of contradiction that there exists an irreducible completely pointed weight module V for which x · V = 0. Then there exists a weight vector v ∈ V such that w = x · v = 0. Since x is homogeneous, w is also a weight vector. Since V is irreducible, there exists some homogeneous element y ∈ U q (sl n+1 ) of degree −β such that y · w = v. Then yx has degree zero, and thus belongs to the centralizer C(U q (sl n+1 )) of K 1 , . . . , K n . Also, yx belongs to ker(π) since it is an ideal in U q . So yx ∈ C(U q (sl n+1 )) ∩ ker(π) which by the previous paragraph implies that (yx) · V = {0}, which contradicts the fact that (yx) · v λ = v λ = 0.Theorem II B. Any infinite-dimensional irreducible completely pointed U q (sl n+1 ) is isomorphic to a direct summand of π * W for some irreducible completely pointed A q n+1 -module W . Proof. Let V be an irreducible completely pointed U q (sl n+1 )-module, where we extend the action to make V become a U q = U q (gl n+1 )-module as in Theorem I.Since I n+1 :=K 1 · · ·K n+1 is central in U q and V is irreducible, it follows that I n+1 acts by some scalar[0]-submodule of W , and can thus be regarded as a U q -module via the map (3.1). By (3.6) and Lemma 3.3, there is a linear mapIt is balanced with respect to the right and left U q -actions and hence induces aSince N is graded, so is W , and(3.11)and It remains to be proved that W is completely pointed. Let λ be a character of V such that V λ = {0} and let v λ ∈ V λ \ {0}. Define m = (ω 1 − λ(K 1 ), . . . , ω n+1 − λ(K n+1 )) ∈ Specm 1 (R). Then the vector (1 ⊗ Uq v λ ) + N is a nonzero R-weight vector of weight m, so by Lemma 3.2, W is a completely pointed A q n+1 -module.AcknowledgmentThe first author is supported in part by the CNPq grant (301320/2013-6) and by the Fapesp grant (2010/50347-9). The second author would like to thank Dennis Hasselstrom Pedersen for interesting discussions. The third author is grateful to 24 the University of São Paulo for hospitality, his supervisor Vyacheslav Futorny, and to the Fapesp for financial support (grant number 2011/12079-5).
Modules with bounded weight multiplicities for simple Lie algebras. G Benkart, D Britten, F Lemire, Mathematische Zeitschrift. 225Benkart, G., Britten, D., Lemire, F. "Modules with bounded weight multiplicities for simple Lie algebras," Mathematische Zeitschrift 225, 333-353, 1997.
A classification of simple Lie modules having a 1-dimensional weight space. D J Britten, F W Lemire, Trans. Amer. Math. Soc. 299D.J. Britten, F.W. Lemire, "A classification of simple Lie modules having a 1-dimensional weight space," Trans. Amer. Math. Soc. 299 (1987), 683-697.
A Guide to Quantum Groups. V Chari, A Pressley, Cambridge University PressCambridgeChari, V., Pressley, A. A Guide to Quantum Groups, Cambridge University Press, Cam- bridge, 1994.
Lie algebra modules with finite dimensional weight spaces, I. S Fernando, Trans. Amer. Math. Soc. 3222Fernando, S., "Lie algebra modules with finite dimensional weight spaces, I," Trans. Amer. Math. Soc. 322 (2), 757-781, (1990).
Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras. J Hartwig, J Öinert, Journal of Algebra. 371Hartwig, J.,Öinert, J., "Simplicity and maximal commutative subalgebras of twisted gen- eralized Weyl algebras," Journal of Algebra, 371 (2012), 312-339.
Q-Analogues of Clifford and Weyl Algebras -Spinor and Oscillator Representations of Quantum Enveloping Algebras. T Hayashi, Commun. Math. Phys. 127Hayashi, T., "Q-Analogues of Clifford and Weyl Algebras -Spinor and Oscillator Repre- sentations of Quantum Enveloping Algebras," Commun. Math. Phys, 127, 129-144 (1990).
Introduction to Quantum Groups and Crystal Bases. J Hong, S.-J Kang, American Mathematical Society PressProvidenceHong, J., Kang, S.-J. Introduction to Quantum Groups and Crystal Bases, American Math- ematical Society Press, Providence, 2002.
Introduction to Lie Algebras and Representation Theory. J Humphreys, Springer-VerlagNew YorkHumphreys, J. Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
Quantum deformation of certain simple modules over enveloping algebras. G Lusztig, Adv. Math. 70Lusztig, G. "Quantum deformation of certain simple modules over enveloping algebras," Adv. Math. 70 (1988), 237-249.
Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. G Lusztig, J. Amer. Math. Soc. 3Lusztig, G. "Finite dimensional Hopf algebras arising from quantized universal enveloping algebras," J. Amer. Math. Soc. 3 (1990), 257-296.
Some associative algebras associated to U (g) and twisted generalized Weyl algebras. V Mazorchuk, M Ponomarenko, L Turowska, Math. Scand. 92Mazorchuk, V., Ponomarenko, M., Turowska, L.,"Some associative algebras associated to U (g) and twisted generalized Weyl algebras," Math. Scand. 92 (2003), 5-30.
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"Nonlocality, correlations, and magnetotransport in spatially modulated two-dimensional electron gas",
"Nonlocality, correlations, and magnetotransport in spatially modulated two-dimensional electron gas"
]
| [
"O E Raichev \nInstitute of Semiconductor Physics\nNational Academy of Sciences of Ukraine\nProspekt Nauki 4103028KyivUkraine\n"
]
| [
"Institute of Semiconductor Physics\nNational Academy of Sciences of Ukraine\nProspekt Nauki 4103028KyivUkraine"
]
| []
| It is shown that the classical commensurability phenomena in weakly modulated two-dimensional electron systems is a manifestation of intrinsic properties of the correlation functions describing a homogeneous electron gas in magnetic field. The theory demonstrates the importance for consideration of nonlocal response and removes the gap between classical and quantum approaches to magnetotransport in such systems.Magnetotransport properties of two-dimensional (2D) electrons in the presence of spatially varying weak electrostatic potential energy U r or magnetic field δB r have been extensively studied in connection with the problem of commensurability phenomena, in particular, Weiss oscillations, in periodically modulated systems . The Weiss oscillations of the resistance of unidirectionally modulated electron gas appear because of periodic dependence of the drift velocity, averaged over the path of cyclotron rotation, on the ratio of cyclotron radius R to modulation period a. Similar commensurability oscillations existing in the case of 2D (bidirectional) modulation have the same origin. Whereas the classical nature of Weiss oscillations has been established [2] very soon after their discovery, the vast majority of theoretical works devoted to this phenomenon are based on application of the quantum linear response (Kubo) theory to calculation of conductivity. Within this approach, the resistance oscillations are explained in terms of modulation-induced transformation of Landau levels into one-dimensional subbands whose bandwidth oscillates as a function of the subband number. The classical analog of the Landau bandwidth is the average of the modulation energy over the path of cyclotron rotation[4,7]. However, the link between quantum and classical approaches to the problem is still incomplete. In the quantum linear response formalism, the oscillating dependence of conductivity appears as a result of direct influence of the modulation on the electron energy spectrum, so the classical origin of the commensurability phenomena is concealed. More important, the results obtained from the linear response theory deviate from the classical Boltzmann equation results[2,18,19]in the region R a corresponding to the high-field part of the oscillations and subsequent transition to the adiabatic regime.In this Letter, the Kubo formalism is applied for calculation of the nonlocal conductivity σ(r, r ′ ) of weakly modulated electron gas. It is shown that this approach is free from the difficulties mentioned above. In the regime of classically strong magnetic fields, relevant for observation of commensurability phenomena, the conductivity tensor is subdivided into the local part that describes the Drude response and the nonlocal one, entirely responsi-ble for the effect of modulation. The nonlocal part is proportional to a product of the field of potential gradients, ∇ γ U r ∇ γ ′ U r ′ , or varying magnetic fields, δB r δB r ′ , by the spatial correlation functions of the homogeneous (unmodulated) 2D electron gas. Remarkably, the correlation functions already contain oscillating dependence on the magnetic field because they account for the cyclotron motion. This observation leads to a general point of view on the classical commensurability phenomena as manifestations of intrinsic properties of homogeneous 2D systems in the presence of modulation. The theory is valid for arbitrary weak and classically smooth U r and δB r , and is applied as well for description of the magnetoresistance due to random modulation. General formalism. Throughout the Letter, the Planck's constant is set at unity. A parabolic spectrum of 2D electrons is assumed, and the Zeeman splitting is neglected. The Hamiltonian of non-interacting electrons in a perpendicular magnetic field B r = (0, 0, B + δB r ) has a standard form,Ĥ = jĤ rj ,Ĥ r = mv 2 r /2 + V r + U r , wherev r = [−i∇ − (e/c)(A r + δA r )]/m is the velocity operator, r is the 2D coordinate, m is the effective mass of electron, A r and δA r are the vector potentials describing the uniform and the modulating magnetic fields, respectively. Next, V r is a random impurity potential varying on a scale much smaller than the cyclotron radius R = v F /ω c , where v F = 2ε F /m is the Fermi velocity expressed through the chemical potential ε F and ω c = |e|B/mc is the cyclotron frequency. Finally, U r is a potential varying on a scale much larger than the magnetic length ℓ = c/|e|B with the amplitude much smaller than ε F . Similar conditions of smoothness and smallness apply for magnetic modulation. It is assumed that U r and δB r have zero average over the sample area.The Kubo-Greenwood formula for the steady-state nonlocal conductivity tensor is written in the exact eigenstate representation as follows:whereÎ r = e j {v xj , δ(x j −r)} is the operator of current density, {, } denotes a symmetrized product, λ → +0, S | 10.1103/physrevlett.120.146802 | [
"https://arxiv.org/pdf/2006.12339v1.pdf"
]
| 13,878,652 | 2006.12339 | 9d2a88c0a20bbd733fafaae926b59b32d261f422 |
Nonlocality, correlations, and magnetotransport in spatially modulated two-dimensional electron gas
22 Jun 2020
O E Raichev
Institute of Semiconductor Physics
National Academy of Sciences of Ukraine
Prospekt Nauki 4103028KyivUkraine
Nonlocality, correlations, and magnetotransport in spatially modulated two-dimensional electron gas
22 Jun 2020(Dated: June 23, 2020)numbers: 7343Qt7363Hs7210Bg
It is shown that the classical commensurability phenomena in weakly modulated two-dimensional electron systems is a manifestation of intrinsic properties of the correlation functions describing a homogeneous electron gas in magnetic field. The theory demonstrates the importance for consideration of nonlocal response and removes the gap between classical and quantum approaches to magnetotransport in such systems.Magnetotransport properties of two-dimensional (2D) electrons in the presence of spatially varying weak electrostatic potential energy U r or magnetic field δB r have been extensively studied in connection with the problem of commensurability phenomena, in particular, Weiss oscillations, in periodically modulated systems . The Weiss oscillations of the resistance of unidirectionally modulated electron gas appear because of periodic dependence of the drift velocity, averaged over the path of cyclotron rotation, on the ratio of cyclotron radius R to modulation period a. Similar commensurability oscillations existing in the case of 2D (bidirectional) modulation have the same origin. Whereas the classical nature of Weiss oscillations has been established [2] very soon after their discovery, the vast majority of theoretical works devoted to this phenomenon are based on application of the quantum linear response (Kubo) theory to calculation of conductivity. Within this approach, the resistance oscillations are explained in terms of modulation-induced transformation of Landau levels into one-dimensional subbands whose bandwidth oscillates as a function of the subband number. The classical analog of the Landau bandwidth is the average of the modulation energy over the path of cyclotron rotation[4,7]. However, the link between quantum and classical approaches to the problem is still incomplete. In the quantum linear response formalism, the oscillating dependence of conductivity appears as a result of direct influence of the modulation on the electron energy spectrum, so the classical origin of the commensurability phenomena is concealed. More important, the results obtained from the linear response theory deviate from the classical Boltzmann equation results[2,18,19]in the region R a corresponding to the high-field part of the oscillations and subsequent transition to the adiabatic regime.In this Letter, the Kubo formalism is applied for calculation of the nonlocal conductivity σ(r, r ′ ) of weakly modulated electron gas. It is shown that this approach is free from the difficulties mentioned above. In the regime of classically strong magnetic fields, relevant for observation of commensurability phenomena, the conductivity tensor is subdivided into the local part that describes the Drude response and the nonlocal one, entirely responsi-ble for the effect of modulation. The nonlocal part is proportional to a product of the field of potential gradients, ∇ γ U r ∇ γ ′ U r ′ , or varying magnetic fields, δB r δB r ′ , by the spatial correlation functions of the homogeneous (unmodulated) 2D electron gas. Remarkably, the correlation functions already contain oscillating dependence on the magnetic field because they account for the cyclotron motion. This observation leads to a general point of view on the classical commensurability phenomena as manifestations of intrinsic properties of homogeneous 2D systems in the presence of modulation. The theory is valid for arbitrary weak and classically smooth U r and δB r , and is applied as well for description of the magnetoresistance due to random modulation. General formalism. Throughout the Letter, the Planck's constant is set at unity. A parabolic spectrum of 2D electrons is assumed, and the Zeeman splitting is neglected. The Hamiltonian of non-interacting electrons in a perpendicular magnetic field B r = (0, 0, B + δB r ) has a standard form,Ĥ = jĤ rj ,Ĥ r = mv 2 r /2 + V r + U r , wherev r = [−i∇ − (e/c)(A r + δA r )]/m is the velocity operator, r is the 2D coordinate, m is the effective mass of electron, A r and δA r are the vector potentials describing the uniform and the modulating magnetic fields, respectively. Next, V r is a random impurity potential varying on a scale much smaller than the cyclotron radius R = v F /ω c , where v F = 2ε F /m is the Fermi velocity expressed through the chemical potential ε F and ω c = |e|B/mc is the cyclotron frequency. Finally, U r is a potential varying on a scale much larger than the magnetic length ℓ = c/|e|B with the amplitude much smaller than ε F . Similar conditions of smoothness and smallness apply for magnetic modulation. It is assumed that U r and δB r have zero average over the sample area.The Kubo-Greenwood formula for the steady-state nonlocal conductivity tensor is written in the exact eigenstate representation as follows:whereÎ r = e j {v xj , δ(x j −r)} is the operator of current density, {, } denotes a symmetrized product, λ → +0, S
It is shown that the classical commensurability phenomena in weakly modulated two-dimensional electron systems is a manifestation of intrinsic properties of the correlation functions describing a homogeneous electron gas in magnetic field. The theory demonstrates the importance for consideration of nonlocal response and removes the gap between classical and quantum approaches to magnetotransport in such systems. Magnetotransport properties of two-dimensional (2D) electrons in the presence of spatially varying weak electrostatic potential energy U r or magnetic field δB r have been extensively studied in connection with the problem of commensurability phenomena, in particular, Weiss oscillations, in periodically modulated systems . The Weiss oscillations of the resistance of unidirectionally modulated electron gas appear because of periodic dependence of the drift velocity, averaged over the path of cyclotron rotation, on the ratio of cyclotron radius R to modulation period a. Similar commensurability oscillations existing in the case of 2D (bidirectional) modulation have the same origin. Whereas the classical nature of Weiss oscillations has been established [2] very soon after their discovery, the vast majority of theoretical works devoted to this phenomenon are based on application of the quantum linear response (Kubo) theory to calculation of conductivity. Within this approach, the resistance oscillations are explained in terms of modulation-induced transformation of Landau levels into one-dimensional subbands whose bandwidth oscillates as a function of the subband number. The classical analog of the Landau bandwidth is the average of the modulation energy over the path of cyclotron rotation [4,7]. However, the link between quantum and classical approaches to the problem is still incomplete. In the quantum linear response formalism, the oscillating dependence of conductivity appears as a result of direct influence of the modulation on the electron energy spectrum, so the classical origin of the commensurability phenomena is concealed. More important, the results obtained from the linear response theory deviate from the classical Boltzmann equation results [2,18,19] in the region R a corresponding to the high-field part of the oscillations and subsequent transition to the adiabatic regime.
In this Letter, the Kubo formalism is applied for calculation of the nonlocal conductivity σ(r, r ′ ) of weakly modulated electron gas. It is shown that this approach is free from the difficulties mentioned above. In the regime of classically strong magnetic fields, relevant for observation of commensurability phenomena, the conductivity tensor is subdivided into the local part that describes the Drude response and the nonlocal one, entirely responsi-ble for the effect of modulation. The nonlocal part is proportional to a product of the field of potential gradients, ∇ γ U r ∇ γ ′ U r ′ , or varying magnetic fields, δB r δB r ′ , by the spatial correlation functions of the homogeneous (unmodulated) 2D electron gas. Remarkably, the correlation functions already contain oscillating dependence on the magnetic field because they account for the cyclotron motion. This observation leads to a general point of view on the classical commensurability phenomena as manifestations of intrinsic properties of homogeneous 2D systems in the presence of modulation. The theory is valid for arbitrary weak and classically smooth U r and δB r , and is applied as well for description of the magnetoresistance due to random modulation. General formalism. Throughout the Letter, the Planck's constant is set at unity. A parabolic spectrum of 2D electrons is assumed, and the Zeeman splitting is neglected. The Hamiltonian of non-interacting electrons in a perpendicular magnetic field B r = (0, 0, B + δB r ) has a standard form,Ĥ = jĤ rj ,Ĥ r = mv 2 r /2 + V r + U r , wherev r = [−i∇ − (e/c)(A r + δA r )]/m is the velocity operator, r is the 2D coordinate, m is the effective mass of electron, A r and δA r are the vector potentials describing the uniform and the modulating magnetic fields, respectively. Next, V r is a random impurity potential varying on a scale much smaller than the cyclotron radius R = v F /ω c , where v F = 2ε F /m is the Fermi velocity expressed through the chemical potential ε F and ω c = |e|B/mc is the cyclotron frequency. Finally, U r is a potential varying on a scale much larger than the magnetic length ℓ = c/|e|B with the amplitude much smaller than ε F . Similar conditions of smoothness and smallness apply for magnetic modulation. It is assumed that U r and δB r have zero average over the sample area.
The Kubo-Greenwood formula for the steady-state nonlocal conductivity tensor is written in the exact eigenstate representation as follows:
σ αβ (r, r ′ ) = i S 2 δδ ′ δ ′ |Î α r |δ δ|Î β r ′ |δ ′ (f ε δ − f ε δ ′ ) (ε δ − ε δ ′ − iλ)(ε δ − ε δ ′ ) ,(1)
whereÎ r = e j {v xj , δ(x j −r)} is the operator of current density, {, } denotes a symmetrized product, λ → +0, S is the normalization area, δ is the eigenstate index, and f ε is the equilibrium Fermi distribution. It is convenient to transform Eq. (1) by using the operator identitŷ
v r = ℓ 2ǫ ∇U r − {v r , δB r }/B − iω −1 cǫ [v r ,Ĥ r ],(2)
where U r = V r + U r is the total potential andǫ is the antisymmetric unit matrix in the Cartesian 2D coordinate space. After substituting Eq. (2) into Eq. (1), the last term in Eq. (2) gives the classical Hall conductivity, the rest of the contributions come from the first two terms.
In the case of purely potential modulation, δB = 0, the dissipative part of the conductivity is
σ d αβ (r, r ′ ) = 2πe 2 ℓ 4 ǫ αγ ǫ βγ ′ dε − ∂f ε ∂ε × (∇ γ U r )(∇ γ ′ U r ′ )A ε (r, r ′ )A ε (r ′ , r) ,(3)
where the angular brackets define the average over the random potential, and A ε (r,
r ′ ) = (2πi) −1 [G A ε (r, r ′ ) − G R ε (r, r ′ )]
is the spectral function in the coordinate representation, expressed through the non-averaged Green's functions G s (s = R, A denotes the retarded and the advanced ones). Since the case of degenerate electron gas is assumed, the energy ε stands in a narrow interval around Fermi level and can be replaced by ε F if the correlation function in Eq. (3) slowly varies with energy, in particular, in the classical transport regime. Evaluating Eq. (3) within the accuracy up to the first power in the random potential correlator w(q) defined as a Fourier transform of the correlation function V 0 V r leads to two contribu-
tions: σ d αβ ≃ σ (1) αβ + σ (2) αβ , σ (1) αβ (r, r ′ ) = 2πe 2 ℓ 4 ǫ αγ ǫ βγ ′ (∇ γ U r )(∇ γ ′ U r ′ ) × dε − ∂f ε ∂ε A ε (r, r ′ )A ε (r ′ , r) ,(4)σ (2) αβ (r, r ′ ) = 2πe 2 ℓ 4 ǫ αγ ǫ βγ ′ dε − ∂f ε ∂ε × dq (2π) 2 q γ q γ ′ w(q)e iq·(r−r ′ ) A ε (r, r ′ )A ε (r ′ , r),(5)
where A ε (r, r ′ ) = A ε (r, r ′ ) is the averaged spectral function. The first contribution describes the conductivity due to the presence of smooth potential gradients. The second one is the leading term in the expansion of the conductivity in powers of the ratio of the scattering rate to cyclotron frequency. Keeping only these contrinutions is sufficient in the case of classically strong magnetic fields, (ω c τ tr ) 2 ≫ 1, where τ tr is the transport time.
The difference between the present technique and previous applications of the Kubo formalism to the problem is a consideration of nonlocal response instead of the local one, which is necessary for correct evaluation of the conductivity, and the application of the identity Eq.
(2), which separates the drift-induced σ (1) and diffusioninduced σ (2) contributions and removes the necessity to specify eigenstates and Green's functions at the early stage of calculations.
To find σ (1) , one needs to calculate the pair correlation function in Eq. (4), which is determined, in the Born approximation, by the particle-hole ladder series. In the case of arbitrary w(q), the problem cannot be solved analytically even in the classical limit. Therefore, the case of white noise random potential is assumed, when w(q) is replaced by a constant. Introducing the correlator C ss ′ ε (r, r ′ ) = w G s ε (r, r ′ )G s ′ ε (r ′ , r) and applying a standard technique of summation leads to the integral equation r) is the "bare" correlator expressed through the averaged Green's functions. It is convenient to rewrite this equation for the double Fourier transforms of C and K:
C ss ′ ε (r, r ′ ) = K ss ′ ε (r, r ′ ) + dr 1 K ss ′ ε (r, r 1 )C ss ′ ε (r 1 , r ′ ), where K ss ′ ε (r, r ′ ) = wG s ε (r, r ′ )G s ′ ε (r ′ ,C ε (q, q ′ ) = K ε (q, q ′ ) + dq 1 (2π) 2 K ε (q, q 1 )C ε (q 1 , q ′ ). (6)
Since only the terms with s = s ′ are important, the repeating s-indices are omitted here and below. The correlators C and K are essentially different. While K ε (r, r ′ ) describes correlations on the 2R scale, C ε (r, r ′ ) has no definite correlation length and logarithmically depends on |r − r ′ |. This is a consequence of the diffusion-pole divergence of C ε (q, q ′ ), as in the limit of small q Eq. (6) can be reduced to a diffusion equation. The long-range behavior of correlations is a general property topologically dictated by the dimensionality 2 [49,50]. In contrast to σ (1) , the contribution σ (2) can be treated locally, because it contains the exponential factor e iq·(r−r ′ ) , where q has meaning of the momentum transferred in the scattering of electrons by the potential V . Since q is comparable to Fermi momentum (except for the scattering on very small angles), the correlation length is much smaller than both R and modulation length, and it is sufficient to consider the local conductivity, σ (2) αβ (r) = d∆rσ (2) αβ (r + ∆r/2, r − ∆r/2).
Classical conductivity. The contribution σ (1) is already proportional to the squared gradient of the smooth potential U r . In the classical case, when the Landau quantization is neglected, accounting for U r in the Green's functions entering C ε leads to an expansion in powers of small parameters U r /ε F and ∇U r R/ε F . Therefore, to calculate σ (1) in the classical limit, it is sufficient to employ the Green's functions of a homogeneous system:
G R,A ε (r, r ′ ) = e iθ(r,r ′ ) 2πℓ 2 ∞ N =0 L 0 N (|∆r| 2 /2ℓ 2 )e −|∆r| 2 /4ℓ 2 ε − ε N ± i/2τ ,(8)
where ∆r = r − r ′ , the sum is taken over the Landau level numbers, L M N is the Laguerre polynomial, ε N = ω c (N + 1/2) is the Landau level spectrum, τ = 1/mw is the scattering time, and θ(r, r ′ ) = (e/c) r r ′ dr 1 ·A r1 . Due to the homogeneity, Eq. (6) is solved analytically:
C ε (q, q ′ ) = C εq (2π) 2 δ(q − q ′ ), C εq = K εq /(1 − K εq ), (9) where K εq = w 2πℓ 2 N,N ′ (−1) N +N ′ e −β L N −N ′ N (β)L N ′ −N N ′ (β) (ε − ε N + i/2τ )(ε − ε N ′ − i/2τ )(10)
and β = q 2 ℓ 2 /2. The classical limit corresponds to treatment of the Landau level numbers as continuous variables and to application of the asymptotic form of L M N (β) at large N . With ε = ε F and q ≪ mv F , this leads to
K εq ≃ K q = ∞ n=−∞ J 2 n (qR) 1 + (nω c τ ) 2 ,(11)
where J n is the Bessel function. If (ω c τ ) 2 ≫ 1, it is sufficient to retain a term with n = 0. As a result,
C εq ≃ C q = J 2 0 (qR)/[1 − J 2 0 (qR)](12)
and
σ (1) αβ (r, r ′ ) = e 2 τ πmω 2 c ǫ αγ ǫ βγ ′ dq 1 (2π) 2 dq 2 (2π) 2 dq (2π) 2 ×e i(q−q1)·r e i(q2−q)·r ′ q 1γ q 2γ ′ U −q1 U q2 J 2 0 (qR) 1 − J 2 0 (qR) ,(13)
where U q is the Fourier transform of U r . Using the Green's functions (8) for calculations of the local contribution σ (2) αβ (r) in the classical limit gives the isotropic Drude conductivity at (ω c τ ) 2 ≫ 1:
σ (2) αβ = δ αβ e 2 n s mω 2 c τ ,(14)
where n s is the electron density. Consideration of higherorder terms (not included in σ (2) ) leads to an additional contribution −σ (2) αβ /[1 + (ω c τ ) 2 ] that complements the conductivity to the full Drude form. A generalization to the case of arbitrary w(q) is straightforward and results in a substitution of the transport time τ tr in place of τ . The effect of U r on σ (2) leads to contributions of the order (ω c τ ) −2 σ (1) and, therefore, is neglected. Magnetic modulation. If the modulation δB r instead of U r is present, σ (1) of Eq. (4) is replaced by
σ (1) αβ (r, r ′ ) = 2πe 2 δB r δB r ′ B 2 dε − ∂f ε ∂ε × ṽ rαṽr ′ β A ε (r, r ′ )A ε (r ′ , r) ,(15)
whereṽ r = [−iν∂/∂r − (e/c)A r ]/m is a differential operator with ν = 1/2 (ν = −1/2) when acting on the first (second) coordinate variable of the Green's functions. The response is determined by the correlator M αβ ε (r, r ′ ) = w ṽ rαṽr ′ β G s ε (r, r ′ )G s ′ ε (r ′ , r) with s = s ′ :
M αβ ε (r, r ′ ) = M αβ ε (r, r ′ ) + dr 1 dr 2ṽrα K ε (r, r 1 )
×[δ(r 1 − r 2 ) + C ε (r 1 , r 2 )]ṽ r ′ β K ε (r 2 , r ′ ),(16)
where M αβ ε (r, r ′ ) = wṽ rαṽr ′ β G s ε (r, r ′ )G s ′ ε (r ′ , r). In the classical case, using Green's functions of Eq. (8) and C εq of Eq. (12), one gets the expression for Fourier transform of M αβ ε (r, r ′ ) at ε = ε F and q ≪ mv F :
M αβ q ≃ ǫ αγ ǫ βγ ′ q γ q γ ′ q 2 v 2 F J 2 1 (qR) 1 − J 2 0 (qR) .(17)
Therefore, σ
αβ (r, r ′ ) of Eq. (15) can be written in the form of Eq. (13), when the latter is modified by the substitution q 1γ q 2γ ′ U −q1 U q2 J 2 0 (qR) → q γ q γ ′ δB −q1 δB q2 (ε F /B) 2 J 2 1 (qR)/(qR/2) 2 . Periodic modulation. In the case of a periodic U r or δB r , the problem becomes macroscopically homogeneous and described by the conductivity tensor
σ αβ = 1 S dr dr ′ σ αβ (r, r ′ ),(18)
which can be also viewed as the average of the local conductivity over the elementary cell of modulation lattice. Application of Eq. (18) to Eq. (13) gives, for potential and magnetic modulation, respectively,
σ (1) αβ = e 2 n s τ m dq Ω q 2q 2 ǫ αγ ǫ βγ ′ q γ q γ ′ 1 − J 2 0 (qR) (qR) 2 J 2 0 (qR) 4J 2 1 (qR)(19)
with Ω q = k1,k2 |u k1,k2 | 2 δ(q − k 1 Q 1 − k 2 Q 2 ), where k 1 and k 2 are integers, Q 1 and Q 2 are the Bravais vectors of the reciprocal lattice, and u k1,k2 are the Fourier coefficients of the relative modulation strength, u(r) = U r /ε F for the potential modulation and u(r) = δB r /B for the magnetic one. For harmonic unidirectional modulation, u(r) = η cos(Qx), the vectors are Q 1 = (Q, 0) and Q 2 = (0, 0), while nonzero coefficients are u 1,0 = u −1,0 = η/2. Thus, only the component σ (1) yy survives, leading to the resistivity ρ
(1) xx ≃ σ (1) yy /σ 2 H ,
where σ H is the classical Hall conductivity. This contribution is identified with the Weiss oscillations term, in full agreement with the results of theories based on the Boltzmann equation [2,18,19]. Previous theories based on the Kubo formula for local conductivity miss the term J 2 0 in the denominator. This would occur if the correlators C q and M αβ q were replaced by the bare correlators K q and M αβ q . Such an approximation is justified at qR ≫ 1, when J 2 l (qR) ≃ (2/πqR) cos 2 (qR − lπ/2 − π/4). In the general case of anharmonic 2D modulation, Eq. (19) gives a superposition of Weiss oscillations with different q in both ρ xx and ρ yy [9]. In the adiabatic limit, qR ≪ 1, ρ (1) ∝ B 2 in agreement with the experiment [10]. Random modulation. In the case of weak modulation by random potential or magnetic field, the problem is again macroscopically homogeneous. The current density averaged over a large area is approximately related to the averaged driving electric field by the local Ohm's law with the conductivity tensor of Eq. (18), averaged over the random modulation distribution. This approximation is valid because of the assumed weakness of modulation, while in the general case the problem of linear response in inhomogeneous media remains very complicated even in the local formulation [51]. The averaging of σ (1) αβ written in the form of Eq. (13) is equivalent to a substitution u −q1 u q2 → Sδ q1,q2 W (q 1 ), where W (q) is the Fourier transform of the correlator u(0)u(r) . This leads to the isotropic conductivity
σ (1) = e 2 n s τ m ∞ 0 dq 8π qW (q) 1 − J 2 0 (qR) (qR) 2 J 2 0 (qR) 4J 2 1 (qR)
. (20) The function W (q) is expected to decrease with q on the scale of inverse mean modulation length r −1 0 . For example, W (q) ∝ e −r0q in the case of remote ionized impurity potential relevant for 2D electrons in high-mobility heterostructures. According to Eq. (20), in the adiabatic limit R ≪ r 0 one has ρ (1) ∝ B 2 for both types of modulation, while at R ≫ r 0 ρ (1) ∝ B for the potential modulation and ρ (1) ∝ B 3 for the magnetic one. Though both V r and u(r) are random, the problem studied here is not equivalent to the problem of electron motion in the presence of two kinds of scatterers, the short-ranged and the long-ranged ones. Indeed, the effect of modulation accounted in σ (1) is electron drift rather than scatteringassisted diffusion, while the diffusion occurs due to the potential V r . The positive magnetoresistance described above is a consequence of the drift motion (although the drift along closed contours is also known to be a cause of localization, which cannot be accounted within the Born approximation). A different model of two-component disorder [52] can lead to a negative magnetoresistance. Finally, one should discuss possible effects of electronelectron (Coulomb) interaction on the magnetoresistance of modulated 2D electron gas. Although this interaction conserves the total momentum of electrons, it does contribute into the Green's functions, modifying the energy spectrum and, consequently, the conductivity. The combined effect of the periodic modulation and the Coulomb interaction is essential in strong magnetic fields, when the interaction changes the shape of the Shubnikov-de Haas oscillations [53,54]. Next, the interaction-induced correction to conductivity [55] generates oscillations in ρ yy [56], which are not related to the Landau quantization and, therefore, are important as well in the classical region of fields studied in this Letter. Apart from that, the interaction-induced imaginary part of selfenergy in the Green's functions, which can be described by the temperature-dependent inelastic scattering time τ in , leads to a cutoff of the diffusion pole in the correlator C q . As a result, one should expect a suppression of the conductivity σ (1) when the modulation length (period) increases and becomes comparable to the diffusion length l D = √ τ in D, where D = R 2 /2τ is the diffusion coefficient. Since l D ≫ R, owing to the assumed τ in ≫ τ at low temperatures, this effect may influence the resistance in the adiabatic limit only.
In summary, the problem of magnetotransport in modulated 2D electron systems requires consideration of nonlocal response. The classical commensurability phenomena are described as a result of mapping of the modulation structure onto the spatial correlation pattern of a homogeneous electron system. The correlation functions responsible for potential and magnetic modulation in the regime of classically strong magnetic fields [Eqs. (12) and (17)] depend only on the cyclotron radius. A random modulation leads to a positive magnetoresistance that is sensitive to the modulation type until the adiabatic limit is reached. It remains a question whether similar conclusions apply to 2D systems with Dirac band spectrum such as graphene and related materials.
PACS numbers: 73.43.Qt, 73.63.Hs, 72.10.Bg
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[
"Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs contributing in the third order",
"Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs contributing in the third order"
]
| [
"Huynh M Hien [email protected] \nDepartment of Mathematics and Statistics\nQuy Nhon University\n170 An Duong VuongQuy NhonVietnam\n"
]
| [
"Department of Mathematics and Statistics\nQuy Nhon University\n170 An Duong VuongQuy NhonVietnam"
]
| []
| In this paper we consider orbit pairs contributing in the third order of the spectral form factor in the Hadamard-Gutzwiller model. We prove that periodic orbits including two 2-encounters in certain structures have partner orbits. The action differences are estimated at ln(1 + u 1 s 1 )(1 + u 2 s 2 ) with explicit error bounds, where (u 1 , s 1 ) and (u 2 , s 2 ) are the coordinates of the piercing points. A new symbolic dynamics for orbit pairs via conjugacy classes is also provided.Keywords.Hadamard-Gutzwiller model, Spectral form factor, Third order, Orbit pair, 2-encounter.factor, which is expressed by a double sum over periodic orbitswhere · abbreviates the average over the energy and over a small time window, T H denotes the Heisenberg time and A γ , S γ , and T γ are the amplitude, the action, and the period of the orbit γ, respectively. The diagonal approximation γ = γ ′ to (1.1) studied by Hannay/Ozorio de Almeida [9] and Berry [2] in the 1980's contributes to the first order term 2τ ; see also[15]. The efforts of researchers have been to understand higher order effects. To the next orders, as → 0, the main term from (1.1) arises owing to those orbit pairs γ = γ ′ for which the action difference S γ − S γ ′ is 'small'. In 2001, an influential heuristic work of Sieber and Richter[22]who predicted that a given periodic orbit with a small-angle self-crossing in configuration space will admit a partner orbit with almost the same action. The original orbit and its partner are then called a Sieber-Richter pair. In phase space, a Sieber-Richter pair contains a region where two stretches of each orbit are almost mutually time-reversed and one addresses this region as a 2-encounter or, more strictly, a 2-antiparallel encounter; the '2' stands for two orbit stretches which are close in configuration space, and 'antiparallel' means that the two stretches have opposite directions. It was shown in[22]that Sieber-Richter pairs contribute to the spectral form factor (1.1) the second order term −2τ 2 , and it turned out that the result agreed with what is obtained using random matrix theory [5], for certain symmetry classes. The work by Sieber and Richter has led to the important and difficult problem of understanding this phenomenon is more detail and more rigorously in particular classes of systems. Until 2012, Gutkin and Osipov [7] analysed Sieber-Richter pairs for the Baker map, which admits very transparent symbolic dynamics, in a combinatorial way.Most contribution in this subject matter is Müller et al. In a series of works[11,[17][18][19], the authors provided an expansion to all orders in τ K(τ ) = 2τ − τ ln(1 + 2τ ) = 2τ − 2τ 2 + 2τ 3 + . . . for the symmetry class relevant for time-reversal invariant systems, by including the higher-order encounters also; see also[8,16]. It was shown in[11] | null | [
"https://arxiv.org/pdf/2202.11991v1.pdf"
]
| 247,084,035 | 2202.11991 | a364d87ca29b680ce4d57a2a84746efca809cb1e |
Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs contributing in the third order
24 Feb 2022
Huynh M Hien [email protected]
Department of Mathematics and Statistics
Quy Nhon University
170 An Duong VuongQuy NhonVietnam
Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs contributing in the third order
24 Feb 2022Hadamard-Gutzwiller modelSpectral form factorThird orderOrbit pair2-encounter
In this paper we consider orbit pairs contributing in the third order of the spectral form factor in the Hadamard-Gutzwiller model. We prove that periodic orbits including two 2-encounters in certain structures have partner orbits. The action differences are estimated at ln(1 + u 1 s 1 )(1 + u 2 s 2 ) with explicit error bounds, where (u 1 , s 1 ) and (u 2 , s 2 ) are the coordinates of the piercing points. A new symbolic dynamics for orbit pairs via conjugacy classes is also provided.Keywords.Hadamard-Gutzwiller model, Spectral form factor, Third order, Orbit pair, 2-encounter.factor, which is expressed by a double sum over periodic orbitswhere · abbreviates the average over the energy and over a small time window, T H denotes the Heisenberg time and A γ , S γ , and T γ are the amplitude, the action, and the period of the orbit γ, respectively. The diagonal approximation γ = γ ′ to (1.1) studied by Hannay/Ozorio de Almeida [9] and Berry [2] in the 1980's contributes to the first order term 2τ ; see also[15]. The efforts of researchers have been to understand higher order effects. To the next orders, as → 0, the main term from (1.1) arises owing to those orbit pairs γ = γ ′ for which the action difference S γ − S γ ′ is 'small'. In 2001, an influential heuristic work of Sieber and Richter[22]who predicted that a given periodic orbit with a small-angle self-crossing in configuration space will admit a partner orbit with almost the same action. The original orbit and its partner are then called a Sieber-Richter pair. In phase space, a Sieber-Richter pair contains a region where two stretches of each orbit are almost mutually time-reversed and one addresses this region as a 2-encounter or, more strictly, a 2-antiparallel encounter; the '2' stands for two orbit stretches which are close in configuration space, and 'antiparallel' means that the two stretches have opposite directions. It was shown in[22]that Sieber-Richter pairs contribute to the spectral form factor (1.1) the second order term −2τ 2 , and it turned out that the result agreed with what is obtained using random matrix theory [5], for certain symmetry classes. The work by Sieber and Richter has led to the important and difficult problem of understanding this phenomenon is more detail and more rigorously in particular classes of systems. Until 2012, Gutkin and Osipov [7] analysed Sieber-Richter pairs for the Baker map, which admits very transparent symbolic dynamics, in a combinatorial way.Most contribution in this subject matter is Müller et al. In a series of works[11,[17][18][19], the authors provided an expansion to all orders in τ K(τ ) = 2τ − τ ln(1 + 2τ ) = 2τ − 2τ 2 + 2τ 3 + . . . for the symmetry class relevant for time-reversal invariant systems, by including the higher-order encounters also; see also[8,16]. It was shown in[11]
Introduction
In quantum chaos, there is considerable interest in understanding statistics associated to periodic orbits since these are related to eigenvalue statistics through trace formulae. Special attention has been given to the spectral form that there are five families of pairs of orbits responsible for the third order τ 3 , namely three families of orbit pairs differing in two 2-encounters and two families of orbit pairs differing in one single 3-encounter. Periodic orbits with encounters have partners obtained by reconnections stretches inside encounter area owing to the hyperbolicity. However, the existence of partner orbits and estimates of the action differences are still missing.
To establish a more detailed mathematical understanding, it is necessary to consider the classical side and try to prove the existence of partner orbits and derive good estimates for the action differences of the orbit pairs. For 2-antiparallel encounters this was done in [12,14], where the authors considered the geodesic flow on compact factors of the hyperbolic plane; in this case the action of a periodic orbit is half of its length/period. It was shown in [12] that a T -periodic orbit of the geodesic flow crossing itself in configuration space at a time T 1 has a unique partner orbit that remains 9| sin(φ/2)|-close to the original one and the action difference between them is approximately equal ln(1 − (1 + e −T 1 )(1 + e −(T −T 1 ) ) sin 2 (φ/2))) with the error bound 12 sin 2 (φ/2)e −T , where φ is the crossing angle, and this proved the accuracy of Sieber/Richter's prediction in [22] mentioned above. For higherorder encounters, Huynh [13] shows that there exist (L − 1)! − 1 partner orbits for a given periodic orbit with an L-parallel encounter such that any two piercing points are not too close and provided estimates for the action differences.
In the present paper we continue considering the geodesic flow on compact factor of the hyperbolic plane, which is a compact Riemann surface of constant curvature of genus at least two. In the physics community this system is often called the Hadamard-Gutzwiller model, and it has frequently been studied [4,11,21]; further related work includes [8,19,23]. We prove the existence of the partner orbit which differs in both encounters for a given periodic orbit including two 2-encounters with piercing points having coordinates (u 1 , s 1 ), (u 1 , u 2 ) in certain distributions. The action differences of orbit pairs of all cases are estimated at ln(1 + u 1 s 1 )(1 + u 2 s 2 ) with explicit error bounds. This paper also provides a new symbolic dynamics for orbit pairs via conjugacy classes.
The paper is organized as follows. In Section 2 we recall background and materials, including Poincaré sections, the Anosov and closing lemmas, conjugacy classes and rigorous definitions of encounters, partners in the Hadamard-Gutwiller model. Section 3 considers periodic orbits with one single 2-antiparallel encounter. In the last section we consider periodic orbits with two 2-antiparallel encounters serial, with two 2-parallel encounter intertwined, and with one 2-parallel encounter and one 2-antiparallel encounter intertwined. In each case, we prove the existence of partner orbits, estimate the action differences as well as provide symbolic dynamics for orbit pairs.
The Hadamard-Gutzwiller model
The Hadamard-Gutzwiller model is the geodesic flow on compact Riemann surfaces of constant negative curvature. It is well-known that any compact orientable surface with constant negative curvature is isometric to a factor Γ\H 2 , where H 2 = {z = x + iy ∈ C : y > 0} is the hyperbolic plane endowed with the hyperbolic metric ds 2 = dx 2 +dy 2 y 2 and Γ is a discrete subgroup of the projective Lie group PSL(2, R) = SL(2, R)/{±E 2 }. The hyperbolic plane has constant Gaussian curvature −1. The group PSL(2, R) acts transitively on H 2 by Möbius transformations z → az+b cz+d . If the action has no fixed points, then the factor Γ\H 2 has a Riemann surface structure. Such a surface is a closed Riemann surface of genus at least 2 and has the hyperbolic plane H 2 as the universal covering; so the natural projection π Γ : H 2 → Γ\H 2 , π Γ (z) = Γz, z ∈ H 2 becomes a local isometry. This implies that Γ\H 2 also has constant curvature −1. The geodesic flow (ϕ X t ) t∈R on the unit tangent bundle X = T 1 (Γ\H 2 ) goes along the unit speed geodesics on Γ\H 2 . On the other hand, the unit tangent bundle T 1 (Γ\H 2 ) is isometric to the quotient space Γ\PSL(2, R) = {Γg, g ∈ PSL(2, R)}, which is the system of right co-sets of Γ in PSL(2, R), by an isometry Ξ. Then the geodesic flow (ϕ X t ) t∈R can be equivalently expressed as the natural "quotient flow" ϕ t (Γg) = Γga t on X = Γ\PSL(2, R) associated to the flow ϕ G t (g) = ga t on G := PSL(2, R) by the conjugate relation
ϕ X t = Ξ −1 • ϕ t • Ξ for all t ∈ R.
Here a t ∈ PSL(2, R) denotes the equivalence class obtained from the matrix
A t = e t/2 0 0 e −t/2 ∈ SL(2, R).
There are some more advantages to work on X = Γ\PSL(2, R) rather than on X = T 1 (Γ\H 2 ). One can calculate explicitly the stable and unstable manifolds at a point x = Γg ∈ X to be
W s X (x) = {Γgb s , s ∈ R} and W u X (x) = {Γgc u , u ∈ R}, where b s = {±B s }, c u = {±C u } ∈ PSL(2, R) denote the equivalence classes obtained from B s = 1 s 0 1 , C u = 1 0 u 1 ∈ SL(2, R). If the space X is compact, then the flow (ϕ t ) t∈R is a hyperbolic flow.
There is a natural Riemannian metric on G = PSL(2, R) such that the induced metric function d G is left-invariant under G. We define a metric function d X on X = Γ\PSL(2, R) by
d X (x 1 , x 2 ) = inf γ 1 ,γ 2 ∈Γ d G (γ 1 g 1 , γ 2 g 2 ) = inf γ∈Γ d G (g 1 , γg 2 ), where x 1 = Γg 1 , x 2 = Γg 2 .
General references for this section are [1,6], and these works may be consulted for the proofs to all results which are stated above
Poincaré sections
It is well-known that the Riemann surface Γ\H 2 is compact if and only if the quotient space X = Γ\PSL(2, R) is compact.
First we recall the definitions of Poincaré sections in [12,13].
Definition 2.1. Let x ∈ X and ε > 0. The Poincaré sections of radius ε at x are defined by P ε (x) = {Γgc u b s : |u| < ε, |s| < ε},
and P ′ ε (x) = {Γgb s c u : |s| < ε, |u| < ε}, where g ∈ G is such that x = Γg. If z = Γgc u b s ∈ P ε (x) (resp. z = Γgb s c u ∈ P ′ ε (x)), we write z = (u, s) x (resp. z = (s, u) ′ x )
. Note that the couple (u, s) are not unique. As we will see below, if X is compact and ε is sufficiently small, then the uniqueness of couple (u, s) is obtained. 13]). If the space X = Γ\PSL(2, R) is compact and ε ∈ (0, σ 0 4 ), then for each z ∈ P ε (x) there exist a unique couple (u z , s z ) ∈ (0, ε) 2 such that z = Γgc uz b sz , where g ∈ PSL(2, R) is such that x = Γg, and we call (u z , s z ) the coordinates of z.
Lemma 2.1. If the space X = Γ\PSL(2, R) is compact, then there exists σ 0 > 0 such that d G (γg, g) > σ 0 for all γ ∈ Γ \ {e}.(2.
Conjugacy classes
Let Γ be a discrete subgroup of PSL(2, R). Definition 2.2. (a) An element γ ∈ Γ is called primitive if γ = ζ m for some ζ ∈ Γ implies that m = 1 or m = −1.
(b) The conjugacy class of γ ∈ Γ is defined by
{γ} Γ = {σγσ −1 : σ ∈ Γ}.
The collection of all conjugacy classes of primitive elements in Γ \ {e} are denoted by C Γ ; here e = [E 2 ] denotes the unity of PSL(2, R).
For g = [G] ∈ PSL(2, R), G = a b c d ∈ SL(2, R)
, the trace of g is defined by tr(g) = |a + d|. If the action of Γ on H 2 is free and the factor Γ\H 2 is compact then all elements g ∈ Γ \ {e} are hyperbolic [20, Theorem 6.6.6], i.e. tr(g) > 2.
Denote by PO X the set of all periodic orbits of the flow (ϕ t ) t∈R . We define a mapping ς : PO X → C Γ (2.2) as follows. Take a periodic orbit c of the flow, any point x on c, and let T > 0 be the prime period for x. Then ϕ T (x) = x, and the definition of the flow implies that there are g ∈ PSL(2, R) and γ ∈ Γ such that x = Π Γ (g) and γ = ga T g −1 , due to Γga T = xa T = ϕ T (x) = x = Γg; note that γ = e, since otherwise a T = e so that T = 0. Then put ς(c) = {γ} Γ . Lemma 2.3. Suppose that all elements in Γ\{e} are hyperbolic. Then the mapping ς defined by (2.2) is a bijection between the periodic orbits PO X of the flow (ϕ t ) t∈R and the collection of all conjugacy classes of primitive elements C Γ in Γ \ {e}.
Anosov closing lemma, connecting lemma
The next two results are illustrated in Figure 1 (a). For proofs, see [12,13].
Lemma 2.4 (Anosov closing lemma I). Suppose that ε ∈ (0, 1 4 ), x ∈ X, T ≥ 1, and ϕ T (x) ∈ P ε (x). If ϕ T (x) = (u, s) x ∈ P ε (x), in the notation from Definition 2.1, then there are x ′ = (σ, η) x ∈ P 2ε (x) and T ′ ∈ R so that
ϕ T ′ (x ′ ) = x ′ and d X (ϕ t (x), ϕ t (x ′ )) < 2|u| + |η| < 4ε for all t ∈ [0, T ]. Furthermore, T ′ − T 2 − ln(1 + us) < 5|us|e −T and |σ| < 2|u|e −T , |η| < 3|s| 2 .
Remark 2.1. According to the proof of the Anosov closing lemma I in [12,Theorem 2.3], x = Γg, g ∈ PSL(2, R) and ζ ∈ Γ is such that ga T = ζgc u b s then gc σ b η a T ′ = ζgc σ b η . This yields that the periodic orbit of x ′ = Γgc σ b η corresponds to the conjugacy class {ζ} Γ , provided that all elements in Γ\{e} are hyperbolic.
Using the other version of Poincaré sections, we have a respective statement for the Anosov closing lemma which will be also useful afterwards. Lemma 2.5 (Anosov closing lemma II). Suppose that ε ∈ (0, 1 4 ), x ∈ X, T ≥ 1, and ϕ T (x) ∈ P ′ ε (x). If ϕ T (x) = (s, u) ′ x ∈ P ′ ε (x), in the notation from Definition 2.1, then there are
x ′ = (η, σ) ′ x ∈ P ′ 2ε (x) and T ′ ∈ R so that ϕ T ′ (x) = x and d X (ϕ t (x), ϕ t (x ′ )) ≤ 2|u| + |η| < 4ε for all t ∈ [0, T ].
Furthermore,
T ′ − T 2 < 4|us|e −T and |σ| < 2|u|e −T , |η| ≤ 3|s| 2 .
Lemma 2.6 (Connecting lemma). Let x j ∈ X be T j -periodic point of the flow (ϕ t ) t∈R for j = 1, 2 and T 1 + T 2 ≥ 1 and let ε ∈ (0, 1 4 ).
If x 2 = (u 1 , s 1 ) x 1 ∈ P ε (x 1 ), then there are x ∈ X and T > 0 such that ϕ T (x) = x, d X (ϕ t (x), ϕ t (x 1 )) < 5ε for all t ∈ [0, T 1 ], (2.3) d X (ϕ t+T 1 (x), ϕ t (x 2 )) < 5ε for all t ∈ [0, T 2 ],(2.
4)
and
T − (T 1 + T 2 ) 2 − ln(1 + us) < 7|us|(e −T 1 + e −T 2 ). (2.5) Furthermore, if x i = Γg i for some g i ∈ PSL(2, R), then x = Γg 1 c ue −T 1 +σ b η , where σ, η ∈ R satisfy |σ| < 2|u|e −T 1 −T 2 , |η| < 3|s| 2 (2.6)
and the orbit of x corresponds to the conjugacy class {γ 1 γ 2 } Γ , where γ 1 , γ 2 ∈ Γ such that g 1 a T 1 = γ 1 g 1 , g 2 a T 2 = γ 2 g 2 , provided that all elements in Γ\{e} are hyperbolic.
See Figure 1 (b) for an illustration.
Proof. For the existence of a periodic point x and (2.3)-(2.6), see [13, Theorem 2.6]. For the last assertion, we can choose g 1 , g 2 ∈ PSL(2, R) such that x = Γg 1 , y = Γg 2 and g 2 = g 1 c u b s . According to the proof of the previous lemma in [13,Theorem 2.6
], if γ 1 , γ 2 ∈ Γ such that g 1 a T 1 = γ 1 g 1 , g 2 a T 2 = γ 2 g 2 then g 1 c ue −T 1 +σ b η a T = γg 1 c ue −T 1 +σ b η with γ = g 2 b −s a −T 1 a T 1 +T 2 b −s(1−e −T 2 ) c −u(1−e −T 1 ) a T 1 c −u g −1 1 = g 2 a T 2 b −se −T 2 b −s(1−e −T 2 ) c −u(1−e −T 1 ) c −ue −T 1 a T 1 g −1 1 = g 2 a T 2 b −s c −u g −1 1 γ 1 = γ 2 g 2 b −s c −u g −1 1 γ 1 = γ 2 γ 1 .
This implies that the orbit of x corresponds to the conjugacy class
{γ 2 γ 1 } Γ = {γ 1 γ 2 } Γ . x ϕT (x) x ′ T T ′ x 1 x 2 T2 T1 (a) (b) L θ τ (c)
Self-crossings
Recall that X = T 1 (Γ\H 2 ) denotes the unit tangent bundle of the factor Γ\H 2 ; see Figure 1 (c) for an illustration for the next result.
Lemma 2.7 (Self-crossings, [12]). Suppose that all elements of Γ \ {e} are hyperbolic and let τ ∈ R, L > 0, θ ∈ (0, π), and x ∈ X be given. The orbit of x under the geodesic flow (ϕ X t ) t∈R crosses itself in configuration space at the time τ , at the angle θ, and creates a loop of length L if and only if
either Γga τ +L = Γga τ d θ or Γg τ +L = Γga τ d −θ (2.7)
holds for any g ∈ PSL(2, R), Γg = Ξ(x). Furthermore,
e −L < cos 2 θ 2 .
(2.8)
Encounters and partner orbits
Definition 2.3 (Time reversal). The time reversal map T : X → X is defined by T (p, ξ) = (−p, ξ) for (p, ξ) ∈ X .
The respective time reversal map on X = Γ\PSL(2, R) is determined by
T (x) = Γgd π for x = Γg ∈ X, where d π ∈ PSL(2, R) is the equivalence class of the matrix D π = 0 1 −1 0 ∈ SL(2, R).
Using Lemma 2.9 below, we have ϕ t (T (x)) = T (ϕ −t (x)) for x ∈ X and t ∈ R.
(2.9)
Next, we recall the notions of orbit pairs and partner orbits. Roughly speaking, two periodic orbits are called an orbit pair if they are close enough to each other in configuration space, not for the whole time, since otherwise they would be identical, but they decompose to the same number of parts and any part of one orbit is close to some part of the other. The following is a rigorous definition of orbit pairs, which is recalled from [13].
T ′ ] : 0 = t 0 < · · · < t L = T and 0 = t ′ 0 < · · · < t ′ L = T ′ , and a permutation σ : {0, 1, . . . , L − 1} → {0, 1, . . . , L − 1} such that for each j ∈ {0, . . . , L − 1}, either d X (ϕ t+t j (x), ϕ t+t ′ σ(j) (x ′ )) < ε for all t ∈ [0, t j+1 − t j ] or d X ϕ t+t j (x), ϕ t−t ′ σ(j)+1 (T (x ′ )) < ε for all t ∈ [0, t j+1 − t j ]
holds for some x ∈ c and x ′ ∈ c ′ . Then c ′ is called an ε-partner orbit of c and vice versa.
Definition 2.5 (Encounter). Let ε > 0 and L ∈ Z, L ≥ 2 be given. We say that a periodic orbit c of the flow (ϕ t ) t∈R has an (L, ε)-encounter if there are
x 1 , . . . , x L ∈ c such that for each j ∈ {2, . . . , L}, either x j ∈ P ε (x 1 ) or T (x j ) ∈ P ε (x 1 ).
The point x 2 , . . . , x L are called piercing points. If either x j ∈ P ε (x 1 ) holds for all 2 = 1, . . . , L or T (x j ) ∈ P ε (x) holds for all j = 2, . . . , L then the encounter is called parallel encounter; otherwise it is called antiparallel encounter.
Auxiliary results
The next result is a decomposition of PSL(2, R).
Lemma 2.8 ( [12]). Let g = [G] ∈ PSL(2, R) for G = a b c d ∈ SL(2, R). (a) If a = 0, then g = c u b s a t for t = 2 ln |a|, s = ab, u = c a . (b) If d = 0, then g = b s c u a t for t = −2 ln |d|, s = b d , u = cd.
Lemma 2.9. The following relations hold for t ∈ R:
a t d π = d π a −t , b t d π = d π c −t , c t d π = d π b −t . (2.10) Proof. In SL(2, R) we calculate A t D π = e t/2 0 0 e −t/2 0 1 −1 0 = 0 e t/2 −e −t/2 0 = 0 1 −1 0 e −t/2 0 0 e t/2 = D π A −t
which upon projection yields the first one. The argument is analogous for the others.
Owing to the hyperbolicity, the flow (ϕ t ) t∈R is expansive, i.e., two orbits cannot stay too close together without being identical; see [3,Lemma 1.5]. For periodic orbits, we have the following property; see [12,Theorem 3.14] for a proof.
Lemma 2.10. Let X = Γ\PSL(2, R) be compact. Then there is ε * > 0 with the following property. If ε ∈ (0, ε * ) and if x 1 , x 2 ∈ X are periodic points of (ϕ X t ) t∈R having the periods T 1 , T 2 > 0 such that |T 1 − T 2 | ≤ √ 2ε and d X (ϕ X t (x 1 ), ϕ X t (x 2 )) < ε for all t ∈ [0, min{T 1 , T 2 }],
then T 1 = T 2 and the orbits of x 1 and x 2 under (ϕ X t ) t∈R are identical.
Periodic orbits with one single 2-antiparallel encounter
Let us first recall from [14] periodic orbits with one single 2-antiparallel encounter. It was shown that a given periodic orbit including one single 2-antiparallel encounter has a partner orbit. The action difference between the orbit pair is estimated with an exponentially small error bound. Periodic orbits having small-angle self-crossing are special cases of this phenomenon. The results in this section will be applied for the main results in Subsection 4.1.
Theorem 3.1. Suppose that X = Γ\PSL(2, R) is compact and let ε ∈ (0, σ 0 8 ). If a periodic orbit c of the flow (ϕ t ) t∈R on X with period T > 1 has a (2, ε)antiparallel encounter, then it has a partner. Furthermore, let x, y ∈ c,
T (y) = (u, s) x ∈ P ε (x) and ϕ T 1 (x) = y, 0 < T 1 < T . Then the partner is ε ′ -partner with ε ′ = ε + 2(|u − se −T 1 | + |s − ue T 1 −T |) < 8ε
and the action difference between the orbit pair satisfies
T ′ − T 2 − ln(1 + us)| < 12ε 2 (e −T 1 + e T 1 −T ), (3.1)
where T ′ is the period of the partner. If ε ∈ (0, ε * 18 ), then the partner orbit is unique.
Proof. For the existence of a T ′ -periodic point v, whose orbit under the flow (ϕ t ) t∈R is a ε ′ -partner orbit, see [14,Theorem 9]. It was shown that the action difference satisfies
T ′ − T 2 − ln 1 + (u − se −T 1 )(s − ue T 1 −T ) ≤ |(u − se −T 1 )(s − ue T 1 −T )|e −T ,
which implies (3.1). For the last assertion, it follows from (3.1) that |T − T ′ | < 30ε 2 . Suppose that there is another partner orbit which has the same property, i.e., it is also 8ε-close to the original one and its period called T ′′ satisfies |T ′′ − T | < 30ε 2 . Then these two partner orbits are ε * -close to each other for the whole time and their periods satisfy
|T ′′ − T ′ | ≤ |T ′′ − T | + |T ′ − T | ≤ 60ε 2 < √ 2ε * .
By Lemma 2.10, the partner orbits must coincide.
T 2 T 1 v x y ′v = Γga −T 2 c ue −T 2 +σ b η , where T 2 = T 1 − T, σ, η ∈ R satisfy |η −s| < 2s 2 |ũ| + 2|s|e −T and |σ| < 2|ũ|e −T (3.2) withs = u − se −T 1 andũ = s − ue −T 2 . ⋄
The next result is a new view of symbolic dynamics. Proof. Note that due to ϕ T 1 (x) = y and ϕ T 2 (y) = x, there are γ 1 , γ 2 ∈ Γ such that γ 1 = ga T 1 h −1 and γ 2 = ha T 2 g −1 as assumption. Since x = Γg is a T -periodic orbit, we have γ = ga T g −1 for some γ ∈ Γ, and by Lemma 2.3, the orbit through x (called c) corresponds to the conjugacy class {γ} Γ :
= h ′ = gc u b s . If γ 1 , γ 2 ∈ Γ such that γ 1 = ga T 1 h −1 and γ 2 = ha T 2 g −1 , where T 2 = T − T 1 ,γ 1 γ 2 = (ga T 1 h −1 )(ha T 2 g −1 ) = ga T g −1 = γ.
This means that the orbit c corresponds to the conjugacy class {γ} Γ = {γ 1 γ 2 } Γ . Next, let ζ ∈ Γ be such that
ζ =ḡa T b −s ′ c −u ′ (ḡ) −1 forḡ = gc u a −T 2 = h ′ b −s a T 2 .
According to the proof of Theorem 3.1 and Remark 2.1, the partner orbit corresponds to the conjugacy class {ζ} Γ . Now,
ζ = h ′ b −s a −T 2 a T b −(u−s −T 1 ) c −(s−ue −T 2 ) a T 2 c −u g −1 = h ′ a T 1 b −se −T 1 b −(u−s −T 1 ) c −(s−ue −T 2 ) c −ue −T 2 a T 2 g −1 = h ′ a T 1 b −u c −s a T 2 g −1 = γ −1 1 γ 2 ,
noting h ′ a T 1 = γ −1 1 g ′ owing to ga T 1 = γ 1 h. Therefore, the partner orbit c ′ corresponds to the conjugacy class {ζ} Γ = {γ −1 1 γ 2 } Γ , completing the proof.
Periodic orbits including 2 encounters responsible for the third order term
Let us first review orbit pairs responsible for the cubic contribution to K(τ ). Note that a sufficiently long periodic orbit has a huge number of self-encounters which may involve arbitrarily many orbit stretches. Heusler, Müller et al. [11,19] show that only orbit pairs differing in two 2-encounters or in one single 3-encounter are responsible for the third order. There are two ways of connections of orbit stretches forming two 2-encounters, namely serial and intertwined, whereas the two stretches of each encounter may be either close in phase space (depicted by nearly parallel arrows ✲ ✲ or ❍ ❍ • ✟ ✟ ✯ ), which is called parallel encounter, or almost mutually time-reversed (like in ✛ ✲ or ❍ ❍ ( ✟ ✟ ✯ ), which is called antiparallel encounter. In each way, therefore, there are three possibilities: both 2-encounters are parallel-encounters , one 2-parallel encounter and one 2-antiparallel encounter, and two 2-antiparallel encounters, i.e. there are totally six cases. Only three of them lead to (genuine) periodic orbits: two 2-antiparallel encounters serial, one parallel-encounter and one antiparallel encounters intertwined and two anti-parallel encounters intertwined, which are responsible in the cubic order to the form factor. The others form pseudo-periodic orbits and do not contribute to the spectral form factor.
In this section we only consider periodic orbits including two 2-antiparallel encounters contributing to the third term of the spectral form factor. The case of one 3-parallel encounter is rigorously done in [13,Section 3.3]. Orbits with a single 3-antiparallel encounter can be done analogously.
Throughout this section, we assume that the space Γ\PSL(2, R) is compact.
Periodic orbits including two 2-antiparallel encounters serial
In this subsection we consider periodic orbits having two 2-antiparallel encounters serial, which is so called antiparallel-antiparallel serial (aas for short) in [19]. Periodic orbits with either two small-angle self-crossings in configuration space ( ❍ ❍ ( ✟ ✟ ✯ and ❍ ❍ ( ✟ ✟ ✯ ) or one small-angle self-crossing and one anti-parallel avoided self-crossing ( ❍ ❍ ( ✟ ✟ ✯ and ✛ ✲ ), or two anti-parallel avoided self-crossings ( ✛ ✲ and ✛ ✲ ) in configuration space are special cases of aas.
Theorem 4.1. Let ε ∈ (0, ε 0 24 ) and let c be a T -periodic orbit of the flow (ϕ t ) t∈R on X having two (2, ε)-antiparallel encounters serial. More precisely, let x, y, z, w ∈ c and T 1 , T 2 , T 3 , T 4 > 0 be such that T 1 + T 2 + T 3 + T 4 = T , ϕ T 1 (x) = y, ϕ T 2 (y) = z, ϕ T 3 (z) = w and ϕ T 4 (w) = x, and T (z) = (u 1 , s 1 ) y ∈ P ε (y), T (w) = (u 2 , s 2 ) x ∈ P ε (x) satisfying
|u 1 | > 6εe −T 2 , |s 1 | > 30ε 3 + 13εe −T 1 + 5εe −T 2 + 3εe −T 3 . (4.1)
Then c has a 20ε-partner orbit which differs in both encounters of the original orbit and the action difference satisfies
T ′ − T 2 − ln(1 + u 1 s 1 )(1 + u 2 s 2 ) < ε 2 (21e −T 1 + 30e −T 2 + 12e −T 3 + 19e −T 4 ).
(4.2) In addition, if ε ∈ (0, ε * 40 ), then the partner is unique. Proof. The sketch of the proof is as follows; see Figure 3 for an illustration. First, we apply Theorem 3.1 for the left encounter to have one partner orbit, which is depicted by the dashed line in Figure 3 (b). Next, we show that the new orbit admits one 2-antiparallel encounter; see Figure 3 (c). Finally, we apply Theorem 3.1 again to get a partner orbit depicted by the dashed line in Figure 3 (d). Let x = Γg, y = Γh, z = Γk, w = Γl for some g, h, k, l ∈ PSL(2, R) and set g ′ = gd π , h ′ = hd π , k ′ = kd π , l ′ = ld π . By hypothesis, ϕ T 123 (x) = w and
T (w) = (s 2 , u 2 ) x ∈ P ε (x), where T 123 = T 1 + T 2 + T 3 . Due to Theorem 3.1, there arex = Γl ′ b −s 2 a −T 4 c σ 2 b η 2 = Γgc u 2 a −T 4 c σ 2 b η 2 ∈ X and T > 0 such that T − T 2 − ln(1 + u 2 s 2 ) ≤ 12ε 2 (e −T 4 + e −T 123 ). (4.3) Furthermore, d X (ϕ t (x), ϕ t (w)) < 7ε for t ∈ [0, T 4 ] (4.4) and d X (ϕ t (x), ϕ t (x ′ )) < 7ε for t ∈ [T 4 , T ]. (4.5)
Next we show that the orbit ofx possesses one 2-antiparallel encounter. We write
T (z) := ϕ T 4 +T 3 (x) = Γl ′ b −s 2 a −T 4 c σ 2 b η 2 a T 4 +T 3 = Γl ′ b −s 2 a T 3 c σ 2 e T 3 +T 4 b η 2 e −T 3 −T 4 = Γl ′ a T 3 b −s 2 e −T 3 c σ 2 e T 3 +T 4 b η 2 e −T 3 −T 4 = Γk ′ b −s 2 e −T 3 c σ 2 e T 3 +T 4 b η 2 e −T 3 −T 4 , using ϕ T 3 (T (w)) = T (z) and b s a t = a t b se −t , c u a t = a t c ue t for all u, s, t ∈ R. Thenz = Γkc s 2 e −T 3 b −σ 2 e T 3 +T 4 c −η 2 e −T 3 −T 4 ; (4.6)
recall (2.9). Now,
T (ŷ) := ϕ T 4 +T 3 +T 2 (x) = Γh ′ b −s 2 e −T 2 −T 3 c σ 2 e T 2 +T 3 +T 4 b η 2 e −T 2 −T 3 −T 4 = (Γkc s 1 b u 1 )b −s 2 e −T 2 −T 3 c σ 2 e T 2 +T 3 +T 4 b η 2 e −T 2 −T 3 −T 4 = (Γkc s 2 e −T 3 b −σ 2 e T 3 +T 4 c −η 2 e −T 3 −T 4 ) c η 2 e −T 3 −T 4 b σ 2 e T 3whereũ 1 = s 1 + η 2 e −T 3 −T 4 − s 2 e −T 3 + σ 2 e T 2 +T 3 +T 4 + 1 1 +ρ 1 × × (s 1 − s 2 e −T 3 )σ 2 e T 2 +T 3 +T 4 (u 1 − s 2 e −T 2 −T 3 ) −(s 1 − s 2 e −T 3 + σ 2 e T 2 +T 3 +T 4 )ρ 1 ,(4.7)s 1 = u 1 + σ 2 e T 3 +T 4 − s 2 e −T 2 −T 3 + η 2 e −T 2 −T 3 −T 4 +ρ 1 (2 +ρ 1 )η 2 e −T 2 −T 3 −T 4 + σ 2 e T 3 +T 4 + u 1 − s 2 e −T 2 −T 3 + (s 1 − s 2 e −T 3 )σ 2 e T 3 +T 4 (u 1 − s 2 e −T 2 −T 3 )(1 +ρ 1 ), (4.8) τ 1 = 2 ln(1 +ρ 1 ),(4.9)
with ρ 1 = σ 2 e T 2 +T 3 +T 4 (σ 2 e T 3 +T 4 +u 1 −s 2 e −T 2 −T 3 )+(s 1 −s 2 e −T 3 )σ 2 e T 3 +T 4 1+σ 2 e T 2 +T 3 +T 4 (u 1 −s 2 e −T 2 −T 3 ) , owing to Lemma 2.8 (a). A short calculation shows that |ũ 1 | < 2ε and |s 1 | < 2ε. This means that T (ỹ) = ϕ −τ 1 (T (ŷ)) = (ũ 1 ,s 1 )z ∈ P 2ε (z), whereỹ := ϕτ 1 (ŷ). Apply Theorem 3.1 to obtain v =za − T 2 cũ 1 e − T 2 +σ 1 b η 1 ∈ X and T ′ ∈ R such that
T ′ − T 2 −ln(1+ũ 1s1 ) ≤ 28ε 2 (e −T 1 −T 3 −T 4 +τ 1 +e T 1 +T 3 +T 4 −τ 1 − T ) < 7ε 2 e −T 4 +30ε 2 e −T 2
(4.10) and
|η 1 −ŝ 1 | < 2ŝ 2 1 |û 1 | + 2|ŝ 1 |e − T < 30ε 3 + 10εe − T , (4.11) |σ 1 | < 6εe − T , (4.12) whereû 1 =s 1 −ũ 1 e T 2 − T ,ŝ 1 =ũ 1 −s 1 e − T 2 . Furthermore, d X (ϕ t (v), ϕ t (ỹ)) < 13ε for t ∈ [0, T 2 ] and d X (ϕ t (v), ϕ t (T (z))) < 13ε for t ∈ [T 2 , T ].
This means that the orbit of v is 13ε-close to the orbit ofx. Recalling from (4.4) and (4.5) that the orbit ofx is 7ε-close to the orbit of x, we deduce that the orbit of v is 20ε-close to the original one. Next, in order to establish an estimate for the action difference, observe that by (4.7), (4.8),
|ũ 1 − s 1 | < 7εe −T 1 + 2εe −T 3 , |s 1 − u 1 | < εe −T 3 (4.13)
after a short calculation. This yields
| ln(1 +ũ 1s1 ) − ln(1 + u 1 s 1 )| ≤ 21ε 2 e −T 1 + 11ε 2 e −T 3
and hence
T ′ − T 2 − ln(1 + u 1 s 1 ) ≤ 21ε 2 e −T 1 + 30ε 2 e −T 2 + 11ε 2 e −T 3 + 7ε 2 e −T 4 ,(4.14)
using (4.10). The estimate (4.2) follows from (4.3) and (4.14).
Next we are going to show that the partner orbit is different from the original one. For, we find a point in the partner orbit which lies in the Poincaré section of y and is different from y and T (z). Letting
T 2 = T 2 −τ 1 , we have ϕ T − T 2 (z) =ỹ and ϕ T 2 (ỹ) =z. Recall v =za − T 2 cũ 1 e − T 2 +σ 1 b η 1 . Using (4.6) and Lemma 2.8 (a), we write v =za − T 2 cũ 1 e − T 2 +σ 1 b η 1 = Γkc s 2 e −T 3 b −σ 2 e T 3 +T 4 c −η 2 e −T 3 −T 4 a −T 2 +τ 1 cũ 1 e − T 2 +σ 1 b η 1 = (Γka −T 2 )c s 2 e −T 2 −T 3 b −σ 2 e T 2 +T 3 +T 4 c −η 2 e −T 2 −T 3 −T 4 +ũ 1 e −T 2 +σ 1 e −τ 1 b η 1 eτ1 aτ 1 = Γhc uv b sv a τv , where u v = s 2 e −T 2 −T 3 + −η 2 e −T 2 −T 3 −T 4 +ũ 1 e −T 2 + σ 1 e −τ 1 1 + ρ v , s v = (−σ 2 e T 2 +T 3 +T 4 + η 1 eτ 1 + η 1 eτ 1 ρ v )(1 + ρ v ), τ v = 2 ln(1 + ρ v ), here ρ v = (−σ 2 e T 2 +T 3 +T 4 )(−η 2 e −T 2 −T 3 −T 4 +ũ 2 e −T 2 + σ 1 e −τ 1 );
recallτ 1 from (4.9). A short calculation shows that
|u v | < 6εe −T 2 , |s v − η 1 | < 6εe −T 1 , (4.15)
which imply thatṽ = ϕ τv (z) = (u v , s v ) y ∈ P 6ε (y). By assumption, ε < σ 0 24 and T (z) = (u 1 , s 1 ) y with
|u 1 | ≥ 6εe −T 2 > |u v |.
As a consequence of Lemma 2.2, we getṽ = T (z). It remains to check that v = y. Recalling thatŝ 1 =ũ 1 −s 1 e − T 2 , it follows from (4.11) and (4.13) that
|η 1 − s 1 | ≤ |η 1 −ŝ 1 | + |ŝ 1 −ũ 1 | + |ũ 1 − s 1 | < 30ε 3 + 7εe −T 1 + 5εe −T 2 + 3εe −T 3 . (4.16)
Using (4.15), (4.16), together with the assumption (4.1) we deduce
|s v | ≥ |s 1 | − |s v − η 1 | − |η 1 − s 1 | > 0,
which showsṽ = y. Consequently, the orbit of v is different from the obit of x and its time reverse. For the last assertion, suppose that there is another 20ε-partner orbit which differs in both encounters of the original orbit. Then the two partner orbits are 40ε-close to each other for the whole time and the period difference |T ′′ −T ′ | < 50ε 2 , where T ′′ is the period of the new partner. Due to ε < ε * 40 , the two partners are ε * -close to each other for the whole time and |T ′′ − T ′ | < ε * ; so they must be identical by Lemma 2.10. The proof is complete.
Remark 4.1. (a)
The orbit ofx is also a partner orbit of the original one, which is depicted by the dotted line in Figure 3 (b). However, this partner orbit differs only in one encounter and this orbit pair only contribute to the second order term of the spectral form factor as a Sieber-Richter pair.
(b) If we first apply Theorem 3.1 for the other encounter, then we have another partner orbit for the original one. This orbit pair also contribute to the second order term of K(τ ). Then, using the same argument of the previous proof, we get the same partner orbit.
The next result provides a new view of symbolic dynamics of orbit pair in the preceding theorem.
Theorem 4.2. In the setting of Theorem 4.1, let γ 1 , γ 2 , γ 3 , γ 4 ∈ Γ be such that ga T 1 = γ 1 h, ha T 2 = γ 2 k, ka T 3 = γ 3 l, la T 4 = γ 4 g. Then the original orbit and the partner orbit correspond to the conjugacy classes {γ 1 γ 2 γ 3 γ 4 } Γ and
{γ −1 1 γ 4 γ −1 3 γ 2 } Γ , respectively. Proof. First observe that a T 1 = g −1 γ 1 h, a T 2 = h −1 γ 2 k, a T 3 = k −1 γ 3 l, a T 4 = l −1 γ 4 g leads to ga T = ga T 1 a T 2 a T 3 a T 4 = g(g −1 γ 1 h)(h −1 γ 2 k)(k −1 γ 3 l)(l −1 γ 4 g) = γ 1 γ 2 γ 3 γ 4 g.
This yields the orbit of x = Γg under the flow (ϕ t ) t∈R , which is the original orbit, corresponds to the conjugacy class {γ 1 γ 2 γ 3 γ 4 } Γ by the definition of the mapping ς in (2.2).
Next, due to ga T 1 +T 2 +T 3 = γ 1 γ 2 γ 3 l and la T 4 = γ 4 g, the orbit ofx corresponds to the conjugacy class
{ζ} Γ = {(γ 1 γ 2 γ 3 ) −1 γ 4 } Γ = {γ −1 3 γ −1 2 γ −1 1 γ 4 } Γ = {γ −1 2 γ −1 1 γ 4 γ −1 3 } Γ .
by Theorem 3.2. Similarly, the orbit of v corresponds to the conjugacy class
{ξ} Γ = {(γ −1 2 ) −1 γ −1 1 γ 4 γ −1 3 } Γ = {γ 2 γ −1 1 γ 4 γ −1 3 } Γ = {γ −1 1 γ 4 γ −1 3 γ 2 } Γ ,
as was to be shown.
The following corollary considers periodic orbits of the geodesic flow having two small-angle self-crossings.
Corollary 4.1. Suppose that all elements in Γ \ {e} are hyperbolic. Suppose that T -periodic orbit of the geodesic flow (ϕ X t ) t∈R on X = T 1 (Γ\H 2 ) crosses itself in configuration space at a time T 1 , at an angle θ 1 , creates a loop of length T 2 and then crosses itself again after a time T 3 at an angle θ 2 , and creates another loop of length T 4 with T 1 + T 2 + T 3 + T 4 = T . If 0 < φ < 1 3 for φ = max{π − θ 1 , π − θ 2 }, then it has a 36| sin(φ/2)|-partner orbit; the period of the partner orbit denoted by T ′ satisfies
T ′ − T 2 − ln 1 + sin 2 (φ 1 /2) 1 + sin 2 (φ 2 /2) < sin 2 (φ/2)(21e −T 1 + 31e −T 2 + 13e −T 3 + 19e −T 4 ). (4.17)
Furthermore, if Γ\H 2 is compact and φ < ε * 20 then the partner is unique. Proof. Let x ∈ X = T 1 (Γ\H 2 ) be a T -periodic point of the flow (ϕ X t ) t∈R and let
ϕ X T 1 (x) = y, ϕ X T 2 (y) = z, ϕ X T 3 (z) = w, ϕ X T 4 (w) = x.(4.18)
Let φ 1 = π − θ 1 , φ 2 = π − θ 2 , φ = max{φ 1 , φ 2 } and assume that |φ| < 1 3 . Then in particular
sin φ i 2 ≤ |φ i | 2 < 1 6 (4.19)
holds for i = 1, 2. Set x = Ξ(x), y = Ξ(y), z = Ξ(z), and w = Ξ(w); recall the isometry Ξ : T 1 (Γ\H 2 ) → Γ\PSL(2, R). It follows from Theorem 2.7 that either Γh = Γkd θ 1 or Γh = Γkd −θ 1 and either Γl = Γgd θ 2 or Γl = Γgd −θ 2 with some g, h, k, l ∈ PSL(2, R) such that Γg = x, Γh = y, Γk = z and Γl = w. We only consider Γh = Γkd θ 1 and Γl = Γgd θ 2 , the other cases are similar. Define z ′ = T (z) and w ′ = T (w); recall the notation T from Definition 2.3. Then z ′ = Γhd π−θ 1 = Γhd φ 1 and w ′ = Γgd π−θ 2 = Γgd φ 2 .
For i = 1, 2, apply Lemma 2.8 (a) to write
d φ i = c u i b s i a τ i ,(4.20)
where
τ i = 2 ln(cos(φ i /2)), u i = tan(φ/2), s i = − sin(φ i /2) cos(φ i /2). (4.21)
Owing to (4.19), we have
cos φ i 2 > 5 6 . (4.22)
Define ε = 6 5 | sin(φ/2). Observe that
|u i | = | tan(φ i /2)| ≤ 6 5 | sin(φ i /2)| ≤ ε, |s i | = | sin(φ i /2) cos(φ i /2)| ≤ | sin(φ i /2)| < ε,(4.
23) and
|τ
i | = | ln(1 − sin 2 (φ i /2))| ≤ 2 sin 2 (φ i /2) ≤ 1 2 ε 2 ,
due to | ln(1 + z)| ≤ 2|z| for |z| ≤ 1/2. Denotez = ϕ τ 1 (z) andw = ϕ τ 2 (w). This leads to
T (z) = ϕ −τ 1 (z ′ ) = Γk ′ a −τ 1 = Γhc u 1 b s 1 = (u 1 , s 1 ) y ∈ P ε (y), T (w) = ϕ −τ 2 (w ′ ) = Γl ′ a −τ 2 = Γgc u 2 b s 2 = (u 2 , s 2 ) x ∈ P ε (x),
using (2.9). Define T 2 = T 2 +τ 1 ,T 3 = T 3 −τ 1 +τ 2 , T 4 = T 4 −τ 2 , T 1 = T 1 . Then
T 1 + T 2 + T 3 + T 4 = T , ϕ T 1 (x) = y, ϕ T 2 (y) =z, ϕ T 3 (z) =w, and ϕ T 4 (w) = x.
We now apply Theorem 3.1 with ε = 6 5 | sin(φ/2)| to have a partner orbit, which is 33| sin(φ/2)|-close to the original one and has period T ′ satisfying
T ′ − T 2 − ln(1 + sin 2 (φ 1 /2))(1 + sin 2 (φ 2 /2)) < ε 2 (21e −T 1 + 30e − T 2 + 12e − T 3 + 19e − T 4 ) < ε 2 (21e −T 1 + 31e −T 2 + 13e −T 3 + 19e −T 4 ),
which is (4.17).
For the last assertion, if φ < ε * 12 , then ε = 6 5 | sin(φ/2)| < ε * 20 and whence the partner orbit is unique according to Theorem 4.1; see Figure 4 for an illustration. Figure 4: A periodic orbit with 2 small-angle-self-crossings has a partner orbit which has 2 avoided crossings.
T 4 T 1 T 3 T 2 φ 1 φ 2
Remark 4.2. The existence of the partner orbit in the preceding theorem does not need the condition that the space is compact. In fact, according [12,Theorem 3.11] and the proof of Theorem 4.1, the partner orbit has a smaller period T ′ < T .
Periodic orbits including two 2-parallel encounters intertwined
In this subsection we consider periodic orbits with two 2-parallel encounters intertwined, which is so called parallel-parallel intertwined (ppi for short) in [19].
Theorem 4.3. Let ε ∈ (0, σ 0 20 ).
Suppose that a T -periodic orbit c of the flow (ϕ t ) t∈R on X with period T has two (2, ε)-parallel encounters intertwined. For instance, suppose that there are x, y, z, w ∈ c and T 1 , T 2 , T 3 , T 4 > 0 such that
T 1 + T 2 + T 3 + T 4 = T , ϕ T 1 (x) = y, ϕ T 2 (y) = z, ϕ T 3 (z) = w and ϕ T 4 (w) = x with z = (u 1 , s 1 ) x ∈ P ε (x), w = (u 2 , s 2 ) y ∈ P ε (y) satisfying |u 2 | > 9εe −T 4 , |s 2 | > 72ε 3 + 5εe −T 1 + 2εe −T 3 . (4.24)
Then c has a 19ε-partner orbit of period T ′ which differs in both encounters and the action difference satisfies
T ′ − T 2 − ln(1 + s 1 u 1 )(1 + s 2 u 2 )| ≤ 54ε 4 + 25ε 2 (e −T 1 + e −T 2 + e −T 3 + e −T 4 ).
(4.25) If ε ∈ (0, ε * 38 ), then the partner orbit is unique.
Proof. The construction of a partner orbit is summarized as follows. We first apply the Anosov closing lemmas for the first encounter to obtain two shorter periodic orbits, which are expressed by dashed line and dotted line in Figure 5 (b). We next show that the obtained orbits create a pseudo-orbit (see Figure 5 (c)). Finally we use the connecting lemma to get a new periodic orbit, which is illustrated by the dashed line in Figure 5 (d). Figure 5: Reconnection to give the partner orbit for orbit with two parallel encounters intertwined.
T 4 T 2 T 1 T 3 T ′ 12 T ′ 34 T ′ 12 T ′ 34 x w y zw ỹ x z (a) (c) (b) (d) v
Denote T 12 = T 1 + T 2 and T 34 = T 3 + T 4 . By assumption, ϕ T 12 (x) = z = (u 1 , s 1 ) x ∈ P ε (x). According to the Anosov closing lemma I, there exist x = Γhb ηx c σx ∈ X and T ′ 12 ∈ R such that ϕ T ′ 12 (x) =x,
T ′ 12 − T 12 2 − ln(1 + u 1 s 1 ) ≤ 5ε 2 e −T 1 −T 2 (4.26) and d X (ϕ t (x), ϕ t (x)) < 4ε, for all t ∈ [0, T 1 + T 2 ]. (4.27)
The orbit ofx is depicted by the dashed line in Figure 5
(b). Observe that ϕ T 34 (z) = x = (−s 1 , −u 1 ) ′ z ∈ P ′ ε (z)
. Apply the Anosov closing lemma II, there existz = Γlb ηz c σz ∈ X and T ′ 34 ∈ R such that ϕ T ′ 34 (z) =z,
T ′ 34 − T 34 2 ≤ 4ε 2 e −T 3 −T 4 (4.28) and d X (ϕ t (z), ϕ t (z)) < 4ε, for all t ∈ [0, T 3 + T 4 ]. (4.29)
The orbit ofz is depicted by the dotted line in Figure 5 (b). Next we show that two orbits ofx andz form a pseudo-orbit, and hence it is possible to connect them to obtain a longer periodic orbit. Using Lemma 2.8 (b), we write
ϕ T 3 (z) = Γlb ηz c σz a T 3 = Γla T 3 b ηze −T 3 c σze T 3 = Γkb ηze −T 3 c σze T 3 = Γhc u 2 b s 2 b ηz e −T 3 c σze T 3 = (Γhc σxe T 1 b ηxe −T 1 )b −ηxe −T 1 c u 2 −σxe T 1 b s 2 +ηze −T 3 c σze T 3 = (Γhc σxe T 1 b ηxe −T 1 )b s c u a τ , where s = η z e −T 1 + s 2 + η z e −T 3 1 + (u 2 − σ x e T 1 )(s 2 + η z e −T 3 ) , u = u 2 − σ x e T 1 + σ z e T 3 + (u 2 − σ x e T 1 )(s 2 + η z e −T 3 )σ z e T 3 1 + (u 2 − σ x e T 1 )(s 2 + η z e −T 3 ) , τ = −2 ln 1 + (u 2 − σ x e T 1 )(s 2 + η z e −T 3 ) . (4.30)
A short calculation shows that |u| < 3ε, |s| < 3ε. If we set
y := ϕ T 1 (x) = Γgc σx b ηx a T 1 = Γga T 1 c σxe T 1 b ηxe −T 1 = Γhc σxe T 1 b ηxe −T 1
andw := ϕ T 3 −τ (z), thenỹ = (−u, −s)w ∈ P 3ε (w). Apply the connecting lemma I to obtain a T ′ -periodic point v =wc −ue −T ′ 12 +σ b η satisfying
T ′ − (T ′ 12 + T ′ 34 ) 2 − ln(1 + us) < 63ε 2 (e −T ′ 12 + e −T ′ 34 ) (4.31)
and
|η + s| < 2s 2 |u| + 2|s|e −T 1 −T 2 < 54ε 3 + 6εe −T 1 −T 2 . (4.32) Furthermore, d X (ϕ t (v), ϕ t (z)) < 15ε for t ∈ [0, T ′ 3,4 ] and d X (ϕ t+T ′ 34 (v), ϕ t (w)) < 15ε for t ∈ [0, T ′ 1,2 ]
. This yields that the orbit of v is 15ε-close to the orbits ofx andz. Together with (4.27) and (4.29), this implies that the orbit of v is 19ε-close to the orbit of x.
Next, we estimate the action difference. According to the Anosov closing lemmas,
|η x | < 3ε 2 , |η z | < 3ε 2 , |σ x | < 2εe −T 1 −T 2 , |σ z | < 2εe −T 3 −T 4 .
Observe that |u−u 2 | < 9ε 3 +2εe −T 2 +2εe −T 4 and |s−s 2 | < 3ε 3 +2εe −T 1 +2εe −T 3 (4.33)
imply | ln(1 + us) − ln(1 + u 2 s 2 )| < 54ε 4 + 18ε 2 e −T 1 + 6ε 2 e −T 2 + 18ε 2 e −T 3 + 6ε 2 e −T 4 , and hence
T ′ − T ′ 12 − T ′ 34 2 − ln(1 + u 2 s 2 ) ≤ 54ε 4 + 24ε 2 e −T 1 + 24ε 2 e −T 2 + 24ε 2 e −T 3 + 24ε 2 e −T 4 ,(4.v =wc ue −T ′ 34 +σ b η = (Γhc σxe T 1 b ηxe −T 1 )b s c u c ue −T ′ 34 +σ b η = Γhc σxe T 1 b ηxe −T 1 +s c u+ue −T ′ 34 +σ b η = Γhc uv b sv a τv = Γhc u 2 b s 2 b ηz e −T 3 c σze T 3 a −τ c ue −T ′ 34 +σ b η = Γhc u 2 b s 2 +ηze −T 3 c σze T 3 +ue −T ′ 34 +τ +σ b ηe −τ a −τ = Γhc uv b sv a τv , where u v = u 2 + σ z e T 3 + ue −T ′ 34 +τ + σ 1 + (s 2 + η z e −T 3 )(σ z e T 3 + ue −T ′ 34 +τ + σ) , s v = (s 2 + η z e −T 3 + ηe −τ + ηe −τ ρ v )(1 + ρ v ), τ v = 2 ln 1 + ρ v − τ,
recalling τ from (4.30); here
ρ v = (s 2 + η z e −T 3 )(σ z e T 3 + ue −T ′ 34 +τ + σ).
A short calculation shows that |s v | < 5ε and |u v | < 5ε. Consequently, v := ϕ −τv (v) = (u v , s v ) y ∈ P 5ε (y). We need to check thatṽ = y andṽ = w.
For, observe that
|u v − u 2 | < 9εe −T 4 , |s v − s 2 − η| < 15ε 3 .
Furthermore, it follows from (4.32) and (4.33) that
|η + s 2 | ≤ |η + s| + |s − s 2 | < 57ε 3 + 5εe −T 1 + 2εe −T 3 .
Using assumption (4.24), we derive
|u v | > |u 2 | − |u v − u 2 | > |u 2 | − 9εe −T 4 > 0 |s v − s 2 | > |s 2 | − |s v − s 2 − η| − |η + s 2 | > 0.
This means u v = 0 and s v = s 2 . Owing to ε < σ 0 20 , we getṽ = y,ṽ = w and hence the orbit of v is different from the orbit of x.
For the last assertion, recall that the orbit of v is 19ε-close to the original orbit. The uniqueness of partner orbit can be done analogously to the proof of Theorem 4.1. The term 33ε 4 appears when we change the coordinates (u, s) to (u 2 , s 2 ) (see (4.33)) and it cannot avoid.
(b) Condition (4.24) can be replaced by
|u 1 | > 9εe −T 3 , |s 1 | > 72ε 3 + 5εe −T 4 + 2εe −T 2 . (4.35)
Theorem 4.4. In the setting of Theorem 4.3, suppose that x = Γg, y = Γh, z = Γk, w = Γl for some g, h, k, l ∈ PSL(2, R). If γ 1 , γ 2 , γ 3 , γ 4 ∈ Γ satisfy ga T 1 = γ 1 h, ha T 2 = γ 2 k, ka T 3 = γ 3 l, la T 4 = γ 4 g, then the original orbit corresponds to the conjugacy class {γ 1 γ 2 γ 3 γ 4 } and the partner orbit corresponds to the conjugacy class {γ 2 γ 1 γ 4 γ 3 }.
Proof. That the original orbit corresponds to the conjugacy class {γ 1 γ 2 γ 3 γ 4 } can be done analogously to Theorem 4.2. For the last assertion, we can choose g, h, k, l ∈ PSL(2, R) such that k = gc u 1 b s 1 and l = hc u 2 b s 2 . Then ϕ T 1 +T 2 (Γg) = Γgc u 1 b s 1 and ga T 1 +T 2 = γ 1 ha T 2 = γ 1 γ 2 k = γ 1 γ 2 gc u 1 b s 1 . Similarly, ϕ T 3 +T 4 (Γk) = Γkb −s 1 c −u 1 and ka T 3 +T 4 = γ 3 la T 4 = γ 3 γ 4 g = γ 3 γ 4 kb −s 1 c −u 1 . By the Remark 2.1, the orbit ofx corresponds to the conjugacy class {γ 1 γ 2 } Γ = {γ 2 γ 1 } Γ and the orbit ofz corresponds to the class {γ 3 γ 4 } Γ = {γ 4 γ 3 } Γ . The partner orbit corresponds to the conjugacy class
{γ 2 γ 1 γ 4 γ 3 } Γ ;
recall the last assertion of Lemma 2.6.
Periodic orbits including one 2-antiparallel encounter and one 2-parallel encounter intertwined
This subsection deals with periodic orbits with one 2-antiparallel encounter and one 2-parallel encounter intertwined, which is so called antiparallel-parallel intertwined (api for short) in [19]. Theorem 4.5. Let ε ∈ (0, σ 0 20 ). Suppose that a periodic orbit c of the flow (ϕ t ) t∈R on X with period T > 1 has one (2, ε)-parallel encounter and one (2, ε)-antiparallel encounter intertwined. More precisely, suppose that there are x, y, z, w ∈ c and T 1 , T 2 , T 3 , T 4 > 0 be such that T 1 + T 2 + T 3 + T 4 = T , ϕ T 1 (x) = y, ϕ T 2 (y) = z, ϕ T 3 (z) = w and ϕ T 4 (w) = x and z = (u 1 , s 1 ) x ∈ P ε (x), T (w) = (u 2 , s 2 ) y ∈ P ε (y) and
|u 1 | > 9εe −T 3 , |s 1 | > 72ε 3 + 5εe −T 4 + 2εe −T 2 .
(4.36)
Then c has a 19ε-partner orbit of period T ′ which differs in both encounters and the action difference satisfies T ′ − T 2 − ln(1 + s 1 u 1 )(1 + s 2 u 2 )| ≤ 54ε 4 + 25ε 2 (e −T 1 + e −T 3 + e −T 4 ).
If ε ∈ (0, ε * 38 ), then the partner orbit is unique.
Proof. The proof of this theorem is similar to that of Theorem 4.3. Note that after obtaining two shorter periodic orbits, the dashed and dotted lines, we consider the time reversal of the former one ( Figure 6 (c)), and apply the connecting lemma to have a partner depicted by dashed line in Figure 6 (d). The assumption of compactness is unnecessary for the existence of partner orbits in all cases above. However, we need it for the uniqueness.
(b) The conditions expressed by the coordinates of the piercing points (4.1), (4.24), (4.36) guarantee that the encounter stretches are separated by non-vanishing loops and whence the partner orbit and the original orbit do not coincide. A similar condition is needed in physics literature; see [16,19].
(c) The approach in the present paper can be applied to consider periodic orbits responsible for all order in τ to the spectral form factor K(τ ).
Figure 1 :
1(
Definition 2. 4 (
4Orbit pair/Partner orbit). Let ε > 0 be given. Two given T -periodic orbit c and T ′ -periodic orbit c ′ of the flow (ϕ t ) t∈R are called an ε-orbit pair if there are L ≥ 2, L ∈ Z and two decompositions of [0, T ] and [0,
Figure 2 :
2Periodic orbit with a single 2-antiparallel encounter has a partner orbit.Remark 3.1. According to the proofs of[14, Theorem 9] and the Anosov closing lemma I, we have
Theorem 3. 2 (
2Symbolic dynamics). In the setting of Theorem 3.1, let g, h ∈ PSL(2, R) be such that x = Γg, y = Γh and hd π
then the original orbit corresponds to the conjugacy class {γ 1 γ 2 } Γ and the partner orbit corresponds to the conjugacy class {γ −1 1 γ 2 } Γ .
Figure 3 :
3Construction of partner orbit for a given periodic orbit with two 2-antiparallel encounters serial.
+T 4 c
4−s 2 e −T 3 +s 1 b u 1 −s 2 e −T 2 −T 3 c σ 2 e T 2 +T 3 +T 4 b η 2 e −T 2 −T 3 −T 4 = (Γkc s 2 e −T 3 b −σ 2 e T 3 +T 4 c −η 2 e −T 3 −T 4 )cũ 1 bs 1 aτ 1 ,
Remark 4 . 3 .
43(a) The period of the partner orbit can be explicitly computed according to the proof of Anosov closing lemmas and the connecting lemma.
Figure 6 :
6Reconnecting encounter stretches to form a partner orbit in api case.
Remark 4. 4 .
4Analogously toTheorem 4.4, in the setting of the previous theorem, if the original orbit corresponds to the conjugacy class {γ 1 γ 2 γ 3 γ 4 } Γ , then the partner orbit corresponds to the conjugacy class{(γ 2 γ 1 ) −1 γ 4 γ 3 } Γ = {γ −1 1 γ −1 2 γ 4 γ 3 } Γ .
Next, we are going to show that the orbit of v is different from the orbit of x. Analogously to the proof of Theorem 4.1, we write34)
owing to (4.31). The estimate (4.25) follows from (4.26), (4.28) and (4.34).
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| []
|
[
"Wave-Style Token Machines and Quantum Lambda Calculi",
"Wave-Style Token Machines and Quantum Lambda Calculi"
]
| [
"Ugo Dal \nUniversità di Bologna\nItaly & INRIA\n",
"Lago \nUniversità di Bologna\nItaly & INRIA\n",
"Margherita Zorzi \nUniversità di Verona\nItaly\n"
]
| [
"Università di Bologna\nItaly & INRIA",
"Università di Bologna\nItaly & INRIA",
"Università di Verona\nItaly"
]
| [
"Third International Workshop on Linearity 2014 (LINEARITY'14) EPTCS"
]
| Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum λ -calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation. | 10.4204/eptcs.176.6 | [
"https://arxiv.org/pdf/1502.04774v1.pdf"
]
| 15,353,622 | 1502.04774 | 8becd7a69676af6e29a8d915cc8f0524262c9af7 |
Wave-Style Token Machines and Quantum Lambda Calculi
2015
Ugo Dal
Università di Bologna
Italy & INRIA
Lago
Università di Bologna
Italy & INRIA
Margherita Zorzi
Università di Verona
Italy
Wave-Style Token Machines and Quantum Lambda Calculi
Third International Workshop on Linearity 2014 (LINEARITY'14) EPTCS
176201510.4204/EPTCS.176.6
Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum λ -calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.
Introduction
One of the strongest trends in computer science is the (relatively recent) interest in exploiting new computing paradigms which go beyond the usual, classical one. Among these paradigms, quantum computing plays an important role. In particular, the quantum paradigm is having a deep impact on the notion of a computationally (in)tractable problem [18].
Even if quantum computing has catalysed the interest of a quite large scientific community, several theoretical aspects are still unexplored. As an example, the definition of a robust theoretical framework for quantum programming is nowadays still a challenge. A number of (paradigmatic) calculi for quantum computing have been introduced in the last ten years. Among them, some functional calculi, typed and untyped, have been proposed [5,6,7,16,19,22], but we are still at a stage where it is not clear whether one calculus could be considered canonical. Since quantum data have to undergo restrictions such as nocloning and no-erasing, it is not surprising that in most of the cited quantum calculi the use of resources is controlled. Linear logic therefore provides an ideal framework for quantum data treatment, since weakening and contraction (to which linear logic gives a special status) precisely correspond to erasing and copying via the Curry-Howard correspondence. But linear logic also offers another tool which has not been widely exploited in the quantum setting: its mathematical model in terms of operator algebras, i.e. the Geometry of Interaction (GoI in the following). Indeed, the latter provides a dynamical interpretation and a semantic account of the cut-elimination procedure as a flow of information circulating into a net structure. This idea can be formulated both as an algebra of bounded operators on a infinite-dimensional Hilbert space [11] or as a token-based machine [12,14]. On the one hand, the Hilbert space on top of which the first formulation of GoI is given is precisely the canonical state space of a quantum Turing machine [2]. On the other hand, the definition of a token machine provides a mathematically simpler setting, which has already found a role in this context [4,13].
In this paper, we show that token machines are also a model of a linear quantum λ -calculus with implicit quantum bits (qubits), called QΛ and defined along the lines of van Tonder's λ q [19]. This allows us to give an operational semantics to QΛ which renders the quantum nature of QΛ explicit: type derivations become quantum circuits on the set of gates occurring in the underlying λ -term. This frees us from the burden of having to define the operational semantics of quantum calculi in reduction style, which is known to be technically challenging in a similar setting [19]. On the other hand, the power of β -style axioms is retained in the form of an equational theory for which our operational semantics can be proved sound. Technically, the design of our token machine for QΛ, called IAM QΛ , is arguably more challenging than the one of classical token machines. Indeed, the principles of quantum computing, and the so-called entanglement in particular, force us to go towards wave-style machines, i.e., to machines where more than one particle can travel inside the program at the same time. Moreover, the possibly many tokens at hand are subject to synchronisation points, each one corresponding to unitary operators of arity greater than 1. This means that IAM QΛ , in principle, could suffer from deadlocks, let alone the possibility of non-termination. We here prove that these pathological situations can never happen. In the present paper we also establish a soundness theorem: we state and prove that the semantics induced by the token machine IAM QΛ is sound with respect to QΛ's equational theory, i.e. it is invariant with respect to term equivalence. The proof, which we only sketch and which can anyway be found in [8], is not trivial, since our notion of term has to deal with quantum superposition [15] and is thus non-standard. Finally, it is mandatory to recall that, even if the possibility of observing quantum data is a useful and expressive programming tool, considering a measurement-free calculus is a theoretically well-founded choice, since measurements can always been postponed [15]. Thus, this is not a limitation when one addresses computability issues.
The calculus QΛ and its token machine IAM QΛ are introduced in Section 2 and Section 3, respectively. Main results about IAM QΛ are in Section 4. An extended version of this paper with more details, proofs and a gentle introduction to quantum computing is available [8].
The Calculus QΛ
An essential property of quantum programs is that quantum data, i.e. quantum bits, should always be uniquely referenced. This restriction follows from the well-known no-cloning and no-erasing properties of quantum physics, which state that a quantum bit cannot in general be duplicated nor canceled [15]. Syntactically, one captures this restriction by means of linearity: if every abstraction λ x.M is such that there is exactly one free occurrence of x in M, then the substitution triggered by firing any redex is neither copying nor erasing and, as a consequence, coherent with the just stated principles. In this Section, we introduce a quantum linear λ -calculus in the style of van Tonder's λ q [19] and give an equational theory for it. This is the main object of study of this paper, and is the calculus for which we will give a wave-style token machine in the coming sections.
The Language of Terms
Let us fix a finite set U of unitary operators, each on a finite-dimensional Hilbert space C 2 n , where n can be arbitrary. To each such U ∈ U we associate a symbol U and call n the arity of U. The syntactic categories of patterns, bit constants, constants and terms are defined by the following grammar: where n ranges over N, b ranges over {0, 1}, and x ranges over a denumerable, totally ordered set of variables V. We always assume that the natural numbers occurring next to bits in any term M are pairwise distinct. This condition, by the way, is preserved by substitution when the substituted variable occurs (free) exactly once. Whenever this does not cause ambiguity, we elide labels and simply write |b for a bit constant. Notice that pairs are formed via the binary operator ⊗.
P ::= x | x, y ;:= x | C | M ⊗ N | MN | λ P.M. terms (a v ) x : A x : A (a q0 ) · |0 : B (a q1 ) · |1 : B (a U ) · U : B n B n Γ, x : A M : B (I 1 ) Γ λ x.M : A B Γ, x : A, y : B M : C (I 2 ) Γ λ x, y .M : (A ⊗ B) C Γ M : A B ∆ N : A (E ) Γ, ∆ MN : B Γ M : A ∆ N : B (I ⊗ ) Γ, ∆ M ⊗ N : A ⊗ B
We will sometime write |b 1 b 2 . . . b k for |b 1 ⊗ |b 2 ⊗ . . . ⊗ |b k (where b 1 , . . . , b n ∈ {0, 1}). We work modulo variable renaming; in other words, terms are equivalence classes modulo α-conversion. Substitution up to α-equivalence is defined in the usual way. Observe that the terms of QΛ are the ones of a λ -calculus with pairs (which are accessed by pattern-matching) endowed with constants for bits and unitary operators. We don't consider measurements here, and discuss the possibility of extending the language of terms in sections 5 and 6.
Judgements and Typing Rules.
We want terms to be non-duplicable and non-erasable by construction and, as a consequence, we adopt a linear type discipline. Formally, the set of types is defined as follows
A ::= B | A B | A ⊗ B,
where B is the ground type of bits. We write B n for the n-fold tensor product n times B ⊗ . . . ⊗ B. Judgements and environments are defined as follows:
• A linear environment Γ is a (possibly empty) finite set of assignments in the form x : A. We impose that in a linear environment, each variable x occurs at most once; • If Γ and ∆ are two linear environments assigning types to distinct sets of variables, Γ, ∆ denotes their union; • A judgement is an expression Γ M : A, where Γ is a linear environment, M is a term, and A is a type in QΛ. Typing rules are in Figure 1. Observe that contexts are treated multiplicatively and, as a consequence, variables always appear exactly once in terms. In other words, a strictly linear type discipline is enforced.
Example 1 (EPR States) Consider the term M EPR = λ x, y .CNOT(Hx ⊗ y). M EPR encodes the quantum circuit on two input qubits which has the ability to produce an entangled state from any element of the underlying computational basis 1 . It can be given the type B ⊗ B B ⊗ B in the empty context. The following is a type derivation π EPR for it:
· CNOT : B ⊗ B B ⊗ B · H : B B x : B x : B (E ) x : B Hx : B y : B y : B (I ⊗ ) x : B, y : B Hx ⊗ y : B ⊗ B (E ) x : B, y : B CNOT(Hx ⊗ y) : B ⊗ B (I 2 ) · M EPR : B ⊗ B
B ⊗ B M EPR and π EPR will be used as running examples in the rest of this paper, together with the following type derivation ρ EPR :
π EPR · M EPR : B ⊗ B B ⊗ B · |0 1 : B · |1 2 : B (I ⊗ ) · |0 1 ⊗ |1 2 : B ⊗ B (E ) · M EPR (|0 1 ⊗ |1 2 ) : B ⊗ B
If π Γ (λ x.M)N : A, one can build a type derivation ρ with conclusion Γ M{x/N} : A in a canonical way, and similarly when π Γ (λ x, y .M)(N ⊗ L) : A. This, as expected, is a consequence of the following: Lemma 1 (Substitution Lemma) If π Γ, x 1 : A 1 , . . . , x n : A n M : B and for every 1 ≤ i ≤ n there is ρ i ∆ i N i : A i , then there is a canonically defined derivation π{x 1 , . . . , x n /ρ 1 , . . . , ρ n } of Γ, ∆ 1 , . . . , ∆ n M{x 1 , . . . , x n /N 1 , . . . , N n } : B.
Proof. Just proceed by the usual, simple induction on π.
An Equational Theory.
The λ -calculus is usually endowed with notions of reduction or equality, both centred around the β -rule, according to which a function λ x.M applied to an argument N reduces to (or can be considered equal to) the term M{N/x} obtained by replacing all free occurrences of x with N. A reduction relation implicitly provides the underlying calculus with a notion of computation, while an equational theory is more akin to a reasoning technique. Giving a reduction relation on QΛ terms directly, however, is problematic. What happens when a n-ary unitary operator U is faced with an n-tuple of qubits |b 1 . . . b n ? Superposition should somehow arise, but how can we capture it?
In this section, an equational theory for QΛ will be introduced. In the next sections, we will show that the semantics induced by token machines is sound with respect to it. The equational theory we are going to introduce will be a binary relation on formal, weighted sums of QΛ terms:
Definition 1 (Superposed Term) A superposed term of type (Γ, A) is a formal sum T = n ∑ i=1 κ i M i 1
The quantum circuit EPR is built out from the unitary gates H (the so-called Hadamard gate) and CNOT. The unary gate H is able to create a superposition of elements of the computational basis |0 and |1 , i.e. a linear combination in the form
1 √ 2 (|0 +|1 ) or 1 √ 2 (|0 −|1 )
. The binary gate CNOT negates its second argument, according to the value of the first one. We provide two simple examples of entangled and non-entangled quantum states. The state |ψ = 1
√ 2 |00 + 1 √ 2 |11 is entangled
whereas any state φ = α|00 + β |01 is not. In fact, it is possible to express the latter in the mathematically equivalent form φ = |0 ⊗ (α|0 + β |1 ). See [8] for a gentle introduction to quantum computing essential notions.
where for every 1 ≤ i ≤ n, κ i ∈ C and there is π i such that that π i Γ M i : A. In this case, we write Γ T : A. Superposed terms will be denoted by metavariables like T or S . Please observe that terms in a superposed term are uniformly typed, i.e., they can be given the same type in the same context. Please, notice that:
• If π · U|b 1 . . . b k : B k , then there is a naturally defined superposed term of type (·, B k ) which is nothing more than the element of C 2 k obtained by applying U to the vector |b 1 . . . b k With a slight abuse of notation, this superposed term will be indicated with U|b 1 . . . b k . • All the term constructs can be generalised to operators on superposed terms, with the proviso that the types match. As an
example if T = ∑ i α i M i where π i Γ M i : A B and ρ ∆ N : A, then T N denotes the superposed term S = ∑ i α i (M i N). Indeed, there exist type derivations σ i Γ, ∆ M i N : B
each obtained applying the rule (E ) to π i and ρ. It is now time to define our equational theory, which will be defined on superposed terms of the same type. Formally, ≈ is a binary relation on superposed terms, indexed by contexts and types. The fact that T ≈ Γ,A S is indicated with Γ T ≈ S : A. The relation ≈ is defined inductively, by the rules in Figure 2. Notice that for each Γ, A, ≈ Γ,A is by construction an equivalence relation. Example 2 As an example, consider the term M EPR (|0 1 ⊗ |1 2 ) from Example 1. The equations in the following chain can all be derived through axioms and context closure rules:
M EPR (|0 ⊗ |1 ) ≈ CNOT(H|0 ⊗ |1 ) ≈ 1 √ 2 CNOT(|0 ⊗ |1 ) + 1 √ 2 CNOT(|1 ⊗ |1 ) ≈ 1 √ 2 |0 ⊗ |1 + 1 √ 2 CNOT(|1 ⊗ |1 ) ≈ 1 √ 2 |0 ⊗ |1 + 1 √ 2 |1 ⊗ |0 .
The context (which is ·) and the type (which is B 2 ) have been elided for the sake of readability. By rule trans, we can derive that
· EPR(|0 ⊗ |1 ) ≈ 1 √ 2 |0 ⊗ |1 + 1 √ 2 |1 ⊗ |0 : B 2 .
In other words, EPR, when fed with |0 ⊗ |1 , produces an entangled pair of qubits. The fourth superposed term in the chain above has the remarkable property of not being homogeneous, i.e., of being the sum of two terms which are not identical up to the value of bit constants. Please observe that the equational theory we have just defined can hardly be seen as an operational semantics for QΛ. Although equations can of course be oriented, it is the very nature of a superposed term which is in principle problematic from the point of view of quantum computation: what is the mathematical nature of a superposed term? Is it an element of an Hilbert Space? And if so, of which one? If we consider a simple language such as QΛ, the questions above may appear overly rhetorical, but we claim they are not. For example, what would be the quantum meaning of linear beta-reduction? If we want to design beta-reduction according to the principles of quantum computation, it has to be, at least, easily reversible (unless measurement is implicit in it). Moving towards more expressive languages, this non-trivial issue becomes more difficult and a number of constraints have to be imposed (for example, superposition of terms can be allowed, but only between homogenous terms [19]). This is the reason for which promising calculi [19] fail to be canonical models for quantum programming languages. This issue has been faced in literature without satisfactory answers, yielding a number of convincing arguments in favour of the (implicit or explicit) classical control of quantum data [5,16]. As observed above, our equational theory permits non-homogeneous superposed terms in a very natural way.
Axioms
Γ (λ x, y .M)(N ⊗ L) : A beta.pair Γ (λ x, y .M)(N ⊗ L) ≈ M{x, y/N, L} : A Γ (λ x.M)N : A beta Γ (λ x.M)N ≈ M{x/N} : A · U|b 1 . . . b k : B k quant · U|b 1 . . . b k ≈ U|b 1 . . . b k : B k Context Closure Γ T ≈ S : A B ∆ M : A l.a Γ, ∆ T M ≈ S M : B Γ M : A B ∆ T ≈ S : A r.a Γ, ∆ MT ≈ MS : B Γ, x : A T ≈ S : B in.λ Γ λ x.T ≈ λ x.S : A B Γ, x : A, y : B T ≈ S : C in.λ .pair Γ λ x, y .T ≈ λ x, y .S : A ⊗ B C Γ T ≈ S : A ∆ M : B l.in.tens Γ, ∆ T ⊗ M ≈ S ⊗ M : A ⊗ B Γ M : A ∆ T ≈ S : B r.in.tens Γ, ∆ M ⊗ T ≈ M ⊗ S : A ⊗ B Γ T ≈ S : A Γ V : A sum Γ αT + V ≈ αS + V : A
Reflexive, Symmetric and Transitive Closure
Γ T : A refl Γ T ≈ T : A Γ T ≈ S : A sym Γ S ≈ T : A Γ T ≈ S : A Γ S ≈ V : A trans Γ T ≈ V : A
A Token Machine for QΛ
In this section we describe an interpretation of QΛ type derivations in terms of a specific token machine called IAM QΛ . Before formally defining IAM QΛ , it is necessary to give some preliminary concepts.
With a slight abuse of notation, a permutation σ : {1, . . . , n} → {1, . . . , n} will be often applied to sequences of length n with the obvious meaning: σ (a 1 , . . . , a n ) = a σ (1) , . . . , a σ (n) . Similarly, such a permutation can be seen as the unique unitary operator on C 2 n which sends |b 1 · · · b n to |b σ (1) · · · b σ (n) .
Suppose given a unitary operator U ∈ U of arity n ∈ N. Now, take a natural number m ≥ n and n distinct natural numbers j 1 , . . . , j n , all of them smaller or equal to m. With U j 1 ,..., j n m (or simply with U j 1 ,..., j n ) we indicate the unitary operator of arity m which acts like U on the quantum bits indexed with j 1 , . . . , j n and leave all the other qubits unchanged. In the following, with a slight abuse of notation, occurrences of types in type derivations are confused with types themselves. On the other hand, occurrences of types inside other types will be defined quite precisely, as follows.
Contexts For every type derivation π, B(π) is the sequence of all these occurrences of B in π which are introduced by the rules (a q0 ) and (a q1 ) (recall Figure 1). Similarly, V (π) is the corresponding sequence of binary digits, seen as a vector in C 2 |B(π)| . Both in B(π) and in V (π), the order is the one induced by the natural number labeling the underlying bit in π. Example 4 Consider the following type derivation, and call it τ:
· |0 2 : B 1 · |1 1 : B 2 (I ⊗ ) · |0 2 ⊗ |1 1 : B 3 ⊗ B 4
There are four occurrences of B in it, and we have indexed it with the first four positive natural numbers, just to be able to point at them without being forced to use the formal, context machinery. Only two of them, namely the upper ones, are introduced by instances of the rules (a q0 ) and (a q1 ). Moreover, the rightmost one serves to type a bit having an index (namely 1) greater than the one in the other instance (namely 2). As a consequence, B(τ) is the sequence B 2 , B 1 . The two instances introduces bits 0 and 1; then V (π) = |1 ⊗ |0 . As another example, one can easily compute B(π EPR ) and V (π EPR ) (where π EPR is from Example 1), finding out that both are the empty sequence. We are finally able to define, for every type derivation π, the abstract machine A π interpreting it: • The states of A π form a set S π and are in the form S = (O 1 , . . . , O n , Q) where:
• O 1 , . . . , O n are occurrences of the type B in π;
• Q is a quantum register on n quantum bits, i.e. a normalised vector in C 2 n . • The transition relation → π ⊆ S π × S π is defined based on π, following the rules from Figure 3. In the last rule, B in the type of U is simply denoted through its index, and for every 1 ≤ k ≤ m, i k is the position of B k in the sequence ϕ. The transition rules induced by (I 2 ) have been elided for the sake of simplicity (see [8]). The number of positive (negative, respectively) occurrences of B in the conclusion of π is said to be the output arity (the input arity, respectively) of π. Suppose, for the sake of simplicity, that π is a type derivation of · M : A. An initial state for a quantum register Q is a state in the form (N (A) · B(π), Q ⊗ V (π)). Given a permutation σ on n elements, a final state for a quantum register Q and σ is a state in the form (ϕ, Q), where ϕ = σ (P(A)). A run of A π is simply a finite or infinite sequence S 1 , S 2 , . . . of states from S π such that S i → π S i+1 for every i. Example 5 (A run of IAM QΛ ) Consider the term M EPR and its type derivation π EPR (see Example 1). Forgetting about terms and marking different occurrences of B with distinct indices, we obtain:
· B 9 ⊗ B 10 B 11 ⊗ B 12 · : B 21 B 22 B 23 B 24 (E ) B 17 B 18 B 19 B 20 (I ⊗ ) B 13 , B 14 B 15 ⊗ B 16 (E ) B 5 , B 6 B 7 ⊗ B 8 (I 2 ) · B 1 ⊗ B 2 B 3 ⊗ B 4
Let us consider the following computation of A π EPR :
(B 1 ,B 2 , Q) → * π (B 5 , B 6 , Q) → * π (B 13 , B 14 , Q) → π (B 17 , B 19 , Q) → * π (B 23 , B 20 , Q) → * π (B 24 , B 16 ) → π (B 24 , B 10 , Q) → π (B 21 , B 10 , Q) → π (B 22 , B 10 , H 1 (Q)) → π (B 18 , B 10 , H 1 (Q)) → π (B 15 , B 10 , H 1 (Q)) → π (B 9 , B 10 , H 1 (Q))
→ π (B 11 , B 12 , CNOT 1,2 (H 1 (Q))) → * π (B 7 , B 8 , CNOT 1,2 (H 1 (Q))) → π (B 3 , B 4 , CNOT 1,2 (H 1 (Q))).
(a v ) x : A 1 x : A 2 ((ϕ, (A 1 , P), ψ), Q) → π ((ϕ, (A 2 , P), ψ), Q) ((ϕ, (A 2 , N), ψ), Q) → π ((ϕ, (A 1 , N), ψ), Q) Γ 1 , x : A 1 M : B 1 (I 1 ) Γ 2 λ x.M : A 2 B 2 ((ϕ, (A 1 , N), ψ), Q) → π ((ϕ, (A 2 B 2 , N B 2 ), ψ), Q) ((ϕ, (A 2 B 2 , P B 2 ), ψ), Q) → π ((ϕ, (A 1 , P), ψ), Q) ((ϕ, (B 1 , P), ψ), Q) → π ((ϕ, (A 2 B 2 , A 2 P), ψ), Q) ((ϕ, (A 2 B 2 , A 2 N), ψ), Q) → π ((ϕ, (B 1 , N), ψ), Q) ((ϕ, (Γ 2 , P), ψ), Q) → π ((ϕ, (Γ 1 , P), ψ), Q) ((ϕ, (Γ 1 , N), ψ), Q) → π ((ϕ, (Γ 2 , N), ψ), Q) Γ 1 M : A 1 B 1 ∆ 1 N : A 2 (E ) Γ 2 , ∆ 2 MN : B 2 ((ϕ, (A 2 , P), ψ), Q) → π ((ϕ, (A 1 B 1 , P B 1 ), ψ), Q) ((ϕ, (A 1 B 1 , N B 1 ), ψ), Q) → π ((ϕ, (A 2 , N), ψ), Q) ((ϕ, (A 1 B 1 , A 1 P), ψ), Q) → π ((ϕ, (B 2 , P), ψ), Q) ((ϕ, (B 2 , N), ψ), Q) → π ((ϕ, (A 1 B 1 , A N), ψ), Q) ((ϕ, (Γ 2 , P), ψ), Q) → π ((ϕ, (Γ 1 , P), ψ), Q) ((ϕ, (Γ 1 , N), ψ), Q) → π ((ϕ, (Γ 2 , N), ψ), Q) ((ϕ, (∆ 2 , P), ψ), Q) → π ((ϕ, (∆ 1 , P), ψ), Q) ((ϕ, (∆ 1 , N), ψ), Q) → π ((ϕ, (∆ 2 , N), ψ), Q) Γ 1 M : A 1 ∆ 1 N : B 1 (I ⊗ ) Γ 2 , ∆ 2 M ⊗ N : A 2 ⊗ B 2 ((ϕ, (A 2 ⊗ B 2 , N ⊗ B 2 ), ψ), Q) → π ((ϕ, (A 1 , N), ψ), Q) ((ϕ, (A 2 ⊗ B 2 , A 2 ⊗ N), ψ), Q) → π ((ϕ, (B 1 , N), ψ), Q) ((ϕ, (A 1 , P), ψ), Q) → π ((ϕ, (A 2 ⊗ B 2 , P ⊗ B 2 ), ψ), Q) ((ϕ, (B 1 P), ψ), Q) → π ((ϕ, (A 2 ⊗ B 2 , A 2 ⊗ P), ψ), Q) ((ϕ, (Γ 1 , N), ψ), Q) → π ((ϕ, (Γ 2 , N), ψ), Q) ((ϕ, (∆ 1 , N), ψ), Q) → π ((ϕ, (∆ 2 , N), ψ), Q) ((ϕ, (Γ 2 , P), ψ), Q) → π ((ϕ, (Γ 1 , P), ψ), Q) ((ϕ, (∆ 2 , P), ψ), Q) → π ((ϕ, (∆ 1 , P), ψ), Q) (a U ) · U : B 1 ⊗ . . . ⊗ B m B m+1 ⊗ . . . ⊗ B 2m (ϕ(B 1 , . . . , B m ), Q) → π (ϕ(B m+1 , . . . , B 2m ), U i 1 ,...,im (Q))
Figure 3: IAM QΛ Transition Rules
Notice that the occurrence of CNOT acts as a synchronisation operator: the second token is stuck at the occurrence B 10 until the first token arrives (from the occurrence B 15 ) as a control input of the CNOT and the corresponding reduction step actually occurs.
What the example above shows, indeed, is that the presence of a potential entanglement in π is intimately related to the necessity of synchronisation in the underlying machine A π : if all unitary operators in π can be expressed as the tensor product of unitary operators of arity one (and, thus, entanglement is not possible), then synchronisation is simply not necessary.
Given a type derivation π, the relation → π enjoys a strong form of confluence:
Proposition 1 (One-step Confluence of → π ) Let S, R, T ∈ S π be such that S → π R and S → π T. Then either R = T or there exists a state U such that R → π U and T → π U.
Proof. By simply inspecting the various rules. Notice that there are no critical pairs in → π .
The way A π is built by following a type derivation π induces the following notion:
Definition 2 Given a type derivation π, the partial function computed by π is denoted as [π], has domain C 2 n and codomain C 2 m (where n and m are the input and output arity of π) and is defined by stipulating that [π](Q) = R iff any initial state for Q rewrites into a final state for S and σ , where S = σ −1 (R).
Given a type derivation π, [π] is either always undefined or always defined. Indeed, the fact any initial configuration (for, say, Q) rewrites to a final configuration or not does not depend on Q but only on π:
Lemma 2 (Uniformity) For every type derivation π and for every occurrences O 1 , . . . , O n , P 1 , . . . , P n , there is a unitary operator U such that whenever (O 1 , . . . , O n , Q) → π (P 1 , . . . , P n , R) it holds that R = U(Q).
Proof. Observe that for every O 1 , . . . , O n , P 1 , . . . , P n there is at most one of the rules defining → π which can be applied. Moreover, notice that each rule acts uniformly on the underlying quantum register.
In the following section, we will prove that [π] is always a total function, and that it makes perfect sense from a quantum point of view.
Main Properties of IAM QΛ
In this section, we will give some crucial results about IAM QΛ . More specifically, we prove that runs of this token machine are indeed finite and end in final states. We proceed by putting QΛ in correspondence to MLL, inheriting its very elegant proof theory and token machines.
A Correspondence Between MLL and QΛ
Let A = {α, β , . . .} be a countable set of propositional atoms. A formula A of Multiplicative Linear Logic (MLL) is given by the following grammar:
A, B ::= α | α ⊥ | A ⊗ B | A`B.
Linear negation can be extended to all formulas in the usual way:
(α ⊥ ) ⊥ = α; A ⊗ B ⊥ = A ⊥`B⊥ ; A`B ⊥ = A ⊥ ⊗ B ⊥ .
This way, A ⊥⊥ is just A. The one-sided sequent calculus for MLL is very simple:
ax A, A ⊥ Γ, A ∆, A ⊥ cut Γ, ∆ Γ, A ∆, B ⊗ Γ, ∆, A ⊗ B Γ, A, B` Γ, A`B
The logic MLL enjoys cut-elimination: there is a terminating algorithm turning any MLL proof into a cut-free proof of the same conclusion. A notion of structural equivalence between two MLL proofs ξ , µ having the same conclusion Γ can be easily defined and holds only if ξ and µ are essentially the same proof modulo renaming of the formulas occurring in ξ and µ. Remarkably, two MLL proofs which are structurally equivalent are actually the same proof, a result which does not hold in more expressive logics like MELL. More details on that can be found in [8].
Any QΛ type derivation π can be put in correspondence with some MLL proofs. We inductively define the map (·) • from QΛ types to MLL formulas as follows:
(B) • = α; (A B) • = (A) •⊥`( B) • ; (A ⊗ B) • = (A) • ⊗ (B) •
Given a judgment J = Γ M : A and a natural number n ∈ N, the MLL sequent corresponding to J and n is the following one:
α ⊥ , . . . , α ⊥ n times , ((B 1 ) • ) ⊥ , . . . , ((B m ) • ) ⊥ , (A) • ,
where Γ = x 1 : B 1 , . . . , x m : B m . For every π, we define now a set of MLL proofs I (π). This way, every type derivation π for J = Γ M : A such that n bits occur in M, is put in relation to possibly many MLL proofs of the sequent corresponding to J and n. One among them is called the canonical proof for π. The set I (π) and canonical proofs are defined by induction on the structure of the underlying type derivation π. The type constructions of QΛ are mapped to the corresponding MLL logical operators, rules (a q0 ) and (a q1 ) are mapped to axioms, and rule (a U ) is mapped to a proof encoding a permutation of the involved atoms. When the latter is the identity, we get the canonical proof for π. For more details, please refer to [8].
Given an MLL proof ξ , let us denote as T ξ the class of all finite sequences of atom occurrences in ξ . The relation → ξ can be extended to a relation on T ξ by stipulating that whenever P → ξ R. As usual, → + ξ is the transitive closure of → ξ . Let us now consider a type derivation π in QΛ, its quantum token machine A π , and any ξ ∈ I (π). States of A π can be mapped to T ξ by simply forgetting the underlying quantum register and mapping any occurrence of π to the corresponding atom occurrence in ξ . This way one gets a map R π,ξ : S π → T ξ such that, given a state S = (O 1 , . . . , O n , Q) in S π , |R π,ξ (S)| = n, i.e., the number of occurrences in S is the same as the length of R π,ξ (S). Each reduction step on the token machine A π corresponds to at least one reduction step in the MLL machine M ξ , where ξ ∈ I (π) is the canonical proof: Lemma 3 Let us consider a token machine A π and two states S, R ∈ S π . If S → π R and ξ ∈ I (π) is canonical, then R π,ξ (S) → + ξ R π,ξ (R). Proof. This goes by induction on the structure of π.
Any (possible) pathological situation on the quantum token machine, then, can be brought back to a corresponding (absurd) pathological situation in the MLL token machine. This is the principle that will guide us in the rest of this section.
Termination, Progress and Soundness
The first property we want to be sure about is that every computation of any token machine A π always terminates. The second one is progress (i.e. deadlock-freedom). In both cases, we use in an essential way the correspondence between QΛ and MLL.
Proposition 2 (Termination) For any quantum token machine A π , any sequence S → π R → π . . . is finite.
Proof. Suppose, for the sake of contradiction, than there exists an infinite computation in A π . This implies by Lemma 3 that there exists an infinite path in the token machine M ξ where ξ is the canonical MLL proof for π. This is a contradiction, because paths in MLL proofs are well-known to be always finite.
Progress (i.e. deadlock-freedom) is more difficult to prove than termination. Given a type derivation π, an argument occurrence is any negative occurrence (A, N) of B in a (a U ) axiom. We extend this definition to the corresponding atom occurrence when ξ ∈ I (π). A result occurrence is defined similarly, but the occurrence has to be positive.
Proposition 3 (Progress)
Suppose π is a type derivation in QΛ and S ∈ S π is initial. Moreover, suppose that S → * π R. Then either R is final or R → π T for some T ∈ S π . Proof. Let us consider a computation S 1 → π . . . → π S k on a quantum token machine A π . Suppose that the state S k is a deadlocked state, i.e. S k is not a final state, and that there exists no S m such that S k → π S m . The fact S k is a deadlocked state means that l ≥ 1 occurrences in S k are argument occurrences, since the latter are the only points of synchronisation of the machine. Let us consider any maximal sequence
R π,ξ (S 1 ) → ξ . . . → ξ R π,ξ (S k ) → ξ Q 1 → ξ . . . → ξ Q n ,(1)
where ξ ∈ I (π) is the canonical proof corresponding to π. Observe that in (1), all occurrences of atoms in ξ are visited exactly once, including those corresponding to argument and result occurrences from π. Notice, however, that the argument and result occurrences of the unitary operators affected by S k cannot have been visited along the subsequence R π,ξ (S 1 ) → ξ . . . → ξ R π,ξ (S k ) (otherwise we would visit the occurrences in S k at least twice, which is not possible). Now, form a directed graph whose nodes are the unitary constants U 1 , . . . ,U h which block S k , plus a node F (representing the conclusion of π), and whose edges are defined as follows:
• there is an edge from U i to U j iff along Q 1 → ξ . . . → ξ Q n one of the l independent computations corresponding to a blocked occurrence in S k is such that a result occurrence of U i is followed by an argument occurrence of U j and the occurrences between them are neither argument nor result occurrences. • there is an edge from U i to F iff along Q 1 → ξ . . . → ξ Q n one of the l traces is such that a result occurrence of U i is followed by a final occurrence of an atom and the occurrences between them are neither argument nor result occurrences. The thus obtained graph has the following properties:
• Every node U i has at least one incoming edge, because otherwise the configuration S k would not be deadlocked. • As a consequence, the graph must be cyclic, because otherwise we could topologically sort it and get a node with no incoming edges (meaning that some of the U i would not be blocked!). Moreover, the cycle does not include F, because the latter only has incoming nodes.
From any cycle involving the U j , one can induce the presence of a cycle in the token machine M µ for some µ ∈ I (π). Indeed, such a µ can be formed by simply choosing, for each U j , the "good" permutation, namely the one linking the incoming edge and the outgoing edge which are part of the cycle. This way, we have reached the absurd starting from the existence of a deadlocked computation.
The immediate consequence of the termination and progress results is that [π] is always a total function. The way A π is defined ensures that [π] is obtained by feeding some of the inputs of a unitary operator U with some bits (namely those occurring in π). U is itself obtained by composing the unitary operators occurring in π, which can thus be seen as a program computing a quantum circuit. In a way, then, token machines both show that QΛ is a truly quantum calculus and can be seen as the right operational semantics for it.
The last step consists in understanding the relation between token machines and the equational theory on superposed terms introduced in Section 2.3. First of all, observe that T = ∑ n i=1 κ i M i has type A in the context Γ, then M 1 , . . . , M n all have type A in the context Γ. But there is more to that: for every 1 ≤ i ≤ n, there is exactly one type derivation π i Γ M i : A. This holds because two such type derivations π i and ρ i are such that the canonical proofs in I (π i ) and I (ρ i ) are structurally equivalent, thus identical. It is then possible to extend the definition of [·] to superposed terms:
if T = ∑ n i=1 κ i M i has type A in Γ, then [T ], when fed with a vector x, returns ∑ n i=1 κ i [π i ](x),
where π i is the unique derivation giving type A to M i in the context Γ. Remarkably, token machines behave in accordance to the equational theory: this is our Soundness Theorem. Proof. We only give a sketch of the proof. More details can been found in [8]. The first step consists in proving that any derivation of Γ T ≈ S : A can be put in normal form, a concept defined by giving an order on the rules in Figure 2. More specifically, define the following two sets of rules: AX = {beta, beta.pair, quant}; CC = {l.a, r.a, in.λ , in.λ .pair, l.in.tens, r.in.tens}.
A derivation of Γ T ≈ S : A is said to be in normal form (and we write Γ T ∼ S : A) iff • either the derivation is obtained by applying rule refl; • or any branch in the derivation consists in instances of rules from AX, possibly followed by instances of rules in CC, possibly followed by instances of sum, possibly followed by instances of sym possibly followed by instances of trans. In other words, a derivation of Γ T ≈ S : A is in normal form iff rules are applied in a certain order. As an example, we cannot apply transitivity or symmetry closure rules too early, i.e., before context closure rules. One may wonder whether this restricts the class of provable equivalences. Infact it does not: ΓΓT ≈ S : A iff ΓΓT ∼ S : A, a result which is not particularly deep although a bit tedious to prove [8]. Once we have this result in our hands, however, proving Soundness becomes much easier, since the difficult and problematic rules, namely those in CC, are applied to superposed terms of a very specific shape, namely those obtained through AX.
Related Work
In [13], a geometry of interaction model for Selinger and Valiron's quantum λ -calculus [16] is defined. The model is formulated in particle-style. In [4] QMLL, an extension of MLL with a new kind of modality, is studied. QMLL is sound and complete with respect to quantum circuits, and an interactive (particle-style) abstract machine is defined. In both cases, adopting a particle-style approach has a bad consequence: the "quantum" tensor product does not coincide with the tensor product in the sense of linear logic. Here we show that adopting the wave-style approach solves the problem. Quantum extensions of game semantics are partially connected to this work. See, for example [10,9]. Purely linear quantum lambda-calculi (with measurements) can be given a fully abstract denotational semantics, like the one proposed by Selinger and Valiron [17]. In their work, closure (necessary to interpret higher-order functions) is not obtained via traces and is not directly related in any way to the geometry of interaction. Moreover, morphisms are just linear maps, and so the model is far from being a quantum operational semantics. A language of terms similar to QΛ has been also studied in [21], where the calculus of proof-nets MLL qm is introduced. MLL qm 's syntax also includes a measurement box-like operator (which models the possibility of "observe" the value of a quantum bit [15]). A multi-token machine semantics for MLL qm proof-nets is defined and proved to be sound, i.e. invariant along reduction of proof nets. Moreover, although a λ -calculus is given, together with a compilation scheme to MLL qm proof-nets, the considered λ -calculus is one with explicit qubits, contrary to QΛ. Finally, Arrighi and Dowek's work shows that turning a sum-based algebraic λ -calculus into a quantum computational model can be highly non-trivial [1].
Conclusions
We have introduced IAM QΛ , an interactive abstract machine which provides a sound operational characterisation of any type derivation in a linear quantum λ -calculus QΛ. This is an example of a concrete wave-style token machine whose runs cannot be seen simply as the asynchronous parallel composition of particle-style runs. Interestingly, synchronisation is intimately related to entanglement: if, for example, only unary operators occur in a term (i.e. entanglement is not possible), synchronisation is not needed and everything collapses to the particle-style. Our investigation is open to some possible future directions. A natural step will be to extend the syntax of terms and types with an exponential modality. The generalisation of the token machine to this more expressive language would be an interesting and technically challenging subject. Moreover, giving a formal status to the connection between wave-style and the presence of entanglement is a fascinating subject which we definitely aim to investigate further. Finally, an interesting proof-theoretical investigation would consist in analysing the possible connections the with the deep inference-oriented graph formalism developed in [3].
Figure 1 :
1Typing Rules.
Figure 2 :
2Equational Theory
(types with a hole) are denoted by metavariables like C or D. A context C is said to be a context for a type A if C[B] = A. Negative contexts (i.e., contexts where the hole is in negative position) are denoted by metavariables like N, M. Positive ones are denoted by metavariables like P, Q. An occurrence of B in the type derivation π is a pair O = (A,C), where A is an occurrence of a type in π and C is a context for A. Sequences of occurrences are indicated with metavariables like ϕ, ψ (possibly indexed). All sequences of occurrences we will deal with do not contain duplicates. Type constructors and ⊗ can be generalised to operators on occurrences and sequences of occurrences, e.g.(A,C) B is just (A B,C B). If a sequence of occurrences ϕ contains the occurrences O 1 , . . . , O n , we emphasise it by indicating it with ϕ(O 1 , . . . , O n ). Given (an occurrence of) a type A, all positive and negative occurrences of B inside A form sequences called P(A) and N (A), respectively. These are defined as follows (where · is sequence concatenation): P(B) = (B, [·]); N (B) = ε; P(A ⊗ B) = (P(A) ⊗ B) · (A ⊗ P(B)); N (A ⊗ B) = (N (A) ⊗ B) · (A ⊗ N (B)); P(A B) = (N (A) B) · (A P(B)); N (A B) = (P(A) B) · (A N (B)). Example 3 As an example, the positive occurrences in the type B B ⊗ B should be the two rightmost ones. And, indeed, P(B B ⊗ B) = (N (B) B ⊗ B) · (B P(B ⊗ B)) = ε · (B P(B ⊗ B)) = B P(B ⊗ B) = (B (P(B) ⊗ B)) · (B (B ⊗ P(B))) = (B, B ([·] ⊗ B)), (B, B (B ⊗ [·])). Similarly, one can prove that N (B B ⊗ B) = (B, [·] (B ⊗ B)).
(O 1
1, . . . , O n−1 , P, O n+1 , . . . , O m ) → ξ (O 1 , . . . , O n−1 , R, O n+1 , . . . , O m )
Theorem 1 (
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| []
|
[
"Iterative Best Response for Multi-Body Asset-Guarding Games",
"Iterative Best Response for Multi-Body Asset-Guarding Games"
]
| [
"Emmanuel Sin ",
"Murat Arcak ",
"Douglas Philbrick ",
"Peter Seiler "
]
| []
| []
| We present a numerical approach to finding optimal trajectories for players in a multi-body, assetguarding game with nonlinear dynamics and non-convex constraints. Using the Iterative Best Response (IBR) scheme, we solve for each player's optimal strategy assuming the other players' trajectories are known and fixed. Leveraging recent advances in Sequential Convex Programming (SCP), we use SCP as a subroutine within the IBR algorithm to efficiently solve an approximation of each player's constrained trajectory optimization problem. We apply the approach to an asset-guarding game example involving multiple pursuers and a single evader (i.e., n-versus-1 engagements). Resulting evader trajectories are tested in simulation to verify successful evasion against pursuers using conventional intercept guidance laws. | null | [
"https://arxiv.org/pdf/2011.01893v1.pdf"
]
| 226,237,160 | 2011.01893 | 99c5874272d726e77d14cc6e187090a9997e85b9 |
Iterative Best Response for Multi-Body Asset-Guarding Games
Emmanuel Sin
Murat Arcak
Douglas Philbrick
Peter Seiler
Iterative Best Response for Multi-Body Asset-Guarding Games
1Index Terms Differential gamesPursuit-evasion gamesMulti-agent systemsIterative best responseSequential convex programming
We present a numerical approach to finding optimal trajectories for players in a multi-body, assetguarding game with nonlinear dynamics and non-convex constraints. Using the Iterative Best Response (IBR) scheme, we solve for each player's optimal strategy assuming the other players' trajectories are known and fixed. Leveraging recent advances in Sequential Convex Programming (SCP), we use SCP as a subroutine within the IBR algorithm to efficiently solve an approximation of each player's constrained trajectory optimization problem. We apply the approach to an asset-guarding game example involving multiple pursuers and a single evader (i.e., n-versus-1 engagements). Resulting evader trajectories are tested in simulation to verify successful evasion against pursuers using conventional intercept guidance laws.
I. Introduction
In his seminal work on differential games, Isaacs describes a pursuit-evasion game that he calls Guarding a target [1]. In this game, the motive of the pursuer P is to guard an asset C from an attack by the evader E. The motive of E is to reach its target while evading P . The payoff in this zero-sum game is the distance between C and the point of P 's capture of E. Assuming simple motion (i.e., constant velocity with the ability to change heading angle instantaneously), the optimal trajectories of P and E are depicted and explained using geometric arguments alone. But when we attempt to build upon this basic game by introducing more complex motion, imposing state and input constraints, or adding more players with non-diametrically opposed objectives, arriving at such analytical solutions becomes difficult. In this work we present a framework to model advanced forms of the asset-guarding game and introduce a numerical solution method to find optimal strategies for the players. In our version of Isaacs' guarding game we have an asset C, a group of n pursuers that we collectively call P = {P 1 , P 2 , . . . , P n }, and a single evader E. The asset C can be specified as a point, surface, or volume to represent, for example, a vehicle, an area of land, or a large structure. We assume that its motion follows a known trajectory (e.g., stationary or moving with constant velocity). The evader E and pursuers P are always active, strategic players in our game. Depending on the application at hand, we may approach the game from the perspective of either E or P . If we approach the problem from the perspective of E, we are concerned with finding a dynamically-feasible trajectory that both reaches C and avoids capture by P during the course of the trajectory. This reach-while-avoiding condition is non-negotiable in E's mission and so we consider it as a constraint that must be satisfied. On the other hand, the specific final time at which E captures C is negotiable -a later arrival time may be acceptable if E is required to perform evasive maneuvers. Hence, it is natural for E to define its objective as the minimum time required to fulfill its mission-critical constraints. E's speed and acceleration capabilities are greater than those of C, but they may be inferior to those of P . In any case, E must contend with the sheer number of pursuers that comprise P . If found, a trajectory for E that can evade numerous, superior pursuers would be remarkable.
If we now switch our perspective to consider the game from P 's point-of-view, we are a group of pursuers, each engineered with the task of intercepting a maneuvering target. If we assume that it only takes one pursuer to disable E, we may employ the other pursuers to assist, perhaps by corralling E in a certain direction. This approach can lead to interesting collaborative behavior but it may also result in our use of pursuer vehicles in a manner other than which they were designed for (e.g., using interceptor missiles with lethal warheads simply as herders). Hence, we assume that each pursuer should be programmed with the intent to intercept E, without regard to objective formulations for other pursuers. A practical advantage of this assumption is that if a pursuer fails during the engagement, other pursuers will still be on feasible trajectories to intercept E. An important temporal constraint on P is that each pursuer's intended intercept time of E should be less than E's predicted intercept time of C. The pursuer that can intercept E in the shortest amount of time decides the performance of P as a whole. An interesting situation to consider is the case where P 's maneuverability is significantly inferior to that of E -to see if quantity can outperform quality. This discussion of the players' objectives and constraints are mathematically expressed as optimal control problems in a following Problem Statement section.
A. Relevant Work
Isaacs initiated the study of differential games through a series of unpublished RAND Corporation reports [2] starting in 1954. Isaacs' approach resembled that of dynamic programming where he derived control laws for the players and conditions for certain outcomes in simple motion games. Soon after Isaacs' initial reports, Berkowitz and Fleming introduced a calculus of variations approach where they constructed necessary and sufficient conditions for a saddle-point that must be satisfied along the trajectories of the players [3]. The approach is applied to a wide class of differential games in [4] but did not treat specific examples or applications. In 1965, Ho, Bryson and Baron used the same variational techniques to derive the well-known proportional navigation intercept guidance law by modeling a pursuit-evasion game [5].
We note two state-of-the-art numerical methods for finding local solutions in a wide-class of differential games involving multiple players, non-zero-sum objectives, nonlinear dynamics and constraints. The first method [6] is inspired by the iterative linear quadratic regulator (iLQR) approach used for optimal control of nonlinear dynamical systems [7]. However, rather than using the closedform solution to the LQR problem, the closed-form solution to the n-player, non-zero-sum, linearquadratic dynamic game [8] is used at each iteration of the algorithm. State and input constraints are implemented as soft constraints, penalizing constraint violations in the objective. The second method [9] also approaches dynamic games by iteratively solving linear-quadratic approximations of the problem. Constraint satisfaction is encouraged by introducing penalty terms in the players' objectives to form augmented Lagrangians. Since a first-order condition of optimality for each player requires that the gradient of its augmented Lagrangian be null at an optimal point, a root-finding problem is formulated and solved using Newton's method. Both methods require initial guesses for all players -either initial feedback control strategies or initial trajectories for states, inputs, and Lagrange multipliers associated with any constraints.
Yet another solution method is inspired by the concept of best response, used in normal-form games [10]- [11], where the optimal strategy for each player is determined by assuming strategies of all other players are known and fixed. The concept has been used to find open-loop Nash equilibria for simple finite-time horizon differential games in [12]- [13]. Recent work has used the best response concept to numerically find Nash equilibria in practical examples involving autonomous car trajectory planning and competitive vehicle racing [14]- [17]. A common theme is to employ an Iterative Best Response (IBR) scheme where players' strategies are sequentially and iteratively updated until all players have converged on their best responses. One drawback of IBR, as pointed out in [6] and [9], is that the application of existing trajectory optimization methods in an IBR framework may be computationally inefficient with long solution times, detracting from its practical use in differential games.
Recent advances in sequential convex programming (SCP) have enabled efficient, real-time trajectory optimization for constrained, nonlinear systems. SCP is an iterative method that repeatedly formulates and solves a convex, finite-dimensional parameter optimization problem that approximates the original non-convex optimal control problem. A convex formulation is typically achieved by linearizing the nonlinear system around a nominal trajectory (i.e., the solution from the previous iteration) and approximating any non-convex constraints with first-order or second-order approximations. In this convex form, fast and reliable Interior Point Method algorithms [18] may be used as solvers. Successive Convexification, a type of sequential convex programming, is used in [19] to solve a minimum-fuel, 6-DOF powered descent Mars landing guidance problem. In [20], a timenormalization procedure allows for the treatment of minimum-time problems. After these initial works that laid the foundation for SCP, subsequent work by the authors and collaborators aimed at improving various aspects of the method. For example, the issue of guaranteed constraint satisfaction in-between temporal nodes of a transcribed problem is addressed and applied to obstacle avoidance constraints [21]. Given the multitude of options for discretizing continuous-time systems, various methods are compared in terms of accuracy and efficiency in [22]. The tutorial [23] places successive convexification in the context of trajectory planning and tracking for aerospace applications and the convergence properties of the method are studied in [24]. Then, the method's applicability is broadened to problems involving mixed integer constraints [25]- [27]. Most recently, [28]- [30] focus on real-time implementations with computation times within fractions of a second.
Even without real-time SCP implementation, the IBR scheme serves as a valuable tool when analyzing certain types of differential games. For example, in our asset-guarding game we are willing to assume that the evader E acts first and the pursuers P respond in turn -without an evader, there is no need for pursuers. If E finds a solution to its optimal control problem (i.e., a feasible trajectory and corresponding strategy) that forces P 's optimal control problem to become infeasible, then E may as well announce its strategy in advance and execute it in open-loop. In a deterministic setting, E has no incentive to change its trajectory as the game evolves since P cannot feasibly respond to its current strategy. Knowing whether such a strategy exists is of obvious value to E but is also important to P and C, who may decide to recruit more pursuers or strategically change initial conditions. Such an analysis can be carried out at the beginning of an asset-guarding engagement and not necessarily executed online.
B. Main Contribution
The main contributions of this work are: 1) Formulation of optimal control problems (OCPs) that capture the nature of the asset-guarding game for the Asset C, Pursuers P , and Evader E. The OCPs allow for nonlinear dynamics and non-convex constraints that arise in multi-body, asset-guarding games involving aerospace vehicles. 2) Demonstration of an Iterative Best Response (IBR) solution method that uses SCP trajectory optimization as a subroutine to find solutions for players in a differential game. In our numerical examples, we demonstrate the ability to apply SCP to dynamical models with data in tabular form. Since many practical aerospace systems require the use of lookup tables (e.g., atmospheric parameters, aerodynamic coefficients), this technique may further the adoption of SCP as a powerful trajectory optimization for aerospace systems. Together, the main contributions provide a framework to model differential games, such as the asset-guarding game, and a solution method to find trajectories that satisfy the players' objectives, constraints, and dynamics. The combination of the IBR algorithm and SCP subroutine provides a practical tool to design and analyze strategies for dynamical systems in conflict.
C. Notation
Bold, lower-case letters are used to indicate (column) vector-valued quantities while unbolded letters refer to scalar quantities. The letters x, u, and t are reserved to denote state, input, and time trajectories, respectively. We use t 1 for the initial time and T for the final time of a problem. Subscript indices are used in shorthand notation to refer the state and input at specified temporal nodes of a trajectory, e.g.,
x 1 := x(t 1 ), x k := x(t k ), x K := x(T )
. Superscript indices reference the player to which the quantity applies, e.g., x P i K refers to the state of player P i at its final time T . In certain expressions we do not use the shorthand notation, e.g., x P i (T C ) is the state of player P i at the final time specified for player C's problem. An overbar signifies that the quantity is known and fixed in a problem formulation, and not a variable to be decided.
II. Problem Statement
For players E and P we formulate the optimal control problems: OCP E , OCP P , below. The goal is to determine optimal trajectories for each player that would render the opponent's problem infeasible.
A. Optimal Control Problem for player E (OCP
E ) Given {x C ,T C }, {x P i ,T P i } ∀ i ∈ I = {1, . . . , n}, and x E (t 1 ), the following problem will produce the solution {x E ,ū E ,T E }, if feasible. minimize x E ,u E ,T E T E t 1 1 dt (1) subject toẋ E (t) = f E (x E (t), u E (t)) ∀ t 1 ≤ t ≤ T E (2) g E (x E (t), u E (t)) ≤ 0 ∀ t 1 ≤ t ≤ T E (3) (OCP E ) H (x E (T E ) −x C (T E )) 2 ≤ r c(4)H (x P i (t) − x E (t)) 2 ≥ r e ∀ t 1 ≤ t ≤ min{T E ,T P i }, i ∈ I(5)T E ≤T C(6)
Note that the final time T E in the minimum-time objective (1) is a decision variable, implicitly defined as the capture time of Asset C in constraint (4), where left multiplication by matrix H isolates the relative positions, and r c is the user-defined capture radius. Constraint (2) represents player E's dynamics while (3) is a general expression for any state/input constraints specific to player E. Constraint (5) describes the condition that E must evade all pursuers with evasion radius r e until the time at which E succeeds in capturing C or until the pursuers no longer exist, whichever is shorter. Constraint (6) puts an upper bound on how much time E has to capture C , e.g., the opportunity to capture the Asset may exist for only a finite length of time.
B. Optimal Control Problem for player P (OCP P ) Given {x E ,T E }, and {x P i (t 1 )} ∀ i ∈ I = {1, .
. . , n}, the following problem will produce the solution
{x P i ,ū P i ,T P i } ∀ i ∈ I, if feasible. minimize {x P i ,u P i ,T P i } n i=1 min i T P i t 1 1 dt (7) subject toẋ P i (t) = f P i (x P i (t), u P i (t)) ∀ t 1 ≤ t ≤ T P i , i = 1, . . . , n (8) (OCP P ) g P i (x P i (t), u P i (t)) ≤ 0 ∀ t 1 ≤ t ≤ T P i , i = 1, . . . , n(9)∃ i ∈ I ( H (x P i (T P i ) −x E (T P i )) 2 ≤ r c ) ∧ T P i <T E(10)
The expression in (7) represents an objective where the pursuer P i that can capture E in the shortest amount of time decides the objective value for P as a whole. Constraints (8) and (9) are general expressions for the pursuers' dynamics and constraints. The constraint (10) describes the condition that there must exist a pursuer that can capture E within a time span that is less than the time it takes E to capture C.
We have captured the nature of the asset guarding game in constraints (5) and (10), where the satisfaction of one constraint implies that the other cannot be satisfied. Hence, either OCP E or OCP P may be feasible. Both cannot be simultaneously feasible when {T P i } n i=1 and T E take on assumed values. This fact is exploited when using the Iterative Best Response method (IBR), to be described in a later section.
III. Trajectory Optimization using Sequential Convex Programming
The work in [19]- [32] describe a process of transcribing a continuous-time optimal control problem OCP into a finite-dimensional, convex parameter optimization problem OPT, where the dynamics are first linearized about a nominal trajectory to produce a time-varying model. This continuoustime model is then discretized so that the values of state and input variables at specified temporal nodes may be solved for. Non-convex constraints on the state and input are also linearized about the trajectory to produce a convex program. A similar transcription process is completed for each of the optimal control problems stated in the Problem Statement: (OCP E → OPT E ) and (OCP P → OPT P ). We note that OCP P may be decoupled into n separate optimal control problems that are then transcribed into n convex formulations, {OPT P i } n i=1 . Each problem is solved using an SCP algorithm and we choose the best optimal value to represent P , i.e.,T P = min{T P i } n i=1 . Since the Pursuers' problems are decoupled, we may solve them in parallel.
When executing an SCP algorithm on a trajectory optimization problem, the approximated convex problems may become infeasible. This artificial infeasibility [20] is frequently encountered in the early iterations of the algorithm where the dynamics are linearized about a poor initial guess. To alleviate this issue, certain implementations of SCP, such as Successive Convexification [24] or Penalized Trust Region [29], add slack variables called virtual controls to the optimization problem formulation. These virtual controls act as dynamic relaxation terms that take on nonzero values when necessary to satisfy state and input constraints. In turn, use of these slack variables is heavily penalized with a term in the objective function. As a condition for convergence in these SCP algorithms, use of these virtual controls must become negligible at later SCP iterations so that the converged solution is dynamically feasible without the aid of these fictitious terms. Both converged (dynamically feasible) and unconverged (dynamically infeasible) SCP solutions are used in the IBR algorithm to be introduced in the following section.
IV. Iterative Best Response
In our IBR algorithm, we use the following notation where a bold, uppercase E with superscript i refers to player E's solution at the i th iteration of the algorithm. Similar notation is used for player P 's solution at a given iteration.
E i := {x E ,ū E ,T E } = {x E 1 , . . . ,x E k , . . . ,x E K }, {ū E 1 , . . . ,ū E k , . . . ,ū E K }, T E(11)
The Iterative Best Response algorithm takes initial guesses for the solutions, E 0 and P 0 , and a known trajectory C for the Asset. Since an asset-guarding engagement begins with the presence of the Evader E, we assume E takes on its given initial guess trajectory and we solve for the pursuers P 's best response to that trajectory using the SCP subroutine on a transcribed OPT P . Once we obtain P 's best response trajectory, we then solve player E's OPT E to determine its best response in return. The sequential solve process for these two optimization problems using the SCP subroutine constitutes a single IBR iteration. We proceed with the iterations until a user-defined number of iterations N IBR is completed. We note that even if the SCP subroutine fails to converge for any of the players, we still use the un-converged solution and proceed with the IBR procedure.
Algorithm 1: Iterative Best Response for Asset Guarding Game with non-strategic Asset C Input : E 0 , P 0 , C Output: recorded E i , P i for i = 1 :
N IBR do P i ← OPT i P (P i−1 ; E i−1 , C) if ( P i not converged ) ∧ (E i−1 converged ) then record E i−1 E i ← OPT i E (E i−1 ; P i , C) if ( P i converged ) ∧ ( E i not converged ) then record P i
As we proceed with the IBR iterations, we keep record of a player's converged solution if the opponent's subsequent SCP subroutine fails to converge on a solution. For instance, given that player E has a converged (dynamically feasible) trajectory, if the opposing player P fails to also find a converged (dynamically feasible) trajectory in response, then player E has found a strategy with the potential to win the game.
V. Numerical Examples
For the examples in this paper, we assume point-mass-with-drag dynamics for all players. That is, the dynamics of (2) and (8) are replaced with (12) below, where we have dropped superscripts denoting the players (i.e., E, P 1 , . . . , P n ). The state vector x ∈ R 6 includes a player's position p := [p x , p y , p z ] and velocity v := [v x , v y , v z ] in 3-dimensional space. Each player can control its motion by inducing an acceleration via its input vector u ∈ R 3 . In addition, each body is affected by the acceleration due to atmospheric drag d ∈ R 3 . The quantities are measured using a Cartesian coordinate system and an inertial reference frame placed at sea level with the z-axis pointing "upwards."
x(t) := ṗ(t) v(t) = v(t) u(t) + d (p(t), v(t)) ∀ t 1 ≤ t ≤ T(12)
The acceleration due to atmospheric drag is a function of position and velocity:
d (p(t), v(t)) := − 1 2 S m ρ (p z (t)) C D (M (p(t), v(t))) v(t) 2 v(t)(13)
where mass m and reference area S of the body are constant parameters. We note that atmospheric density ρ is a function of altitude and that the drag coefficient C D is a function of the Mach number M . The Mach number depends on both the magnitude of the velocity and the speed of sound a, which in turn is also a function of altitude.
M (p(t), v(t)) := v(t) 2 a (p z (t))(14)
The atmospheric parameters (ρ, a, C D ) are determined using look-up tables represented in Fig. (2). We use atmospheric density and speed of sound data from the 1976 U.S. Standard Atmosphere [33] and assume drag coefficient data representative of the V-2 rocket [34]. We use a piecewise cubic Hermite interpolator [35] in this work as it is relatively inexpensive to evaluate but also provides continuous first derivatives. Since the dynamics and any non-convex constraints must be linearized about a nominal trajectory when using SCP, the partial derivatives of the atmospheric parameters with respect to the states must be approximated. One approach is to use the analytical partial derivative of the piecewise interpolating polynomials. Another method is to use finite differencing [18] on the interpolation. In this work, we use the finite differencing approach as it provides sufficient accuracy compared to analytical partial derivatives. Furthermore, for applications where a parameter is a function of multiple states (e.g., maximum thrust as a 2D lookup table of both Mach number and altitude), finite-differencing may be more convenient than the analytical derivative approach, which requires storing and then locating polynomials on a multi-dimensional grid.
We replace the general constraint expressions of (3) and (9) with the following box constraints on the input accelerations and a lower bound on the Mach number:
u(t) ∞ ≤ u max ∀ t 1 ≤ t ≤ T (15) M min ≤ M (p(t), v(t)) ∀ t 1 ≤ t ≤ T(16)
The lower bound on the Mach number is an important and practical constraint to include since endoatmospheric vehicles (e.g., aircraft, cruise missiles) generally have a lower bound on speed to avoid stall and maintain lift force. We note that (16) is a non-convex constraint and we use the procedure described in [20] to approximate it with a convex relaxation.
In Table I Fig. 3 where the Evader starts 1,000 [ft] above the Asset and Pursuers. The direction of the velocity vectors imply head-on engagements (as opposed to a tail-chase). We verify by simulation that these initial conditions describe a well-posed engagement where each player can nominally intercept its target using the proportional navigation (PN) guidance law with navigation ratio N = 3 [36]. As described in Table II, the Asset is assumed The IBR-SCP method requires an initial solution guess that consists of the state and input trajectories of all players. We may use the PN-guided trajectories described above, however, we have observed that crude straight-line state trajectories and zero input trajectories converge to trajectories similar to those warm-started by PN-based initial guesses within one IBR iteration.
After running the IBR-SCP solution method for each of the examples, we verify the solutions in simulation against opponents using conventional guidance laws, i.e., PN and Augmented PN laws with navigation ratios N = 3, 4, 5. For example, if the algorithm's result implies that the Evader successfully evades its Pursuers and reaches the Asset, then we pit the Evader's open-loop strategy against six instances of each Pursuer in simulation (two guidance laws, three navigation ratio values). This verification step checks if the solution provided by the IBR-SCP method is indeed feasible for the optimal control problem formulations listed in the Problem Statement.
VI. Results
We apply the IBR-SCP solution method to the four asset-guarding game examples described in the prior section. In Fig. 4, we show a subset of twenty IBR iterations we run for Example 1. In the first IBR iteration we observe that the SCP subroutine has converged on a blue Pursuer trajectory that intercepts the initial straight-line guess provided for the red Evader. After five IBR iterations, we observe that the Evader's SCP subroutine has failed to converge on a dynamically feasible solution, signaled by the dashed line. Nevertheless, we solve for the Pursuer's response to that trajectory and find that it can still intercept the Evader's artificial trajectory. After five more IBR iterations we observe that neither the Evader's nor the Pursuer's SCP subroutines have converged on dynamically feasible solutions. The trajectories still intercept but are only feasible given the virtual controls that the SCP subroutine employs. After a few more IBR iterations we find a solution where the Evader is dynamically feasible but the Pursuer is not. We continue until the twentieth IBR iteration where we observe that the Evader's state trajectory has not changed significantly from that shown in IBR Iteration 16. where it is appears that the Evader is intercepted mid-course by the Pursuer. However we recall that the blue dashed line implies that this Pursuer trajectory is not dynamically feasible. The Evader then dives in altitude and performs a "turn-around" maneuver to reach the Asset at the red point. We expect that if the Evader commits to the open-loop strategy described by this solution, then the Pursuer will be unable to intercept the Evader and defend the Asset. The Evader's state trajectories are shown in Fig. 6 where we observe that the Evader reaches the Asset's location in 24 expect it to maintain a high Mach number (or speed) throughout the engagement. Counterintuitive to our expectation, the Evader maintains the lowest possible Mach number for a significant duration of the engagement. This behavior is attributable to the turnaround maneuver that the Evader conducts after evading the Pursuer. Rather than taking a large-radius turn back to the Asset, the minimumtime strategy is to reduce speed drastically so that it may turn more sharply into the direction of its target. This observation is in line with minimum-time turn maneuvers by supersonic aircraft [37]. In fact, when we remove the lower bound constraint on the Mach number and run the IBR algorithm, we find that the Evader reduces its speed significantly from an initial value of 3,000 [ft/s] down to approximately 60 [ft/s] to conduct a high-agility turn maneuver. Although interesting to observe in simulation, such a maneuver may result in loss of lift for conventional aircrafts. In the same plot we note that the Evader endures the penalty of a high drag coefficient as it reduces its Mach number below the drag-divergence value of 1.2. After completing the turnaround maneuver, the Evader speeds up and once again passes through the high drag coefficient regime to reach the Asset in minimum-time.
In Fig. 8 we show the effective (control and drag) accelerations profiles for both the Evader and the Pursuer. The Pursuer's predicted accelerations stop at 2.67 seconds, when we infer that the Evader has successfully evaded the Pursuer. Considering the initial high speeds of the players in this head-on engagement, the players' strategies are to apply maximum deceleration in their respective attempts to evade and capture. The "hard-braking" by the Evader in the x-axis direction forces the Pursuer to follow suit in its direction of motion. We note that both players actually surpass their acceleration bounds by optimally taking advantage of atmospheric drag. In the y-axis and z-axis directions we observe that the Evader's strategy causes the Pursuer to hit and ride its acceleration bounds. The Evader has found a strategy that forces the Pursuer to apply maximum effort with all of its ability, and yet we know that the Pursuer's trajectory is artificially supported (since the SCP subroutine failed to converge on a solution that has weaned itself off of virtual controls). In all remaining Examples 2-4, where we add pursuers to the engagement, the IBR algorithm finds successful Evader trajectories by the 17 th iteration. Using the notation introduced in (11) Fig. 9 and observe that the Evader trajectory avoids all twenty-four instances of the Pursuers (four pursuers, two guidance laws, three navigation ratios).
We note that the (solid, red) open-loop, simulated Evader trajectories do not exactly match the (dashed, red) predicted trajectories found by the IBR-SCP method (see first plot of Fig. 9). We attribute this error to the various approximations made by the SCP subroutine. Using the linear, time-varying model corresponding to the trajectory found at the last IBR iteration (produced as a byproduct of the SCP method), we design a finite-horizon, discrete-time LQR tracking control law [38] to be used online. By choosing the terminal state weights to be relatively larger than the running state and input weights, we produce closed-loop, simulated Evader trajectories that not only evade Pursuers but also reach the Asset. Recall that we had buffered the Evader's input acceleration to be 1[G] less in magnitude compared to that of the Pursuers in the IBR-SCP method (see Table II). The acceleration buffer may be used to realize corrective control actions produced by the feedback law.
VII. Conclusion
We have implemented an Iterative Best Response (IBR) algorithm to find solutions in a multibody asset guarding game. In particular, we have found open-loop trajectories for an Evader to evade multiple, more-maneuverable Pursuers and reach its target in minimum-time given state and input constraints. We capture the nature of the game by formulating a set of constrained optimal control problems that are coupled based on the roles taken by players in the game. The generalized modeling framework allows us to consider a wide variety of applications that can be modeled as assetguarding games, with an arbitrary number of Pursuers, different dynamical models, objectives and constraints. Leveraging recent advances in Sequential Convex Programming (SCP), we efficiently perform constrained trajectory optimization for the players as a subroutine in the IBR solution method. The players may be modeled as nonlinear dynamical systems with data in tabular form, opening up the potential to apply the IBR-SCP solution method to a wide class of practical applications. By implementing a tracking feedback control law about the solution trajectories that we find, we may mitigate the effects of model mismatch caused by the approximations used in the approach. Future work will apply the IBR-SCP solution method to other, more general differential games. In doing so, we will also attempt to find and characterize different solutions, including Nash equilibria. Furthermore, efforts should be made to understand the convergence properties of the algorithm and determine if any guarantees can be made.
Fig. 1 :
1Asset-guarding game with Evader E, Pursuers P , and Asset C E. Sin is with the Department of Mechanical Engineering, University of California, Berkeley, CA. M. Arcak is with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA. D. Philbrick is with the Naval Air Warfare Center Weapons Division, U.S. Naval Air Systems Command, China Lake, CA. P. Seiler is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI. Respective email addresses: ([email protected]), ([email protected]), ([email protected]), ([email protected]) arXiv:2011.01893v1 [eess.SY] 3 Nov 2020
Fig. 2 :
2Tabular data for atmospheric density, speed of sound, drag coefficient
Fig. 3 :
3Initial positions and velocity vectors to remain stationary while the Evader and Pursuers begin with the same speed. We restrict the Evader's input acceleration to be 1[G] less in magnitude compared to that of the Pursuers.
Figures 4-8 pertain to Example 1 (E versus P 1 ) where we discuss the solution in depth. The final figure 9 shows simulation results for all four examples where we verify the open-loop Evader trajectories against PN-guided Pursuer trajectories.
Fig. 4 :
4Trajectories for E and P 1 evolve after each IBR iterationFig. 5 shows a scaled plot of the 3-D position trajectories found at the last IBR iteration,
.45 seconds with a terminal speed of 1,593.1 [ft/s]. Fig. 7 plots the Evader's Mach number on the left-hand axis and the drag coefficient on the righthand side axis as functions of time. Considering the Evader's minimum-time objective we would
Fig. 5 :
53-D trajectories for E and P 1 after final IBR iteration Fig. 6: Evader's position and velocity trajectories
Fig. 7 :
7Evader's Mach Number and Drag Coefficient
Fig. 8 :
8Effective acceleration of Evader E and Pursuer P 1
Fig. 9 :
9Top to bottom: Evader (red, solid open-loop trajectory, dashed predicted trajectory) successfully evades against 1-, 2-, 3-, 4-Pursuers (blue, dashed PN law guided trajectories)
, we list the engagement parameters used for all examples. We assume that a Pursuer has captured the Evader (or the Evader has captured the Asset) if their relative distance falls under 1 [ft]. We enforce the Evader to maintain a comfortable 500 [ft] distance from the Pursuers to avoid capture. All players have a mass of 1,000 [slugs] and a cross-sectional diameter of 5 [ft], representative of the V-2 rocket. Each player's Mach number is constrained to stay above 0.5, or about 500 [ft/s].Parameter Value
Units
Description
First Mention
rc
1
[ft]
Capture Radius
Eqns (4), (10)
re
500
[ft]
Evasion Radius
Eqn (5)
m
1,000
[slugs] Mass
Eqn (13)
S
π
4 (5) 2
[ft 2 ]
Reference Area
Eqn (13)
Mmin
0.5
[ ]
Minimum Mach Number Eqn (16)
G
32.174 [ft/s 2 ] Standard Gravity
TABLE I :
IEngagement ParametersWe study four example engagements, each subsequent example including an additional Pursuer, i.e., Example 1 includes E versus P 1 , Example 2 includes E versus {P 1 , P 2 }, Example 3 includes E versus {P 1 , P 2 , P 3 } and Example 4 includes E versus {P 1 , P 2 , P 3 , P 4 }. The initial positions and velocity vectors are depicted with a birds-eye view in
TABLE II :
IIInitial conditions and acceleration bounds, i.e., u ∞ ≤ u max
, we may measure the change in the Evader's solution after each iteration by evaluating the Frobenius norm of the relative state trajectories between iterations:{x E } i − {x E } i−1F . If this quantity falls under some tolerance value IBR and remains under for subsequent iterations, we may consider this as a condition for IBR algorithm convergence. In each of the four examples we find that this quantity takes on large values in early iterations but drops significantly to stay under IBR =1e-2 by the IBR iteration numbers listed inTable III. With this definition of convergence and tolerance value, we may say that the IBR algorithm converges on Evader solutions for all four examples by the 19 th iteration.Example
Players
IBR Iteration i
{x E } i − {x E } i−1
F
1
E vs. P1
16
0.06e-2
2
E vs. {P1,P2}
12
0.18e-2
3
E vs. {P1,P2,P3}
19
0.05e-2
4
E vs. {P1,P2,P3,P4}
13
0.35e-2
TABLE III :
IIIIn each example, the change in Evader's state trajectory is minimal in later IBR iterations, implying IBR algorithm convergence These four examples highlight the ability to find successful Evader strategies. For different initial conditions we do find situations where the IBR algorithm converges on solutions where the Pursuers are successful. Furthermore, there are initial conditions where the algorithm does not converge on a solution within the user-defined number of IBR iterations N IBR . Fig. 9 illustrates our trajectory verification step (described in the prior Numerical Examples section) where the Evader's input trajectories found via the IBR-SCP method are applied as open-loop strategies in simulation against Pursuers guided by conventional guidance laws. In all four examples, the (solid, red) simulated Evader trajectories are successful in evading the (dashed, blue) Pursuers guided by either PN or APN laws. To illustrate, we consider the fourth plot in
VIII. acknowledgementsThe authors gratefully acknowledge support from the Office of Naval Research under grant N00014-18-1-2209. We thank our collaborators Jyot Buch from the University of Michigan, and Kate Schweidel, Alex Devonport, He Yin at the University of California, Berkeley.
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| []
|
[
"Creation of equal-spin triplet superconductivity at the Al/EuS interface",
"Creation of equal-spin triplet superconductivity at the Al/EuS interface"
]
| [
"S Diesch \nDepartment of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany\n",
"P Machon \nDepartment of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany\n",
"M Wolz \nDepartment of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany\n",
"C Sürgers \nPhysikalisches Institut\nKarlsruhe Institute of Technology (KIT)\nWolfgang Gaede Straße 1D-76131KarlsruheGermany\n",
"D Beckmann \nInstitute of Nanotechnology\nKarlsruhe Institute of Technology (KIT)\nHermann-von-Helmholtz-Platz 1D-76344Eggenstein-LeopoldshafenGermany\n",
"W Belzig \nDepartment of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany\n",
"E Scheer \nDepartment of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany\n"
]
| [
"Department of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany",
"Department of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany",
"Department of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany",
"Physikalisches Institut\nKarlsruhe Institute of Technology (KIT)\nWolfgang Gaede Straße 1D-76131KarlsruheGermany",
"Institute of Nanotechnology\nKarlsruhe Institute of Technology (KIT)\nHermann-von-Helmholtz-Platz 1D-76344Eggenstein-LeopoldshafenGermany",
"Department of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany",
"Department of Physics\nUniversity of Konstanz\nUniversitätsstraße 10D-78457KonstanzGermany"
]
| []
| In conventional superconductors, electrons of opposite spins are bound into Cooper pairs. However, when the superconductor is in contact with a non-uniformly ordered ferromagnet, an exotic type of superconductivity can appear at the interface, with electrons bound into three possible spin-triplet states. Triplet pairs with equal spin play a vital role in lowdissipation spintronics. Despite the observation of supercurrents through ferromagnets, spectroscopic evidence for the existence of equal-spin triplet pairs is still missing. Here we show a theoretical model that reveals a characteristic gap structure in the quasiparticle density of states which provides a unique signature for the presence of equal-spin triplet pairs. By scanning tunnelling spectroscopy we measure the local density of states to reveal the spin configuration of triplet pairs. We demonstrate that the Al/EuS interface causes strong and tunable spin-mixing by virtue of its spin-dependent transmission. | 10.1038/s41467-018-07597-w | null | 54,451,516 | 1811.08825 | 00420e38acbd3869bdccfb5e1f52b8a62eaa902b |
Creation of equal-spin triplet superconductivity at the Al/EuS interface
S Diesch
Department of Physics
University of Konstanz
Universitätsstraße 10D-78457KonstanzGermany
P Machon
Department of Physics
University of Konstanz
Universitätsstraße 10D-78457KonstanzGermany
M Wolz
Department of Physics
University of Konstanz
Universitätsstraße 10D-78457KonstanzGermany
C Sürgers
Physikalisches Institut
Karlsruhe Institute of Technology (KIT)
Wolfgang Gaede Straße 1D-76131KarlsruheGermany
D Beckmann
Institute of Nanotechnology
Karlsruhe Institute of Technology (KIT)
Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-LeopoldshafenGermany
W Belzig
Department of Physics
University of Konstanz
Universitätsstraße 10D-78457KonstanzGermany
E Scheer
Department of Physics
University of Konstanz
Universitätsstraße 10D-78457KonstanzGermany
Creation of equal-spin triplet superconductivity at the Al/EuS interface
10.1038/s41467-018-07597-wARTICLE OPEN. Correspondence and requests for materials should be addressed to W.B. (
In conventional superconductors, electrons of opposite spins are bound into Cooper pairs. However, when the superconductor is in contact with a non-uniformly ordered ferromagnet, an exotic type of superconductivity can appear at the interface, with electrons bound into three possible spin-triplet states. Triplet pairs with equal spin play a vital role in lowdissipation spintronics. Despite the observation of supercurrents through ferromagnets, spectroscopic evidence for the existence of equal-spin triplet pairs is still missing. Here we show a theoretical model that reveals a characteristic gap structure in the quasiparticle density of states which provides a unique signature for the presence of equal-spin triplet pairs. By scanning tunnelling spectroscopy we measure the local density of states to reveal the spin configuration of triplet pairs. We demonstrate that the Al/EuS interface causes strong and tunable spin-mixing by virtue of its spin-dependent transmission.
H ybrid superconductor-ferromagnet (S/F) heterostructures are fundamental building blocks for next-generation, ultralow power computers. In these devices, Cooper pairs of equal spin would carry spin information without dissipation 1,2 , reducing power consumption by several orders of magnitude 3 . Due to these promising applications, such exotic electron states have attracted considerable interest in recent years: after the initial prediction of triplet superconductivity in conventional s-wave superconductors, the theory has been elaborated to cover several scenarios, including multilayers and microscale devices [4][5][6][7][8][9][10] . The Pauli principle can be fulfilled for both singlet and triplet Cooper pairs by adjusting the symmetry in the time argument. Thus, spin triplet pairs in s-wave superconductors with even spatial symmetry must have a pair correlation function that is odd in the time argument, called odd frequency spin triplets 5,11 . Triplet pairs are created at interfaces between superconductors and ferromagnetic materials, and their properties have been studied theoretically intensively 12,13 . When a single magnetisation direction is present, solely m = 0 triplet pairs 1 ffiffi 2 p "# j i þ #" j i ½ are created by spin-mixing, i.e., spindependent phase shifts for the electrons scattering at this interface 11 . These triplet pairs can be converted into Cooper pairs with parallel spins, m ¼ ± 1 "" j i= ## j i ð Þ by a second, non-collinear magnetisation direction defining a new spin quantisation axis 10 . We will refer to these different kinds of pairings as mixed-spin (m = 0) or equal-spin (m = ±1) state.
Experimentally, the existence of equal-spin pairing has been shown by measuring the penetration depth of the supercurrent in ferromagnetic materials [14][15][16] , where the term long-ranged spintriplet superconductivity has been coined. More experimental indication for spin-triplet superconductivity has been reported from measurements of the critical temperature on S/F/F' spin valves [17][18][19][20] , from measurements of the supercurrent through S/F/ F′/F″/S junctions as a function of the relative magnetisation directions of the three ferromagnetic layers 21,22 and by studies of the paramagnetic Meissner effect in S/F bilayers 23 . These works, however, do not consider non-collinear magnetisations and can therefore, in principle, not distinguish different types of spintriplet superconductivity. Obtaining experimental support for the creation of triplet pairs from measurements of the quasiparticle density of states by scanning tunnelling spectroscopy (STS) has been suggested 24,25 . In a previous study, STS has been performed on Nb superconducting films proximity coupled to the chiral ferromagnet Ho 26 . In that work a Nb/Ho bilayer has been probed from the superconducting side and the obtained spectra were compared to a specialised theoretical model relying on the chiral magnetic state of Ho and pinning effects occurring at the Nb/Ho interface. Both zero-bias peaks and double peaks in the spectra were observed and interpreted as signatures of triplet superconductivity consistent with theory 25 . However, only very qualitative agreement between experimental and theoretical spectra could be achieved, most likely because the full boundary conditions were not used 27 .
As a result, it has not been revealed whether these triplets are equal-spin or mixed-spin pairs due to a missing unique signature, which would allow to distinguish between these two possibilities experimentally. Furthermore, the influence of the spin-dependent phase shifts are expected to be much weaker on the S side of the interface, resulting in only small amplitudes of the subgap features. To address these two issues, we study here the local density of states (LDOS) of an S/FI/N trilayer, where FI is a ferromagnetic insulator with non-collinear magnetisation. We first predict theoretically the formation of a small spectral gap, henceforth called triplet gap, in the LDOS and show how it is related to the creation of equal-spin triplet pairs. We then support the theory through experimental STS studies on the normal side of the trilayer. To our knowledge, such a method for clearly identifying the triplet states spectroscopically has not been reported before, neither theoretically nor experimentally. The fact that equal-spin and mixed-spin states result in distinctly different structures in the LDOS is a novel observation. The triplet gap develops around zero energy, resulting in a symmetric double-peak structure around zero-bias voltage in the LDOS. The width of the triplet gap monotonically depends on the ratio of equal-spin to mixedspin states in the pairing amplitude. Such a formation of an additional gap within the superconducting gap, solely depending on the magnetic structure in the proximity of the superconductor represents a new signature of equal-spin triplet Cooper pairs. Our experimental evidence presented below strongly supports the formation of an equal-spin triplet state, thus making a strong case to pursue superconducting spintronics.
Results
Circuit theory. In our theory based on the language of circuit theory 27-33 , a ferromagnetic insulator separating an s-wave superconductor and a normal metal can be represented by the circuit diagram depicted in Fig. 1a. Each conducting layer is represented by one node, characterised by its conductance G N/S (the index N/S labels the normal/superconductor side) and its size-dependent Thouless energy ϵ Th;N=S . The superconducting layer is specified by the pseudo terminal characterised by the pair potential Δ, which constitutes a source of coherence that has to meet the self-consistency relation 34 . The ferromagnetic insulator is described by a connector representing a tunnel barrier with the conductance G T . The ferromagnetic nature is accounted for by parameters G P andG P resulting from the spin polarisation of the tunnel probabilities 27,32 , and the so-called spin-mixing term G ϕ30 , which all depend on the respective magnetisation directions. We use two G ϕ terms, one in the ferromagnetic connector and one in the superconducting node, to account for the magnetic texture in the system (which we will discuss later). One advantage of the circuit theory approach is that it starts from very basic concepts and can easily be extended to describe many other realisations of S, F, N heterostructures. Note that in order to realistically represent the investigated circuit with the specific model chosen here, the layer thicknesses must not exceed the coherence length of the superconductor. Fig. 1 Illustration of the theoretical and experimental setup. a Circuit diagram representing the theoretical model. A superconducting S-node with a Δ-pseudo-terminal is connected to a normal conducting N-node (both with conductances G S/N and Thouless energies ϵ N=S ) via a tunnelling connector. The connector has the spin-dependent parametersG P ; G P and G ϕ , which depend on the relative magnetisation direction θ to the spindependent G ϕ term of the S-node. The relative angle θ between those magnetisations is the main free parameter in our fits. b Schematic of the tunnel contact. A PtIr tip (normal metal) is brought into tunnel contact with a trilayer sample of an EuS layer sandwiched between a normal conducting Ag and a superconducting Al film
The circuit diagram in Fig. 1a represents the discretised version of the Usadel equation 28,35 and has to be expanded with a spindependent boundary condition, details of which are shown in the Methods section and a full derivation of which can be found in previous publications 27,32,33 . Solving the Usadel equation (see Eq. (1) in the Methods section) allows us to calculate the LDOS in the N-node. Changing the direction of magnetisation in the ferromagnetic connector parametrised by the angle θ between the magnetisations of the ferromagnetic insulator's interior and the interface spins, results in an evolution of features inside the gap shown in Fig. 2a. For θ close to 0 and π, the LDOS shows a peak at zero bias (corresponding to the Fermi energy). For all other angles, i.e., situations where not all magnetisation directions are collinear, a gap opens symmetrically centered at the Fermi energy. This non-collinear orientations of magnetisations in the system has been identified as a mandatory prerequisite of equalspin triplet pairing 11 . Accordingly, the gap is most pronounced at θ = π/2, corresponding to a maximal equal-spin pairing. This is visualised in Fig. 2c, d, where the equal-spin triplet components of the anomalous Green's functions of the superconductor are plotted in the z-basis projected onto the magnetisation direction of the S-node m S . It is important to stress that the equal-spin triplet pairing F "" j i= ## j i has a distinctly different energy dependence than the mixed-spin pairing F "# j iþ #" j i shown in Fig. 2b. Hence, a full interpretation of the LDOS requires a simultaneous consideration of the energy-dependent pair amplitudes.
Scanning tunnelling spectroscopy. From the calculated LDOS, the theoretical differential conductance dI/dV can be calculated by including experimental parameters like non-zero temperature and amplitude of the voltage modulation added to the bias in order to perform lock-in measurements. These calculated curves are compared to dI/dV tunnel spectra, measured in lock-in technique between a normal metal tip and an Al/EuS/Ag trilayer sample (see Fig. 1b and Methods section) in a scanning tunnelling microscope (STM) at 290 mK, far below the superconducting critical temperature of the Al layer (T c = 1.7 K). As we discuss in Supplementary Note 2, there is strong evidence for the formation of an oxide layer at the interface between Al and EuS. As we will argue below, this oxide layer might be important for the formation of the non-collinear magnetisation arrangement, which itself is crucial for the formation of the triplet pairs. The changing direction of magnetisation in the tunnel connector is experimentally realised by exposing the sample to an external magnetic field parallel to the sample plane.
Spatial dependence of subgap features. In order to characterise the trilayer film sample, we record tunnel spectra (see Supplementary Note 1) by scanning the tip over the sample with a step size of 12.5 nm. In Fig. 3 we show a colour-coded map of typical dI/dV spectra. All spectra are normalised to the conductance value G b far outside the gap. We categorised the spectra in four distinct groups, each characterised by a specific shape ( Fig. 3b-e). Category (b) corresponds to tip locations where the tunnel contact is too noisy for spectroscopy or where superconductivity is being suppressed. Spectra of these types are only rarely observed. We attribute these spectra to surface contamination or defects in the film. Category (c) shows a spectrum with a hard gap as known −0.4 Bias (mV)
0 1 = 0 = /2 = = = = 0 = /2 = 0 = /2 LDOS (norm) a b F ⎥↑↑ 〉 c F ⎥↓↓ 〉 d F ⎥↑↓ 〉 + ⎥↓↑ 〉 0.0 0.4 −0.4 Bias (mV) 0.0 0.4 −0.4 Bias (mV) 0.0 0.4 −0.4 Bias (mV) 0.0 0.4 Fig. 2
Dependence of the superconducting properties on the magnetic configuration. a Evolution of the local density of states (LDOS) as a function of the relative angle θ between the magnetisation of the ferromagnetic connector and the G ϕ term of the S-node, showing zero-bias peaks with varying amplitude for parallel (θ = 0) and anti-parallel (θ = π) alignment and the appearance of the triplet gap around perpendicular alignment. b Pair amplitude of the mixedspin and c, d the equal-spin components as a function of θ. The mixed-spin component is prominent for θ ≈ 0 and θ ≈ π and almost vanishes around θ = π/ 2, whereas for the equal-spin component it is opposite. All curves have been offset vertically for better visibility NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07597-w ARTICLE from BCS theory, which is the result known for spin-independent tunnelling and is seen in all Al/Ag bilayer reference samples (see Supplementary Fig. 1). The gap size Δ ≈ 250 ± 20 μeV is slightly enhanced with respect to the bulk value of Δ 0 = 180 μeV as usual for thin Al films 36 . The most relevant spectra for this work are of type (d) and (e), in the following referred to as triplet gap and zero-bias peak spectra, respectively. They show coherence peaks at the gap edge similar to BCS theory, however, inside the gap the dI/dV remains finite everywhere with a minimum value ≈0.25-0.5G b . The triplet gap spectra (d) feature two peaks located symmetrically around zero bias, which are not necessarily symmetric in height. Category (e) shows a maximum at zero bias the amplitude of which can be larger than G b . While the majority of the spectra recorded on all measured samples show the BCS-like spectra (c), some areas show reproducible clusters of zero-bias peaks or triplet-gap features when addressing the same spot repeatedly. The fact that spectra can swap from triplet gap to zero-bias peak and to BCS like within one pixel reveals that the phenomenon responsible for the nature of the features inside the gap can change on short length scales, corresponding to, e.g., the grain size of the EuS thin film (8-18 nm) 37 .
Non-collinear magnetic model. To explain our findings we propose the following model: According to our theoretical studies, the appearance of spectra with triplet gap features corresponds to areas with at least two magnetisation directions which are non-collinear. We assume these two distinct magnetic areas are given by the bulk EuS film on the one hand, and by the interface to the Al layer on the other hand ( Fig. 4a-c), as we will explain in more detail below and in Supplementary Note 3. Second, the ferromagnetic interface must provide some degree of spin mixing between similarly oriented magnetic domains. These are two non-trivial requirements explaining why only a fraction of all spectra reveals these features. The dominance of BCS-like spectra indicates that the domain size of the ferromagnetic film is small. As wide areas of the EuS film have proven to be nanocrystalline in transmission electron microscopy (TEM) measurements (see Supplementary Fig. 3), the large number of domains probed simultaneously results in a minimised average magnetisation. Areas that do show zero-bias peak and triplet-gap features thus hint at locally enlarged domain sizes, thereby reducing the effective number of domains being studied at one spot and resulting in a net magnetic moment. Such an extreme sensitivity of the superconducting state to the local domain structure is well known 12 . As no voids or areas with reduced thickness of the EuS film are observed in the TEM images, we consider the possibility, that no ferromagnetic material was present in large parts of the trilayer, to be unlikely.
Magnetic field dependence. To further verify this model, we performed STS measurements with applied magnetic field, see Fig. 4. We assume that the magnetic configuration of a sample in the zero-field cooled (ZFC) state consists of domains with independent magnetisations pointing in random directions 38 (black arrows) and of moments found at the interface of the ferromagnet, that do not necessarily align with the direction of magnetisation of the underlying bulk domains 26 (grey arrows). Once exposed to external magnetic fields, those interface moments follow the direction of the external field only at higher fields compared to the magnetically softer bulk moments. These interface moments can be magnetically harder due to a local variation of the interface 39,40 or surface effects 41 . The exact nature of the interface moments is irrelevant for the functional principle of the creation of spin-triplet pairs by non-collinear magnetisation. As mentioned above, we observed an oxide layer at the interface between Al and EuS. In Supplementary Note 2, we show that this layer contains grains of EuO. EuO is a ferromagnet with a higher Curie temperature than EuS. Together with the small size of the grains they may be magnetically harder than the bulk EuS film and may help creating locally a non-collinear magnetisation configuration.
We can use the different coercive fields of the oxide interface and the bulk moments of the EuS layer to control the angle between them by applying an external magnetic field (Fig. 4a-c). In order to better understand the relationship between the features inside the gap and the magnetic properties of the F/S system, we show a complete up and down sweep of the in-plane magnetic field at a position on the sample, where a zero-bias peak appears in the ZFC state (for the evolution of a triplet gap feature in magnetic fields see Supplementary Note 4 and Supplementary Fig. 9), and we fit differential conductance spectra calculated using the model described above to the experimental data ( Fig. 4d-m). This fitting is done under the constraint that only the relative magnetisation angle θ between the oxide interface moments and the bulk moments can change between different set points of the external field. The parameters characteristic for the sample geometry G S=N ; G ϕ S=N ; P and the materials used (T c = 1.84 K, and derived from these self-consistently Δ = 280 μeV), were fit to the zero-field spectrum and then kept constant. We note, that the quality of the fits could be substantially improved by varying these parameters individually for every field independently, which would, however, not be justified by physical arguments. Due to the complexity of the model there are several combinations of spin-dependent parameters (see Supplementary Note 6 and Supplementary Table 2), which fit the experimental spectra almost equally well. However, all these parameter sets correspond to the same evolution of the relevant physical properties (see Supplementary Figs. 11 and 12), i.e., the same magnetisation configurations. The solutions share a stronginduced exchange field in the Al (here measured by G ϕ S ), and a large spin-polarisation P n of the tunnel current of at least 60%. For distinctly smaller values of either G ϕ S or P n the hallmarks of equal-spin triplets vanish. The material choice is thus crucial for the creation of equal-spin triplet pairs.
The observed field dependence can be consistently interpreted when assuming the tip to be located at an area on the sample, where the magnetic configuration in the ZFC state is anti-parallel. Anti-parallel configurations between interface moments and bulk magnetisation might be energetically favoured because of their reduced stray-field. Microscopically this could be realised by a magnetically harder layer of interface moments 41 or by the formation of a ferromagnetic oxidised state of EuS at the EuS/Al interface (see Supplementary Note 2). In the circuit theory model, this anti-parallel configuration results in a strong zero-bias peak in the LDOS, which corresponds to the creation of mixed-spin triplet pairs as visible by the peak in the mixed-spin pairing amplitude (Fig. 2b). The experimental data follow this prediction closely. As the magnetic field is increased, the bulk magnetisation readily follows the field direction, and at 10 mT already the angle between the interface moments (grey arrows) and the bulk magnetisation (black arrows) is decreased. Fitting this misorientation to our experimental data results in an angle of 75°, which means that the initial bulk magnetisation was rotated inplane by 105° (Fig. 4e). In our theory model, this rotation from anti-parallel to non-collinear magnetisations opens up a gap in the LDOS, which directly corresponds to the creation of equalspin triplet Cooper pairs, as signalled by the increasing equal-spin triplet pairing amplitudes (Fig. 2c, d). The experimental data clearly reflect this trend. As the magnetic field is further increased, the oxide interface moments finally also follow the field direction at around 150 mT. The triplet gap and its confining double peaks disappear to reveal again a zero-bias peak, corresponding to collinear magnetisations according to our fits. This behaviour, the appearance, disappearance, and reappearance of a double peak around zero bias, cannot be explained by a simple Zeeman shift of the LDOS (see Supplementary Note 8 and Supplementary Fig. 11). Further increasing the external field does not substantially change the features inside the gap, but suppression of the superconducting gap by the magnetic field Fig. 4 Model of the magnetisation behaviour of the EuS layer. a The sample in the zero-field cooled (ZFC) state consists of magnetically soft domains (black arrows) with an overall magnetic moment that is random in direction, and harder interface magnetic moments (grey arrows). b The internal domains are expected to follow the external magnetic field (directed into the plane of projection) more readily, aligning at smaller magnetic fields. c The interface moments follow the applied field only for higher field values. d-m Experimental dI/dV spectra (blue circles) recorded for the same tunnel contact in varying magnetic fields at 290 mK and theoretical spectra (black curves) fitted to the data. A full up and down sweep is performed to show that the observed curves depend on the magnetisation behaviour of the F layer. The black and grey arrows indicate the fitted relative angle between the different magnetisations. Direction of the external field is indicated in red. Data were recorded on sample EuS-1 starts at around |B| ≈ 300 mT. As expected, here the theoretical model does not describe the experimental spectra any more, as this suppression is most likely due to the onset of orbital depairing, which would requires yet another fit parameter in the theory. As we decrease the external field, no triplet gap opens up and no double peaks reappear, supporting our assumption that we started with a magnetic configuration in the anti-parallel state, which we cannot recover by decreasing the field. However, the zero-bias peak starts reappearing at around 300 mT and is fully developed at 100 mT. This model for the as-cooled magnetisation configuration also explains why finding such a transition is so rare-most in-gap features show a much less pronounced field dependence under varying the external magnetic field.
In conclusion, we have shown combined theoretical and experimental evidence that a non-collinear magnetic configuration of the S/FI interface leads to the appearance of a novel type of gap in the superconducting density of states. This triplet gap is closely related to the creation of equal-spin triplet Cooper pairs because it goes along with a significant increase of the equal-spin triplet pair amplitudes. Zero-bias peaks, contrary to earlier claims, do not hallmark equal-spin triplets, but short-ranged mixed-spin triplets. By selectively tuning the relative magnetisation direction between magnetic moments trapped at the interface and the softer magnetisation of the bulk domains, we are able to significantly influence the LDOS of the system. Our experiments provide spectroscopic evidence for the superconducting state induced by an FI interlayer with non-collinear magnetic texture. Here, the FI is realized by EuS that not only provides high spin polarisation and effective creation of spin splitting in the superconductor Al, but also builds up an oxide layer between the FI and the superconductor.
Our study also reveals that local variation from a collinear magnetisation arrangement is mandatory to form equal-spin triplet Cooper pairs. The FI thus fulfills several functions: By coupling it to the superconductor it creates spin-triplet correlations, promotes spin-dependent tunnelling of the pair amplitude, as well as non-collinear texture on small length scales. In EuS the texture is given by the grain size and by the formation of an oxide layer between EuS and Al. The texture can be elegantly tuned by an external magnetic field, and thereby the magnitude of the spinpolarised Cooper pairs can be adjusted, thus opening up the possibility for controlling dissipation-less transport of spin information in spintronics devices.
Methods
Circuit theory. The spin-dependent boundary condition of the Usadel equation for an arbitrary contact depends on the transmission probability T n , polarisation P n , the spin mixing angle ϕ n 42 and the magnetisation axes with unit vector m n , wherein the index n labels the transport channels. The number of channels depends on the size and the shape of the tunnel contact. For simplicity, we assume a smooth contact plane, thus the number of channels N is given as N ¼ Ak 2 F =ð4πÞ, where A is the tunnelling contact area and k F the Fermi wave vector.
We work in the tunnel limit T n ( 1 ð Þand assume small spin mixing ϕ n ( 1 À Á . In this case, the combination of the discrete Usadel equation and the boundary condition leads to two coupled equations, one for each node 29 , expressing the matrix current conservation
I S!N þ I L N ¼ 0 ¼ I N!S þ I L S :ð1Þ
The matrix currents are given by
I L N ¼ Àiϵ G N ϵ Th;N τ 3 ; G N " # ; I L S ¼ x' G S ϵ Th;S Àiϵτ 3 þ Δ À Á À iG ϕ S κ S ; G S " # I S!N ¼ 1 2 G T 2 G S ; κ È É κ þG P 2 G S ; κ Â Ã κ þ G P G S ; κ È É À iG ϕ κ; G N ! I N!S ¼ 1 2 G T 2 G N ; κ È É κ þG P 2 G N ; κ Â Ã κ þ G P G N ; κ È É À iG ϕ κ; G S ! :
The equations are closed by demanding the normalisation conditions
G 2 S ¼ 1 ¼ G 2 N .
The conductances are defined from the experimental interface parameters. The spin-independent parameter is G T ¼ 2 P n T n and the spindependent parameters areG P ¼ 2 P n T n ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À P 2 n p , G P ¼ P n T n P n , and G ϕ ¼ 2 P n δϕ n . We defined κ ¼ τ 3 mσ, with τ i and σ i being Pauli matrices in Nambu and spin space, respectively. For the theoretical curves shown in Figs. 2 and 4, the following parameters were used:
G S = G T ϵ Th;S = 4.1/(k B T c ), G ϕ S =G T ¼ 5, G N = G T ϵ Th;S = 0.07/(k B T c )
, G ϕ /G T = −0.061, and P n = 0.6.
The directions of the respective magnetisation directions are denoted by m. The gap matrix is defined by Δ ¼ Δτ 1 . Note that the LDOS measured in the experiment does not depend on the lateral system size and, hence, Eq. (1) can be normalised e.g. by the conductance of the connector G T , thus giving a solution independent of the size of the interface area. The leakage parameters can be further related to sample parameters via G N=S =ϵ Th;N=S NG 0 = 8π 2 N 0;S=N d S=N =k 2 F;S=N~dS;N = hv F;S=N with the thickness of the layer d, the density of states at the Fermi energy N 0,S/N and the Fermi wave number/velocity k F,S/N , v F,S/N . We also note, that the order parameter Δ has been calculated self-consistently according to the standard BCS relations.
Sample fabrication and characterisation. For samples EuS-1 to EuS-4, a 25 nm Al layer, followed by an EuS film of varying thickness, and a 5 nm capping Ag layer are deposited by e-beam evaporation on a silicon (111) chip. Sample EuS-5 had a thinner Al layer of ≈10 nm. For this work, five batches of samples were fabricated, with the exact parameters shown in Supplementary Table 1. The substrate and the sample holder are cooled to below 100 K using liquid nitrogen to grow an Al film of homogeneous thickness with relatively small grain size. At 25 nm, the surface RMS roughness of Al is <0.6 nm, thus forming a smooth surface for the subsequent EuS and Ag layers. The EuS layer is evaporated onto the substrate at higher temperatures to provide a compound with a Curie temperature (see Supplementary Fig. 5) close to the value reported in the literature (T Curie = 16.7 K 43 ). At room temperature, EuS is semiconducting with an indirect energy gap, the conduction band minimum at 300 K is 1.64 eV 44 . At cryogenic temperatures, EuS is insulating with resistivity of around ρ = 10 4 Ω cm for high-quality single crystals 45 . Disorder at the atomic level and unintentional doping can lower this resistivity by several orders of magnitude and at the same time increase the Curie temperature (as observed by SQUID magnetometry measurements in Supplementary Fig. 6) due to interactions between charge carriers and the Eu 2+ ions 45 . Because of the presence of an additional insulating oxide film under the EuS layer (see Supplementary Figs. 3 and 4), we assume the ferromagnetic insulator layer to have very low conductance, i.e. enabling only tunnel transport. The surface RMS roughness of this oxide layer is ≈0.52 nm for line profiles recorded on TEM lamellas.
The deposition rate for the EuS film is 0.01 nm s −1 , and the final thickness of the film is varied from sample to sample (each chip holding samples with four different thicknesses) using a sample holder with a movable shutter that allows parts of the chip to be covered during evaporation. The surface RMS roughness of the EuS layer is ≈0.59 nm for line profiles recorded on TEM lamellas. However, during STS measurements we often observe variations of the spectra taken on different locations on the same sample to be more pronounced than the differences between the various film thicknesses. This supports our theory that the EuS film has only vanishing magnetic influence on a proximitised superconductor when nanocrystalline, whereas regions with more uniformly magnetised domain clusters or locally enlarged domains can strongly influence the superconducting state. Thus, the lateral extent of the magnetically ordered region seems to play a much larger role than the film thickness.
Some of our EuS films were found to be conductive, allowing for a proximity coupling between Al and Ag that is much stronger than anticipated. These samples generally displayed triplet gap features only for very few locations on the film, or often not at all. Al is a suitable superconductor for this study because of its long spin relaxation time 46 . Thus, electrons that have experienced spin-mixing at the EuS interface can carry that information far into the superconductor. The bulk critical field of Al (μ 0 H c = 10 mT) is significantly increased in thin films (see Supplementary Fig. 2). For in-plane fields at 25 nm thickness it is around 800 mT 46 . In order to protect the EuS layer from contamination and to have a normal metal layer providing clean metallic surface necessary to perform STM and STS, Ag is an ideal choice for the top layer, as the proximity effect of diffusive Al/Ag bilayers is well understood and is reliably described by the Usadel equation 47 . The surface RMS roughness of the entire Ag-capped multilayer film is <0.7 nm. The Al (25 nm)/EuS(5 nm)/Ag(5 nm) (Fig. 1a) multilayers have a critical temperature of T c ≈ 1.7 K (Supplementary Fig. 7), similar to the critical temperature of the Al (25 nm)/Ag(5 nm) reference sample.
High-resolution transmission electron microscopy (TEM) images ( Supplementary Fig. 3) show all films (including the oxide layer between the Al and EuS films) to be nanocrystalline with visible lattice planes within most grains. SQUID magnetometry (Supplementary Fig. 5) and further subgap features (Supplementary Note 7 and Supplementary Fig. 13) are shown in Supplementary Information.
Scanning tunnelling microscopy. Scanning tunnelling spectroscopy with an IrPt tip is performed in a 3 He cryostat at 290 mK, which results in differential conductance dI/dV vs. voltage V spectra of the types shown throughout this work. All spectra are normalised to the conductance value far outside the gap. Spectra were recorded in lock-in technique across a 10 MΩ tunnelling gap, the STM set point was set to a tunnelling current of 400 pA at 4 mV tip voltage and the feedback loop was then stopped before voltage sweeps. The STM used to conduct this study was home built in Konstanz 48 and optimised for high energy resolution spectroscopy at low temperatures 49 . The STM controller used was a commercially available SPM1000 by RHK with a proprietary current amplifier (IVP-300) for STM studies. The amplitude of the AC voltage modulation added to the tip bias voltage was set between 7 and 20 μV at a frequency of 733 Hz.
Data availability
The datasets generated during and analysed during the current study are available from the corresponding authors on reasonable request. The compiled custom computer code applied during the current study is available from the corresponding author W.B. on reasonable request.
Received: 1 October 2018 Accepted: 12 November 2018
Fig. 3
3Spatial dependence of the differential conductance. a Map of locations on the sample surface where different types of spectra shown in b-e have been observed. The different colours in a correspond to the different types of spectra observed and match the spectra shown in b-e. The shape of the spectra changes on length scales corresponding to the grain size of the EuS films (8-18 nm)37 . Data were recorded on sample EuS-3
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07597-w
NATURE COMMUNICATIONS | (2018) 9:5248 | DOI: 10.1038/s41467-018-07597-w | www.nature.com/naturecommunications
© The Author(s) 2018
AcknowledgementsWe are grateful to A. di Bernardo, F. Giazotto, H. v. Löhneysen, P. Leiderer, J. Moodera, O. Millo, J.W.A. Robinson, R. Schneider, and C. Strunk for fruitful discussion. We thank M. Wolf for preparing early sample batches, M. Krumova and R. Schneider for providing the TEM data, S. Andreev and G. Fischer for carrying out the SQUID magnetometry measurements. This work was partially funded by a scholarship according to the Landesgraduiertenförderungsgesetz, by the Baden-Württemberg foundation in the framework of research network Functional Nanostructures, by the Deutsche Forschungsgemeinschaft (DFG) through SPP 1538 Spincaloric Transport and a Deutsch-Iraelisches Projekt, as well as by the Leverhulme Trust.Author contributions E.S. and W.B. conceived and developed the project. C.S. fabricated the samples, C.S. and D.B. performed sample characterisation studies. S.D. and M.W. performed the tunnel experiments. P.M. and W.B. analysed and described the theoretical model and carried out the numeric implementation of the theory. S.D. analysed the data and performed the fitting procedure. S.D., P.M., C.S., W.B., and E.S. wrote the paper, all authors discussed the results and the manuscript.Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-07597-w.Competing interests: The authors declare no competing interests.Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.
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"Evaluating a Methodology for Increasing AI Transparency: A Case Study",
"Evaluating a Methodology for Increasing AI Transparency: A Case Study"
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"John Richards \nIBM Research AI\nUSA\n",
"Michael Hind \nIBM Research AI\nUSA\n"
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| In reaction to growing concerns about the potential harms of artificial intelligence (AI), societies have begun to demand more transparency about how AI models and systems are created and used. To address these concerns, several efforts have proposed documentation templates containing questions to be answered by model developers. These templates provide a useful starting point, but no single template can cover the needs of diverse documentation consumers. It is possible in principle, however, to create a repeatable methodology to generate truly useful documentation. Richards et al.[25]proposed such a methodology for identifying specific documentation needs and creating templates to address those needs. Although this is a promising proposal, it has not been evaluated.This paper presents the first evaluation of this user-centered methodology in practice, reporting on the experiences of a team in the domain of AI for healthcare that adopted it to increase transparency for several AI models. The methodology was found to be usable by developers not trained in user-centered techniques, guiding them to creating a documentation template that addressed the specific needs of their consumers while still being reusable across different models and use cases. Analysis of the benefits and costs of this methodology are reviewed and suggestions for further improvement in both the methodology and supporting tools are summarized. | null | [
"https://arxiv.org/pdf/2201.13224v1.pdf"
]
| 246,430,679 | 2201.13224 | fc3a6af9c12f690a2b72c6e4e6e1514d45dc7f2d |
Evaluating a Methodology for Increasing AI Transparency: A Case Study
David Piorkowski
IBM Research AI
USA
John Richards
IBM Research AI
USA
Michael Hind
IBM Research AI
USA
Evaluating a Methodology for Increasing AI Transparency: A Case Study
CCS Concepts: • Software and its engineering → Documentation• Computing methodologies → Artificial intelligence• Human-centered computing → Field studies Additional Key Words and Phrases: AI Transparency
In reaction to growing concerns about the potential harms of artificial intelligence (AI), societies have begun to demand more transparency about how AI models and systems are created and used. To address these concerns, several efforts have proposed documentation templates containing questions to be answered by model developers. These templates provide a useful starting point, but no single template can cover the needs of diverse documentation consumers. It is possible in principle, however, to create a repeatable methodology to generate truly useful documentation. Richards et al.[25]proposed such a methodology for identifying specific documentation needs and creating templates to address those needs. Although this is a promising proposal, it has not been evaluated.This paper presents the first evaluation of this user-centered methodology in practice, reporting on the experiences of a team in the domain of AI for healthcare that adopted it to increase transparency for several AI models. The methodology was found to be usable by developers not trained in user-centered techniques, guiding them to creating a documentation template that addressed the specific needs of their consumers while still being reusable across different models and use cases. Analysis of the benefits and costs of this methodology are reviewed and suggestions for further improvement in both the methodology and supporting tools are summarized.
The FactSheets methodology is a documentation process centered on three roles: content producers, content consumers, and a FactSheet team (FS team). The FS team follows the methodology and shepherds the creation of a FactSheet. During this process, they collaborate with both producers, who have the information to be captured, and consumers, who have specific documentation needs. The FS team determines consumers' needs and records them as fields to be completed in a FactSheet template. Then, the FS team iterates with producers and consumers to fill out and refine the template, producing a FactSheet.
The goal of this paper is to evaluate this methodology via a case study of an independent team that built FactSheets for multiple real AI models and systems over three months. We interviewed 16 participants to address the following research questions: RQ1 Is the methodology usable by FS team members not trained in human-centered practices?
RQ2 How well did the resulting FactSheets address the needs of different consumers?
RQ3 What did consumers and producers of FactSheets see as the benefits and costs?
BACKGROUND
AI transparency requires the availability of model and system documentation that is understandable and trustworthy.
Documenting any software artifact well requires effort. AI models, and the systems in which they are embedded, pose additional documentation challenges [13]. First, AI development is highly collaborative. People with diverse skills, specialized vocabularies, and unique tools all contribute to the final deployed model or system. Second, AI development is highly iterative. Model development often begins with a series of lightweight experiments. Multiple threads are followed with frequent false starts and backtracking. Along the way, important decisions about training data and algorithm refinement are rarely captured. Finally, tooling to support traditional software development is not easily adapted to AI development and documentation. Proposals for new mechanisms for collecting important facts about AI development throughout the lifecycle [13], may lead to improved tooling in the future. Currently, only the most disciplined teams can reliably capture information needed for complete and accurate documentation.
FactSheets and FactSheets Methodology
To tackle the complexity of creating AI documentation that affords effective transparency, Richards et al. [25] have proposed a human-centered methodology for creating the form of documentation called AI FactSheets. FactSheets [1,13,14] are collections of important facts about the development, testing, and deployment of AI models and systems.
These include facts about model purpose, training data selection and cleaning, algorithm selection and tuning, testing for accuracy, bias, privacy risks, adversarial attacks, etc. Different consumers of FactSheets (data scientists, business owners, system integrators, deployment engineers, risk officers, regulators, end users, affected subjects, etc.) will have different skills and different information needs. By engaging with these consumers, it is possible to discover both common and unique needs and, thereby, create FactSheets that meet those needs. Those needs may include how to render the documentation, providing for example, a condensed table view for a quick overview or slides to aid presentation of the model [14]. By engaging with fact producers, it is also possible to discover what facts they can create for consumption by others and how to help them create these facts efficiently. Earlier work [25] has posited that executing the steps of this methodology will lead to useful FactSheets, contributing to model transparency.
The methodology consists of seven steps. The following list presents a somewhat idealized view insofar as there may be iterations within and between steps. In other cases, some of the steps may be collapsed due to informants playing both fact consumer and fact producer roles within the lifecycle [13]. Further details on these steps, including example questions to explore with informants, can be found in [25].
(1) Know Your FactSheet Consumers. FactSheets (and by extension, any form of AI documentation) are produced so that they can be consumed. Understanding the information needs of FactSheet consumers is the first and most important task. This initial exploration need not be formal. Working with even one representative informant from each major process in the AI lifecycle will provide useful insights into the overall set of consumer needs.
(2) Know Your FactSheet Producers. Some facts can be automatically generated by tooling. Some facts can only be produced by a knowledgeable human. Both kinds of facts will be considered during this step. Again, working with even one representative producer of relevant facts from each major process in the AI lifecycle will provide the information needed to proceed to the next step.
(3) Create a FactSheet Template. What is learned in the first two steps leads directly to the most important part of creating FactSheets, namely the creation of a FactSheet template. A FactSheet template will contain what can be thought of as questions or, alternatively, the fields in a form. Each individual FactSheet will contain the answers to these questions. For example a template may start with the question "What is this model for?". It may then expand on that question by asking where the model is well-suited and where the model is ill-suited.
(4) Fill In FactSheet Template. This step is where the creator or creators of the FactSheet template attempt to fill it in for the first time. As this is done, it is important to informally assess the quality of the template itself by reflecting on what was learned about both consumers and producers, their skills, and information needs in the first two steps. While this assessment is not a substitute for further work with them (to follow), it may quickly highlight where improvements are needed.
(5) Have Actual Producers Create a FactSheet. In this step, actual fact producers fill in the template for their part of the lifecycle. For example, if there is a question in the template about model purpose, find someone who would actually be providing that information and have them answer the question. Ask a data scientist to answer the questions related to the development and testing of the model. If this model was validated, the model validator would enter information about that process. Similarly, a person responsible for model deployment would answer questions related to deployment and ongoing monitoring. If the lifecycle is not that structured, the person responsible for most of the work could fill in this template.
(6) Evaluate Actual FactSheet with Consumers. In this step an assessment is conducted of the quality and completeness of the actual FactSheet produced in the previous step. If the FactSheet is intended to be used by multiple roles (not uncommon), it should be evaluated separately for each role. To ground this assessment properly, each consumer should be asked to reflect and comment on how this FactSheet would actually help them perform their work or provide information to address their concerns.
(7) Devise Other Templates for Other Audiences and Purposes. This step returns to the beginning and is the only defined iteration in the methodology. There may be other consumers that need to be supported. Or, if the work so far has focused on the process of creating and deploying AI models, it would be worthwhile to consider consumers beyond this such as internal or external review boards or regulators, sales personnel, and end users or others affected by the product or service.
RELATED WORK
The drive to improve AI transparency via documentation has been motivated by the numerous high-profile examples of AI risk in society. This has sparked research groups [1,3,11,13,17,18,24] and government agencies [7,19,28] to explore what form this documentation should take. The main focus of these works has been on what should be included in this AI documentation for near universal use. Several of these efforts have included feedback loops, where society and industries have been able to comment on the proposals. Essential stakeholders for AI transparency are citizens who want to know more about the algorithmic decisions that are affecting them. This is challenging because the average citizen is not familiar with the technical details of AI. In recent work, Domagala and Spiro [9] engaged with citizens to better understand their needs regarding information about algorithmic systems deployed by the UK government. This was one way to approach Step 1 of the methodology described in Section 2. Earlier, Hind et al. [13] explored the needs of documentation producers via interviews and a documentation creation exercise lasting a few hours. This paper differs from these works since it examines AI documentation practice in situ through a months-long case study involving both producers and consumers.
Software documentation, in general, describes important characteristics of a system, application, or module. Despite its necessity, developers and software engineers find the task of documenting uninteresting and its creation often falls to technical writers who do not know all the details of the software they are documenting [20]. Not surprisingly, consumers of documentation generally dislike what is produced because such documentation is often incomplete, difficult to understand, or out of date [4]. But documentation is essential as it remains one of the few channels of communication between the developers of software and its various users [15].
For conventional software, researchers have developed sets of rules [20], evaluation critieria [22], and assessment methodologies [8,23,27]. Quality attributes such as accuracy, concreteness, writing style, and understandability have been offered as useful dimensions of quality [23]. Other dimensions such as completeness, unambiguity, conciseness, and ease of access have been proposed [20]. Yet other dimensions include consistency, traceability, reusability, format, trustworthiness, and retrievability [8,30]. This paper draws on many of these dimensions to assess the quality of AI FactSheets.
EVALUATION METHODOLOGY
To answer our research questions, we partnered with an AI organization in the healthcare domain that was piloting the FactSheets methodology [25] for several models as a possible solution to their AI documentation and transparency needs. Data for the evaluation reported here consisted of two sets of interviews for each model: one for the FS team who followed the methodology to create the FactSheets, and one for FactSheet consumers to assess how the resulting FactSheets were meeting their needs. Together, these two perspectives provide a holistic view of the usefulness of the methodology.
Study Participants and Case Study Context
We divided participants into two roles: consumers and FactSheet (FS) team members 1 . As in the FactSheets methodology, consumers are the intended end users of the created FactSheet. FactSheet team members are the ones driving the methodology for each of the models, defining the FactSheet template, and iteratively making the FactSheet. In total, we ran 17 interviews over 16 participants: seven FS team interviews and ten consumer interviews. One participant was Table 1. Throughout this paper, we identify participants using a three-part code. The first character indicates whether a participant is a FS The participants in this case study are part of an AI organization responsible for creating and maintaining several different kinds of models for a variety of health and medical use cases. During the case study period, the team worked on documentation for four models (
Interview Protocols
For the FS team, we conducted approximately one-hour, semi-structured interviews that focused on how well the methodology worked or did not work in practice. More specifically, we focused on five main topics: (1) the context of, and motivations for, creating the FactSheet, (2) how they used the available resources, (3) how they implemented the FactSheets methodology within their actual context, (4) perceived benefits, and (5) perceived costs.
Interviews with the consumers lasted approximately 30 minutes. Unlike the FS team interviews, that focused mainly on the process of creating a FactSheet, consumer interviews focused on whether the created FactSheet met their needs, and if not, where additional or different content was needed. Topics in consumer interviews focused on: (1) evaluating the quality of the created FactSheet and (2) how the FactSheet has changed the way they work compared to the past.
Both FS team and consumer interviews were similarly run. In each interview, participants were given access to the finished (or in progress) FactSheet that they could refer to as needed. Interview sessions were run remotely, recorded, and transcribed for later analysis.
Data Analysis
To analyze participants' responses, we used two approaches. For questions that expected more direct responses such as identifying missing content in the FactSheet, we identified and aggregated participants' answers. For more complex questions such as the ones around perceived benefits and costs, we conducted thematic analysis [2], focusing on themes that addressed our research questions. First, participants' responses were extracted from each interview transcript such that a complete response was available for each question, including any relevant context. One author extracted responses, annotating any references a participant made to the FactSheet as necessary.
All authors participated in the thematic analysis. The thematic analysis was an iterative and inductive process, with new themes emerging and collapsing as the authors worked through the data. Using the set of interview responses, each of the authors coded any sentences with a theme or topic relevant to the research or interview questions. Extracted codes and their definitions were iteratively added, consolidated, and removed as needed to form the codes. Upon completion, all three authors together discussed codes and consolidated codes into themes and subthemes. The emergent themes and subthemes contributed to our findings.
RESULTS
Our results are structured by research question. We address if the methodology was usable by a real-world FactSheets team not trained in human-centered practices (RQ1), if the resulting FactSheets met the needs of their various consumers (RQ2), and what the benefits and costs of this approach were seen to be (RQ3).
RQ1: Usability of Methodology by FS Team Members
Practitioners skilled in human-centered practices will recognize that the methodology applies these practices to the creation of FactSheet templates (and the resulting FactSheets). But is it reasonable to expect data scientists, engineers, and others not trained in these practices to execute the methodology and derive the expected benefits? In this section we report on how well FS team members were able to apply the methodology. Each subsection looks at a different aspect of the methodology's usability: the value of available educational resources; how well individual steps of the methodology could be followed and how useful they were; and the ways in which the methodology helped in documenting AI models.
How FactSheet Teams Used Educational
Resources. The FactSheets 360 website [14] provides an overview of the methdology, a number of illustrative FactSheets, and links to more detailed instructional content. When we asked the four FS team members who used the website to identify the resources that were the most useful to them, responses included the example FactSheets on the website (T1-A and T5-B) and the methodology summary (T2-A). When asked for more details, T1-A described how one example showed how disparate kinds of documentation could be brought together into a single place saying "It was self-contained. It provided all the info I needed to understand. " T5-B described how the different views in the example FactSheets, such as the table view and slide view, helped him understand how FactSheets could be tailored to the needs of different consumers saying "Something clicked (for me) on one of the (example) pages about how the same facts can be represented in different ways, which was the key idea for me to really see the value of this approach. I haven't thought about documentation in that way before". T6-D echoed this sentiment explaining how many of the discussions around what to document revolved around this perspective on documentation saying "There were two critical things on my mind. It was understanding the players. What do we mean by producers?
What do we mean by consumers?" By framing documentation as something to be used by others, FS teams changed their orientation from just reporting on what a model did to focusing on how to make the description useful for specific consumers. More details on this observation follow in Section 5.1.2.
FS team members did not report that any resources were unhelpful. They did identify some gaps and stated that there was too much content to digest it all. T1-A identified procedural guidance that he would have liked to have such as a "minimum set of fields" required for a FactSheet template. T2-A described a desire to have concrete guidance for how long the methodology should take and which facts to start with. To address this, T2-A developed a schedule based on his team's first attempt at creating a FactSheet template, which provided guidance for subsequent efforts. "We needed something to explain the process... Starting with the methodology of the site and then breaking it out into 'Here's a six-week schedule' basically. To give (the rest of the team) something concrete." Similarly, to address T1-A's concern, the team shared the template (tailored through their work for the healthcare domain) and FactSheet for future teams to use as a starting point. about, but never documented in any way. " Although FS team members did not consider any specific steps of the methodology to be unhelpful, they did adapt the methodology to better fit their team's needs and timeline. FS teams chose, for example, to minimize the seeming seriality of Steps 5 (have actual producers create a FactSheet) and 6 (evaluate the FactSheet with actual consumers). They preferred a more iterative approach where FactSheet content was evaluated as soon as there was enough for a specific consumer role, even if the rest of the information was not yet ready. These changes were implemented during the first pilot. Since the subsequent pilots were informed from the first, T3-B, T4-B, T5-B, T4-C and T6-D all followed the altered protocol, further compressing the timeline by starting with the template from model A.
How the Methodology Helped FactSheet
Teams Document AI Models. FS team members noted how the methodology provided specific benefits over their previous documentation practices. Table 3 summarizes the questions about specific benefits that we asked about and participants' responses. All FS team members agreed that the FactSheets methodology would likely enhance documentation consistency, primarily by consolidating what was previously scattered in multiple documents into a single place. T5-B summarized, "We'll be able to consolidate in ways that make sense so that there's one place for facts, rather than, lots of places to keep things up to date. " T5-B and T6-D described how completing the FactSheet template had caused them to reflect on the usefulness of what they were writing in the FactSheet fields. T5-B described how a consumer need for making the problem description reusable encouraged conciseness. He said, "One of the best (consumer) feedbacks that we received was in the way we tried to condense the problem description in the FactSheet so that they could copy and paste it to engage with customers". Importantly, five of the FS team members agreed that the methodology helped their team identify new documentation needs. They discovered several additional template fields as a direct result of following the methodology. Table 4 shows most of the fields from the FactSheet template for model A. Newly-discovered fields are italicized. Examples included regulatory requirements, model maturity, usage considerations, and run time requirements. T6-D summarized the value stating, "There was a lot of information here that we wouldn't perhaps otherwise made available or collected or even thought of. "
RQ2: FactSheet Usefulness for Consumers
To determine whether the FactSheets generated by the FS teams were useful to consumers, we asked consumers to (1) evaluate the FactSheet's quality along several dimensions and describe any gaps and (2) to describe if and how the FactSheet differed from the AI documentation they had encountered previously.
Evaluation of Usefulness.
We asked consumers to evaluate the FactSheets, along multiple quality dimensions, on a seven-point Likert scale ranging from "strongly disagree" to "strongly agree". The dimensions were drawn from work on assessing AI documentation quality [21] along with prior research on the pragmatics of effective communication.
The dimensions and prompts are shown in Table 5. We found that the overall quality of the FactSheets produced was excellent, scoring high marks across all the quality dimensions as shown in Figure 1. These results suggest that the
Things That Were Missing.
Even with the high quality-dimension scores, consumers were able to identify things that were still missing. For any response to a prompt with a score of less than 'strongly agree', we asked the respondent what was missing or needed to be changed for that dimension. Table 6 summarizes their replies.
A closer look at this reveals that missing content tends to be quite role-specific. For example, C2-A's suggestions are related to how the model fits into the larger business context. He wanted to see more information about which customers are using the model along with pointers to additional materials useful in discussions with potential customers.
He stated, "It might be good to include something about where has the model been deployed?" The requests from C3-A, whose role involved regulatory concerns, focused on risk: the model context, risk evaluations, and risk mitigation. C4-A and C10-B, both in customer-facing roles, focused on how to determine model quality and how the quality compares to Table 5. Likert-scale prompts given to consumers to assess FactSheet quality
Quality Dimension Prompt Completeness
The FactSheet has all the information that I require for my use case. Evidence
The FactSheet's information is well supported with additional evidence provided where needed. Vocabulary
The FactSheet is written using appropriate vocabulary and word choice. Understandability
The FactSheet's content and information is easy to understand. Layout
The FactSheet's structure and layout is intuitive.
Representation
The FactSheet's information is presented in the expected way with text, tables, and figures appropriately chosen. Organization
The FactSheet's information was well organized and easy to locate. further evidence that a one-size-fits-all solution for AI documentation will likely not suffice.
KEY
The combination of high quality scores for the FactSheet along with several suggestions for further improvement tell a somewhat mixed story. On the one hand, as indicated by the high scores, participants seemed to be satisfied with the content of the FactSheets. On the other hand, they still had suggestions for how to improve them. There are potentially several reasons for this. FS teams may not have iterated enough with consumers to gather all the relevant documentation needs (on average going through only two iterations). Or FS teams intentionally did not address all these needs to keep FactSheet size manageable (a role-based filtering mechanism, see 6.2, may help manage this trade off). Another interesting possibility is that by asking consumers in our interviews to reflect on a FactSheet from the perspective of the quality dimensions, gaps surfaced that they did not see before. If true, this latter interpretation may have implications for eliciting consumer needs more completely. Customer-Facing Customer-specific model evaluation
RQ3: Methodology Benefits and Costs
To better understand the value proposition of both the methodology and FactSheets themselves, we asked FS team members and consumers what specific pros and cons they experienced, further reflecting on how this compared with their documentation experiences from before. T1-A T2-A T3-B T4-B T5-B T4-C T6-D 15 hours 24 hours 10 hours 8 hours 15 hours 24 hours 6-8 hours and then I can understand where to go to find more information. " C3-A and C6-A explained that the FactSheet served as a pointer to the right people to contact for further follow up, potentially reducing the time spent gathering information about a model. C3-A summarized, "I think it reduces the amount of time that will be spent of clarifying issues... It would allow me to do pre-work without having to do 3 meetings with a group of 5 to 6 people. " C5-A and C7-A likewise agreed that it reduced the number of meetings required to understand the current state of a model. C1-A, C4-A and C10-B echoed similar sentiments for discussions with customers.
In addition to serving as a jumping-off point, FactSheets acting as the sole source (or at least sole anchor) of truth, enabled FS teams and consumers to reduce work in several ways. Previously, there would be several sources of documentation that sometimes provided conflicting information. The FactSheet as the single source of truth allowed content to be reliably copied for other creations such as customer-facing descriptions, reports, or presentations. C8-B recalled a time that she was able to reuse content in the FactSheet for a client. She said, "This is actually really nice that they've got this [documentation field] added in here because now I can lift this paragraph and put it into a client document... And in an approved definition or language for how to describe [it]. " One FS team member, T3-B, likewise used the FactSheet as a text source for additional materials. Simply knowing where the most current documentation is also enabled more effective employee on boarding, especially when the people who worked on the model were no longer available. C6-A summarized, "You would talk to [someone] who'd go talk to someone else who has a link somewhere, or sends you some old weird spreadsheet that says, 'go talk to this person'. And then someone would have left the company and then they don't have access to something. So this is much much better, because it puts everything in one place." Finally, FS team members also reported benefits centered around the theme of improved organization. They reported how
Methodology
Costs. The most obvious cost faced by participants was simply the time required to create the template and FactSheet. FS team members reported times ranging from 6 to 24 hours (Table 7). This estimate includes all the steps of the methodology, including meetings with others and time spent filling out the fields. Of course, creating documentation of any sort takes time, and it is not clear that this is outside the norm. Scheduling time with consumers was the most common cost mentioned by FS team members, specifically, T1-A, T2-A, T5-B, and T6-D.
Aside from the challenge of getting people together, some steps of the methodology required additional effort from the FS team. One mentioned by FS team members T1-A, T3-B and T6-D was Step 4, filling in the FactSheet template for the first round of consumer feedback. This approach is meant to make the feedback sessions more streamlined as it asks consumers to critique something that exists instead of creating it on the spot. However, this approach passed the burden off to the FS team, leaving them to fill out fields they may be unfamiliar with.
Another challenge in preparing that first draft was locating some of the documentation that existed from past work on the model. Both T3-B and T6-D were working with a models that had scattered documentation. T6-D described how much of her time was spent just finding the source that had the information she needed for the FactSheet. She said, "(The difficulty) is not knowing where to look, right? It's reading through documents that you potentially didn't have to read through. "
Other than the missing fields, noted above, consumers reported few specific costs to the methodology. The one exception was consumer C10-B who did not see the benefit of a FactSheet over the documentation that already existed for model B (with which she was already familiar). C10-B may be a bit of an outlier, however, as all nine of the other consumers found FactSheets to be better than their prior documentation.
DISCUSSION
The current case study looks at the early process of adopting the FactSheets methodology within an organization. It examines whether the methodology can be followed by those with no particular training in human-centered design, and whether the near-term benefits outweigh the costs. We believe this early picture is promising and expect even more gains from long-term adoption due to reuse of templates, growing awareness of what facts matter to consumers, and increased automation of fact collection.
Enabling Reuse
Despite our belief that there is no universal template or checklist for AI transparency, there is a strong and quite understandable desire to reuse an existing template, for example, one from [14], rather than developing one from scratch. What we have observed in the present work is that a core set of facts did apply in a very different domain from the ones considered to date. These included descriptions of model purpose, training data, and model inputs and outputs.
As shown in Table 4, this core set was augmented with domain-specific facts, such as therapeutic-area definition (for these models in the healthcare space) and market differentiator for models that reached a level of maturity where they are competing for sales within a segment. This kind of "base plus extensions" form of reuse is likely to become a dominant form.
An additional observation from the present work is that once a template has been tailored for a particular organization, it can be used with very little additional work by others within that organization. While the models in this study were all in the healthcare space, they had very different types, use cases, and maturities. Even so, the template developed for the first model was readily reused for subsequent models.
Design Implications for Tools
One of the key gaps identified by FS team participants was the lack of tool support for gathering facts. Facts needed to be found in existing documentation (large, dispersed, and created for a variety of purposes for a variety of audiences), then adapted for use within the much more compact FactSheet, or were obtained by tracking down and talking with others in the organization. We believe this situation will rapidly improve in several ways:
• Fact automation: many kinds of facts can be captured automatically by suitably instrumented tools. Code repositories adapted for use in AI development can record key facts about training data and model versioning.
Lightweight tools such as Python notebooks can include fact-capture mechanisms with very little overhead. As the field identifies the kinds of facts that best support transparency, this sort of automation will likely appear.
• Fact elicitation: some facts, such as model purpose or non-quantifiable aspects of potential bias, must be entered by a knowledgeable human. A good library of examples, wizards, and other elicitation tools will make this both more efficient and help make facts more consistent (supporting comparison) and consumable.
• Fact explanations: there will always be a wide range of expertise among the various consumers of FactSheets. Automatic mechanisms for tailoring fact content to skill level are not infeasible. In addition, a library of reusable links to definitions of terms and supporting material will allow consumers to better understand potentially complex aspects of AI without requiring fact producers to author this supplemental information repeatedly.
• Fact filtering: either through customized templates or filtering controls, consumers will eventually be able to customize documentation to their needs.
A Broader View of Transparency
A key focus of the research on AI transparency views it from the perspective of AI's impacts on individual citizens and society. This is essential work. However, we must acknowledge that there are many stakeholders involved in the creation of facts about AI that collectively flow (or fail to flow) into documentation that may or may not be suitably transparent.
AI transparency must grapple with this fairly untidy reality. The development of models is not a solitary activity. It involves cross-functional teams with different expertise, iteratively building, testing, refining, deploying, and monitoring AI models [22,29]. Critically, these roles drop in and out at different points of the AI model development lifecycle [22].
Along the way, intermediate consumers of important facts about model development attempt to understand, and contribute to, the growing body of facts (the quality of their contributions being dependent on the quality of the facts they consume). Problematically, AI documentation is not designed to meet the needs of its various consumers. And while a single, high-level template may provide a level of basic transparency for some consumers, it will generally fail to generate the transparency needed by all.
We believe that documentation needs to be designed with a clear understanding of its various consumers. In the end, such documentation may have a multi-tiered structure with a near universal top-level view, supplemented by more detailed and specialized information in further tiers. This is a common approach for structuring complex information, for example, in areas such as product comparisons where the top view may include capacities and energy ratings with additional views providing detailed information on modes of operation, compatibility with other products, and multi-dimensional ratings. Our study suggests that at these more detailed levels, consumers' documentation needs are going to be specific and unlikely to overlap across roles as shown in Table 6. This requires a human-centered approach to understanding and capturing these needs in templates and, eventually, supporting tools.
CONCLUSION
Our results suggest that the FactSheets methodology [25] worked as intended, that is, FS team members were able to successfully use the methodology to create AI documentation that was useful to its intended audience. Specifically, we found that RQ1 Methodology usable without human-centered training? Even without training in human-centered design methodologies, FS team members adopted the methodology, successfully elicited needs from their consumers, and encoded those needs into a FactSheet template. Furthermore, the template addressed needs specific to the AI healthcare domain, but were still general enough to be reused across several, quite different, models.
RQ2 FactSheets addressed needs of consumers? Consumers found the resulting FactSheets met their needs, giving them high quality scores along several dimensions. Not all needs were met, however, and the identification of missing content further reinforced a core motivation for the FactSheets methodology: that AI documentation must be tailored to the specific needs of each consumer.
RQ3 Methodology benefits and costs? FS team members and consumers benefited from documentation that was authoritative, in a single location, and supported additional detailed exploration for specific needs. In 16 of 17 interviews, participants agreed that the FactSheets were an improvement over earlier documentation practices.
In summary, the benefits of following the FactSheets methodology to improve AI transparency seemed to outweigh the costs, and are best expressed by T2-A who closed their interview saying the following about the methodology, "I think it's going to be very useful for us going forward and really it's becoming a core part of what we're doing. "
team member (T) or consumer (C). The second part is a participant identification number. The last character notes the model that the participant produced or consumed. For example, 'C2-B' indicates the second consumer participant for model B. We interviewed T4 about two models: model B and D. We distinguish between the two using the identifier 'T4-B' when the participant talks about model B and 'T4-C' when talking about model C.
5.1. 2
2Methodology Steps. To better understand how well the individual steps of the methodology could be applied, we asked FS team members to reflect on the steps one by one. FS team members responded most often that Step 1 of the methodology, Know your FactSheet Consumers, was the most useful of the seven steps. Understanding their consumers encouraged FS teams to consider who they would be writing for, and what their documentation for them should include.Half the FS team members (T1-A, T2-A, T5-B) decided this step was the most important one. T1-A said, "Knowing your consumers was a step that was particularly useful for me... Once I had a particular consumer identified I realize that there are things that I need to think more about, or I need to reach out to people who know about this. So, it was useful." Revealingly, T1-A first tried skipping Step 1 to save time, opting to go directly to getting feedback from consumers on a FactSheet based on an example in the FactSheets 360 website. He recalled, " We started filling it out without the right [consumer needs]... and then we kind of realized a lot of this stuff isn't right. So we went back and rewrote quite a lot of it basically. " After this setback, the other FS team members decided to speak with potential consumers first.Participants T2-A and T5-B noted the benefit of creating a first version of a FactSheet (Step 4) before gathering feedback from consumers on just the template (produced in Step 3). While creating this first version might be seen as adding unnecessary time, the conversations with consumers focused on making changes to this version instead of generating content from scratch. Since one of largest costs reported by participants in the FactSheets methodology was validating the FactSheet with consumers, FS teams appreciated this time-saver. T6-D noted the value of this step as well and added detail about how the act of filling out the FactSheet template spurred reflection on aspects of the model they previously had not considered saying, "It's forcing us to answer questions about our model that we may have thought
Fig. 1 .
1Consumer responses for evaluating the FactSheet quality. The responses on the right side of the dark vertical line represent responses of 'slightly agree' or higher.competitors. C5-A, like C4-A, not having a deep data science background, wanted more guidance on how to interpret the model evaluation metrics, both in terms of how the model relates to the business, but also including the reasonable question of what makes a good metric score. P5-A said, "I think I understand what these (metric) scores might tell me, but I have to do a lot of thinking to connect it back to the business problem... What do they mean for me and the situation that I'm trying to evaluate? ... What's a good range? What's a bad range?" C7-A, a data scientist, wanted to see information that would help him improve the model, along with a history of model evaluations over time. These consumers' role-specific requests underscore the point that different consumers have markedly different documentation needs, and provide
5.3. 1
1Methodology Benefits. As mentioned earlier, one key benefit of FactSheets was to consolidate disparate documentation. T1-A expressed how bringing in the different viewpoints gave a more holistic picture of the model and its context. He said, "When [you] have the document in front of [you] from all points of view, you start joining things together and asking questions... It gives a broader overall picture and fills gaps in your own knowledge." This broader context would be difficult to capture without input from all the consumers. Another benefit of consolidation was how the FactSheet facilitated additional exploration. T1-A, T3-B and T6-D discussed how the more broadly written content of the FactSheet coupled with its pointers to where more details can be found enabled such exploration. T1-A said, "I think this is useful documentation for the reason that everyone can go in and find what they are looking for, maybe not at the level of detail that [they need], but there is links and hyperlinks to everything." T6-D referred to the FactSheet as a "one-stop shop for the model". Seven consumers (C2-A, C3-A, C4-A, C5-A, C6-A, C8-B and C9-B) agreed that the FactSheet acted as an effective point of entry for understanding the aspects of the model relevant to their role. Consumers elaborated on the kinds of further exploration that the FactSheet enabled. For example, C2-A and C8-B explained how the FactSheet eased navigation to other relevant documents. C2-A said, "It's a single document that makes it easy for me to navigate to the next level. (Previously), I'd have to hunt down multiple documents or ask people what was available... This seems to bring it together at least in a single launch point where I can read a bit of detail about it,
FactSheets helped them get a better understanding of documentation needs earlier (T1-A, T2-A), reduce repetition (T2-A, T6-D), improve awareness of other documentation (T3-B), and help identify overlaps in existing documentation (T4-C, T5-B).
Table 1 .
1The FS team (T#) and consumer (C#) participants and their roles for each model. The first character in each column header indicates the model under consideration, referred to as model A, model B, model C, and model D.A Role
T1 Data Scientist
T2 Lead Software Architect
C1 Domain Expert
C2 Product Manager
C3 Regulatory
C4 Customer
C5 Business Analyst
C6 Research
C7 Data Scientist
B
Role
T3 Data Scientist
T4 Data Scientist
T5 Data Scientist Manager
C8 Consultant
C9 Database Administrator
C10 Customer
C Role
T4 Data Scientist
D Role
T6 ML Researcher
Table 2 .
2The four models documented during the case study model to identify populations that are likely to have improved health outcomes if their social factors are improved interviewed twice. Participant details and which models each participant discussed are provided inModel ID Model Description
A
A set of models to assist Medicaid Fraud, Waste, and Abuse investigators in retrieving relevant
information from Medical Insurance policy
B
A mature model (version 21) to predict relative mortality risk of a hospitalization stay
C
A model (in development) to identify potential risks of surgical complications to proactively mitigate
these potential issues
D
A
Table 2 )
2and completed the FactSheets for models A and B. FactSheets for models C and D were in progress and had at least a first draft. Because of this, we were only able to interview consumers for models A and B. The FS team used the publicly available resources on the AI FactSheets 360 site[14] to help them follow the methodology. Resources included explanations of what FactSheets are and their intended applicability; adescription of the methodology for creating FactSheets; example FactSheets; and links to research papers, videos and a
Slack community.
Table 3 .
3FS team member responses to if the methodology helped them in various ways. A response of 'n/a' indicates that the FS team member was unsure or unable to give a response.Did the methodology help...
T1-A T2-A T3-B T4-B T5-B T4-C T6-D # Yes
... Improve documentation consistency?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
7
... Evaluate documentation usefulness?
Yes
n/a
Yes
Yes
Yes
Yes
Yes
6
... Identify new documentation needs?
Yes
Yes
No
No
Yes
Yes
Yes
5
... Facilitate creating useful documentation? Yes
No
n/a
No
Yes
Yes
Yes
4
... Improve documentation practices?
Yes
n/a
Yes
No
No
n/a
Yes
3
... Identify new consumers?
Yes
Yes
No
No
No
n/a
No
2
Table 4 .
4The template created by the FS team for model A. Italicized fields indicate new fields that were added during the case study. Related Areas Where Models May Be Useful Other potential use cases for this model work of the FS teams, perhaps especially the iterative evaluation of FactSheet content with consumers, paid off and resulted in highly relevant, useful, and usable documentation.Overview
Table 6 .
6Summary of missing documentation information per consumer. Italicized entries were mentioned by multiple consumers. Although market differentiation content existed in the FactSheets, participants wanted more detail than what was available, so it is included below.Consumer Role
Missing documentation fields
C1-A
Domain Expert
None
C2-A
Product Manager
Market differentiation; existing deployments; customers; pricing informa-
tion; model maturity; links to marketing materials, presentations, and
demos
C3-A
Regulatory
Market differentiation; Process model is replacing; benefits and costs of
deployment; access information; model evaluation summary; known risks;
applied risk mitigations; how to interpret model metrics; productization
details
C4-A
Customer-Facing
Market differentiation; Mapping between model metrics to business use
case; how to interpret model metrics; good/bad ranges for model metrics;
business context
C5-A
Business Analyst
Good/bad ranges for model metrics
C6-A
Research
Data sources; data cleaning details; data limitations and constraints; model
training details; background of domain experts; definitions for bias; defini-
tions for explainability
C7-A
Data Scientist
Existing deployments; opportunities for model improvement; customer
feedback; history of model evaluations
C8-B
Consultant
None
C9-B
Database Administrator None
C10-B
Table 7 .
7The number of hours each FS team member spent working on the FactSheet
The FS team members were generally also the knowledgeable producers of facts in this case study. We ignore the distinction between FS team members and producers in what follows.
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| []
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[]
| [
"María J Martín ",
"Jukka Tuomela "
]
| []
| [
"Mathematics Subject Classification. 76B03, 35Q31, 13P10"
]
| There are not too many known explicit solutions to the 2-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the 19th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -in the 1980s-obtained new explicit solutions with a similar feature.We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family.In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important. | 10.3934/dcds.2019187 | [
"https://arxiv.org/pdf/1810.01475v2.pdf"
]
| 119,151,766 | 1810.01475 | bf98fdfcc025e339db2941a9632811e469a546d7 |
October 4. 2018. 2010
María J Martín
Jukka Tuomela
Mathematics Subject Classification. 76B03, 35Q31, 13P10
DateOctober 4. 2018. 20102D INCOMPRESSIBLE EULER EQUATIONS: NEW EXPLICIT SOLUTIONSand phrases Explicit solutionsEuler equationsfluid mechanics
There are not too many known explicit solutions to the 2-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the 19th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -in the 1980s-obtained new explicit solutions with a similar feature.We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family.In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.
Introduction
There are two standard frameworks for analyzing the motion of an ideal homogeneous fluid. The Eulerian description observes at fixed locations the flow properties as the particles go by. The motion is then obtained by imposing the law of mass conservation ∇ · u = 0 (1) and the conservation of momentum law
u t + u∇u + ∇p = 0 ,(2)
where u = u(t, x) is the velocity field in the time and space variables (t, x) and the scalar function p represents the pressure. An alternative representation of the flow is provided by the (material) Lagrangian coordinates, in which the observer follows the fluid by picking out a particular particle and keeping track of where it goes. There exists a precise Eulerian state corresponding to a Lagrangian state and vice versa (see [8]).
Let us briefly review how the equations of the motion look like in Lagrangian coordinates.
Starting with a domain Ω 0 ⊂ R n and using x to denote the Eulerian (spatial) coordinates and β to denote the Lagrangian coordinates, we see that each coordinate β ∈ Ω 0 identifies by means of the map
β → x(t, β) = Φ(t, β) = Φ t (β)(3)
the evolution in time of a specific particle. Here Φ t is a diffeomorphism, Ω t = Φ t (Ω 0 ), and Φ 0 is the identity. Within these terms the Eulerian and Lagrangian descriptions of the flow are related by the equation
Φ ′ (t, β) = u(t, Φ(t, β)) ,(4)
where Φ ′ (t, β) = ∂ ∂t Φ(t, β) .
(The reader may be advised of the fact that, unless otherwise explicitly stated, this is the notation we will use in this paper; namely, the derivative of any map f with respect to t will be denoted by f ′ .) It is easy to check that differentiating (4) with respect to t and using (2) we have Φ ′′ (t, Φ(t, β)) = u t + u∇u = −∇p(t, Φ(t, β)) ,
which, in terms of Lagrangian coordinates, is equivalent to
(dΦ) T Φ ′′ + ∇p = 0 .
The incompressibility condition (1) gives then the equation det(dΦ t ) = 1 for all t.
Finding explicit solutions in this form, when the initial mapping Φ 0 satisfies det(dΦ 0 ) = 1, is unnecessarily hard (cf. Section 3 below). However, it is possible to introduce a further modification as follows. Let D ⊂ R n be a domain and assume that there is a diffeomorphismφ : D → Ω 0 . We call D the labelling domain or parameter domain, and denote the labels, i.e. the coordinates of D, by α. We can then look for the solutions to the following problem. Problem 1. Find diffeomorphisms ϕ = ϕ(t, α) = ϕ t (α) with α ∈ D and t ≥ 0 such that (dϕ) T ϕ ′′ + ∇p = 0 (5) and with det(dϕ t ) = det(dϕ 0 ) = 0 for all such t.
If such a ϕ can be found then we can define Φ t = ϕ t •φ −1 which then gives the Lagrangian description of the fluid properly speaking.
Though the non-linearity of the problem makes rather difficult to construct interesting explicit solutions in either Eulerian coordinates or Lagrangian coordinates, it seems that the most complete description of a flow is attained within the Lagrangian framework. Some of the celebrated explicit solutions to the two-dimensional incompressible Euler equations in Lagrangian variables are in fact quite old: Gerstner's flow [12], found in 1809 (rediscovered in 1863 by Rankine [15]); Kirchhoff's elliptical vortex [13], found in 1876. More recently Abrashkin and Yakubovich [1] found some new solutions in 1984. Somewhat surprisingly, apparently these examples (and small variations of them) were all known explicit solutions up to short time ago.
In the construction of all these classical flows, harmonic maps are essential, as all of them present a labelling by harmonic functions. A. Aleman and A. Constantin [3] proposed a complex analysis approach aimed at classifying all such flows. With the aim of complementing the work in [3], a new different approach, based on ideas from the theory of harmonic mappings, was used in [9], where the authors explicitly provide all solutions, with the specified structural property, to the incompressible 2-dimensional Euler equations (in Lagrangian variables).
It is not difficult to check that the map (3) corresponding to the classical solutions due to Gerstner and Kirchhoff (and those ones obtained by Abrashkin and Yakubovich in [1]) as well as those in [3] and [9] are as follows. Up to an additive constant, either
ϕ(t, α) = ϕ t (α) = A(t)v(α) ,(6)
where A belongs to the special linear group SL(2) of 2×2 matrices with determinant 1 and v is a vector field whose coordinates are harmonic functions and such that det(dv) = 0, or
ϕ(t, α) = ϕ t (α) = M 1 (t)v(α) + M 2 (t)w(α) ,(7)
where M 1 , M 2 belong to the group O(2) of 2 × 2 orthogonal matrices and v and w are, again, vector fields whose coordinates are harmonic functions and satisfy det(dv) = 0 and det(dw) = 0.
In this article, instead of considering those solutions to the 2-dimensional incompressible Euler equations for which the map (3) is harmonic for all times t, we will focus on analyzing those solutions for which the labelling map ϕ takes one of the forms described by (6) or (7) without any assumption on harmonicity of the vector fields involved in this description. The methods used show that perhaps curiously the harmonicity of the maps is in fact not essential (cf. Sections 4 and 5 below). Harmonic maps simply provide one family of solutions to a certain PDE system which has plenty of other solutions as well. This PDE system allows us to construct new families of solutions (using the Lagrangian framework). In the old and our new solutions the classical Lie groups SO(n) and SL(n) appear naturally. The geodesics in these groups seem to be also important. These facts might be somehow related to the theory developed in the books [4] or [5], for instance. Our method for constructing new solutions extends also to the three dimensional case, though this case will not be treated in this article but in a forthcoming paper.
Preliminaries and notation
Unfortunately, for questions of space, it is not possible to include all the details related to the theory involved in the approach developed in this paper to obtain new explicit solutions to the 2-dimensional incompressible Euler equations, which is the main goal in this article. Nevertheless, with the hope to make this paper self-contained, we now review the main tools, concepts, and results used in our standpoint. We also include the main references with the hope that the reader can figure out the key points in the proofs of our results, though we should point out that it is possible to check directly that the functions ϕ obtained in our main Theorems 4 and 5 -or in Theorems 2 and 3-satisfy the requirements stated in Problem 1, thus they provide new explicit solutions to the problem considered.
2.1. Geometry. We will need some elementary notions related to Riemannian geometry; one standard reference is [14].
Recall that given a Riemannian manifold M , the curve a : R → M is a geodesic if it satisfies the differential equation
(a ′′ ) k + Γ k ij (a ′ ) i (a ′ ) j = 0 .
Here we are using the Einstein summation convention: in those cases when in a single term an index appears twice (once up and once down) and is not otherwise defined, it implies summation of that term over all the values of the index. The symbols Γ k ij are the Christoffel symbols of second kind and, as above, a ′ represents the derivative of the curve a with respect to the parameter (the time t, say) it depends on.
In those cases when M is an n−1 dimensional submanifold of R n with the induced Riemannian metric, it is sometimes convenient to express the geodesic equations in ambient coordinates. More concretely, let ν be any non-zero normal vector field of M . Then a : R → R n is a geodesic if there is some function λ :
M → R such that a ′′ + λν = 0 , a ′ , ν = 0 , a(t) ∈ M .
2.2. Algebra. In this section, we refer the reader to [10] for more information about questions related to computational commutative algebra. We should also mention that the theory about the relationship between varieties and ideals is much more developed in the complex case than in the real case. However, in our context only real varieties are of interest. For general information about real algebraic geometry we refer the reader to [7]. Let us consider polynomials of variables x 1 , . . . , x n with coefficients in the field K and let us denote the ring of all such polynomials by A = K[x 1 , . . . , x n ]. The given polynomials f 1 , . . . , f k ∈ A generate an ideal
I = f 1 , . . . , f k = {f ∈ A : f = h 1 f 1 + . . . + h k f k , where h i ∈ A}.
We say that the polynomials f i are generators of the ideal I and as a set they are the basis of I. Notice that we do not assume that the polynomials {f 1 , . . . , f k } are independent in any way so that if f k+1 ∈ I, then {f 1 , . . . , f k , f k+1 } is also a basis of I.
Not all the bases of an ideal are equally good; the good ones are known as Gröbner bases. Gröbner bases depend on the monomial order, but once we have chosen a particular order the Gröbner basis is essentially unique and one can actually compute it with the Buchberger algorithm. It may be convenient as well to stress that in actual computations below (related to Gröbner bases), we use Singular [11].
Let I ⊂ A and let G = {g 1 , . . . , g ℓ } be a Gröbner basis of I. Then given any f ∈ A we can compute the representation f = ℓ j=1 a j g j + R such that R (the remainder) is unique. This remainder is also known as the normal form of f with respect to I and in this case we can write R = NF(f, I) or R = NF(f, G). The normal form can also be interpreted as an element of the residue class ring A/I.
The radical of I is
√ I = {f ∈ A : f m ∈ I for some integer m ≥ 1}.
An ideal I is a radical ideal if I = √ I. The real radical R √ I is the set of polynomials f ∈ A which satisfy the condition that
f 2r + m i=1 g 2 i ∈ I
for some positive integers r and m and some functions g i , i = 1, . . . , m, in A. An ideal is a real ideal if R √ I = I. In particular, if I is a real ideal and
f 2 1 + · · · + f 2 m ∈ I , then f j ∈ I for all j.
An ideal I is prime if the condition that f g ∈ I implies that, necessarily, either f ∈ I or g ∈ I. Every prime ideal is a radical ideal.
Let I ⊂ A. Then k-th elimination ideal of I equals
I k = I ∩ K[x k+1 , . . . , x n ].
Geometrically, the concept of k-th elimination ideal is related to the projections π k : L n → L n−k defined by
π k ((x 1 , . . . , x n )) = (x k+1 , . . . , x n ).
A Gröbner basis of an elimination ideal can also be computed using a suitable product in the Buchberger algorithm.
To each ideal I we can associate the corresponding variety V(I). There are various ways to define the associated variety depending on the desired level of abstraction. For us, the following procedure is the most convenient.
Let L be some extension field of K (typically, in applications, one has K = Q and L = R). Then we set
V(I) = a ∈ L n : f (a) = 0 for all f ∈ I ⊂ L n . Note that V(I) = V( √ I).
The connection between all these notions is that if L is algebraically closed then
π k V(I) = V(I k ) ,
where the overline denotes the Zariski closure.
Many, or perhaps even most, computational applications of commutative algebra are based on the so-called Hilbert basis theorem, which states that every ideal I ⊂ A is finitely generated. Buchberger's algorithm is a constructive proof of this fact.
2.3.
PDEs. Let ν be a multi-index and, as usual, let |ν| = ν 1 + · · · + ν n . It is well-known that any linear PDE can be written as
Lu = |ν|≤q c ν ∂ ν u = f ,
where c ν are some known matrices, not necessarily square.
The principal symbol σL of the operator L is defined by
σL = |ν|=q c ν ξ ν , ξ ∈ C n ,
and we say that L is elliptic if σL is injective for all ξ ∈ R n , ξ = 0.
In the context of formal theory of PDEs [16] it is convenient to use the so-called jet notation for derivatives. More concretely, given a vector field u = (u 1 , . . . , u n ), we use
u j ν = ∂ ν u j = ∂ |ν| u j ∂x ν1
1 · · · ∂x νn n to denote the derivatives of the components of u. Also it is convenient to denote by u |ν| the vector which contains all derivatives of u of order |ν|.
One final remark must be mentioned at this point. There are two natural conventions regarding the sign in the Cauchy-Riemann equations. Namely, one which is used in complex analysis and the other which comes from the de Rham complex. In the present context, it is more natural to use the one provided from the latter theory which allows us to use the same definition in any dimension. Hence, we say that a map or vector field u is a Cauchy-Riemann map (or just CR map) if du is symmetric and tr(du) = div(u) = 0.
A non-existence result
The aim of this section is to show that the additional assumption that the maps considered in Problem 1 are area-preserving (so that det(dϕ t ) = 1 for all t ≥ 0) and harmonic in the domain D is in a sense too restrictive, as Theorem 1 below shows. We use the formal theory of PDEs. We only briefly outline the relevant concepts and refer the reader to [16] for the details.
Let E = R n × R n be a bundle with coordinates (x, y) and let the projection π : E → R n be defined by π(x, y) = x. The corresponding jet bundle of order q is denoted by J q (E).
Let a system of PDEs of order q be given by a map F : J q (E) → R k . To this map we can associate a set R q = F −1 (0) ⊂ J q (E). Two fundamental operations come up naturally: projection and prolongation.
To prolong a given system of PDEs, one just differentiates all the equations. More concretely, if we have system R q ⊂ J q (E) given by some map F , then its first prolongation is R q+1 = ρ 1 (R q ) ⊂ J q+1 (E), given by
R q+1 = ρ 1 (R q ) ≡ F = 0 , ∂F ∂xj = 0, 1 ≤ j ≤ n .
The projections π q+r q : J q+r (E) → J q (E) are simply the maps which forget the highest derivatives or jet variables. When this map is restricted to the differential equation, we obtain the map π q+r q : R q+r → R q . The image of this map is denoted by R (r) q . Note that we always have R (r) q ⊂ R q . The fact that the inclusion is strict means that by differentiating and eliminating, we have found integrability conditions; that is, equations of order q which are algebraically independent of the original equations and which are also satisfied by the solutions of the system.
The goal of this type of analysis is thus to find all integrability conditions. The theory behind this intuitive idea is too involved to be developed in this paper; as mentioned above, the details can be found in [16]. For the result given below the general theory is not needed. Note that when one finds integrability conditions, the formal solution space remains the same.
Let now y = (y 1 , y 2 ) : R 2 → R 2 be some (smooth) map and define
F = det(dy) − 1 = y 1 10 y 2 01 − y 1 01 y 2 10 − 1 ,
where, as usual, the first sub-index denotes the number of derivatives with respect to the first variable and the second sub-index denotes the number of derivatives with respect to the second variable.
The requirement that y is area preserving (F = 0) and harmonic (∆y 1 = ∆y 2 = 0) produces the initial system
R 2 ≡ f 1 = y 1
10 y 2 01 − y 1 01 y 2 10 − 1 = 0 , f 2 = y 1 20 y 2 01 + y 1 10 y 2 11 − y 1 11 y 2 10 − y 1 01 y 2 20 = 0 , f 3 = y 1 11 y 2 01 + y 1 10 y 2 02 − y 1 02 y 2 10 − y 1 01 y 2
11 = 0 , f 4 = y 1 20 + y 1 02 = 0 , f 5 = y 2 20 + y 2 02 = 0 .(8)
Hence we can write
R 2 = f −1 (0) ⊂ J 2 (E), where f = (f 1 , f 2 , f 3 , f 4 , f 5 ).
Theorem 1. All solutions of the system (8) are, in fact, affine.
Proof. Notice that the equations in (8) are polynomials in jet variables so that we can write
R 2 ≃ R 2 × R 2 × V 2 , where V 2 = V(I 2 ) ⊂ R 10 . Here, I 2 = f 1 , f 2 , f 3 , f 4 , f 5 ⊂ Q[y 1 , y 2 ],
where, as explained in the previous section, we use y |ν| to denote the vector which contains all derivatives of y of order |ν|.
The prolongation can then be written as
R 3 ≡ R 2 , ∂ ν f 1 = 0 , |ν| = 2 , ∂ ν f 4 = 0 , |ν| = 1 , ∂ ν f 5 = 0 , |ν| = 1 .
There are thus 7 equations of order 3. The relevant ideal can be written as
I 3 = I 2 + f 6 , . . . , f 12 ⊂ Q[y 1 , y 2 , y 3 ] .
To construct the projection we must then compute the 8th elimination ideal I 3,8 of I 3 which is
I 3,8 = I 3 ∩ Q[y 1 , y 2 ] .
Now choosing an appropriate product order, we obtain the Gröbner basis {f 1 , . . . , f 5 , g 1 , g 2 , g 3 , g 4 } of I 3,8 , where g 1 = y 1 11 y 2 20 − y 2 11 y 1 20 , g 2 = (y 1 11 ) 2 + (y 1 20 ) 2 , g 3 = y 2 11 y 1 11 + y 2 20 y 1 20 , and g 4 = (y 2 11 ) 2 + (y 2 20 ) 2 . However, since we are only interested in real varieties, we get (from the conditions g 2 = g 4 = 0) that, necessarily, These conditions imply that y = y(x) = Ax + b for some constant matrix A with determinant equal to 1 and some constant vector b.
Note that in the proof of the previous theorem, we have essentially constructed the real radical of the elimination ideal I 3,8 which coincides with R I 3,8 = y 1 10 y 2 01 − y 1 01 y 2 10 − 1 + y k ν , |ν| = 2 , k = 1, 2 . In general, computing the real radical is actually very difficult, and searching for good algorithms is an active research topic [6]. This shows that constructing the projection is sometimes quite tricky.
We should also mention that R 2 , defined by (8), is of finite type. This can be characterized by the property that after prolonging "far enough" one can express all partial derivatives of certain order in terms of lower order derivatives which means that the formal solution space is finite dimensional. In other words, there is only a finite number of degrees of freedom in the general solution. Here in R (The dimension of the variety can be computed once the Gröbner basis is available.)
Perhaps it is convenient to remark the dimension of the variety V(I 3,8 ) considered in the proof of Theorem 1 as a complex variety is higher than its dimension as a real variety. More concretely,
dim V(I 3,8 ) = 4 > 3 = dim V( R I 3,8 ) .
New solutions to the 2D incompressible Euler equations, I
As mentioned in the introduction, in this paper instead of considering those solutions to the 2-dimensional incompressible Euler equations for which the map (3) is harmonic for all times t, we will focus on analyzing those solutions for which the labelling map ϕ takes one of the forms described by (6) or (7) without any assumption on harmonicity of the vector fields involved in this description.
We now consider the cases when ϕ is given by (6). Our first result is the following lemma, which works in any dimension n. Lemma 1. Let A ∈ SL(n). Then the map of the form (6) provides a solution to the incompressible Euler equations if A T A ′′ is symmetric and v is any diffeomorphism. In this case the pressure is given by
p = − 1 2 v, A T A ′′ v . Proof.
It is clear that det(dϕ t ) = det(dϕ 0 ) = 0. A straightforward calculation shows that (5) holds.
Several remarks are now in order. The first one is about the assumption on the symmetry of A T A ′′ in the previous lemma. We would like to point out that this is rather a mild condition since the dimension of SL(n) is n 2 − 1 and the condition on the symmetry (A T A ′′ ) T = A T A ′′ gives 1 2 n(n − 1) differential equations of second order. So when n = 2 we expect that there are 2 arbitrary functions in the solution and this is precisely what happens, as will become apparent below.
Notice that there is no need to suppose that the domain is simply connected. This is because the existence of the pressure is a consequence of the fact that in Eulerian coordinates the velocity field is divergence free. In fact, one of the standard problems in the applications of fluid mechanics is the 2-dimensional flow of the air around the airplane wing and in this case the domain is not simply connected.
It may seem strange that there is no condition on the vector field u in the previous lemma, except that it is a diffeomorphism. Let us outline the reasons that justify this observation.
Recall that the labelling coordinates are completely arbitrary, they do not have any physical significance. Let now ϕ have the form (6). Then, by writing
ϕ(t, α) = A(t)A −1 (0) (A(0)v(α)) = A(t)V (α) ,
if needed, we can assume without loss of generality that A(0) = I and v is a diffeomorphism from the labelling domain D to the Lagrangian domain Ω 0 . Now, we have from (3) that we can write
x = Φ t (β) = ϕ t • v −1 = A(t)β ,(9)
so that Eulerian coordinates depend linearly on the Lagrangian coordinates, and since the map v does not appear anymore in the physical description of the flow when considering the Eulerian coordinates, it makes sense that there is no need to have conditions on it. Let us also see how the motion looks like in Eulerian coordinates in this case. Using (9), it is obvious that
A ′ (t)β = x ′ (t) = u(t, x(t)) = u(t, A(t)β) , so that u(t, x) = A ′ A −1 x.
That is, the velocity vector field is a linear vector field. Since A ∈ SL(n), the matrix that determines the velocity vector field, A ′ A −1 , has the property trace(A ′ A −1 ) = 0. Hence, A ′ A −1 is an element of the Lie algebra sl(n) corresponding to the Lie group SL(n). If we take a smaller group and require that A ∈ SO(n) then A ′ A −1 = A ′ A T is skew symmetric, i.e. an element of so(n). Moreover, now we easily obtain
u t + u∇u = A ′′ A −1 x = −∇p .
Note also that A T A ′′ is symmetric implies that A ′′ A −1 is symmetric so that the pressure is given by
p = − 1 2 β, A T A ′′ β = − 1 2 x, A ′′ A −1
x . Now, let us focus on the 2-dimensional case. In order to present the results, it is convenient to remark that any matrix A ∈ SL(2) can be parametrized using the function ψ :
R 3 → SL(2) defined by ψ(s, µ, θ) = cosh(s)R 1 + sinh(s)R 2 ,(10)
where R 1 = cos(µ) − sin(µ) sin(µ) cos(µ) and R 2 = cos(θ) sin(θ) sin(θ) − cos(θ)
.
Let now A = A(t),
where t belongs to some interval I ⊂ R, be a curve in SL(2). Then there are scalar functions s, µ, and θ on I such that
A(t) = cosh(s(t))
cos(µ(t)) − sin(µ(t)) sin(µ(t)) cos(µ(t)) + sinh(s(t)) cos(θ(t)) sin(θ(t)) sin(θ(t)) − cos(θ(t)) .
A straightforward calculation shows that A T A ′′ is symmetric when
sinh 2 (s)θ ′′ − cosh 2 (s)µ ′′ + 2 sinh(s) cosh(s)s ′ (θ ′ − µ ′ ) = 0 ,(11)
which implies (after integrating)
cosh 2 (s)µ ′ − sinh 2 (s)θ ′ = constant .(12)
In fact, it may be convenient to stress that Kirchhoff's solution is of the form (6) with s constant, µ(t) = µ 0 t, where µ 0 is a non-zero real number, and θ ≡ 0.
An easy consequence of Lemma 1 is the following.
Theorem 2. Let ϕ have the form (6), where v is a diffeomorphism. Assume that A(t) are parametrized via the function ψ above where s, µ, and θ satisfy (12). Then, the function ϕ provides a solution to the incompressible Euler equations.
We would like to remark that the previous theorem recovers all the solutions in [9] corresponding to what was classified as solutions of type 1 and solutions of type 2 (i) in that paper. Of course, the harmonicity condition on the coordinates of v is not assumed in our case.
There is an interesting connection to the geodesics that we now describe. Recall that the group of orthogonal matrices O(2) has two disjoint components. Considering the Riemannian metric induced by the embedding O(2) ⊂ R 4 one can check that both components are in fact isometric (up to a constant factor) to the unit circle with the standard embedding S 1 ⊂ R 2 . Now let us look for "pure"rotations among the curves A = A(t) = ψ(s(t), µ(t), θ(t)) in SL(2) such that s, µ, and θ satisfy (12), that is, we assume that the matrices A ∈ SO(2). Then, necessarily s ≡ 0 and hence µ ′ is constant. We can interpret this observation by saying that the curve t → A(t) must be a geodesic in this case.
In the same way one could think about geodesics in SL(2). We consider SL(2) as a Riemannian manifold where the metric is induced by the embedding SL(2) ⊂ R 4 and A is a curve in SL (2). We then obtain the last result in this section. Proof. The geodesic equations in terms of the parametrization in (10) are
2 cosh(2s)s ′′ + sinh(2s) 2(s ′ ) 2 − (θ ′ ) 2 − (µ ′ ) 2 = 0 , cosh(s)µ ′′ + 2 sinh(s)s ′ µ ′ = 0 , sinh(s)θ ′′ + 2 cosh(s)s ′ θ ′ = 0 .(13)
Therefore, (11) and hence (12) hold. A direct application of Theorem 2 ends the proof.
Note that, however, not all matrices A = A(t) for which (12) holds satisfy (13), so that they are not necessarily geodesics.
New solutions, II
In this section we consider those solutions to the 2-dimensional incompressible Euler equations for which the labelling map ϕ is of the form (7), again without any assumption on harmonicity of the vector fields involved in this description. That is, we assume that
ϕ(t, α) = M 1 (t)v(α) + M 2 (t)w(α) ,(14)
where M 1 (t) = cos(µt) − sin(µt) sin(µt) cos(µt) and M 2 (t) = cos(θt) − sin(θt) sin(θt) cos(θt)
for certain real constants µ and θ (with µ = θ) and sufficiently smooth vector fields v and w.
Notice that the matrices M 1 and M 2 are not just any curve of O(2) but actually geodesics in the sense explained above.
The following important example as well as the solutions obtained in [1] and those classified as solutions of type 2 (ii) in [9] are of this form.
Example 1. Gerstner's flow [12] is of the form (14) with M 1 = I, v(α) = α = (α 1 , α 2 ) T ,
M 2 = cos(kt) − sin(kt) sin(kt) cos(kt) , and w(α) = e kα2 k sin(kα 1 ) − cos(kα 1 ) ,
where k is a non-zero real number.
We should mention that, in contrary to those cases considered in the previous section, where the (global) injectivity of the vector field v implies the global injectivity of the functions ϕ, now some extra conditions must be satisfied in order to guarantee the injectivity of the functions in (14) in a similar way as what occur in the known solutions of this type (see, for instance, [9,Thm. 4]). With the hope to make our exposition more clear, we have decided to make a local analysis in the proofs of our main results in this section and do not include the details on the conditions for the global univalence of the mappings in (14).
Notice that the vector fields v and w in the previous example satisfy that their coordinates are harmonic, though there are other solutions with the same structure as in (14) which are not harmonic, as our next theorem shows.
and det(dv) + det(dw) = 0.
Before stating the proof of Theorem 4, let us see how this result relates to the previously known solutions and make some comments.
Perhaps one can say that Theorem 4 explains why harmonic maps have played a prominent role in the analysis of Euler equations. If we choose v as a CR map (so that, in particular, the coordinates of v are harmonic functions), then the system in Theorem 4 becomes v 1 10 w 1 10 − w 2 01 + v 1 01 w 1 01 + w 2 10 = 0 , v 1 10 w 1 01 + w 2 10 + v 1 01 w 1 10 − w 2 01 = 0 . Hence w must be a CR map with the sign convention of complex analysis. These are precisely the solutions found in [1], and this choice of v and w is called solutions of type 2 (ii) in [9].
However, there are plenty of solutions of (15) which are not harmonic: we have 4 unknowns and only 2 equations so that we can, for example, choose w arbitrarily and then solve the corresponding system, which can be written in the form Lv = 0, for v. In fact the two linear equations obtained in this way are rather special, as the following result shows.
Lemma 2. If det(dw) = 0, then L is elliptic.
Proof. To show that L is elliptic, we need to check that its principal symbol σL defined by (2.3) is injective for all ξ ∈ R 2 , ξ = 0. It suffices, then, to check det(σL) = w 1 01 ξ 1 − w 1 10 ξ 2 ) 2 + w 2 01 ξ 1 − w 2 10 ξ 2 ) 2 = 0 for all real non-zero vector ξ = (ξ 1 , ξ 2 ) ∈ R 2 , which holds if det(dw) = 0.
When considering elliptic boundary value problems, one needs to impose appropriate boundary conditions. Since det(σL) is a second order polynomial, we need to impose one boundary condition. The relevant criterion is known as Shapiro-Lopatinskij condition [2], which is easy to check in our case since rather amazingly we have the following explicit factorization det(σL) = 1 |w 10 | 2 |w 10 | 2 ξ 2 − ( w 10 , w 01 + i det(dw))ξ 1 × |w 10 | 2 ξ 2 − ( w 10 , w 01 − i det(dw))ξ 1 .
We should also point out that in (14) we have assumed that both matrices M 1 and M 2 are rotations. We could as well choose one or both of the matrices to be reflections. The computations would be precisely the same as above, except a few changes in the signs. The overall conclusion is the same: given one vector field one obtains an elliptic system for the other.
Let us then turn to the proof of the theorem. To this end, notice that the statement of Theorem 4 as well as its conclusion already make it clear that it is not essential to consider v and w as distinct objects, it is more natural to consider a map with four components. Hence let us introduce a more convenient way to analyze the problem. We will look for solutions in the following form:
ϕ(t, α) = A(t)u(α) ,(16)
where A = (M 1 | M 2 ) : R → R 2×4 is 1 × 2 block matrix with first entry M 1 and second entry M 2 and and u :
R 2 → R 4 is the vector u = (v | w) = (v 1 , v 2 , w 1 , w 2 ).
By the Cauchy-Binet formula we can write
det(dϕ) = 6 i=1 p i g i ,(17)
where p i are the 2 × 2 minors of A and g i are the 2 × 2 minors of dV . Let now B = A T A ′′ and let y = dϕ T ϕ. If the map ϕ given by (16) solves Problem 1, then we must have
N = dy − dy T = 0 .
Note that N is skew symmetric so we have only one condition: N 12 = −N 21 = 0. Let us analyze further this condition.
First, let us write N with index notation of Riemannian geometry, where it matters if indices are up or down (though in our context we can simply arrange the expressions so that the summation convention works). Let us use the notation
y i = V ℓ ;i B ℓk V k to get N ij =y i;j − y j;i = u ℓ ;i B ℓk u k ;j − u ℓ ;j B ℓk u k ;i =B ℓk u ℓ ;i u k ;j − u ℓ ;j u k ;i = u ℓ ;i B ℓk − B kℓ u k ;j .
A straightforward calculation shows that since the parameters µ and θ that determine the matrix A are different, then the matrix B is not symmetric (which would imply N = 0). We can then write f = N 12 as
f = 6 i=1 f i g i = B ℓk u ℓ ;1 u k ;2 − u ℓ ;2 u k ;1 ,(18)
where f i depend only on t and g i are, again, the 2 × 2 minors of du.
The strategy of the proof of Theorem 4 is now clear: we compute the formulas (17) and (18) -the use of symbolic computation programs is fundamental to this end-and then try to choose A and u such that det(dϕ) does not depend on time and f is identically zero. Note that a priori we have 12 conditions and 12 independent functions. However, as we will see below, there are plenty of possibilities to choose the appropriate functions. (16), where M 1 and M 2 are two given matrices as in (14). Let us introduce the variables c 1 = cos(µt), s 1 = sin(µt), c 2 = cos(θt), and s 2 = sin(θt); and the ideal
Proof. (Theorem 4) We now choose
A = (M 1 | M 2 ) inI = c 2 1 + s 2 1 − 1, c 2 2 + s 2 2 − 1 ⊂ A = Q(µ, θ)[c 1 , s 1 , c 2 , s 2 ]
, where Q(µ, θ) is the field of rational functions of µ and θ.
Let us first use the division algorithm to compute
det(dϕ) = 6 i=1 p i g i ,
where p i ∈ A and g i are the 2 × 2 minors of du. Then, we compute the normal forms of p i with respect to I, which gives det(dϕ) = u 1 10 u 2 01 − u 1 01 u 2 10 + u 3 10 u 4 01 − u 3 01 u 4 10 + (c 1 c 2 + s 1 s 2 )q 1 + (s 1 c 2 − s 2 c 1 )q 2 , where q 1 = u 3 10 u 2 01 −u 3 01 u 2 10 −u 4 10 u 1 01 +u 4 01 u 1 10 and q 2 = u 4 10 u 2 01 −u 4 01 u 2 10 +u 3 10 u 1 01 −u 3 01 u 1 10 . Therefore, we see that det(dϕ) is independent of t if q 1 = q 2 = 0.
Moreover, by dividing the polynomial f in (18) with respect to q 1 , q 2 gives
f = µ 2 − θ 2 (s 2 c 1 − c 2 s 1 )q 1 + (c 1 c 2 + s 1 s 2 )q 2 = 0 if q 1 = q 2 = 0.
The condition that q 1 = q 2 = 0 is precisely (15) since u = (v|w). This ends the proof.
Instead of considering matrices A = (M 1 | M 2 ), where M 1 and M 2 are as in (14), we could consider more general 2 × 4 matrices A = (a ij ) in (16). The systematic analysis of all cases would take too much space so we will simply give a particular construction which indicates the general idea and produces a result which we hope can be of some interest to the reader.
We just impose the matrix A in (16) the following constraints:
The first two conditions in (19) are rather natural since one suspects that SL(2) will anyway be essential in the analysis and they are obviously satisfied by any matrix
and suppose that the vector u : R 2 → R 4 is a solution of −u 3 10 u 2 01 + u 3 01 u 2 10 − u 3 10 u 1 01 + u 3 01 u 1 10 = 0 , −u 4 10 u 3 01 + u 4 01 u 3 10 − u 3 10 u 1 01 + u 3 01 u 1 10 = 0 . Using again the division algorithm we have
det(dϕ) = 6 i=1 p i g i ,
where p i ∈ A and g i are, again, the 2 × 2 minors of du. The computation of the normal forms of p i with respect to I gives det(dϕ) = g 1 + g 2 + g 3 − g 4 + p 5 (g 4 + g 5 ) + g 6 .
Hence, det(dϕ) is independent of t if we require g 4 + g 5 = 0, which is precisely the first equation in (21) since g 4 + g 5 = −u 3 10 u 2 01 + u 3 01 u 2 10 − u 3 10 u 1 01 + u 3 01 u 1 10 . Let us now identify the matrix A (resp. A ′′ ) with the vector a (resp. a ′′ ) which contains all the elements of A and consider the ideal
I 1 = I + a ′′f = 6 i=1 f i g i ,
where f i ∈ A 1 . Computing the normal forms we find NF(f 2 , I 1 ) = NF(f 3 , I 1 ) = NF(f 6 , I 1 ) = a ′′ 21 a 22 − a ′′ 22 a 21 + a ′′ 11 a 12 − a ′′ 12 a 11 . Then we define a bigger ideal I 2 = I 1 + a ′′ 21 a 22 − a ′′ 22 a 21 + a ′′ 11 a 12 − a ′′ 12 a 11 and reduce f with respect to I 2 , which yields f =f 1 g 1 +f 4 g 4 +f 5 g 5 , wheref 1 +f 4 −f 5 = 0 . Now using the relation g 4 + g 5 = 0, we get f =f 1 (g 1 − g 4 ). Hence, the assumption that g 1 − g 4 = 0 (which is equivalent to the second PDE for u in (21)) gives f = 0.
Notice that in the system (21) there are two equations for four unknowns. We can then, for instance, consider two given arbitrary functions u 1 and u 3 to obtain a decoupled system for u 2 and u 4 . This resulting system for u 2 and u 4 is a kind of transport equation which can be solved with the method of characteristics in the usual way. (In particular, again the harmonicity or lack of harmonicity of u plays no role here.)
The system (20) is also underdetermined so that there are also plenty of possibilities for choosing the time dependence of the system. For instance, we can choose the submatrix =Â(t) = a 11 a 12 a 21 a 22 ∈ SL (2) such that is a geodesic. Then the first and last equations in (20) are automatically satisfied. Then, we get (a 14 , a 24 ) from the third and fourth equations given in (20) and, finally, we just need to consider the linear relation a 13 a 24 − a 14 a 23 − 1 = 0 to obtain (a 13 , a 23 ) and produce a 2 × 4 matrix A that solves the system (20).
second order derivatives are given, and indeed there are only 5 arbitrary constants in the general solution. This number comes directly from our computations if we consider R I 3,8 as an ideal in the ring Q[y, y 1 , y 2 ] since in this case dim V( R I 3,8 ) = 5 .
Theorem 3 .
3If A = A(t) is a geodesic in SL(2), then the function ϕ(t, α) = A(t)v(α),where v is a diffeomorphism, provides a solution to the incompressible Euler equations.
Theorem 4 .
4Let the map ϕ have the form(14). Then ϕ provides a solution to the 2-dimensional Euler equations if v and w satisfy the system of PDEs
a
11 a 22 − a 12 a 21 − 1 = 0 , a 13 a 24 − a 14 a 23 − 1 = 0 , a 14 − a 12 + a 11 = 0 , a 24 − a 22 + a 21 = 0 .
A = (M 1 | M 2 ) as above. The other two conditions in (19) are not a priori clear, though the 4 conditions together lead to the following family of solutions to the 2-dimensional Euler equations. Theorem 5. Assume that the 2 × 4 matrix A = (a ij ) a 22 − a 12 a 21 − 1 = 0 , a 13 a 24 − a 14 a 23 − 1 = 0 , a 14 − a 12 + a 11 = 0 , a 24 − a 22 + a 21 = 0 , a ′′ 21 a 22 − a ′′ 22 a 21 + a ′′ 11 a 12 − a ′′ 12 a 11 = 0 ,
( 21 )
21Then, (16) produces a solution to the 2-dimensional incompressible Euler equations. Proof. Let us introduce the ideal I = a 11 a 22 − a 12 a 21 − 1, a 13 a 24 − a 14 a 23 − 1, a 14 − a 12 + a 11 , a 24 − a 22 + a 21 ⊂ A = Q[a 11 , a 12 , a 13 , a 14 , a 21 , a 22 , a 23 , a 24 ] .
14 − a ′′ 12 + a ′′ 11 , a ′′ 24 − a ′′ 22 + a ′′ 21 ⊂ A 1 = Q[a, a ′′ ] . Next, we compute the formula (18) which yields
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E-mail address: mariaj.martin@uam. Matemáticas Departamento De, es Department of Physics and Mathematics. Madrid. Spain111Universidad Autónoma de Madrid. Campus de Cantoblanco ; University of Eastern FinlandFacultad de Ciencias. FI-80101 Joensuu, Finland. E-mail address: [email protected] de Matemáticas, Facultad de Ciencias (Módulo 17), Universidad Autó- noma de Madrid. Campus de Cantoblanco, 28049, Madrid. Spain. E-mail address: [email protected] Department of Physics and Mathematics, University of Eastern Finland. P.O. Box 111, FI-80101 Joensuu, Finland. E-mail address: [email protected]
| []
|
[
"Instanton Numbers and Exchange Symmetries in N = 2 Dual String Pairs",
"Instanton Numbers and Exchange Symmetries in N = 2 Dual String Pairs"
]
| [
"Gabriel Lopes Cardoso \nTheory Division\nCERN\nCH-1211Geneva 23Switzerland\n",
"Gottfried Curio [email protected] \nHumboldt-Universität zu Berlin\nInstitut für Physik\nD-10115BerlinGermany\n",
"Dieter Lüst \nHumboldt-Universität zu Berlin\nInstitut für Physik\nD-10115BerlinGermany\n",
"Thomas Mohaupt [email protected] \nHumboldt-Universität zu Berlin\nInstitut für Physik\nD-10115BerlinGermany\n"
]
| [
"Theory Division\nCERN\nCH-1211Geneva 23Switzerland",
"Humboldt-Universität zu Berlin\nInstitut für Physik\nD-10115BerlinGermany",
"Humboldt-Universität zu Berlin\nInstitut für Physik\nD-10115BerlinGermany",
"Humboldt-Universität zu Berlin\nInstitut für Physik\nD-10115BerlinGermany"
]
| []
| In this note, we comment on Calabi-Yau spaces with Hodge numbers h 1,1 = 3 and h 2,1 = 243. We focus on the Calabi-Yau space W P 1,1,2,8,12 (24) and show how some of its instanton numbers are related to coefficients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain N = 2 dual models to six dimensional heterotic/heterotic string duality. | 10.1016/0370-2693(96)00668-5 | [
"https://arxiv.org/pdf/hep-th/9603108v2.pdf"
]
| 17,896,071 | hep-th/9603108 | f9d165b73fc139e33e3292adcd9dab98cfa4eb20 |
Instanton Numbers and Exchange Symmetries in N = 2 Dual String Pairs
Apr 1996 March 1996
Gabriel Lopes Cardoso
Theory Division
CERN
CH-1211Geneva 23Switzerland
Gottfried Curio [email protected]
Humboldt-Universität zu Berlin
Institut für Physik
D-10115BerlinGermany
Dieter Lüst
Humboldt-Universität zu Berlin
Institut für Physik
D-10115BerlinGermany
Thomas Mohaupt [email protected]
Humboldt-Universität zu Berlin
Institut für Physik
D-10115BerlinGermany
Instanton Numbers and Exchange Symmetries in N = 2 Dual String Pairs
Apr 1996 March 1996arXiv:hep-th/9603108v2 9 1
In this note, we comment on Calabi-Yau spaces with Hodge numbers h 1,1 = 3 and h 2,1 = 243. We focus on the Calabi-Yau space W P 1,1,2,8,12 (24) and show how some of its instanton numbers are related to coefficients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain N = 2 dual models to six dimensional heterotic/heterotic string duality.
Introduction
Recently, there has been an enormous progress in the understanding of non-perturbative effects in supersymmetric field theories and in superstring theories. In particular, various types of strong-weak coupling duality symmetries are by now quite well established, such as S-duality of the four-dimensional N = 4 heterotic string [1,2,3], string/string dualities [4,5,6] between the heterotic and type II strings and heterotic/heterotic duality in D = 6 [7,8]. It now seems that most or even all non-perturbative duality symmetries originate from underlying theories in higher dimensions (from M-theory [5,9] in D = 11 or from F -theory [10,11] in D = 12).
In the following we will be concerned with N = 2, D = 4 string theories which have both a heterotic and a type II description [6,12]. In this context, there is a particularly interesting class of models, which exhibits a non-perturbative symmetry which exchanges the heterotic dilaton S with one of the vector moduli fields. In the context of heterotic/type II string/string duality this type of exchange symmetry was first discussed in [12], where this exchange symmetry was related to a a remarkle symmetry property of rational instanton numbers. Subsequent work connected this symmetry to the monodromy group of the CY compactification [13,14] and discussed [15] the action of this exchange symmetry on BPS spectra and higher derivative gravitational couplings. In this paper, we will extend the previous work of [6,12,13,16,17,18] in several directions. We will focus on a class of type II theories based on elliptically fibered CY spaces. We will, in particular, discuss the CY space described by a hypersurface of degree 24 in weighted projective space W P 1,1,2,8,12 (24) with, in heterotic language, three vector moduli S, T, U. We will investigate the rational (genus 0) as well as the elliptic (genus 1) instanton numbers for this class of models. We will show that the genus 0 as well as the genus 1 instanton numbers can, in the heterotic weak coupling limit, be precisely expressed by the coefficients of the q expansion of certain modular forms. This means that these instanton numbers are nothing else than the multiplicities of positive roots of some generalized Kac-Moody algebra recently discussed in [19]. (For the rational instanton numbers this relation was already anticipated in [19].) Hence one can expect that the non-perturbative S − T exchange symmetries are reflected in a nice symmetry structure of some non-perturbative infinite-dimensional algebra. In order to relate these instanton numbers to the expansion coefficients of certain modular forms, we will have to work out the precise identification of the heterotic vector moduli with the corresponding type II Kähler class fields. We will also discuss the action of the S − T exchange symmetry in this context. At the end, we will comment on the relation of the four-dimensional exchange symmetry to the six-dimensional heterotic/heterotic duality symmetry [20,7,8] in this class of models.
Instanton numbers and exchange symmetries
The higher derivative couplings of vector multiplets X to the Weyl multiplet W of conformal N = 2 supergravity can be expressed as a power series [21,22] F (X,
W 2 ) = ∞ g=0 F g (X)(W 2 ) g .(1)
In the context of the type II string, F II g only receives perturbative contributions at genus g. In the heterotic context, F het g is perturbatively determined at the tree and at the one loop level; in addition, F het g also receives non-perturbative corrections. For models with heterotic/type II duality one expects that F II g (t i ) = α g F het g (S, T m ), where α g denotes a normalisation constant. The t i (i = 1, . . . , h) denote the Kähler class moduli on the type II side, whereas S and T m (m = 1, . . . , h − 1) denote the dilaton and the vector moduli on the heterotic side.
First consider the prepotential F 0 which determines the gauge couplings. The two prepotentials F II 0 and F het 0 should match up upon a suitable identification of the t i with S and T m . On the type II side, the Yukawa couplings are given by [23]
F II klm = F 0 klm + d 1 ,...,d h n r d 1 ,...,d h d k d l d m 1 − h i q d i i h i=1 q d i i ,(2)
where q i = e −2πt i . The F 0 klm denote the intersection numbers, whereas the n r d 1 ,...,d h denote the rational instanton numbers of genus zero. These instanton numbers are expected to be integer numbers. We will, in the following, work inside the Kähler cone σ(K) =
{ i t i J i |t i > 0}.
For points inside the Kähler cone σ(K), one has for the degrees d i that
d i ≥ 0.
Integrating back yields that
F II 0 = F 0 − 1 (2π) 3 d 1 ,...,d h n r d 1 ,...,d h Li 3 ( h i=1 q d i i )(3)
up to a quadratic polynomial in the t i . F 0 is cubic in the t i . Here one has used that
∂ t k ∂ t l Li 3 = (−2π) 2 d k d l Li 1 , where Li 1 (x) = − log(1 − x).
In the following we will be focusing on a specific type IIA model, namely the S-T-U model [6] based on the Calabi-Yau space W P
F 0 t 1 t 1 t 1 = 8 , F 0 t 1 t 1 t 2 = 2 , F 0 t 1 t 1 t 3 = 4 , F 0 t 1 t 2 t 3 = 1 , F 0 t 1 t 3 t 3 = 2.(4)
It follows that
F 0 = 4 3 t 3 1 + t 2 1 t 2 + 2t 2 1 t 3 + t 1 t 2 t 3 + t 1 t 2 3 .(5)
Some of the instanton numbers n r d 1 ,d 2 ,d 3 can be found in [23]. When investigating the prepotential F II 0 [12], two symmetries become manifest, namely
t 1 → t 1 + t 3 , t 3 → −t 3 for t 2 = ∞,(6)
and
t 2 → −t 2 , t 3 → t 2 + t 3 .(7)
These symmetries are true symmetries of F II 0 , since the world-sheet instanton numbers n r enjoy the remarkable properties [12] n r d 1 ,0,d 3 = n r
d 1 ,0,d 1 −d 3 and n r d 1 ,d 2 ,d 3 = n r d 1 ,d 3 −d 2 ,d 3 .(8)
Observe that F 0 is completely invariant under the symmetry (7).
Next, consider the heterotic prepotential F het 0 . N = 2, D = 4 heterotic strings can be constructed by compactifying the ten-dimensional heterotic string on T 2 × K 3 . A generic compactification of the E 8 × E 8 heterotic string on K 3 , with equal SU(2) instanton number in both E 8 factors, gives rise to D = 6 model with gauge group E 7 × E 7 . For general vev's of the massless hyper multiplets this gauge group is completely broken, and one is left with 244 hyper multiplets and no massless vector multiplets. Upon a T 2 compactification down to four dimensions, one gets a model with 244 hypermultiplets and with three vector multplets S, T and U, where S denotes the heterotic dilaton and T, U denote the moduli of T 2 . This model is the heterotic dual of the type IIA model considered above. The heterotic prepotential has the following structure
F het 0 = −ST U + h (1) (T, U) + F non−pert. (e −S , T, U).(9)
In the following we will consider the semiclassical limit S → ∞, i.e. F non−pert = 0, and we will concentrate on the one-loop corrected prepotential. The heterotic semiclassical prepotential [24,25,27] has nontrivial monodromy properties under the perturbative target space duality symmetries SL(2, Z) T × SL(2, Z) U × Z T ↔U
2
. The singularities of the semiclassical prepotential at the lines/points T = U, T = U = 1 or T = U = e iπ/6 reflect the perturbative gauge symmetry enhancement of U(1) 2 to SU(2) × U(1), SU(2) 2 or SU(3) respectively. Derivatives of the semiclassical prepotential can be nicely expressed in terms of automorphic functions of T and U. The semiclassical prepotential can be written in the following explicit form [19] F het
0 = −ST U + 1 384π 2d 2,2 ABC y A y B y C − 1 (2π) 4 k,l≥0 c 1 (kl)Li 3 (e −2π(kT +lU ) ) − 1 (2π) 4 Li 3 (e −2π(T −U ) ),(10)
where y = (T, U) and where the constants c 1 (n) are related to the positive roots of a generalized Kac-Moody algebra. These constants are determined by
E 4 E 6 η 24 = n≥−1 c 1 (n)q n = 1 q − 240 − 141444q − 8529280q 2 − 238758390q 3 − 4303488384q 4 + . . .(11)
The function F het 0 has a branch locus at T = U. F het 0 given in (10) is defined in the fundamental Weyl chamber T > U. 2 The cubic coefficientsd 2,2 ABC will be determined below. We have ignored a possible constant term as well as a possible additional quadratic polynomial in T and U. The cubic terms cannot be uniquely fixed, since the prepotential contains an ambiguity [24,25] which is a quadratic polynomial in the period vector (1, T, U, T U). Hence, the ambiguity is at most quartic in the moduli and at most quadratic in T and in U. It follows that the third derivative in T or in U is unique; ∂ 2 h (1) ∂T ∂U , however, is still ambiguous. Specifically, in the chamber T > U, the cubic terms have the following general form [19]
d 2,2 ABC y A y B y C = −32π 3(1 + β)T 2 U + 3αT U 2 + U 3 .(12)
The cubic term in U is unique, whereas the parameters α and β correspond to the change induced by adding a quadratic polynomial in (1, T, U, T U). As discussed in [24], it is convenient to introduce a dilaton field S inv , which is invariant under the perturbative T -duality transformations at the one-loop level. It is defined as follows
S inv = S − 1 2 ∂h (1) (T, U) ∂T ∂U − 1 8π 2 log(j(T ) − j(U)) = S + 1 4π (1 + β)T + α 4π U + 1 8π 2 k,l≥0 klc 1 (kl)Li 1 (e −2π(kT +lU ) ) − 1 8π 2 Li 1 (e −2π(T −U ) ) − 1 8π 2 log(j(T ) − j(U)).(13)
2 It is meant here that the real part of T is larger than the real part of U .
In the decompactification limit to D = 5 [26], obtained by sending T, U → ∞ (T > U), the invariant dilaton S inv has a particularly simple dependence on T and U. Namely, by using log j(T ) → 2πT , one obtains that
S inv → S ∞ inv = S + β 4π T + α 4π U.(14)
Substituting S ∞ inv back into the heterotic prepotential (10) yields that
F het 0 = −S ∞ inv T U − 1 12π U 3 − 1 4π T 2 U − 1 (2π) 4 k,l≥0 c 1 (kl)Li 3 (e −2π(kT +lU ) ) − 1 (2π) 4 Li 3 (e −2π(T −U ) ).(15)
Note that the ambiguity in α and β is hidden away in S ∞ inv . Let us now compare the heterotic and the type II prepotentials and identify the t i (i = 1, 2, 3) with S, T and U. In the following, we will actually match −4πF het 0 with F II 0 . First compare the cubic terms in (3) and (10). By assuming that the t i and S, T and U are linearly related, the following identification between the Kähler class moduli and the heterotic moduli is enforced by the cubic terms
t 1 = U t 3 = T − U t 2 = 4πS ∞ inv =Š + βT + αU(16)
whereŠ = 4πS. Recall that we are working inside the Kähler cone
σ(K) = { i t i J i |t i >
0}. Now, in the heterotic weak coupling limit one has that indeed t 2 > 0. Demanding t 3 > 0 implies that one is choosing the chamber T > U on the heterotic side. The identification of t 1 and t 3 agrees, of course, with the one of [12]. The identification of 4πS ∞ inv with the Kähler variable t 2 becomes very natural when performing the map to the mirror Calabi-Yau compactification with complex structure coordinates x, y, z. Here, since y is invariant under the CY monodromy group, y should be identified [12] with e −8π 2 S inv . Thus, equation (13) provides the explicit mirror map; for large T, U the Kähler variable q 2 = e −2πt 2 and the complex structure field y completely agree.
Next, consider the exponential terms in the prepotential F 0 . In the heterotic weak coupling limit S → ∞, one has that t 2 → ∞ and, hence, q 2 = e −2πt 2 → 0. Then,
becomes
F II 0 = F 0 − 1 (2π) 3 d 1 ,d 3 n r d 1 ,0,d 3 Li 3 (q d 1 1 q d 3 3 ).(17)
Some of the instanton coefficients contained in (17)
Note that the fact that n r d 1 ,0,0 = n r d 1 ,0,d 1 is a reflection of the T ↔ U exchange symmetry. Now rewriting kT +lU = (l+k)U +k(T −U) = (l+k)t 1 +kt 3 and matching F II 0 = −4πF het 0 yields the following identifications
d 1 = k + l , d 3 = k n r d 1 ,0,d 3 = n r k+l,0,k = −2c 1 (kl).(19)
Note that d 3 = k ≥ 0 for points inside the Kähler cone. Also, if d 3 = k = 0, then
d 1 = l > 0. On the other hand, if d 3 = k > 0, then d 1 ≥ 0, that is l ≥ −k.
Comparison of the instanton coefficients listed above with the c 1 -coefficients ocurring in the q-expansion of F (q) = E 4 E 6 η 24 in equation (11), shows that the relation (19) is indeed satisfied.
Let us now determine the action of the symmetries (6) and (7) on the heterotic variables.
Clearly the perturbative symmetry (6) corresponds to the exchange T ↔ U for S → ∞.
The non-perturbative symmetry (7) corresponds to
S → −(1 + β)S − α(2 + β) 4π U − β(2 + β) 4π T T → 4πS + (1 + β)T + αU U → U.(20)
There is one very convenient choice for the parameters α and β, in which the nonperturbative symmetry (20) takes a very simple suggestive form. Namely, for α = 0 and β = −1, this transformation becomes
4πS ↔ T,(21)
that is, it just describes the exchange of the heterotic dilaton S with the Kähler modulus T of the two-dimensional torus. This choice for α and β is very reasonable, since it is only in this case that the real parts of S and T remain positive after the exchange (20). At the end of this paper, by considering [8] some six-dimensional one-loop gauge couplings, we will give some further arguments indicating that the choice β = −1 is the physically correct one. So, for the time being, we will set α = 0 and β = −1 and discuss a few issues related to the exchange symmetry 4πS ↔ T .
The non-perturbative exchange symmetry 4πS ↔ T is true for arbitrary U in the chamber S, T > U. As already discussed in detail in [12], at the fixed point t 2 = S ∞ inv = 0 of this transformation, one has that S = T > U, the complex structure field y takes the value y = 1, and the discriminant of the Calabi-Yau model vanishes. The locus S = T > U corresponds to a strong coupling singularity with additional massless states. In the model based on the Calabi-Yau space W P 1,1,2,8,12 (24), a non-Abelian gauge symmetry enhancement with an equal number of massless vector and hypermultiplets takes place at S = T > U, such that the non-Abelian β-function vanishes [28,29].
On the other hand, the non perturbative exchange symmetry 4πS ↔ T implies that for T → ∞ there is a 'perturbative' 4πS ↔ U exchange symmetry. This symmetry is nothing but the T −S transformed perturbative symmetry (6). Furthermore, for T → ∞, there is a modular symmetry SL(2, Z) S ×SL(2, Z) U and the corresponding 'perturbative' monodromy matrices of the prepotential can be computed in a straightforward way.
Hence, for T → ∞, there is a 'perturbative' gauge symmetry enhancement of either U(1) 2 to SU(2) × U(1) or to SU(2) 2 or to SU(3) at the points S = U, S = U = 1 or S = U = e iπ/6 , respectively, with no additional massless hyper multiplets [15].
Let us now investigate the gravitational coupling F 1 , again first in the context of type II compactifications. It can be expressed in terms of the Kähler moduli fields t i as the following instanton sum [30]
F II 1 = −i h i=1 t i c 2 · J i − 1 π n 12n e d 1 ,...,d h log(η( h i=1 q d i i )) + n r d 1 ,...,d h log(1 − h i=1 q d i i ) .(22)
Hereη(q) = ∞ m=1 (1 − q m ), and the n e d 1 ,...,d h denote the elliptic genus one instanton numbers. We will again specialize to the Calabi-Yau space W P 1,1,2,8,12 (24) with h = 3.
Then the non-exponential piece, which dominates for large t i , reads [28] − i
3 i=1 t i c 2 · J i = 92t 1 + 24t 2 + 48t 3 .(23)
This expression is explicitly invariant under the non-perturbative symmetry (7). Furthermore, by also explicitly checking some of the elliptic instanton numbers n e d 1 ,d 2 ,d 3 , one discovers that, just like in the case of n r ,
n e d 1 ,0,d 3 = n e d 1 ,0,d 1 −d 3 and n e d 1 ,d 2 ,d 3 = n e d 1 ,d 3 −d 2 ,d 3 .(24)
It follows that F II 1 is symmetric under the two exchange symmetries (6) and (7). In the heterotic case the holomorphic gravitational coupling at the one-loop level is given by F het
1 = 24S inv + b grav 8π 2 log η −2 (T )η −2 (U) + 2 4π 2 log(j(T ) − j(U)).(25)
For the model we are discussing one has that b grav = 48 − χ = 528. Inserting S inv given
in (13) into F het 1 yields [15] F het 1 = 24 S − 1 768π 2 ∂ T ∂ U d 2,2 abc y a y b y c − 1 8π 2 log(j(T ) − j(U)) + 1 8π 2 k,l≥0 klc 1 (kl)L i 1 (e −2π(kT +lU ) ) − 1 8π 2 L i 1 (e −2π(T −U ) ) + b grav 8π 2 log η −2 (T )η −2 (U) + 2 4π 2 log(j(T ) − j(U)).(26)
Let us now compare the heterotic and the type II gravitational couplings. 4 We will match 4πF het 1 with F II 1 . First take the decompactification limit to D = 5, i.e. the limit T, U → ∞ (T > U). This eliminates all instanton contributions, i.e. all exponential terms. In the heterotic case we get in this limit
F het 1 → F ∞ 1 = 24S ∞ inv + 12 π T + 11 π U = 24S + 12 + 6β π T + 11 + 6α 4π U.(27)
By comparing this expression with the type II large t i limit given in (23), one finds that (23) and (27) match up precisely for the identification given in (16) between heterotic and type II moduli. When choosing α = 0 and β = −1, it follows that F ∞ 1 is symmetric under the exchange 4πS ↔ T . 5 This symmetry implies that in the limit T → ∞, F het 1 can be written [15] in terms of SL(2, Z) S modular functions j(4πS) and η(4πS) by just replacing 4πS with T in equation (26).
Next, let us compare the exponential terms in F II 1 and F het 1 . In the type II case we have to consider the weak coupling limit q 2 → 0; hence only the terms with the instanton numbers n r,e d 1 ,0,d 3 contribute to the sum. We will see that, when comparing with the heterotic expression, one gets a very interesting relation between the rational and elliptic instanton numbers for d 2 = 0. In order to do this comparison, we have to recall that Li 1 (e −2π(kT +lU ) ) = − log(1 − e −2π(kT +lU ) ). The difference j(T ) − j(U) can be written in 4 It was already, to some extent, shown in [16,18,31] that the heterotic and type gravitational couplings agree. 5 In [15] a different choice was made for these two parameters, namely α = −11/6 and β = −2. Hence it follows that F ∞ 1 = 24S.
the following useful form (in the chamber T > U) [32,19] log(j(T ) − j(U)) = 2πT + k,l c(kl) log(1 − e −2π(kT +lU ) ), (28) where the integers k and l can take the following values [19]: either k = 1, l = −1 or k > 0, l = 0 or k = 0, l > 0 or k > 0, l > 0. The universal constants c(n) are defines as follows:
j(q) − 744 = ∞ n=−1 c(n)q n = 1 q + 196884q + 21493760q 2 + 864299970q 3 + 20245856256q 4 + . . .(29)
First consider the terms with k = 1, l = −1 on the heterotic side. Matching the term
log(1 − e −2π(T −U ) ) contained in 4πF het 1 with F II 1 requires that 10c(−1) − 12c 1 (−1) = 12n e 0,0,1 + n r 0,0,1 .(30)
This is indeed satisfied, since c(−1) = c 1 (−1) = 1 and n e 0,0,1 = 0, n r 0,0,1 = −2. Next, consider the terms in the sum with k > 0, l = 0 (and analogously k = 0, l > 0).
Since c(0) = 0, only the term bgrav 8π 2 log η −2 (T ) contributes on the heterotic side (b grav = 528). Matching 4πF het 1 with F II 1 yields the following relation among the instanton numbers (d 1 = d 3 = k):
12 s i=1 n e k i ,0,k i + n r k,0,k = b grav = 528.(31)
The k i (i = 1, . . . , s) are the divisors of k (k 1 = k, k s = 1). Using Klemm's list of explicit instanton numbers, we checked that this relation is indeed true up to k = 4 (n e 1,0,1 = 4, n e k,0,k = 0 for k > 1, n r k,0,k = −χ = 480). Finally, consider the case where k > 0, l > 0. By comparing the heterotic and type II expressions we derive the following interesting relation (d 1 = k + l, d 3 = k):
which, together with equations (11) and (29), confirms the above relation. For k = 2 and l = 1 one finds that 12n e 3,0,2 + n r 3,0,2 = 10c(2) + 24c 1 (2) is indeed satisfied, since n e 3,0,2 = −568640 and n r 3,0,2 = 17058560.
And finally, for k = l = 2 for instance, one finds that the relation 12 n e 4,0,2 + n e 2,0,1 + n r 4,0,2 = 10c(4) + 48c 1 (4) indeed holds due to n e 4,0,2 = −1059653772 , n e 2,0,1 = −948 and n r 4,0,2 = 8606976768.
Now consider the relation E 6 E 4
η 24 ′ = − 2π 6 ( E 2 E 4 E 6 η 24 + 2 E 2 6 η 24 + 3 E 3 4 η 24 )
. From this we can, for n > 0, derive the useful equation 12nc 1 (n) + 10c(n) = −2c 1 (n), where thec 1 (n) are defined as follows
E 2 E 6 E 4 η 24 (q) = ∞ n=−1c 1 (n)q n = 1 q − 264 − 135756q − 5117440q 2 + . . .(36)
It follows that one can rewrite equation (32) as
12 s i=1 n e d i 1 ,0,d i 3 + n r d 1 ,0,d 3 = 10c(kl) + 12klc 1 (kl) = −2c 1 (kl) , k > 0, l > 0.(37)
This corresponds to the following integral 6 of [19]
(with b grav = 48−χ = −2(c 1 (0)−24) = −2c 1 (0))Ĩ 2,2 = −1 2 F d 2 τ τ 2 [− i η 2 T r R J 0 (−1) J 0 q L 0 −22/24qL 0 −9/24 (E 2 − 3 πτ 2 ) − b grav ] = −1 2 F d 2 τ τ 2 [−2Z 2,2 E 4 E 6 η 24 (E 2 − 3 πτ 2 ) − (−2c 1 (0))](38)
Let us briefly summarize our results obtained so far. The type II prepotential F II 0 is determined by rational (genus 0) instanton numbers n r . Comparison with the semiclassical heterotic prepotential F het 0 relates a subset of the rational instanton numbers n r (d 2 = 0) to the coefficients of the modular function E 4 E 6 η 24 of modular weight -2. The type II gravitational coupling F II 1 depends, in addition, on the elliptic (genus 1) instanton numbers n e . A subset of those can be expressed in terms of the coefficients of the modular functions E 2 E 4 E 6 η 24 and E 4 E 6 η 24 . For the higher F g (cf. [33]) we conjecture the following. Under modular transformations T → 1 T the F g transform at weak coupling as 6 The precise relation of this integral to F het 1 was worked out in [15].
F g → T 2(g−1) F g ,(39)
i.e. F g has modular weight 2(g − 1). Thus we are tempted to conclude that the higher (genus g) instanton numbers are determined by the coefficients of a modular form of modular weight 2(g −1). Since the ring of modular functions, together with η −24 is finite, only a finite number of different types of instanton numbers seem to be independent.
Note that such a fact is known for the case of a one dimensional Calabi-Yau target space, that is an elliptic curve, where according to (also cf. [34,35,36]) the F g are quasimodular forms of weight 6g − 6 for g ≥ 2 , i.e. F g ∈ Q[E 2 , E 4 , E 6 ]. Also note that one has F 1 = − log η [30]. This is here conjecturally extended to the elliptically fibered Calabi-Yau space considered above.
Comments on other Calabi-Yau models
At the end, let us briefly consider different Calabi-Yau spaces and also comment on the relation to the heterotic/heterotic duality in six dimensions [7,8], with the 6-dimensional heterotic string compactified on K 3 . (The decompactification limit from D = 4 to D = 6 is obtained by sending T → ∞ with U finite; as discussed in [20,7], the D = 6 heterotic/heterotic duality becomes an exchange symmetry of S with T in D = 4.) We will concentrate on three families of CY's with Hodge numbers (3,243), which are elliptic fibrations over F n with n = 0, 1, 2 [11,37]. Being elliptic fibrations, they can be used to compactify F -theory to six dimensions. In the D = 6 heterotic string, the integer n is related to the number s of SU(2) instantons in one of the two E 8 's by n = s − 12 [11,37].
First consider the case of an elliptic fibration over F 0 , corresponding to the symmetric embedding of the SU(2) bundles with equal instanton numbers s = s ′ = 12 into E 8 × E ′ 8 . This leads to a D = 6 heterotic model with gauge group E 7 × E ′ 7 withṽ α =ṽ ′ α = 0 [7]. There are 510 hypermultiplets transforming as 4(56, 1) + 4(1, 56) + 62(1, 1). The heterotic/heterotic duality originates from the existence of small instanton configurations [38]. The model is, however, not self-dual, since the non-perturbative gauge groups appear in different points of the hyper multiplet moduli space than the original gauge groups [7].
For generic vev's of of the hyper multiplets the gauge group is completely broken and one is left with 244 hyper multiplets and no vector multiplets. Upon compactification to D = 4 on T 2 one arrives at the heterotic string with gauge group U(1) 4 , which is the dual to the considered type II string on the CY W P 1,1,2,8,12 (24). Semiclassically, at special points in the hypermultiplet moduli space, this gauge group can be enhanced to a non-Abelian gauge group, inherited from the E 7 × E ′ 7 with N = 2 β-function coefficient b α = 12(1 +ṽ α vα ) = 12 [8].
Next, consider embedding the SU(2) bundles in an asymmetric way into the two E 8 's [39,8]: s = 14, s ′ = 10. This corresponds to the elliptic fibration over F 2 [11,37]. Note that F 0 and F 2 are of the same parity (even N), so they are connected by deformation [11], as we will discuss in the following. Then, in this case, one has [8] a gauge group E 7 × E ′ 7 withṽ α = 1/6 andṽ ′ α = −1/6 and hyper multiplets transforming as 5(56, 1) + 3(1, 56) + 62 (1,1). The second E 7 can be completely Higgsed away, leading to a D = 6 heterotic model with gauge group E 7 and hypermultiplets 5(56) + 97 (1). As explained in [8], this model also possesses a heterotic/heterotic duality, however without involving non-perturbative small instanton configurations. Hence in this sense, this model is really self-dual. Just like in the case of the symmetric embedding, the gauge group E 7 is spontaneously broken for arbitrary vev's of the gauge non-singlet hyper multiplets and one is again left with 244 hyper multiplets and no vector multiplet. When compactifying on T 2 to D = 4, one obtains the same heterotic string model with U(1) 4 gauge group as before. For special values of the hyper multiplets a non-Abelian gauge group is obtained, now however with β-function coefficient b α = 12(1 +ṽ α vα ) = 24 [8]. In summary, the symmetric (12,12) model and the asymmetric (14,10) model should be considered as being the same [8,11], since both are related by the Higgsing and both lead to the same heterotic string in D = 4.
As already mentioned, we would like to provide a six-dimensional argument for why β = −1 is the physically correct choice for one of the cubic parameters. We will directly follow the discussion given in [8] and consider the one-loop gauge coupling for the enhanced non-Abelian gauge groups that are inherited from the six-dimensional gauge symmetries.
Specifically, the gauge kinetic function is of the form [24,8]
f α = S inv − b α 8π 2 log(η(T )η(U)) 2 .(40)
Using equation (14) this then becomes in the decompactification limit T → ∞ to D = 6
f α → S + 1 + β +ṽ α vα 4π T.(41)
By comparing this expression with the six-dimensional gauge coupling [40], it then follows that β = −1.
Let us also make some remarks on the third model with Hodge numbers (3,243), the (13,11) embedding [11,37]. This is now elliptically fibered over F 1 . In going to the Higgs branch [37] one reaches the Calabi-Yau W P 1,1,1,6,9 (18) with h 1,1 = 2, h 2,1 = 272. Note that in the D = 6 interpretation of F-theory on this Calabi-Yau, this corresponds to loosing a tensor multiplet and gaining 29 hyper multiplets. Thus, in four dimensions, the two vector multiplets correspond to T and U. No dilaton S is present, reflecting the fact that this CY is not a K 3 fibration, and no heterotic dual (at weak coupling)
exists. This CY is now elliptically fibered over P 2 (the exceptional curve of F 1 was blown down) [37]. According to [41], the rational instanton numbers n r j,0 of this CY are all equal to 540 = −χ CY . Thus, compared to the CY W P 1,1,2,8,12 (24), the corresponding modular form is now simply a constant. For q 2 = 0, the Yukawa coupling y 111 of the W P 1,1,1,6,9 (18) model is given by E 4 [41]. This can be nicely compared with the following Yukawa coupling [24] of the CY W P 1,1,2,8,12 (24) model in the limit T → ∞
∂ 3 U h (1) ∼ E 4 (U) j(T ) − j(U) E 4 (T )E 6 (T ) η 24 (T ) → E 4 (U).(42)
The elliptic instanton numbers in the W P 1,1,1,6,9 (18) model satisfy the following relation:
12 n e j,0 + n r j,0 = 12 · 3 + 540. Here, n e j,0 = 3 (versus n e j,0 = 4 in the W P 1,1,2,8,12 (24) model) is determined by the elliptic fibration base with χ(P 2 ) = 3 (versus χ(F 1 ) = 4 in the W P 1,1,2,8,12 (24) model). This difference in the n e j,0 corresponds to the loss of one h 1,1 class (of the elliptic fibration base or equally well of the whole Calabi-Yau) in the blowing down process. Accordingly, the expression b grav = 48 − χ(CY ) is modified.
It is instructive to consider the mirror map of this model. Using the complex structure variable Y 1 = − 1 X 1 (cf. chapter 7.2 in [41]) the mirror map becomes 1 Y 1 (1−432Y 1 ) = j(U) for T → ∞, which corresponds to the elliptic family P 1,2,3 (6) in [12]. This also corresponds to the T → ∞ limit (at S → ∞) of the S − T − U Calabi-Yau model W P 1,1,2,8,12 (24), which can also be considered to be elliptically fibered by the elliptic family P 1,2,3 (6) over F 0 , when one considers [42,11] the one nonpolynomial deformation of W P 1,1,2,8,12 (24), which deforms the base F 2 to F 0 .
Let us close with the following remark. The S −T exchange symmetry is also present [12] in the S − T model based on the CY W P 1,1,2,2,6 (12) with Hodge numbers h 1,1 = 2, h 2,1 = 128. This model, however, falls out of the class of the CY's considered above, since, even though being a K 3 -fibration, it does not correspond to an elliptic fibration. This model is obtained [6] by first performing a toroidal compactification to D = 8 on a torus with T = U and enhanced SU(2) gauge group, and subsequently going down to D = 4 by a K 3 compactification. Like in the case of [7], there is again a symmetric embedding of the SU(2) gauge bundle into E 8 × E ′ 8 × SU(2): (s, s ′ , s ′′ ) = (10, 10, 4). The S − T exchange symmetry, however, is not related to a six-dimensional heterotic/heterotic duality or to F -theory on a CY.
Acknowledgement
We are grateful to A. Klemm for providing us with a list of instanton numbers for the CY model W P 1,1,2,8,12 (24). We would also like to thank P. Berglund, A. Klemm, R. Minasian and especially F. Quevedo and S.J.-Rey for fruitful discussions.
is the number of common divisors m i (i = 1, . . . , s) of d 1 = k + l and d 3 = k with d i 1 = d 1 /m i and d i 3 = d 3 /m i (where m 1 = 1). Again we can explicitly check the non-trivial relation (32) for the first few terms. For example, for k = l = 1 one has n e 2,0,1 = −948 and n r 2,0,1 = 282888,
We are grateful to A. Klemm for providing us with a list of instanton numbers for this model.
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| []
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[
"The countable existentially closed pseudocomplemented semilattice",
"The countable existentially closed pseudocomplemented semilattice"
]
| [
"Joël Adler \nPädagogische Hochschule Bern\nSwitzerland\n"
]
| [
"Pädagogische Hochschule Bern\nSwitzerland"
]
| []
| As the class PCSL of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a ℵ 0 -categorical model companion PCSL * . As PCSL is inductive the models of PCSL * are exactly the existentially closed models of PCSL. We will construct the unique existentially closed countable model of PCSL as a direct limit of algebraically closed pseudocomplemented semilattices. * [email protected] | 10.1007/s00153-017-0527-x | [
"https://arxiv.org/pdf/1203.6700v12.pdf"
]
| 27,017,847 | 1203.6700 | 167bd5129e02c490f73a7b6e224647dad9bbb90e |
The countable existentially closed pseudocomplemented semilattice
16 Jul 2014 July 17, 2014
Joël Adler
Pädagogische Hochschule Bern
Switzerland
The countable existentially closed pseudocomplemented semilattice
16 Jul 2014 July 17, 2014
As the class PCSL of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a ℵ 0 -categorical model companion PCSL * . As PCSL is inductive the models of PCSL * are exactly the existentially closed models of PCSL. We will construct the unique existentially closed countable model of PCSL as a direct limit of algebraically closed pseudocomplemented semilattices. * [email protected]
Introduction
For a first-order language L and an L-structure M with universe M the language L(M ) is obtained by adding a constant symbol for every m ∈ M . To define the notion of model companion we first have to define the notion of model completeness. An L-theory T is said to be model complete if for every model M of T the set of L-sentences T ∪ diag(M) is complete, where diag(M) is the set of atomic and negated atomic L(M )-sentences that hold in M. T * is said to be a model companion of T if (i) every model of T * is embeddable in a model of T and vice versa and (ii) T * is model complete.
A theory T need not have a model companion as is the case for the theory of groups and theory of commutative rings, see Wheeler [9]. However, if T is a set of Horn sentences and the class Mod(T ) of its models is finitely generated then T has a model companion T * as was shown by Burris and Werner [4].
If T is additionally inductive-that is Mod(T ) is closed under the union of chains-then we have the characterization Mod(T * ) = Mod(T ) ec , that is the models of T * are the existentially closed models of T , see Macintyre [7]. A definition of the notions of algebraically and existentially closed can be found in [1]. Finally, if Mod(T ) is generated by single finite structure then Mod(T * ) is ℵ 0 -categorical, see Burris [3].
Horn and Balbes [2] proved that PCSL is equational, Jones [6] showed that it is generated by a single finite structure. Thus PCSL * is ℵ 0categorical and its only countable model is the countable existentially closed pseudocomplemented semilattice.
In Section 2 we provide the basic properties and algebraic notions concerning pseudocomplemented semilattices-p-semilattices for short-while in Section 3 the countable existentially closed p-semilattice is constructed.
2 Basic properties of pseudocomplemented semilattices and notation A p-semilattice P ; ∧, * , 0 is an algebra where P ; ∧ is a meet-semilattice with least element 0, and for all x, y ∈ P , x ∧ a = 0 if and only if x ≤ a * .
1 := 0 * is obviously the greatest element of P . x y is defined to hold if neither x ≤ y nor y ≤ x holds. An element d of P satisfying d * = 0 is called dense, and if additionally d = 1 holds, then d is called a proper dense element. For P ∈ PCSL the set D(P) denotes the subset of dense elements of P, D(P); ∧ being a filter of P ; ∧ . An element s is called skeletal if s * * = s. The subset of skeletal elements of P is denoted by Sk(P). The abuse of notation Sk(x) for x ∈ Sk(P) and D(x) for x ∈ D(P)should not cause ambiguities. Obviously, Sk(P) = {x * : x ∈ P }.
For any p-semilattice P the p-semilattice P is obtained from P by adding a new top element. The maximal dense element of P different from 1 is denoted by e. Furthermore, the p-semilattices B with B being a boolean algebra are exactly the subdirectly irreducible p-semilattices. Moreover, let 2 denote the two-element boolean algebra and A the countable atomfree boolean algebra interpreted as p-semilattices.
An equational set Σ of axioms for PCSL can be found in [1], for more background on p-semilattices in general consult Frink [5] and [6].
In Schmid [8] the following characterization of algebraically closed psemilattices is established: Theorem 2.1. A p-semilattice P is algebraically closed if and only if for any finite subalgebra F ≤ P there exists r, s ∈ ω and a p-
semilattice F ′ isomorphic to 2 r × A s such that F ≤ F ′ ≤ P.
In [1] the following (syntactic) description of existentially closed psemilattices is given:
Theorem 2.2
. A p-semilattice P is existentially closed if and only if P is algebraically closed and satisfies the following list of axioms:
(EC1) if (∀b 1 , b 2 ∈ Sk(P))(∃b 3 ∈ Sk(P))(b 1 < b 2 −→ b 1 < b 3 < b 2 )), (EC2) if (∀b 1 , b 2 ∈ Sk(P), d ∈ D(P))(∃b 3 ∈ Sk(P))( (b 1 ≤ b 2 < d < 1 & b * 1 d) −→ (b 2 < b 3 < 1 & b * 1 ∧ b 3 d & b 1∨ b * 3 < d)), (EC3) if (∃d ∈ D(P))(d < 1), (EC4) if (∀d 1 , d 2 ∈ D(P))(∃d 3 ∈ P )(d 1 < d 2 −→ (d 1 < d 3 < d 2 )), (EC5) if (∀b ∈ Sk(P), d 1 ∈ D(P))(∃d 2 ∈ D(P))(0 < b < d 1 −→ (d 2 < d 1 & b d 2 & d 1 ∧ b * = d 2 ∧ b * )).
Constructing the unique countable model of the model companion Σ * of PCSL thus amounts to constructing a countable algebraically closed p-semilattice that satisfies (EC1)-(EC5).
The construction
As the objects of the direct limit we are going to construct we take
{G n : n ∈ N \ {0}}, where G n := A n .
In view of Theorem 2.1 G n is algebraically closed for all n ∈ N \ {0}. We have to define embeddings f n : G n → G n+1 , n ≥ 1, such that the direct limit of the directed family { G m , g m,n : m, n ∈ N, 1 ≤ m ≤ n} where g i,j := f j−1 • · · · • f i for i < j and g i,i = id Gi additionally satisfies (EC1)-(EC5) of Theorem 2.2. We will see that this is obtained if all f n : G n → G n+1 have the following properties:
(P1) For every anti-atom d of D(G n ) there is a k ∈ N such that g n,n+k (d)
is not an anti-atom of G n+k anymore. This will imply that the order restricted to the dense elements of the direct limit is dense.
(P2) For every a ∈ Sk(G n ) \ {0, 1} and d ∈ D(G n ) with a < d there is an m ∈ N such that π n+m (g n,n+m (a)) = π n+m (g n,n+m (d)) = 1.
Let d 1 , . . . , d n be an enumeration of the anti-atoms of D(G n ), where D(G n ) = {e, 1} n . For every anti-atom d i ∈ D(G n ) let σ n (i) ∈ {1, . . . , n} be such that π k (d i ) = e if and only if k = σ n (i), 1 ≤ k ≤ n. That is, σ n is the permutation that assigns i the place of the component of d i that is e. Furthermore, let a σn(i)1 , a σn(i)2 , . . . be an enumeration of the elements of π σn(i) (Sk(G n )) \ {0, 1}. To define f n we use the following notation: For x = (x 1 , . . . , x n ) ∈ G n and u ∈ A we put ( − → x , u) = (x 1 , . . . , x n , u) ∈ G n+1 . We distinguish between n being even and n being odd.
• For n even we put f n (
x) = − → x , x σn(1) for x = (x 1 , . . . , x n ) ∈ G n . f n obviously is an embedding. f n (d 1 ) = ( − → d 1 , π σn(1) (d 1 )) = ( − → d 1 , e) is not an anti-atom of G n+1 anymore, whereas f n (d i ) = ( − → d i , π σn(1) (d i )) = ( − → d i , 1)
still is an anti-atom of G n+1 for i = 2, . . . , n. These antiatoms of G n+1 are numbered 1 to n − 1 whereas the two anti-atoms of D (G n+1 \ f n (G n )) are numbered n and n + 1 according to the place of the e-component. This guarantees that for every anti-atom d of G n there is k ∈ N such that g n,n+k (d) is not an anti-atom of G n+k anymore. Thus (P1) is satisfied. The enumeration of π i (Sk(G n+1 )) \ {0, 1} is the same as the enumeration of π i (Sk(G n ))\{0, 1} for 1 ≤ i ≤ n+1. π n+1 (Sk(G n+1 ))\{0, 1} can be enumerated arbitrarily.
• For n odd we define
f n (x) = ( − → x , 1), π σn(1) (x) ∈ U σn(1) ∪ {e}, ( − → x , 0), otherwise,(1)
where U σn(i) is an ultrafilter on π σn(i) (Sk(G n )) containing a σn(i)1 . a σn(1)1 is denoted as the distinguished element for n.
Then f n (d) = ( − → d , 1) for all d ∈ D(G n ): π σn(1) (d) ∈ {e, 1} ⊂ U σn(1) ∪ {e}. Thus f n (d i ), i = 1, .
. . , n, are still anti-atoms of G n+1 . They are numbered 1 to n, the anti-atom (1, . . . , 1, e) is numbered n + 1. We obtain f n (min(D(G n ))) = min(D(G n+1 )). The enumeration of π i (Sk(G n+1 )) \ {0, 1}, 1 ≤ i ≤ n + 1, is as follows: For i ∈ {σ n (1), n + 1} the enumeration is the same as for π i (Sk(G n )) \ {0, 1}. π n+1 (Sk(G n+1 )) \ {0, 1} can be enumerated arbitrarily. Let now x be an element of π σn(1) (Sk(G n+1 )) \ {0, 1} = π σn(1) (Sk(G n )) \ {0, 1}. Therefore, x = a σn(1)j is the jth element of π σn(1) (Sk(G n )) \ {0, 1} for a j ∈ N. We distinguish three cases depending on the value of j. If j = 1 then x receives the number 2 n . If 2 ≤ j ≤ 2 n , x receives the number j − 1. Finally, if 2 n < j, then x receives the number j. This guarantees that every x ∈ π i (Sk(G n )) \ {0, 1}, 1 ≤ i ≤ n, becomes the distinguished element for some n ′ ≥ n. Thus (P2) is satisfied in G n ′ .
Claim. The direct limit G of the directed family { G m , g m,n : m, n ∈ N, 1 ≤ m ≤ n} of p-semilattices is countable and existentially closed.
Proof. G is countable since a countable union of countable sets is countable. That G is algebraically closed follows from Theorem 2.1: Let S be a finite subalgebra of G. By the construction of G, there is an n ∈ N such that that the carrier S of S is a subset of G n . Therefore, there is a subalgebra S ′ of G isomorphic to G n = ( A) n containing S.
By Theorem 2.2 it remains to show that G satisfies (EC1)-(EC5). (EC1) is satisfied as it is satisfied in A. (EC3) is obviously satisfied. To prove the remaining three axioms we denote for x ∈ ∞ n=1 G n with [x] ∈ G the equivalence class of x.
For (EC2) consider arbitrary b 1 , b 2 ∈ Sk(G) and d ∈ D(G) such that b 1 ≤ b 2 < d and b * 1 d. There is n ∈ N and x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ), w = (w 1 , . . . , w n ) ∈ G n such that b 1 = [x], b 2 = [y], d = [w], Sk(x), Sk(y), D(w), x ≤ y < w and x * w. We first assume that w is not an anti-atom of G n . Then without loss of generality we can assume x 1 = 0, w 1 = w 2 = e. Then put z = (1, z 2 , 1, . . . , 1) with y 2 < z 2 < e to obtain y < z, z w, x * ∧ z w and x∨ z * < w. The last inequality follows from x 2 ≤ y 2 < z 2 , which implies 0 < x * 2 ∧z 2 , thus e > (x *
2 ∧z 2 ) * = x 2∨ z * 2 . Putting b 3 = [z] yields what is requested in (EC2).
If w is an anti-atom there is by (P1) an l ∈ N such that g n,n+l (w) is not an anti-atom of G n+l anymore. For x ′ := g n,n+l (x), y ′ := g n,n+l (y) and w ′ := g n,n+l (w) we find as above z ∈ G n+l such that y ′ < z, x ′ * ∧ z w ′ and x ′∨ z * < w ′ .
Putting b 3 = [z] yields what is requested in (EC2) because [x] = [x ′ ], [y] = [y ′ ], [w] = [w ′ ].
For (EC4) consider arbitrary d 1 , d 2 ∈ D(G) such that d 1 < d 2 . There is n ∈ N and x, y ∈ G n such that d 1 = [x], d 2 = [y]. There are l ∈ N and z ∈ D(G n+l ) such that g n,n+l (x) < z < g n,n+l (y): We have x = j∈Jx x j , y = j∈Jy x j for subsets J y J x ⊆ {1, . . . , n}, x j being an anti-atom of D(G n ) for j ∈ J x . For j 0 ∈ J x \ J y there is by (P1) a least l ∈ N such that g n,n+l (x j0 ) is not an anti-atom of G n+l anymore, that is, there is are anti-atoms u 1 , u 2 ∈ G n+l with g n,n+l (x j0 ) < u 1 ∧ u 2 and g n,n+l (x j ) u i for all j ∈ J y , i = 1, 2. Because D(G n+l ) is boolean, that is, a ∧-reduct of a boolean algebra, we obtain g n,n+l (x) = j∈Jx\{j0} g n,n+l (x j ) ∧ u 1 ∧ u 2 < j∈Jx\{j0} g n,n+l (x j ) ∧ u 1 ≤ j∈Jy g n,n+l (x j ) ∧ u 1 < j∈Jy g n,n+l (x j ), which implies g n,n+l (x) < g n,n+l (y) ∧ u 1 < g n,n+l (y).
We have For (EC5) consider an arbitrary b ∈ Sk(G) and d 1 ∈ D(G) such that 0 < b < d 1 . There is n ∈ N and x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ G n such that b = [x], d 1 = [y], Sk(x), D(y), 0 < x < y. Let us assume that there is no z ∈ D(G n ) such that z < y, x||z and x * ∧ y = x * ∧ z, since otherwise we put d 2 = [z].
By (P2) there is an l ∈ N such that π n+l (g n,n+l (x)) = π n+l (g n,n+l (y)) = 1. Defining z ∈ G n+l by putting π j (z) = π j (g n,n+l (y)) for 1 ≤ j ≤ n+l −1 and π n+l (z) = e we can then choose d 2 = [z].
d 1 = [x] = [g n,n+l (x)] < [g n,n+l (y) ∧ u 1 ] < [g n,n+l (y)] = [y] = d 2 ,and we can choose d 3 = [g n,n+l (y) ∧ u 1 ].
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[
"SMALE SPACES WITH VIRTUALLY NILPOTENT SPLITTING",
"SMALE SPACES WITH VIRTUALLY NILPOTENT SPLITTING"
]
| [
"Volodymyr Nekrashevych "
]
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| We show that if (X , f ) is a locally connected Smale space such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then (X , f ) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus. | null | [
"https://arxiv.org/pdf/1210.7383v1.pdf"
]
| 117,516,090 | 1210.7383 | 981b9e7619f5ecd4d5e62495d1eb9e9f88143f53 |
SMALE SPACES WITH VIRTUALLY NILPOTENT SPLITTING
27 Oct 2012
Volodymyr Nekrashevych
SMALE SPACES WITH VIRTUALLY NILPOTENT SPLITTING
27 Oct 2012
We show that if (X , f ) is a locally connected Smale space such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then (X , f ) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus.
Introduction
Smale spaces were introduced by D. Ruelle (see [Rue78]) as generalizations of basic sets of Axiom-A diffeomorphisms. They were also extensively studied before as "spaces with hyperbolic canonical coordinates" by R. Bowen [Bow70,Bow71].
A Smale space is a compact metric space X with a homeomorphism f : X −→ X such that there exists a local direct product structure on X with respect to which f is expanding in one and contracting in the other direction. For example, every Anosov diffeomorphism is a Smale space.
Smale spaces are classical objects of the theory of dynamical systems, but many basic questions about them (and even about Anosov diffeomorphisms) remain to be open.
For example, it seems that the following question is open.
Question 1. Is it true that if (X , f ) is a Smale space such that X is connected and locally connected, then (X , f ) is topologically conjugate to an Anosov diffeomorphism?
Many well studied examples of Smale spaces are such that one of the factors of the local direct product structure is totally disconnected, e.g., the Smale solenoid (see [BS02,Section 1.9] ). See more examples in [Wil74,AP98]. See also [Wie12], where it is proved that all such Smale spaces are inverse limits of iterations of one self-map, i.e., are natural generalizations of solenoids.
Note that a question similar to Question 1 for expanding maps has a positive answer. Namely, the following theorem follows from Gromov's theorem on groups of polynomial growth [Gro81], [Nek05, Theorem 6. 1.6], [Nek08,Theorem 5.9].
Theorem 1.1. Let f : X −→ X be a self-covering map of a locally connected and connected compact metric space. Suppose that there exists a covering map π : X −→ X such that f can be lifted by π to an expanding homeomorphism of X . (This is true, for example, if f : X −→ X is locally expanding and X is semilocally simply connected.) Then (X , f ) is topologically conjugate to an expanding infra-nilmanifold endomorphism.
Here an infra-nilmanifold endomorphism is a map φ : G\L −→ G\L, where L is a simply connected nilpotent Lie group, G is a subgroup of the affine group Aut(L) ⋉ L acting on L freely, properly, and co-compactly, and φ is induced by an automorphism Φ : L −→ L. If the automorphism Φ is expanding, i.e., if all eigenvalues of DΦ have absolute value greater than one, then we say that the corresponding endomorphism φ is expanding. If DΦ has no eigenvalues of absolute value one, then we say that φ is hyperbolic.
Note, that the case when f : X −→ X in Theorem 1.1 is an expanding endomorphism of a Riemannian manifold, is a result of M. Gromov [Gro81] (based on results of M. Shub [Shu70]).
All known examples of Anosov diffeomorphisms, and hence apparently all known examples of locally connected Smale spaces are hyperbolic automorphisms of infranilmanifolds. See [Sma67], and Problem 3 in the additional list of problems in [Sma00]. It was proved by A. Manning in [Man74] that every Anosov diffeomorphism of an infra-nilmanifold is topologically conjugate to a hyperbolic automorphism of an infra-nilmanifold. Another result in this direction is a theorem of J. Franks [Fra70] stating that if (X , f ) is an Anosov diffeomorphism such that stable or unstable manifolds of X are one-dimensional, then (X , f ) is topologically conjugate to a hyperbolic linear automorphism of the torus R n /Z n .
One of the main difficulties of proving that every Anosov diffeomorphism is an automorphism of an infra-nilmanifold is showing that the foliations of X into stable and unstable manifolds when lifted to the universal covering X of X come from a direct product decomposition of X .
Definition 1. 1. We say that a Smale space (X , f ) is splittable if there exists a covering map π : M −→ X and a direct product decomposition M = A × B of M such that π maps plaques {a} × B and A × {b} of the direct product decomposition of M bijectively to stable and unstable leaves of X .
Here a stable (resp. unstable) leaf of X is the equivalence class with respect to the equivalence relation lim n→+∞ d(f n (x), f n (y)) = 0 (resp. lim n→−∞ d(f n (x), f n (y))).
If X is locally connected and connected, then every splitting is a Galois covering with a finitely generated group of deck transformations, see Proposition 5.2.
One of the main results of our paper is the following theorem.
Theorem 1. 2. Let (X , f ) be a Smale space such that X is connected and locally connected. Suppose that it has a splitting with a virtually nilpotent group of deck transformations. Then (X , f ) is topologically conjugate to a hyperbolic infranilmanifold automorphism. [Bri77,Bri78] gave a "pinching" condition on the Mather spectrum of an Anosov diffeomorphism (X , f ) (i.e., spectrum of the operator induced by f on the Banach space of vector fields on X ) ensuring that (X , f ) has a splitting with a virtually nilpotent group of deck transformations. In the case of Anosov diffeomorphisms the splitting map π : M −→ X is necessarily the universal covering map, so that the group of deck transformations is the fundamental group of X . M. Brin and A. Manning proved then in [BM81] that all Anosov diffeomorphisms satisfying the Brin's pinching condition are hyperbolic automorphisms of infranilmanifolds.
M. Brin in
We generalize the results of M. Brin and A. Manning. Of course, we can not use the original pinching condition, since we do not have vector fields on Smale spaces. We find, however, a purely topological condition, which follows from Brin's condition in the case of Anosov diffeomorphism. In fact, we even slightly improve the Brin's spectral pinching condition for Anosov diffeomorphisms.
Here is an informal description of our condition. Consider a finite covering R of X by small open rectangles. The covering will induce coverings of the stable and unstable leaves by the plaques of the elements of R. Define, for a stable leaf V and x, y ∈ V , combinatorial distance d R (x, y) equal to the smallest length m of a chain x ∈ R 0 , R 1 , . . . , R m ∋ y, R i ∩ R i+1 = ∅, of plaques of the elements of R. Then d R (f n (x), f n (y)) grows exponentially for x = y. We say that α 0 > 0 and α 1 > 0 are stable lower and upper exponents if there exists C > 1 such that C −1 e α0n ≤ d R (f n (x), f n (y)) ≤ Ce α1n for all stably equivalent x, y such that the distance between x and y inside their stable leaf belongs to some fixed interval [ǫ 1 , ǫ 2 ] for 0 < ǫ 1 < ǫ 2 . Stable upper and lower critical exponents are the infimum and the supremum of all stable upper and lower exponents, respectively. We prove that stable critical exponents are uniquely determined by the topological conjugacy class of the Smale space and are positive and finite. Unstable upper and lower critical exponents are defined in a similar way (they are stable upper and lower critical exponents of (X , f −1 )). For more details, see Sections 3 and 4. Theorem 1. 3. Suppose that (X , f ) is a Smale space such that X is connected and locally connected. Let a 0 , a 1 , b 0 , b 1 be the stable lower and upper, and the unstable lower and upper critical exponents, respectively. If
a 0 a 1 + b 0 b 1 > 1
then (X , f ) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.
We show that Brin's pinching condition on the Mather spectrum of an Anosov diffeomorphism implies our condition on the critical exponents.
As an application of Theorem 1.3, we show that the theorem of J. Franks on co-dimension one Anosov diffeomorphisms is true for all locally connected Smale spaces.
Theorem 1. 4. Let (X , f ) be a Smale space such that X is connected and locally connected, and either stable or unstable equivalence classes of (X , f ) are homeomorphic (with respect to their intrinsic topology) to R. Then (X , f ) is topologically conjugate to a hyperbolic linear automorphism of a torus R n /Z n .
Here intrinsic topology of a leaf is the direct limit topology coming from decomposition of a leaf into the union of plaques of rectangles of X .
Remark.
A more general notion of an endomorphism of an infra-nilmanifold is discussed in [Dek11b,Dek11a]. It is also noted there that some of the results of [Fra70] and [Shu70] are based on a false result. The proof of Theorem 1.2 shows that it is enough to consider the narrower notion of an automorphism of an infranilmanifold in the classification of Smale spaces and Anosov diffeomorphisms up to topological conjugacy. We do not use the results of [Fra70] (except for his proof of Theorem 2.2, which we repeat for our setting). The results of [Shu70] are not used in the proof of Theorem 1.1, where also the narrower notion of an endomorphism of an infra-nilmanifold is used, see [Nek05, Theorem 6. 1.6].
Structure of the paper. In Section 2, we collect basic facts and definitions related to Smale spaces, and fix the related notations . We study lower exponents of a Smale space, and a family of metrics associated with lower exponents in Section 3. We also recall there properties of the SRB measures on leaves of Smale spaces.
Locally connected Smale spaces are studied in Section 4. We show that the following conditions for a Smale space (X , f ) are equivalent (see Theorem 4.1):
(1) The space X is locally connected.
(2) All stable and unstable leaves of X are locally connected.
(3) All stable and unstable leaves of X are connected.
(4) X has finite stable and unstable upper exponents. In Section 5 we study splittings of locally connected Smale spaces. We show that for any splitting π : M −→ X of a locally connected Smale space there exists a well defined group of deck transformations G, that G is finitely generated, and that there exists a lift F : M −→ M of f to M, which is unique up to compositions with elements of G.
The lift F defines then, for any point x 0 ∈ M, an automorphism φ of G by the rule F (g(x 0 )) = φ(g)(F (x 0 )).
Question 2. Does the pair (G, φ) uniquely determine the topological conjugacy class of (X , f )?
We do not know the answer to this question, but we show that we can reconstruct (X , f ) if we add an extra piece of information to (G, φ).
Definition 1.2. Let π : M −→ X be a splitting of a locally connected and connected Smale space (X , f ). Let W + and W − be stable and unstable plaques of a fixed point of a lift of f . We say that Σ + and Σ − are coarse stable and unstable plaques if the Hausdorff distances between Σ + (x 0 ) and W + and between Σ − (x 0 ) and W − are finite.
Here the distance in M is measured with respect to a G-invariant metric.
Theorem 1. 5. The quadruple (G, φ, Σ + , Σ − ) uniquely determines the topological conjugacy class of (X , f ).
We prove Theorem 1.5 by representing W + and W − as boundaries of Gromov hyperbolic graphs constructed using the quadruple. These graphs are quasi-isometric to Cayley graphs of the Ruelle groupoids associated with the Smale space. A general theory of Cayley graphs of hyperbolic groupoids is developed in [Nek11a] . We get the following corollary of Theorem 1.5.
Theorem 1. 6. Let (X i , f i ) for i = 1, 2 be connected and locally connected Smale spaces. Let π i : M i −→ X i be splittings, and let F i : M i −→ M i be lifts of f i . Suppose that F i have fixed points, and that the groups of deck transformations of π i are both isomorphic to a group G. If there exists a continuous map Φ : X 1 −→ X 2 such that Φ(g(x)) = g(Φ(x)) and Φ(F 1 (x)) = F 2 (Φ(x)) for all x ∈ M 1 and g ∈ G, then the Smale spaces (X 1 , f 1 ) and (X 2 , f 2 ) are topologically conjugate.
Section 6 is devoted to the proof of Theorem 1.2. As the first step we prove the following.
Proposition 1.7. Let π : M −→ X be a splitting of a locally connected and connected Smale space (X , f ) such that the group G of deck transformations is torsion free nilpotent. Then f has a fixed point, and the associated automorphism φ : G −→ G is hyperbolic (i.e., its unique extension Φ to a simply connected nilpotent Lie group containing G as a lattice is hyperbolic).
We prove both statements of Proposition 1.7 by induction on the nilpotency class of G.
We show at first that the automorphism φ induces a hyperbolic automorphism of the center Z(G) ∼ = Z n of G. Then we construct an action of R n on M naturally extending the action of Z n , using the direct product structure on M. The action induces an action of the torus R n /Z n on X and agrees with the local product structure, metric on X , and the dynamics, in such a way that the map induced by f on (R n /Z n )\X is a Smale space with a splitting with the group of deck transformations isomorphic to G/Z(G). This provides us the necessary inductive steps to prove Proposition 1.7.
Let now π : M −→ X be a splitting of a Smale space (X , f ) such that its group of deck transformations G is torsion free nilpotent. The associated automorphism φ : G −→ G is hyperbolic. Let L be the unique simply connected nilpotent Lie group such that G is its lattice (L is called the Malcev completion of G). Then φ uniquely extends to an automorphism Φ of L. Then G acts on L by left translations, and we get an Anosov diffeomorphism f L of G\L induced by Φ.
We use the arguments of [Fra70, Theorem 2.2] to show that there exists a Gequivariant map h : M −→ L. Then it follows from Theorem 1.6 that (G\L, f l ) and (X , f ) are topologically conjugate.
Suppose now that the group of deck transformations G is virtually nilpotent. Then there exists a normal φ-invariant torsion free nilpotent subgroup G 1 of G. Let L be its Malcev completion. We know that there exists a homeomorphism h : M −→ L conjugating the actions of G 1 and the lifts of f and f L . Transfer by h the action of G on M to an action of G on L. We prove in Proposition 6.21 that G acts then on L by affine transformations, which finishes the proof of Theorem 1.2. Theorem 1.3 generalizing the Brin's pinching condition to Smale spaces is proved in Section 7. We prove at first that every Smale space satisfying conditions of Theorem 1.3 has a splitting (Theorem 7.1). Then we prove that the group of deck transformations of the splitting is virtually nilpotent (Theorem 7.2) using Gromov's theorem on groups of polynomial growth. Both proofs are similar to the original proofs of M. Brin, except that in the proof of Theorem 7.1 we use results of Section 3 on lower exponents of a Smale space, which allows us to get slightly better pinching condition, and to prove the theorem for all locally connected Smale space, and not only for Anosov diffeomorphisms.
In Section 8, we show how our condition on critical exponents is related to M. Brin's pinching condition on the Mather spectrum of a diffeomorphism. We show that M. Brin's condition implies the condition of Theorem 1.3.
Section 9 is devoted to the proof Theorem 1.4 on co-dimension one Smale spaces. We prove it using Theorem 1. 3. In fact, a more direct proof can be obtained using the theory of Cayley graphs of hyperbolic groupoids, but at this stage Theorem 1.3 gives us a shorter proof. We show that if stable leaves of a Smale space are homeomorphic to R, then the stable lower and upper critical exponents coincide (and are equal to the entropy of the Smale space). But this obviously implies that such Smale spaces (if they are locally connected) satisfy the conditions of Theorem 1. 3. The fact that co-dimension one hyperbolic automorphisms of an infra-nilmanifold are hyperbolic automorphism of a torus is classical.
Figure 1. Rectangle (1) [x, x] = x for all x ∈ R; (2) [[x, y], z] = [x, z] and [x, [y, z]] = [x, z] for all x, y, z ∈ R.
We call a space with a direct product structure on it a rectangle.
If R = A × B is a decomposition of R into a direct product of two topological spaces, then the corresponding direct product structure is given by the operation (1)
[(x 1 , y 1 ), (x 2 , y 2 )] = (x 1 , y 2 ).
Let R be a rectangle. For x ∈ R the corresponding plaques are the sets
(2) P 1 (R, x) = {y ∈ R : [x, y] = x}, P 2 (R, x) = {y ∈ R : [x, y] = y}.
See Figure 1. If R = A × B with the corresponding direct product structure (1), then the plaques are given by
P 1 (R, (a, b)) = A × {b}, P 2 (R, (a, b)) = {a} × B.
The map P 1 (R, x) × P 2 (R, x) −→ R given by
(y 1 , y 2 ) → [y 1 , y 2 ]
is a homeomorphism. For any pair x, y ∈ R the natural maps P 1 (R, x) −→ P 1 (R, y) and P 2 (R, x) −→ P 2 (R, y) given by z → [y, z] and z → [z, y], respectively, are called holonomy maps inside R, and are homeomorphisms.
These homeomorphism agree with the homeomorphisms P 1 (R, x) × P 2 (R, x) −→ R, so that we get a canonical decomposition of R into the direct product of two spaces A and B, which can be identified with P 1 (R, x) and P 2 (R, x), respectively.
Definition 2.2. Let X be a topological space. A local product structure on X is given by a covering R of X by open sets R with a direct product structure [·, ·] R on each of them, such that for any R 1 , R 2 ∈ R, and for every x ∈ X there exists a neighborhood U of x such that [y 1 , y 2 ] R1 = [y 1 , y 2 ] R2 for all y 1 , y 2 ∈ U ∩ R 1 ∩ R 2 .
Two coverings of X by open rectangles define the same local product structures if their union defines a local product structure, i.e., satisfied the above compatibility condition.
If X is a space with a local direct product structure, then an open subset R ⊂ X with a direct product structure [·, ·] is a (sub-)rectangle of X if the union of {R} with a covering defining the local product structure satisfies the compatibility conditions of Definition 2.2.
We say that a continuous map f : X 1 −→ X 2 between spaces with local product structures preserves the local product structures if every point of X 1 has a rect-
angular neighborhood U such that f (U ) is a rectangle of X 2 , and f ([x, y] U ) = [f (x), f (y)] f (U) for all x, y ∈ U .
Definition 2. 3. Let X be a space with a local product structure. We say that a metric d on X agrees with the local product structure if for every point x ∈ X there exists an open rectangular neighborhood R = A × B of x and metrics d A and d B on A and B, respectively, such that restriction of d onto R is bi-Lipschitz equivalent to the metric
d R ((x 1 , y 1 ), (x 2 , y 2 )) = d A (x 1 , x 2 ) + d B (y 1 , y 2 ).
If a metric d agrees with the local product structure, then for every point x ∈ X there exists a rectangular neighborhood R of x such that all holonomy maps inside R are bi-Lipschitz with respect to the metric d with a fixed Lipschitz constant (depending only on R). Conversely, it is easy to see that a metric d agrees with the local product structure if for every x ∈ X there exists a rectangular neighborhood R of x such that the holonomies inside R are uniformly bi-Lipschitz, and
d(y, z) is bi-Lipschitz equivalent to d([x, y], [x, z]) + d([y, x], [z, x]).
Smale spaces.
If X is a compact space with a local product structure on it, then there exists a finite covering of X by rectangles such that the direct product structures on the elements of the covering agree on their intersections. It follows that the local product structure on X can be defined by one map (x, y) → [x, y] defined on a neighborhood of the diagonal in X × X and satisfying conditions of Definition 2.1.
Definition 2.4. A Smale space is a compact metrizable space X together with a homeomorphism f : X −→ X such that there exists a metric d on X , constants λ ∈ (0, 1) and C > 0, and a local product structure on X such that f preserves the local product structure and for every x ∈ X there exists a rectangular neighborhood R of x such that for all n ≥ 0 and y, z ∈ P 1 (R, x) we have d(f n (y), f n (z)) ≤ Cλ n d(y, z), and for all n ≥ 0 and y, z ∈ P 2 (R, x) we have
d(f −n (y), f −n (z) ≤ Cλd(y, z).
We will denote P 1 (R, x) = P + (R, x) and P 2 (R, x) = P − (R, x).
Definition 2. 5. A homeomorphism f : X −→ X of a compact space X is said to be expansive if there exists a neighborhood U of the diagonal in X × X such that (f n (x), f n (y)) ∈ U for all n ∈ Z implies x = y.
Note that if U satisfies the conditions of the definition, then {(x, y) ∈ X 2 : (x, y), (y, x) ∈ U } also satisfies the conditions of the definition. Consequently, we may assume that U is symmetric.
Proposition 2.1. Every Smale space is an expansive dynamical system. Proof. We can find a finite covering R of X by rectangles satisfying the conditions of Definition 2.4. Let ǫ > 0 be a Lebesgue's number of the covering. There exists δ > 0 such that for any points x, y ∈ X such that d(x, y) < δ and any rectangle R ∈ R such that x, y ∈ R we have d(x, [x, y]) < C −1 ǫ and d(y, [x, y]) < C −1 ǫ.
Let x, y ∈ X be such that d(f n (x), f n (y)) < δ for all n ∈ Z. Then for every n ∈ Z there exists a rectangle R n ∈ R such that f n (x), f n (y) ∈ R n .
Then
d(f n (x), [f n (x), f n (y)]) < C −1 ǫ. Note that f n (x) and [f n (x), f n (y)] belong to one plaque P − (R n , f n (x)). It follows that d(f n−k (x), [f n−k (x), f n−k (y)]) ≤ Cλ k d(f n (x), [f n (x), f n (y)]) < Cλ k C −1 ǫ = λ k ǫ for all k ≥ 0 and all n ∈ Z. In par- ticular, d(x, [x, y]) < λ k ǫ for all k ≥ 0, i.e., x = [x, y].
It is shown in the same way that y = [x, y], which implies that x = y. Therefore, the set U ⊂ X ×X equal to the set of pairs (x, y) such that d(x, y) < δ satisfies the conditions of Definition 2.5.
Definition 2. 6. A log-scale on a set X is a function ℓ : X × X −→ R ∪ {∞} satisfying the following conditions:
(1) ℓ(x, y) = ℓ(y, x) for all x, y ∈ X;
(2) ℓ(x, y) = ∞ if and only if x = y;
(3) there exists ∆ > 0 such that
ℓ(x, z) ≥ min{ℓ(x, y), ℓ(y, z)} − ∆ for all x, y, z ∈ X.
We say that two log-scales ℓ 1 , ℓ 2 are bi-Lipschitz equivalent if the difference |ℓ 1 (x, y) − ℓ 2 (x, y)| is uniformly bounded for all x = y.
Let us describe the natural class of metrics on expansive dynamical systems defined in [Fri83], using log-scales.
Let (X , f ) be an expansive dynamical system. Let U be a symmetric neighborhood of the diagonal, satisfying the conditions of Definition 2.5. Define ℓ(x, y) for x, y ∈ X to be maximal n such that (f k (x), f k (y)) ∈ U for all k ∈ [−n, n].
Lemma 2.2. The defined function ℓ is a log-scale. It does not depend, up to bi-Lipschitz equivalence, on the choice of U .
We call ℓ the standard log-scale of the expansive dynamical system.
Proof.
We have ℓ(x, y) = ℓ(y, x), since we assume that U is symmetric. We also have ℓ(x, y) = ∞ if and only if x = y, by Definition 2. 5.
It remains to show that there exists ∆ such that ℓ(x, z) ≥ min{ℓ(x, y), ℓ(y, z)}−∆ for all x, y, z ∈ X .
Since a compact set has a unique uniform structure consisting of all neighborhoods of the diagonal (see [Bou71]), there exists a neighborhood of the diagonal V ⊂ X 2 such that (x, y), (y, z) ∈ V implies (x, z) ∈ U .
Note that the sets U n = {(x, y) : ℓ(x, y) ≥ n} = n k=−n f k (U ) are neighborhoods of the diagonal, U n+1 ⊆ U n for all n, and n≥1 U n is equal to the diagonal. In particular, by compactness of X , there exists ∆ > 0 such that U ∆ ⊂ V .
Denote by V n = n k=−n f k (V ) the set of pairs (x, y) such that (f k (x), f k (y)) ∈ V for all k = −n, . . . , n. Then (x, y), (y, z) ∈ V n implies (x, z) ∈ U n .
Then for every n > ∆ we have U n ⊂ V n−∆ , since the conditions that (f k (x), f k (y)) ∈ U for all |k| ≤ n implies (f k (x), f k (y)) ∈ U ∆ ⊂ V for all |k| ≤ n − ∆.
Let min{ℓ(x, y), ℓ(y, z)} = m. Then (x, y), (y, z) ∈ U m ⊂ V m−∆ , hence (x, z) ∈ U m−∆ , i.e., ℓ(x, z) ≥ m − ∆.
Let us show that ℓ does not depend on the choice of U . Let U ′ and U ′′ be two neighborhoods of the diagonal, satisfying the conditions of Definition 2.5. Then, as above, there exists C > 0 such that U ′ C ⊂ U ′′ and U ′′ C ⊂ U ′ . By the same arguments as above, we conclude that U ′ n+C ⊂ U ′′ n and U ′′ n+C ⊂ U ′ n for all n ≥ 0. But this implies that the values of the log-scales defined by U ′ and U ′′ differ from each other not more than by C.
Definition 2.7. Let (X , f ) be an expansive dynamical system, and let ℓ be the standard log-scale. We say that x, y ∈ X are stably equivalent (denoted x ∼ + y) if ℓ(f n (x), f n (y)) → +∞ as n → +∞. They are unstably equivalent (denoted x ∼ − y) if ℓ(f −n (x), f −n (y)) → +∞ as n → +∞. We call stable and unstable equivalence classes stable and unstable leaves.
Note that ℓ(x n , y n ) → ∞ is equivalent to d(x n , y n ) → 0 for any pair of sequences x n , y n ∈ X and for any metric d on X .
Two points x, y ∈ X are stably equivalent if and only if (f n (x), f n (y)) ∈ U for all n big enough. Denote, for x ∈ X and n ∈ Z, by W n,+ (x) the set of points y ∈ X such that (f k (x), f k (y)) ∈ U for all k ≥ −n. Similarly, we denote by W n,− (x) the set of points y ∈ X such that (f k (x), f k (y)) ∈ U for all k ≤ n.
Then W n,+ (x) and W n,− (x) are decreasing sequences of sets, and W + (x) = n∈N W −n,+ (x) and W − (x) = n∈N W −n,− (x) are equal to the stable and unstable leaves of x, respectively.
Note that for all n ∈ N, x ∈ X , * ∈ {+, −}, and y 1 , y 2 ∈ W n, * (x) we have ℓ(y 1 , y 2 ) ≥ n.
If W is a stable leaf, then we denote, for y 1 , y 2 ∈ W , by ℓ + (y 1 , y 2 ) or ℓ W (y 1 , y 2 ) the biggest n 0 such that (f n (y 1 ), f n (y 2 )) ∈ U for all n ≥ −n 0 .
The following properties of ℓ + follow directly from the definitions.
• ℓ(y 1 , y 2 ) ≤ ℓ + (y 1 , y 2 ) for all stably equivalent y 1 , y 2 ;
• if y 1 , y 2 are stably equivalent and ℓ + (y 1 , y 2 ) > 0, then ℓ(y 1 , y 2 ) = ℓ + (y 1 , y 2 ); • for all stably equivalent y 1 , y 2 we have
(3) ℓ + (f (y 1 ), f (y 2 )) = ℓ + (y 1 , y 2 ) + 1.
Similarly, if W is an unstable leaf, then ℓ − (y 1 , y 2 ) = ℓ W (y 1 , y 2 ), for y 1 , y 2 ∈ W is the biggest n 0 such that (f n (y 1 ), f n (y 2 )) ∈ U for all n ≤ n 0 . We also have ℓ(y 1 , y 1 ) ≤ ℓ − (y 1 , y 2 ), ℓ(y 1 , y 2 ) = ℓ − (y 1 , y 2 ) if ℓ − (y 1 , y 2 ) > 0, and
(4)
ℓ − (f (y 1 ), f (y 2 )) = ℓ − (y 1 , y 2 ) − 1 for all pairs y 1 , y 2 of unstably equivalent points.
Lemma 2.3. Let W be a stable or unstable leaf. Then the corresponding function ℓ + or ℓ − is a log-scale on W .
Proof. If ℓ + (x, y), ℓ + (y, z), ℓ + (x, z) are all positive, then they are equal to the corresponding values of ℓ, hence, by Lemma 2.2, ℓ + (x, z) ≥ min(ℓ + (x, y), ℓ + (y, z)) − ∆ for ∆ not depending on x, y, z.
If they are not positive, then we can find n ≥ 0 such that ℓ + (f n (x), f n (y)) = ℓ + (x, y) + n, ℓ + (f n (y), f n (z)) = ℓ + (y, z) + n, and ℓ + (f n (x), f n (z)) = ℓ + (x, z) + n are positive, and applying the above argument for f n (x), f n (y), and f n (z) conclude that ℓ + (x, z) + n ≥ min(ℓ + (x, y) + n, ℓ + (y, z) + n) − ∆, which is equivalent to ℓ + (x, z) ≥ min(ℓ + (x, y), ℓ + (y, z)) − ∆.
We call the log-scales ℓ + and ℓ − the internal log-scales on the respective leaf. The internal topology on a leaf is the topology defined by the corresponding logscale ℓ + or ℓ − . Here topology defined by a log-scale ℓ on a set X is given by the the basis B(n, x) = {y ∈ X : ℓ(x, y) ≥ n} of neighborhoods of points x ∈ X. Note that B(n, x) is not necessarily open or closed.
Equivalently, the internal topology of a leaf W is equal to the direct limit topology of representation of W as the union of the sequence W −n, * (x) for n ∈ N and x ∈ W .
Note also that leaves of a Smale space are locally compact, since neighborhoods of points of a leaf are continuous images of neighborhoods of points of X .
Lower exponents
3.1. Lower exponents of log-scales. We say that d is a metric associated with a log-scale ℓ, if there exist constants α > 0 and C > 1 such that
C −1 e −αℓ(x,y) ≤ d(x, y) ≤ Ce −αℓ(x,y) .
The number α is called the exponent of the metric. Topology defined by an associated metric obviously coincides with the topology defined by the log-scale.
Note that if d is a metric associated with ℓ of exponent α, then for any 0 < r < 1 the function (d(x, y)) r is a metric associated with ℓ of exponent rα. It follows that the set of exponents α for which there exists a metric associated with a given logscale is an interval of the form (0, α 0 ) or (0, α 0 ], where α 0 ∈ [0, ∞]. We will see later that α 0 > 0 (see also [Nek11a]). The number α 0 is the metric critical exponent of the log-scale.
Let X be a set with a log-scale ℓ. Let Γ n , for n ∈ R, be the graph with the set of vertices X in which two points x, y are connected by an edge if and only if ℓ(x, y) ≥ n. Denote then by d n the combinatorial distance in Γ n (we assume that d n (x, y) = ∞ if x and y belong to different connected components of Γ n ).
Proposition 3.1. Let ∆ be such as in Definition 2. 6. There exist C > 0 such that
d n (x, y) ≥ Ce α(n−ℓ(x,y))
for all x, y ∈ X and all n ∈ N, where α = log 2 ∆ . Proof. If (x 0 , x 1 , x 2 ) is a path in Γ n , then ℓ(x 0 , x 2 ) ≥ n − ∆, hence (x 0 , x 2 ) is a path in Γ n−∆ . It follows that d n−∆ (x, y) ≤ 1 2 (d n (x, y) + 1). In other terms:
d n+∆ (x, y) ≥ 2d n (x, y) − 1.
If ℓ(x, y) = m, then d m+1 (x, y) ≥ 2, and hence
d m+1+k∆ (x, y) ≥ 2 k+1 − 2 k−1 − 2 k−2 − · · · − 1 = 2 k + 1 Note that d m (x, y) ≥ d n (x, y) whenever m ≥ n. It follows that for k = n−ℓ(x,y)−1 ∆ ≥ n−ℓ(x,y)−1 ∆ − 1 we have d n (x, y) ≥ d ℓ(x,y)+1+k∆ (x, y) > 2 k .
Consequently, d n (x, y) ≥ 2 (n−ℓ(x,y)−1−∆)/∆ = Ce α(n−ℓ(x,y))
for all x, y ∈ X and n ∈ R, where C = 2 (−1−∆)/∆ and α = ln 2 ∆ .
Definition 3.1. We say that α is a lower exponent of a log-scale ℓ if there exists
C > 0 such that d n (x, y) ≥ Ce α(n−ℓ(x,y))
for all x, y ∈ X and n ∈ R. Supremum of all lower exponents is called the lower critical exponent.
The proof of the following proposition is straightforward.
Proposition 3.2. Let ℓ 1 and ℓ 2 be bi-Lipschitz equivalent log-scales on X. A number α > 0 is a lower exponent of ℓ 1 if and only if it is a lower exponent of ℓ 2 .
Theorem 3. 3. The metric critical exponent of a log-scale ℓ is equal to its lower critical exponent. In particular, the metric critical exponent is positive.
Proof. Let d be a metric on X of exponent α associated with ℓ, and let C 1 > 1 be such that
C −1 1 e −αℓ(x,y) ≤ d(x, y) ≤ C 1 e −αℓ(x,y) for all x, y ∈ X.
Then for every n the inequality ℓ(x, y) ≥ n implies d(x, y) ≤ C 1 e −αn , hence
d(x, y) ≤ C 1 d n (x, y)e −αn
for all x, y. It follows that
d n (x, y) ≥ C −1 1 d(x, y)e αn ≥ C −2 1 e αn−αℓ(x,y)
, for all x, y ∈ X and n ∈ N, i.e., α is a lower exponent.
Let α be a lower exponent. Let β be an arbitrary number such that α > β > 0. It is enough to show that there exists a metric on X of exponent β associated with ℓ.
Define, for x, y ∈ X, d β (x, y) as the infimum of m i=1 e −βℓ(xi−1,xi) over all sequences x 0 = x, x 1 , x 2 , . . . , x m = y. The function d β (x, y) obviously satisfies the triangle inequality, is symmetric, and
d β (x, y) ≤ e −βℓ(x,y)
for all x, y ∈ X.
It remains to prove that there exists a constant C 2 such that for any sequence
x 0 = x, x 1 , x 2 , . . . , x m = y we have m i=1 e −βℓ(xi−1,xi) ≥ C 2 e −βℓ(x,y) .
Let C be such that 0 < C < 1 and d n (x, y) ≥ Ce α(n−ℓ(x,y)) for all x, y ∈ X and all n. Let us prove our statement by induction on m for C 2 = exp β(log C−2α∆)
α−β .
The statement is true for m = 1, since C 2 < 1. Suppose that we have proved it for all k < m, let us prove it for m.
Lemma 3. 4. Let x 0 , x 1 , . . . , x m be a sequence such that ℓ(x i , x i+1 ) ≥ n for all i = 0, 1, . . . , m−1. Let n 0 ≤ n. Then there exists a sub-sequence y 0 = x 0 , y 1 , . . . , y t−1 , y t = x m of the sequence x i such that n 0 − 2∆ ≤ ℓ(y i , y i+1 ) < n 0 for all i = 0, 1, . . . , t − 1.
Proof. Let us construct the subsequence y i by the following algorithm. Define y 0 = x 0 . Suppose we have defined y i = x r for r < m. Let s be the largest index such that s > r and ℓ(x r , x s ) ≥ n 0 . Note that since ℓ(x r , x r+1 ) ≥ n ≥ n 0 , such s exists.
If s < m, then ℓ(x r , x s+1 ) < n 0 , and
ℓ(x r , x s+1 ) ≥ min{ℓ(x r , x s ), ℓ(x s , x s+1 )} − ∆ ≥ min{n 0 , ℓ(x s , x s+1 )} − ∆ = n 0 − ∆.
Define then y i+1 = x s+1 . We have
n 0 − ∆ ≤ ℓ(y i , y i+1 ) < n 0 .
If s + 1 = m, we stop and get our sequence y 0 , . . . , y t , for t = i + 1. If s = m, then ℓ(x r , x m ) = ℓ(y i , x m ) ≥ n 0 , and
ℓ(y i−1 , x m ) ≥ min{ℓ(y i−1 , y i ), ℓ(y i , x m )} − ∆ ≥ min{n 0 − ∆, n 0 } − ∆ = n 0 − 2∆ and ℓ(y i−1 , x m ) < n 0 ,
since y i was defined and was not equal to x m . Then we redefine y i = x m and stop the algorithm.
In all the other cases we repeat the procedure. It is easy to see that at the end we get a sequence y i satisfying the conditions of the lemma. . . , x m = y be an arbitrary sequence of points of X. Let n 0 be the minimal value of ℓ(x i , x i+1 ). Let y 0 = x, y 1 , . . . , y t = y be a sub-sequence of the sequence x i satisfying conditions of Lemma 3. 4.
Let x 0 = x, x 1 , .
Suppose at first that
n 0 < ℓ(x, y) + 2α∆ − log C α − β . Remember that n 0 = ℓ(x i , x i+1 ) for some i, hence m i=1 e −βℓ(xi−1,xi) ≥ e −βn0 > exp −βℓ(x, y) − β(2α∆ − log C) α − β = C 2 e −βℓ(x,y) ,
and the statement is proved. Suppose now that n 0 ≥ ℓ(x, y) + 2α∆−log C α−β , which is equivalent to
(5) (α − β)n 0 − (α − β)ℓ(x, y) − 2α∆ + log C ≥ 0. If t = 1, then n 0 − 2∆ ≤ ℓ(x, y) < n 0 , hence n 0 ≤ ℓ(x, y) + 2∆ = ℓ(x, y) + 2α∆ − 2β∆ α − β < ℓ(x, y) + 2α∆ − log C α − β ,
since log C < 0 < 2β∆. But this contradicts our assumption. Therefore t > 1, and the inductive assumption implies
m i=1 e −βℓ(xi−1,xi) ≥ t−1 i=0 C 2 e −βℓ(yi,yi+1) > tC 2 e −βn0 . We have t ≥ d n0−2∆ (x, y) ≥ Ce α(n0−2∆−ℓ(x,y)) , hence m i=1 e −βℓ(xi−1,xi) ≥ CC 2 e −βn0+αn0−2α∆−αℓ(x,y) = C 2 exp (log C − βn 0 + αn 0 − 2α∆ − αℓ(x, y)) = C 2 exp (−βℓ(x, y) + (α − β)n 0 − (α − β)ℓ(x, y) − 2α∆ + log C) ≥ C 2 e −βℓ(x,y) ,
by (5).
Lower exponents of Smale spaces.
Definition 3.2. Let (X , f ) be a Smale space. A number α > 0 is a stable (resp. unstable) lower exponent of the Smale space if there exists a constant C > 0 such that for any stable (resp. unstable) leaf W and any x, y ∈ W we have
d n (x, y) ≥ Ce α(n−ℓW (x,y))
for the internal log-scale on W . The supremum of the stable (resp. unstable) lower exponents is called the stable (resp. unstable) lower critical exponent.
Note that by Proposition 3.1 lower stable and unstable exponents exist and are positive for any Smale space. Proposition 3.2 implies that the lower critical exponents of a Smale space depend only on the topological conjugacy class of the Smale space.
Proposition 3.5. Let l ∈ R. A number α > 0 is a lower stable (resp. unstable) exponent of (X , f ) if and only if there exists C l > 0 such that for every stable (resp. unstable) leaf W and for every two points x, y ∈ W such that ℓ W (x, y) ≤ l we have
(6) d n (x, y) ≥ C l e αn
for all n.
Proof. Let us assume that W is a stable leaf (the proof for an unstable leaf is the same). If α is a lower exponent and C is as in Definition 3.2, then for any x, y ∈ W such that ℓ W (x, y) ≤ l we have
d n (x, y) ≥ Ce α(n−ℓW (x,y)) ≥ Ce −αl · e αn ,
and we can take C l = Ce −αl .
Suppose now that C l > 0 is such that d n (x, y) ≥ C l e αn for all x, y belonging to one stable leaf W and such that ℓ W (x, y) ≥ l.
Let x and y be arbitrary stably equivalent points of X . Let W 0 be their stable leaf.
Denote n 0 = ℓ W0 (x, y). Then ℓ W (f l−n0 (x), f l−n0 (y)) = l, where W is the stable leaf of f l−n0 (x) ∼ + f l−n0 (y). Consequently, d n (f l−n0 (x), f l−n0 (y)) ≥ C l e αn for all n.
The map z → f l−n0 (z) transforms every path in Γ n (W 1 ) to a path in Γ n+l−n0 (W 2 ), where W 1 and W 2 are the stable leaves of z and f l−n0 (z), see (3).
It follows that
d n (x, y) ≥ d n+l−n0 (f l−n0 (x), f l−n0 (y)) ≥ C l e α(n+l−n0) = C l e l · e α(n−ℓW 0 (x,y)) ,
which shows that α is a lower exponent.
Metric properties of leaves.
Let (X , f ) be a Smale space, and let ℓ, ℓ + , and ℓ − be the standard log-scale on X , and the internal log-scales on the stable and unstable leaves of X . Let U be a neighborhood of the diagonal satisfying the conditions of Definition 2.5.
The following theorem describes the classical theory of Sinai-Ruelle-Bowen measure on Smale spaces, see [Bow71]. See its exposition in [Nek11b], which is notationally close to our paper.
Denote by d, d + , and d − metrics associated with the log-scales ℓ, ℓ + , and ℓ − , respectively. Denote by B * (r, x) the ball of radius r, with respect to the metric d * , with center in x, where * ∈ {+, −}.
Theorem 3. 6. There exists a number η > 0 (called the entropy of (X , f )), and a family of Radon measures µ + and µ − on the stable and unstable leaves of X satisfying the following properties.
(1) There exists a number C > 1 such that
C −1 r η/α * ≤ µ * (B * (r, x)) ≤ Cr η/α * for all r ≥ 0, x ∈ X , and * ∈ {+, −}.
(2) The measures are preserved under holonomies.
(3) The measures are quasi-invariant with respect to f , and df * (µ+)
dµ+ = e η , df * (µ−) dµ− = e −η .
It follows from condition (1) of the theorem, that µ + and µ − are equivalent to the Hausdorff measures of the metrics d + and d − of dimension η α+ and η α− , respectively. (1) The space X is locally connected.
Locally connected Smale spaces and upper exponents
(2) All stable and unstable leaves are locally connected.
(3) All stable and unstable leaves are connected.
(4) The graphs Γ 0 (W ) are connected for every (stable or unstable) leaf W .
(5) The graphs Γ n (W ) are connected for every leaf W and every n.
(6) There exist α > 0 and C > 0 such that for every leaf W we have
d n (x, y) ≤ Ce α(n−ℓW (x,y))
for all x, y ∈ W and all n ≥ ℓ W (x, y).
Recall that for a stable or unstable leaf W , we denote by Γ n (W ) the graph with the set of vertices W in which two vertices x, y are connected by an edge if and
only if ℓ W (x, y) ≥ n, where ℓ W is the corresponding (ℓ + or ℓ − ) internal log-scale on W .
Let us start by proving equivalence of conditions (1) and (2).
Proposition 4.2.
Let (X , f ) be a Smale space. The space X is locally connected if and only if each leaf is locally connected.
Proof.
Each point x ∈ X has a neighborhood homeomorphic to the direct product of the neighborhoods of x in the corresponding stable and unstable leaves. It follows that if x has bases of connected neighborhoods in the leaves, then x has a basis of connected neighborhoods in X .
In the other direction, if x has a basis of connected neighborhoods in X , then for any rectangular neighborhood R of x there exists a connected neighborhood U ⊂ R of x. Its projection onto the direct factors of R will be connected, hence the point x has bases of connected neighborhoods in its leaves.
Proposition 4.3. If every stable leaf of (X , f ) is locally connected, then every stable leaf of (X , f ) is connected.
Proof. For every point x ∈ X there exists a connected neighborhood U of x in its stable leaf and a rectangular neighborhood R of x in X such that P + (R, x) = U . Then each plaque of R will be homeomorphic to U , hence will be connected. It follows that every point of X has a rectangular neighborhood R such that all its stable plaques are connected. Since X is compact, there exists a finite covering R = {R i } of X by open rectangles with connected stable plaques.
Let W be a stable leaf, and let x, y ∈ W . By Lebesgue's covering lemma, there exists n such that f n (x) and f n (y) belong to one plaque V of a rectangle R i ∈ R. Then f −n (V ) is a connected subset of W containing x and y. We have shown that any two points of W belong to one connected component of W , i.e., that W is connected.
Let R = {R i } i∈I be a finite covering of X by open rectangles. Let R + be the set of all stable plaques of elements of R.
Every stable leaf W is a union T ∈R+,T ⊂W T of stable plaques contained in W . Each plaque is an open subset of W and the internal topology on W coincides with the direct limit topology of the union of the plaques.
Denote by Γ ′ n (W ) the graph with the set of vertices W in which two vertices are connected by an edge if they belong to one set of the form f n (T ), T ∈ R + .
The map f : (3) and (4) for the first isomorphism. Lemma 4.4. There exists a number k 0 such that if x, y ∈ W are adjacent in Γ n (W ), then they are adjacent in Γ ′ n−k0 (W ), and if x, y ∈ W are adjacent in Γ ′ n (W ), then they are adjacent in Γ n−k0 (W ). Proof. There exists k 1 such that for every plaque V ∈ R + and every pair
W −→ f (W ) induces isomorphisms Γ n (W ) −→ Γ n+1 (W ) and Γ ′ n (W ) −→ Γ ′ n+1 (W ), seex, y ∈ V we have (f k (x), f k (y)) ∈ U for all k ≥ k 1 (where U is a neighborhood of the diagonal defining ℓ, ℓ + , and ℓ − ). If x, y ∈ W are connected by an edge in Γ ′ n (W ), then x, y ∈ f n (T ) for T ∈ R + , hence (f k−n (x), f k−n (y)) ∈ U for all k ≥ k 1 , hence ℓ + (x, y) ≥ n − k 1 , i.e.,
x and y are connected by an edge in Γ n−k1 (W ).
By Lebesgue's covering lemma, there exists k 2 such that if x, y ∈ W are such that ℓ W (x, y) ≥ k 2 , then x and y belong to one plaque V ∈ R + . Then every edge of Γ k2 (W ) is an edge in Γ ′ 0 (W ). Consequently, every edge of Γ n (W ) is an edge in Γ n−k2 (W ).
Proposition 4.5. The following conditions are equivalent.
(1) The graph Γ 0 (W ) is connected for every stable leaf W .
(2) The graph Γ n (W ) is connected for every stable leaf W and for every n ∈ Z.
(3) The graph Γ ′ 0 (W ) is connected for every stable leaf W . (4) The graph Γ ′ n (W ) is connected for every stable leaf W and for every n ∈ Z.
Proof. Since the map f k : W −→ f k (W ) induces isomorphisms Γ n (W ) −→ Γ n+k (W ) and Γ ′ n (W ) −→ Γ ′ n+k (W ),
(1) is equivalent to (2), and (3) is equivalent to (4). Let k 0 be as in Lemma 4.4. If all graphs Γ n (W ) are connected, then all graphs
Γ ′ n−k0 (W ) ⊇ Γ n (W ) are connected. If all graphs Γ ′ n (W ) are connected, then all graphs Γ n−k0 (W ) ⊇ Γ ′ n (W ) are connected.
This shows that all conditions (1)-(4) are equivalent to each other.
Proposition 4.6. If a stable leaf W is connected, then the graph Γ ′ n (W ) is connected for every n.
Proof. Suppose that W is a connected stable leaf. Let A be a connected component of Γ ′ 0 (W ). Let W A be the union of the plaques V ∈ R + containing vertices of A. It follows from the definition of the graph Γ ′ 0 (W ) that every plaque V ∈ R + is either contained in W A , or is disjoint with it. Consequently, W A is clopen, which implies that W = W A , hence A = W , and Γ ′ 0 (W ) is connected.
Proposition 4.7. Suppose that the graphs Γ ′ n (W ) are connected for all stable leaves W and all n. Then there exists A ≥ 1 such that any two adjacent vertices in
Γ ′ 0 (W ) are on distance at most A in Γ ′ 1 (W ). Proof.
Let E be the closure in X × X of the set pairs of points (x, y) such that there exists a plaque V ∈ R + such that x, y ∈ V . It is easy to see that E is compact. It contains the set of edges of every graph Γ ′ 0 (W ), and is contained in the stable equivalence relation.
Any pair of points x, y such that (x, y) ∈ E is connected by a path in Γ ′ 1 (W ), as all graphs Γ ′ n (W ) are connected. It means that there exists a sequence of rectangles R 1 , . . . , R n ∈ R, and a sequence of points x i ∈ R i such that x and x 1 belong to the same stable plaque of R 1 , y and x n belong to the same stable plaque of R n , and the stable plaque of x i in R i intersects with the stable plaque of x i+1 in R i+1 . This sequence R 1 , . . . , R n will define a path in Γ ′ 1 (W ) connecting any two points (x ′ , y ′ ) belonging to a neighborhood of (x, y) in E. It follows then from compactness of E that we can find a finite upper bound on the length of a path connecting any two points of E, which finishes the proof.
Recall that d n (x, y) is the distance in the graph Γ n (W ).
Proposition 4. 8. Suppose that all graphs Γ ′ n (W ) are connected. Then there exist positive constants α and C such that for any two points x, y ∈ W we have
d n (x, y) ≤ Ce α(n−ℓW (x,y))
for all n ≥ ℓ W (x, y).
Proof. Let
A and k 0 be as in Propositions 4.7 and 4.4, and let x, y ∈ W be arbitrary.
Denote n 0 = ℓ W (x, Y ).
The points x and y are connected by an edge in Γ n0 (W ), hence they are connected by an edge in Γ ′ n0−k0 (W ). It follows from Proposition 4.7 that for every k ≥ 0, distance between x and y in Γ ′ n0−k0+k (W ) is not greater than A k . As the set of edges of Γ ′ n0−k0+k (W ) is contained in the set of edges of
Γ n0−2k0+k (W ), we have d n0−2k0+k (x, y) ≤ A k . for all k ≥ 0. Consequently, d n (x, y) ≤ A n−n0+2k0 = A 2k0 · A n−ℓW (x,y) for all n ≥ ℓ W (x, y) − 2k 0 .
Proposition 4. 9. Suppose that all graphs Γ ′ n (W ) are connected. Let d be a metric on W associated with ℓ W . There exists a constant C such that for any two points x, y ∈ W there exists a curve γ : [0, 1] −→ W connecting x to y and such that the diameter the range of γ is not larger than Cd(x, y).
Proof. Let
A and k 0 be as in Propositions 4.7 and 4.4. Let C 1 > 1 and α > 0 be such that
C −1 1 e −αℓW (x,y) ≤ d(x, y) ≤ C 1 e −αℓW (x,y) for all x, y ∈ W . Take arbitrary x, y ∈ W . Let n 0 = ℓ W (x, y)
. Then x and y are adjacent in Γ ′ n0−k0 (W ), hence they are connected by a path γ 1 :
{x = x 1,1 , x 1,2 , . . . , x 1,m1 = y} of length at most A in Γ ′ n0−k0+1 (W ). Each pair of points x 1,i , x 1,i+1 is connected by a path of length at most A in Γ ′ n0−k0+2 (W ). We get then a path γ 2 = {x = x 2,1 , x 2,2 , . . . , x 1,m2 } of length at most A 2 in Γ ′ n0−k0+2 (W )
, containing γ 1 . We get then inductively defined sequence of paths γ n = {x = x n,1 , x n,2 , . . . , x n,mn = y} in Γ ′ n0−k0+n (W ) such that each next path γ n is obtained from γ n−1 by inserting at most A − 1 points between each pair of neighbors of γ n .
Every pair of points ) . In particular, for every point of γ n+1 there exists a point of γ n on distance less than AC 1 e −α(n+1+n0−2k0) .
x n,j , x n,j+1 is adjacent in Γ ′ n0−k0+n (W ), hence ℓ W (x n,j , x n,j+1 ) ≥ n 0 − 2k 0 + n, hence d(x i,j , x i,j+1 ) ≤ C 1 e −α(n+n0−2k0
It follows that diameter of the set γ n is not greater than
C 1 e −α(n0−2k0) + 2 n i=1 AC 1 e −α(i+n0−2k0) < 2C 1 Ae −α(n0−2k0) · 1 1 − e −α = 2C 1 Ae 2αk0 1 − e −α · e −αℓW (x,y) ≤ 2C 2 1 Ae 2αk0 1 − e −α · d(x, y).
Since d(x n,j , x n,j+1 ) ≤ C 1 e −α(n0−2k0+n) , the closure of ∞ n=1 γ n is the image of a continuous curve connecting x to y. Diameter of the image of the curve is not greater than
2C 2 1 Ae 2αk 0 1−e −α · d(x, y).
Let us summarize now the proof of Theorem 4.1. Equivalence of (1) and (2) is shown in Proposition 4.2. Implication (2)⇒(3) is given in Proposition 4.3. Equivalence of (4) and (5) is contained in Proposition 4.5. Proposition 4.6 shows then that (3) implies (4). Proposition 4.8 proves that (5) implies (6). Condition (6) obviously implies (5). Proposition 4.9 shows that (5) implies connectivity and local connectivity of the leaves, i.e., that (5) implies (2) and (3).
This finishes the proof of Theorem 4.1.
4.2.
Local product structure on locally connected Smale spaces.
Proposition 4. 10. Let R be a sub-rectangle of a Smale space (X , f ). If R is connected and locally connected, then the direct product structure on R compatible with the local product structure on X is unique.
Proof. Suppose that, on the contrary, there exist two different direct product structures [·, ·] 1 and [·, ·] 2 , both compatible with the local product structure on X . By Definition 2.2, there exists a covering R of X by open rectangles such that for any U ∈ R and x, y
∈ R ∩ U we have [x, y] U = [x, y] 1 = [x, y] 2 .
Then for every U ∈ R the intersection U ∩ R is a (possibly empty) sub-rectangle of R with respect to both direct product structures; and restrictions of the direct products structures
[·, ·] i , i = 1, 2, onto U ∩ R coincide.
Since R is connected, all plaques of R (with respect to both direct products structures) are connected. Let P + be a stable plaque of (R, [·, ·] 1 ). Let x, y ∈ P + . Since P + is connected, there exists a sequence of points x 0 = x, x 1 , . . . , x n = y and a sequence of rectangles U 0 , U 1 , . .
. , U n ∈ R such that x i ∈ U i , and U i ∩U i+1 ∩P + = ∅.
The set U i ∩ P + is a plaque of the rectangle U i ∩ R, hence it is a subset of the stable plaque of (R, [·, ·] 2 ). We get a sequence U i ∩ P + of subsets of plaques of (R, [·, ·] 2 ) such that (U i ∩ P + ) ∩ (U i+1 ∩ P + ) = ∅. But it means that U i ∩ P + are subsets of one plaque of (R, [·, ·] 2 ). We have shown that if two points belong to one stable plaque of (R, [·, ·] 1 ), then they belong to one stable plaque of (R, [·, ·] 2 ). The converse is proved in the same way. Consequently, the stable plaques of R with respect to [·, ·] 1 are the same as the stable plaques of R with respect to [·, ·] 2 . The same statement is obviously true for the unstable plaques, which implies that the direct product structures [·, ·] 1 and [·, ·] 2 on R coincide.
Upper exponents.
Definition 4.1. A positive number α > 0 is a stable (resp. unstable) upper exponent of the Smale space if there exists C > 0 such that for any stable (resp. unstable) leaf W and every pair of points x, y ∈ W we have d n (x, y) ≤ Ce α(n−ℓW (x,y)) for all n ≥ ℓ W (x, y).
Note that changing ℓ W to a bi-Lipschitz equivalent log-scale, one does not change the set of upper exponents, i.e., this notion is well defined and depends only on the topological conjugacy class of the Smale space (see Lemma 2.2).
By Theorem 4.1 a finite upper exponent exists if X is locally connected. The proof of the next proposition is analogous to the proof of Proposition 3.5.
Proposition 4.11. Fix l ∈ R. A number α > 0 is a stable (resp. unstable) upper exponent if and only if there exists a constant C l > 0 such that for any stable (resp. unstable) leaf W and any x, y ∈ W such that ℓ W (x, y) ≥ l we have
d n (x, y) ≤ C l e αn
for all n ≥ 0.
Splittings of Smale spaces
Groups of deck transformations.
Definition 5.1. Let (X , f ) be a Smale space. A splitting of (X , f ) is a covering map π : M −→ X , where M is a space with a (global) direct product structure, such that (1) π agrees with the local product structures on M and X ;
(2) restriction of π onto every plaque P 1 (M, x) of M is a homeomorphism with the stable leaf W + (π(x)), and restriction of π onto every plaque P 2 (M, x) of M is a homeomorphism with the unstable leaf W − (π(x)), with respect to their intrinsic topology.
Proposition 5.1. Suppose that X is connected and locally connected. Let π 1 : M 1 −→ X and π 2 : M 2 −→ X be splittings of (X , f ). If x 1 ∈ M 1 and x 2 ∈ M 2 are such that π 1 (x 1 ) = π 2 (x 2 ), then there exists a unique homeomorphism F : M 1 −→ M 2 preserving the local product structures and such that π 1 = π 2 • F and
F (x 1 ) = x 2 .
Proof. Since the leaves of locally connected Smale spaces are connected and locally connected, the spaces M i are connected and locally connected. Denote x = π 1 (x 1 ) = π 2 (x 2 ). Restriction of π i onto the plaques P 1 (M i , x i ) and P 2 (M i , x i ) are homeomorphisms with the leaves W + (x) and W − (x), respectively. Therefore, the only possible way to define F is by the equality
F ([y 1 , y 2 ] M1 ) = [z 1 , z 2 ] M2 ,
where y 1 ∈ P 1 (M 1 , x 1 ), y 2 ∈ P 2 (M 1 , x 1 ) are arbitrary, while z 1 ∈ P 1 (M 2 , x 2 ), z 2 ∈ P 2 (M 2 , x 2 ) are uniquely determined by the condition π 1 (y 1 ) = π 2 (z 1 ) and π 1 (y 2 ) = π 2 (z 2 ).
The defined map F is a homeomorphism, since it is a direct product of two homeomorphisms. Consequently, π 2 • F : M 1 −→ X is a covering map. Since F and π 2 agree with the local product structures of M i and X , their composition π 2 • F agrees with the local product structures, i.e., the image of the direct product structure on M 1 by F defines the same local product structure on M 2 as the direct product structure [·, ·] M2 . By the same arguments as in the proof of Proposition 4.10, the direct product structure on M 2 is uniquely determined by the corresponding local product structure. Consequently, F preserves the direct product structures, i.e., F ([y 1 , y 2 ] M1 ) = [F (y 1 ), F (y 2 )] M2 for all y 1 , y 2 ∈ M 1 . It follows that if π 1 (y) = π 2 (F (y)) for y ∈ M 1 , then π 1 = π 2 • F on a rectangular neighborhood of y. Consequently, the set of points y ∈ M 1 such that π 1 (y) = π 2 (F (y)) is open and closed, it contains x, hence it is equal to M 1 .
We assume now that (X , f ) is a locally connected and connected Smale space. Let π : M −→ X be a splitting. Let G be the set of all homeomorphisms g : M −→ M preserving the direct product structure on M and such that π = π•F . Then G is obviously a group. By Proposition 5.1 (for the case π 1 = π 2 ), the action of the group on M is free and transitive on π −1 (x) for every x ∈ X . We call G the group of deck transformations of the splitting. It acts properly on M, since every point of M has a neighborhood U such that π(U ) is evenly covered, i.e., such that
g(U ) ∩ U = ∅ for all not-trivial g ∈ G.
Note that if π : M −→ X is a splitting, then π • f is also a splitting. Choose x 1 , x 2 ∈ M such that f • π(x 1 ) = π(x 2 ), and apply Proposition 5.1 to π 1 = f • π and π 2 = π. We get that there exists a unique homeomorphism F : M −→ M preserving the direct product structure such that F (x 1 ) = x 2 and π • F = f • π.
Definition 5.2. Let π : M −→ X be a splitting. We say that a homeomorphism
F : M −→ M preserving the direct product structure is a lift of f if π • F = f • π.
It follows from the above arguments that lifts of f exist, and if F 1 and F 2 are two lifts, then F −1 1 F 2 and F 1 F −1 2 belong to G.
It also follows that for every g ∈ G, and every lift F of f , we have F −1 gF ∈ G. The map g → F −1 gF is an automorphism of G. We say that it is induced by f . Any two automorphisms induced by f on G differ from each other by an inner automorphism.
Proposition 5.2. The group of deck transformations of a splitting of a connected and locally connected Smale space is finitely generated.
Proof. Let π : M −→ X be a splitting. Consider a finite covering U of X by open connected evenly covered by π subsets. Let U M be the union of the sets of connected components of π −1 (U ) for all U ∈ U. Then U M is a covering of M. Since M is connected, every two elements U, V ∈ U M are connected with each other by a chain of elements
U 0 = U, U 1 , . . . , U n = V such that U i ∩ U i+1 = ∅. It follows that there exists a connected finite union V of elements of U M such that π(V ) = X . Note that V is compact. Let S be the set of elements g ∈ G such that g(V ) ∩ V = ∅. It is finite, since the action of G on M is proper. Let g ∈ G and x 0 ∈ V . Since M is connected and M = g∈G g(V ), there exists a sequence g i ∈ G, i = 0, 1, . . . , k such that g 0 = 1, g k = g, and g i (V )∩g i+1 (V ) = ∅ for all i = 0, 1, . . . , k − 1. Note that g i (V ) ∩ g i+1 (V ) = ∅ implies V ∩ g −1 i g i+1 (V ) = ∅, hence g −1 i g i+1 ∈ S.
We see that g = g −1 0 g 1 · g −1 1 g 2 · · · g −1 k−1 g k is a product of k elements of S.
Splittable Smale spaces and hyperbolic graphs. A connection between
Smale spaces and Gromov hyperbolic graphs described in this subsection is a particular case of the theory of Cayley graphs of hyperbolic groupoids, described in [Nek11a]. Since the theory for Smale spaces is simpler than the general case, and in order to make our paper more self-contained, we describe them directly.
Let (X , f ) be a Smale space with locally connected and connected space X , and let π : M −→ X be a splitting. We denote by [·, ·] the direct product structures on M and X . Let G be the group of deck transformations of the splitting. Let Let d be a metric on X associated with the standard log-scale ℓ. We will denote by d + and d − metrics on the stable and unstable leaves of X associated with the respective standard log-scales ℓ + and ℓ − . We assume that exponents of the metrics d, d + , and d − are equal. Then we have the following corollary of [Nek11a, Lemma 7.8], see also [Fri83].
Proposition 5.3. There exist constants ǫ, L > 1 such that for every point x ∈ X and every rectangle R contained in the ǫ-neighborhood of x restriction of d onto R is L-bi-Lipschitz equivalent to the metric given by
d x (y 1 , y 2 ) = d + ([y 1 , x], [y 2 , x]) + d − ([x, y 1 ], [x, y 2 ]).
Let ǫ > 0 be such that it satisfies the conditions of Proposition 5.3 and for every x ∈ X the ǫ-neighborhood of x is evenly covered by π.
Define then d M (x, y) as the infimum of the sum m−1 i=0 d(π(x i ), π(x i+1 )) over all sequences x 0 , x 1 , . . . , x m such that x 0 = x, x m = y, and d(π(x i ), π(x i+1 )) < ǫ for all i = 0, 1, . . . , m − 1.
Then d M is a G-invariant metric on M such that d M (x, y) = d(π(x), π(y)) for all x, y such that d M (x, y) < ǫ.
The map π bijectively identifies the plaques W + (x) and W − (x) of M with their images W + (π(x)) and W − (π(x)), respectively. We get hence metrics d + and d − on W + (x) and W − (x), respectively. Then Proposition 5.3 holds when we replace X by M and d by d M .
Let R ⊂ M be an open relatively compact rectangle such that g∈G g(R) = M. Let W + be a stable plaque of M. Denote by Ω n (W + , R), for n ∈ Z, the set of elements g ∈ G such that of F n (g(R)) intersects W + .
Denote by Ξ(W + , R) the graph with the set of vertices n∈Z Ω n (W + , R) × {n} in which two vertices are connected by an edge if and only if they are either of the form (g 1 , n), (g 2 , n), where g 1 (R) ∩ g 2 (R) = ∅, or of the form (g 1 , n), (g 2 , n + 1), where g 1 (R) ∩ F (g 2 (R)) = ∅. In other words, we connect two vertices of Ξ(W + , R) if and only if they belong to the same or neighboring levels Ω n (W + , R) and the corresponding rectangles intersect.
Theorem 5. 4. The graph Ξ = Ξ(W + , R) is Gromov hyperbolic. All paths of the form (g n , −n) ∈ Ξ, n ≥ 0, converge to one point ω ∈ ∂G. The map mapping the limit in ∂Ξ of a path (g n , n), n ≥ 0 to the limit of the intersections of F n (g n (R)) with W + induces a homeomorphism between ∂Ξ \ {ω} and W + .
Proof. Consider two vertices (g, 0) and (h, 0). Let m be the distance between them in Ω 0 (W + , R).
Let (g k , −k) and (h k , −k) for k ≥ 0 be arbitrary paths in Ξ such that g 0 = g and h 0 = h.
There exists ǫ > 0 such that for every x ∈ M there exist g ∈ G such that the ball of radius ǫ with center x is contained in g(R). It follows that for every m ∈ N there exists N m > 0 such that for any path (a 0 , 0), (a 1 , 0), . . . , (a m , 0) of length m in Ω 0 (W + , R) there exists g ∈ G such that F Nm (a 0 (R))∪· · · ∪F Nm (a m (R))∩W + ⊂ g(R) ∩ W + . It follows that (g Nm , −N m ) and (h Nm , −N m ) are on distance at most 2 in Ω −Nm (W + , R).
It follows now from [Nek11a, Theorem 2.10] that the graph Ξ is Gromov hyperbolic, that the level function (g(R), n) → n is a Busemann function of a point ω ∈ ∂Ξ, and that every path (g n , n) for n ≥ 0 converges to a point of ∂Ξ \ ω, whereas every path of the form (g n , −n), n ≥ 0, converges to ω.
Lemma 5.5. Every point of ∂Ξ \ ω is the limit of a sequence of the form (g n , n).
Proof. Note that since (g, n) → n is a Busemann function associated with ω ∈ ∂Ξ, every point ξ ∈ ∂Ξ \ ω is the limit of a sequence of the form (g n , n) (which is not necessarily a path). For every n ≥ 0 there exists a path (h n,k , k), k ≤ n, such that h n,n = g n . All these paths converge to ω, and by the above arguments, for any n 1 , n 2 , distance from (h n1,k , k) to (h n2,k , k) is not more than 2 for all k smaller than some k(n 1 , n 2 ). Moreover, k(n 1 , n 2 ) → ∞ as n 1 , n 2 → ∞. It follows then by compactness arguments (since Ξ has bounded valency) that there exists a path (h n , n), n ≥ 0, converging to ξ.
Let (g n , n), for n ≥ 0, be a path in Ξ. The sets V n = F n (g n (R)) ∩ W + are compact, their diameters decrease exponentially, and we have V n ∩ V n+1 = ∅ for every n. It follows that the sequence V n converges (in the Hausdorff metric) to a point x ∈ W + . Let us show that the map Λ : lim n→∞ (g n , n) → lim n→∞ F n (g n (R)) is a homeomorphism between ∂Ξ \ ω and W + .
The arguments basically repeat the proof of [Nek11a, Theorem 6.9]. Let us show that the map is well defined. If (g n , n) and (h n , n) converge to the same limit in ∂Ξ, then the distance between g n and h n in Ω n (W + , R) is uniformly bounded. But this implies that the Hausdorff distance between F n (g n (R)) and F n (h n (R)) is exponentially decreasing, hence lim n→∞ F n (g n (R)) = lim n→∞ F n (h n (R)). The same argument shows that the map Λ is continuous, since if ξ 1 and ξ 2 are close to each other, then the sequences (g n , n) and (h n , n) are close to each other for an initial interval n = 0, . . . , L, where L is big but then the limits lim n→∞ F n (g n (R)) and lim n→∞ F n (h n (R)) are close to each other.
The map Λ is onto, since for every point x ∈ W + there exists a path (g n , n) defined by the condition F n (g n (R)) ∋ x.
Using Lebesgue's covering lemma, we show that if x and y are close to each other, then there exists a sequence (g n , n) such that F n (g n (R)) ⊃ {x, y} for all n = 0, . . . , L, where L is big. This shows that Λ −1 exists and is continuous.
Suppose now that the map F : M −→ M has a fixed point x 0 . Let φ : G −→ G be the automorphism defined by the condition F (g(x 0 )) = φ(g)(x 0 ). Let W + and W − be the stable and the unstable plaques of M containing x 0 . Let R, Ω n (W + , R), and Ξ(W + , R) be as above. We assume that R is connected and x 0 ∈ R.
Note that F n (g(R))∩W + = ∅ is equivalent to g(R)∩F −n (W + ) = g(R)∩W + = ∅. It follows that the set Ω n (W + , R) does not depend on n.
The graph Ξ(W + , R) is isomorphic then to the graph with the set of vertices Ω 0 (W + , R) × Z in which two vertices are connected by an edge if and only if they are either of the form (g 1 , n) and (g 2 , n), where g 1 , g 2 ∈ Ω 0 (W + , R) and g 1 (R) ∩ g 2 (R) = ∅, or of the form (g 1 , n) and (g 2 , n + 1), where g 1 , g 2 ∈ Ω 0 (W + , R) and F n (g 1 (R)) ∩ F n+1 (g 2 (R)) = ∅, which is equivalent to g 1 (R) ∩ φ(g 2 )(F (R)) = ∅. Note that (g, n) → (g, n + 1) is an automorphism of Ξ(W + , R).
Let A ⊂ G be a finite set containing the identity, and let S be a finite generating set of G. We assume that S contains all elements g ∈ G such that R ∩ g(R) = ∅ or R ∩ g(F (R)) = ∅ and that A ⊂ S.
Let Ω ′ 0 ⊂ G be any set such that Ω 0 (W + , R) ⊂ Ω ′ 0 ⊂ Ω 0 (W + , R)A. Denote then by Ξ ′ the graph with the set of vertices Ω ′ 0 × Z with edges of two kinds: vertical and horizontal. The horizontal edges connect two vertices (g 1 , n), (g 2 , n) if and only if g −1 1 g 2 ∈ S. The vertical edges connect a vertex (g 1 , n) to a vertex (g 2 , n + 1) if and only if g −1 1 φ(g 2 ) ∈ S. Note that if g 1 (R) ∩ g 2 (R) = ∅, then R ∩ g −1 1 g 2 (R) = ∅. If g 1 (R) ∩ F (g 2 (R)) = ∅, then g 1 (R) ∩ φ(g 2 )(F (R)) = ∅, hence R ∩ g −1 1 φ(g 2 )(F (R)) = ∅. It follows that Ξ(W + , R) is a sub-graph of Ξ ′ .
Proposition 5. 6. The inclusion Ξ(W + , R) ֒→ Ξ ′ is a quasi-isometry.
Proof. Let us prove at first the following lemmas.
Lemma 5.7. There exists n 1 > 0 such that φ −n1 (Ω 0 (W + , R)) ⊂ Ω 0 (W + , R).
For every such n 1 there exists D n1 such that distance from (g, n) to (φ −n1 (g), n+ n 1 ) in Ξ(W + , R) is less than D n1 .
Proof. By Lebesgue's covering lemma, there exists ǫ > 0 such that for every x ∈ M there exists g ∈ G such that the ǫ-neighborhood of x is contained in g(R). It follows that if distance from x to W + is less than ǫ, then there exists g ∈ Ω 0 (W + , R) such that x ∈ g(R).
There exists an upper bound (equal to the diameter of R) on the distance from g(x 0 ) to W + for all g ∈ Ω 0 (W + , R). Consequently, there exists n 1 > 0 such that φ −n1 (Ω 0 (W + , R)) ⊂ Ω 0 (W + , R).
Let us prove the second part of the lemma. Let (g, n) ∈ Ξ(W + , R), and let x ∈ g(R) ∩ W + . Choose for k = 1, . .
. , n 1 , h k ∈ Ω 0 (W + , R) such that F −k (x) ∈ h k (R). Then F −1 (x) ∈ h 1 (R) and x ∈ g(R), hence F (h 1 (R)) ∩ g(R) = ∅, which implies that (g, n) is connected to (h 1 , n + 1). Similarly, F −k (x) ∈ h k (R) and F −(k+1) (x) ∈ h k+1 (R), hence F (h k+1 (R)) ∩ h k (R) = ∅, so that (h k , n + k) is connected to (h k+1 , n+ k + 1). We have g(x 0 ) ∈ g(R), hence F −n1 (x) ∈ h n1 (R) and F −n1 (x), φ −n1 (g)(x 0 ) ∈ F −n1 (g(R)) = φ −n1 (g)(F −n1 (R)). The set F −n1 (g(R))
is connected, hence there exists a path f 1 , f 2 , . . . , f m in Ω 0 (W + , R) × {n + n 1 } connecting (h n1 , n + n 1 ) to (φ −n1 (g), n + n 1 ). Since F −n1 (R) is relatively compact, there exists a uniform bound M such that we may assume that m ≤ M . It follows that the distance in Ξ(W + , R) from (g, n) to (φ −n1 (g), n + n 1 ) is not more than n 1 + m − 1.
Lemma 5.8. For every finite set B ⊂ G there exists D > 0 such that if g −1 1 g 2 ∈ B for g 1 , g 2 ∈ Ω 0 (W + , R), then distance between (g 1 , n) and (g 2 , n) in Ξ(W + , R) is not greater than D.
Proof. Let R B be a compact connected rectangle of M containing B(x 0 ). Note that it follows from Proposition 5.3 that there exists a uniform upper bound on the d − -diameter of the unstable plaques of R B . Then it is also a uniform upper bound on the d − -distance from W + of a point of g(R B ) for g ∈ Ω 0 (W + , R).
It follows that for every ǫ > 0 there exists n 2 > 0 such that for every g ∈ Ω 0 (W + , R) the set F −n2 (g(R B )) belongs to the ǫ-neighborhood of W + , hence (if ǫ is small enough) it is covered by the sets h(R) for h ∈ Ω 0 (W + , R). Since F −n2 (g(R B )) is connected, for every two points x, y ∈ g(R B ) there exists a sequence h 1 , h 2 , . .
. , h m ∈ Ω 0 (W + , R) such that F −n2 (x) ∈ h 1 (R), F −n2 (y) ∈ h m (R), h i (R) ∩ h i+1 (R) = ∅ for all i = 1, . . . , m − 1, and h i are pairwise different. Since F −n2 (g(R B )) = φ −n2 (g)(F −n2 (R B )) belong to the G-orbit of F −n2 (R B )
, there exists a uniform upper bound M (n 2 ) on the length m − 1 of the corresponding path h 1 , . . . , h m in Ω 0 (W + , R) × {0}. We can choose n 2 bigger than the number n 1 from Lemma 5.7.
Let g 1 , g 2 ∈ Ω 0 (W + , R) be such that g −1 1 g 2 ∈ B. Since g −1 1 g 2 ∈ B, g 1 (x 0 ), g 2 (x 0 ) ∈ g 1 (R B ). It follows that there exists a path h 1 , . . . , h m ∈ Ω 0 (W + , R) such that m < M , and F −n2 (g 1 (x 0 )) ∈ h 1 (R), F −n2 (g 2 (x 0 )) ∈ h m (R). The last two conditions are equivalent to φ −n2 (g 1 )(x 0 ) ∈ h 1 (R) and φ −n2 (g 2 )(x 0 ) ∈ h m (R), which imply that (φ −n2 (g 1 ), 0) and (φ −n2 (g 2 ), 0) are connected to (h 1 , 0) and (h m , 0), respectively, by horizontal edges. By Lemma 5.7, we have a uniform bound on the distances from (g 1 , n) to (φ −n2 (g 1 ), n + n 1 ) and from (g 2 , n) to (φ −n2 (g 2 ), n + n 1 ), which finishes the proof.
The proof of the following lemma is analogous.
Lemma 5.9. For every finite set B ⊂ G there exists D > 0 such that if g −1 1 φ(g 2 ) ∈ B for g 1 , g 2 ∈ Ω 0 (W + , R), then the distance between (g 1 , n) and (g 2 , n + 1) in Ξ(W + , R) is not greater than D.
Let us go back to proving Proposition 5.6. The image of Ξ(W + , R) under the inclusion map is a 1-net in Ξ ′ . Distance between vertices in Ξ(W + , R) is not less than the distance between them in Ξ ′ .
Let us show that there exists a constant D > 1 such that distance between (g 1 , n 1 ), (g 2 , n 2 ) in Ξ(W + , R) is not more than D times the distance from (g 1 , n 1 ) to (g 2 , n 2 ) in Ξ ′ . Let (g 1 , n 1 ) = v 0 , v 1 , . . . , v n = (g 2 , n 2 ) be a geodesic path in Ξ ′ . Since Ξ(W + , R) is a net in Ξ ′ , there exists a constant C > 1 such that every such geodesic path can be replaced by a path (g 1 , n 1 ), v ′ 1 , . . . , v ′ n−1 , (g 2 , n 2 ), where v ′ i ∈ Ξ(W + , R), and distance from v ′ i to v ′ n−1 in Ξ ′ is bounded from above by C. Moreover, we may assume that each v ′ i belongs to the same level Ω ′ n as v i . Then v ′ i and v ′ i+1 either belong to one level, or to two neighboring levels. Then Lemmas 5.8 and 5.9 finish the proof. Recall that two subsets A 1 , A 2 of a metric space (X, d) are on a finite Hausdorff distance from each other if there exists D > 0 such that for every x ∈ A 1 there exists y ∈ A 2 , and for every y ∈ A 2 there exists x ∈ A 1 such that d(x, y) < D. Let Ω 0 (W + , R) be, as before, the set of elements g ∈ G such that g(R) ∩ W + = ∅.
For a set Σ ⊂ G and a finite generating set S of G, denote by Ξ(Σ, S) the graph with the set of vertices Σ × Z in which two vertices are adjacent either if they are of the form (g, n) and (gs, n) for g, gs ∈ Σ and s ∈ S, or of the form (g, n) and (φ −1 (gs), n+ 1) for g, φ −1 (gs) ∈ Σ and s ∈ S. Note that the map (g, n) → (g, n+ 1) is an automorphism of Ξ(Σ, S).
If A is big enough, then Σ + A contains Ω 0 (W + , R). Then by Proposition 5.6 the identical embedding Ξ(W + , R) ֒→ Ξ(Σ + A, S) is a quasi-isometry, provided S is big enough. It follows then from Theorem 5.4 that Ξ(Σ + A, S) is Gromov hyperbolic, and that the boundary of Ξ(Σ + A, S) minus the common limit ω of quasi-geodesic paths of the form (g n , −n), n ≥ 1, is homeomorphic to W + . Moreover, it follows directly from Theorem 5.4 that the natural homeomorphism Φ : ∂Ξ(Σ + A, S)\ω −→ W + maps the limit of a sequence (g n , n) ∈ Ξ(Σ + A, S) to the limit of the sequence [F n (g n (x 0 )),
x 0 ] = [φ n (g n )(x 0 ), x 0 ] ∈ W + .
Consequently, the homeomorphism Φ conjugates F : W + −→ W + with the map on the boundary of Ξ(Σ + A, S) induced by the automorphism (g, n) → (g, n + 1). This shows that the dynamical system (F, W + ) is uniquely determined by (G, φ, Σ + ).
It remains to show that for every g ∈ G the homeomorphism g : x → [g(x), x 0 ] of W + is uniquely determined by (G, φ, Σ + ) and g.
Let ξ be the limit of a sequence (g n , n) ∈ Ξ(Σ + A, S), where s n = g −1 n φ(g n+1 ) ∈ S for all n ≥ 0. Note that every point of ∂Ξ(Σ + A, S) can be represented in this way, provided S is big enough (see Theorem 5.4).
There exists ǫ > 0 such that for every x ∈ M there exists g ∈ G such that the ǫ-neighborhood of x is contained in g(R). Let x ∈ W + . Choose for every n ≥ 0 an element g n ∈ G such that the ǫ-neighborhood of F −n (x) is contained in g n (R). Then x is contained in F n (g n (R)). In particular, F n (g n (R)) ∩ F n+1 (g n+1 (R)) = ∅, i.e, the sequence (g n , n) is a path in Ξ(W + , R), and its limit in ∂Ξ(W + , R) is mapped by the natural homeomorphism to x.
The rectangles gF n (g n (R)) contain g(x) for all n. Since F is expanding in the unstable direction, the sets gF n (g n (R)) intersects W + (i.e., φ −n (g)g n ∈ Ω 0 (W + , R)) for all n big enough.
The limit of the intersections of gF n (g n (R)) with W + is equal to [x 0 , g(x)]. It follows that (φ −n (g)g n , n), where n is big enough, is a path in Ξ(W + , R) converging to the point of ∂Ξ(W + , R) corresponding to [x 0 , g(x)] ∈ W + , i.e., to the image of x under the action of g on W + .
Note that left multiplication by g preserves the distances between the vertices of the graph Ξ(Σ + A, S) (when the images of the vertices belong to the graph). It also follows from the classical properties of Gromov hyperbolic graphs that there exists a constant ∆ 1 such that if two paths (g n , n) and (h n , n) of Ξ(Σ + A, S) converge to the same point of the boundary, then distance between (g n , n) and (h n , n) is less than ∆ 1 for all n big enough.
It follows that the action of g on W + can be modeled on the boundary of Ξ(Σ + A, S) by the following rule. Take a path (g n , n) ∈ Ξ(Σ + A, S) converging to a point ξ ∈ ∂Ξ(Σ + A, S). If (gg n , n) for n big enough belong to Ξ(Σ + A, S), then its limit is g(ξ). Since Ξ(W + , R) ⊂ Ξ(Σ + A, S), this rule will determine the action of g on ∂Ξ(Σ + A, S) \ {ω}.
Theorem 5.11. Let (X 1 , f 1 ) and (X 2 , f 2 ) be connected and locally connected Smale spaces. Suppose that they exist fixed points of f i and splittings π i : M i −→ X i . Let G i be the groups of deck transformations of the splittings. Let F i be lifts of f i , with fixed points x i ∈ M i . If there exists a continuous map Φ : M 1 −→ M 2 and an isomorphism Ψ : G 1 −→ G 2 such that Φ(x 1 ) = x 2 , and
Φ(F 1 (x)) = F 2 (Φ(x)), Φ(F 1 (g(x))) = F 2 (ψ(g)(Φ(x)))
for all x ∈ M 1 and g ∈ G 1 , then (X 1 , f 1 ) and (X 2 , f 2 ) are topologically conjugate.
Proof. The map Φ is proper as an equivariant map between two proper actions (see, for instance [Nek08, Lemma 5.2]).
Let U be a compact neighborhood of x 2 ∈ M 2 . Then n≥1 k≥n F k 2 (U ) is equal to the unstable plaque W − (x 2 ) in M 2 . Similarly,
n≥1 k≥n F k 1 (Φ −1 (U )) = Φ −1 n≥1 k≥n F k 2 (U ) is equal to the unstable plaque W − (x 1 ) in M 1 . It follows that Φ −1 (W − (x 2 )) = W − (x 1 ). Similarly, Φ −1 (W + (x 2 )) = W + (x 1 ).
Let K 2 be a compact subset of M 2 such that K 2 and K 1 = Φ −1 (K) are G itransversals. They exists, since the actions of G i are co-compact, proper, and the map Φ is continuous and proper.
Then
g 2 (K 2 ) ∩ W − (x 2 ) = ∅ for g 2 ∈ G 2 is equivalent to ψ −1 (g 2 )(K 1 ) ∩ W − (x 1 ) = ∅.
The same is true for the stable plaques W + (x 1 ) and W + (x 2 ). It follows then from Theorem 5.10 that (X 1 , f 1 ) and (X 2 , f 2 ) are topologically conjugate.
Smale spaces with virtually nilpotent splitting
Let L be a simply connected nilpotent Lie group. Let G be a finitely generated subgroup of Aut L ⋉ L such that the action of G on L is free, proper, and cocompact. Here we identify the elements of A with the transformations g → α(g) · h of L, where α ∈ Aut L and h ∈ L.
Let F ∈ Aut L be a hyperbolic automorphism of L (i.e., such that its differential DF at the identity of L has no eigenvalues of absolute value one). Then F induces an automorphism φ of Aut ⋉L by conjugation. Suppose that G is invariant under this automorphism. Then F induces an Anosov homeomorphism f : G\L −→ G\L. Such homeomorphisms are called hyperbolic infra-nilmanifold automorphisms.
The aim of this section is to prove the following description of locally connected Smale spaces that have a splitting with a virtually nilpotent group of deck transformation.
Theorem 6.1. Let (X , f ) be a Smale space such that X is connected and locally connected, and there exists a splitting π : M −→ X with a virtually nilpotent group of deck transformations. Then (X , f ) is topologically conjugate to a hyperbolic infranilmanifold automorphism. Proof. Let (X , f ) satisfy conditions of the theorem. Let us assume at first that the group G of deck transformations is nilpotent and torsion free.
Let F be a lift of f , and let F (x 0 ) = x 1 . Then the map g → φ(g) defined by F (g(x 0 )) = φ(g)(x 1 ) is an automorphism of G.
Denote by Z(G) the center of G, i.e., the set of elements of G that commute with every element of G.
The group Z(G) is obviously abelian and torsion free. It is finitely generated, since all subgroups of a finitely generated nilpotent group are finitely generated (see [Rob96,5.2.17]). Consequently, Z(G) is isomorphic to Z d for some d.
Lemma 6.2. Let g ∈ Z(G). There exist positive constants D − and D + such that for every stable (resp. unstable) plaque V of M and any x ∈ V , y ∈ g(V ) we have
d − (x, [x, y]) ≤ D − (resp. d + (x, [y, x]) ≤ D + ).
Proof. Let us prove the lemma for stable plaques. Note that [x, y] is equal to the intersection of g(V ) with W − (x), and so does not depend on the choice of y ∈ g(V ). Therefore, it is enough to show that d − (x, [x, g(x)]) is bounded for all x ∈ M, see ) for x ∈ R and y ∈ g(R). It is finite, since there exists a compact rectangle P such that P ⊃ R∪g(R) (see also Proposition 5.3).
For
every x ∈ M there exists h ∈ G such that h(x) ∈ R. Then d − (x, [x, g(x)]) = d − (h(x), [h(x), hg(x)]) = d − (h(x), [h(x), gh(x)]) < D.
Denote for g ∈ Z(G)
D − (g) = sup x∈M d − (x, [x, g(x)]), D + (g) = sup x∈M d + (x, [g(x), x]),
which are finite by Lemma 6.2. Note that we obviously have (7) D + (g 1 g 2 ) ≤ D + (g 1 ) + D + (g 2 ),
D − (g 1 g 2 ) ≤ D − (g 1 ) + D − (g 2 )
for all g 1 , g 2 ∈ Z(G).
Let λ ∈ (0, 1) and C > 1 be such that for any two stably (resp. unstably) equivalent points x, y ∈ X we have d + (f n (x), f n (y)) ≤ Cλ n d + (x, y) (resp. d − (f −n (x), f −n (y)) ≤ Cλ n d − (x, y)) for all n ≥ 0. Then the same estimates will hold for F and x, y ∈ M belonging to one stable (resp. unstable) plaque. Proposition 6.3. For every g ∈ Z(G) and n ≥ 0 we have
D + (φ n (g)) ≤ Cλ n D + (g), D − (φ −n (g)) ≤ Cλ n D − (g).
The center Z(G) is characteristic (i.e., invariant under automorphisms of G), hence φ(Z(G)) = Z(G).
Proof. Let us prove the first inequality. The second is proved in the same way.
Let V be an unstable plaque of M, and let x ∈ V and y ∈ φ n (g)(V ) be such that x and y belong to the same stable plaque. It is enough to prove that d + (x, y) ≤ Cλ n D + (g).
The points F −n (x) and F −n (y) belong to one stable plaque, and F −n (x) ∈ F −n (V ), F −n (y) ∈ F −n (φ n (g)(V )) = g(F −n (V )), hence d + (F −n (x), F −n (y)) ≤ D + (g). But this implies d + (x, y) ≤ Cλ n d + (x, y).
Proposition 6.4. For every finite set S ⊂ Z(G) there exists a constant D S > 0 satisfying the following condition. For every finite set A ⊂ Z(G) there exists n 0 such that for all n ≥ n 0 , g 1 , g 2 ∈ φ n (A)φ n−1 (S)φ n−2 (S) · · · φ(S)S, and every unstable plaque V we have d + (x, [y, x]) < D S for all x ∈ g 1 (V ) and y ∈ g 2 (V ).
Proof. Let ∆ S and ∆ A be upper bounds on D + (g) for g ∈ S and g ∈ A, respectively.
Then, by (7) and Proposition 6.3, we have, for all h ∈ A, g i ∈ S, and all n big enough,
D + (φ n (h)φ n−1 (g 1 ) · · · φ(g n−1 )g n ) ≤ Cλ n ∆ A + C(λ n−1 + · · · + λ + 1)∆ S < 1 + C∆ S 1 − λ .
It follows that we can take D S = 2 + 2C∆S 1−λ . Proposition 6.5. The automorphism φ of Z(G) ∼ = Z d is hyperbolic, i.e., has no eigenvalues of absolute value 1. Proof. Suppose that on the contrary, there exists an eigenvalue cos α + i sin α of φ of absolute value 1. Suppose at first that α / ∈ π · Z. Then there exists a twodimensional subspace L ≤ R d and a Euclidean structure on it such that φ acts on L as a rotation by the angle α.
Denote K = {(x i ) d i=1 ∈ R d : |x i | < 1}, and let S be the set of elements g ∈ Z(G) = Z d such that φ(K) ∩ (K + g) = ∅ or φ −1 (K) ∩ (K + g) = ∅. The set S is obviously finite.
Let R > 0 be arbitrary, and consider the circle γ of radius R in L with center in the origin. Then φ(γ) = γ. Let A R be the set of elements g ∈ Z(G) such that K + g ∩ γ = ∅. It is finite and non-empty. Note that union of the sets A R for all R > 0 is infinite.
Let h be an arbitrary element of A R , and let x ∈ K + h ∩ γ. Then φ −1 (x) ∈ γ, and there exists g ∈ Z d such that φ −1 (x) ∈ K + g. Then g ∈ A R , and x ∈ φ(K)+φ(g)∩K +h. It follows that K +h−φ(g)∩φ(K) = ∅, so that h−φ(g) ∈ S. We see that h = φ(g) + (h − φ(g)) ∈ φ(A R ) + S. We have proved that A R ⊂ φ(A R ) + S. It is proved in the same way that A R ⊂ φ −1 (A R ) + S. By induction we conclude that A R ⊂ φ n (A R ) + φ n−1 (S) + · · · + φ(S) + S and A R ⊂ φ −n (A R ) + φ −(n−1) (S) + · · · + φ −1 (S) + S for all n ≥ 1.
Fix an arbitrary point x 0 ∈ M. Since φ n (A R ) + φ n−1 (S) + · · · φ(S) + S ⊃ A R for all n, it follows from Proposition 6.4 that there exists D S > 0, not depending on
R, such that d + (x 0 , [g(x 0 ), x 0 ]) < D S and d − (x 0 , [x 0 , g(x 0 )]) < D S for all g ∈ A R .
It follows that g(x 0 ) belongs to the rectangle [B + , B − ], where B ± are the balls of radius D S with center in x 0 in the corresponding plaque containing x 0 . Note that the set {g ∈ G : g(x 0 ) ∈ [B + , B − ]} is finite, does not depend on R, and contains A R . But this is a contradiction.
The case when the eigenvalue is equal to ±1 is similar (with one-dimensional space L).
Let E + (resp. E − ) be the sum of the root subspaces of R d of the eigenvalues λ of φ such that |λ| < 1 (resp. |λ| > 1). We have R d = E + ⊕ E − . Denote by P + and P − = 1 − P + the projections onto E + and E − , respectively.
Denote K = {(x i ) d i=1 ∈ R d : |x i | < 1}. Let S ⊂ Z(G)
be a finite set containing all elements g ∈ Z(G) such that K + g ∩ (φ(K) ∪ φ −1 (K)) = ∅.
Proposition 6.6. For every point x ∈ E + there exists a sequence g i ∈ S, i = 1, 2, . . ., and an element g 0 ∈ Z(G) such that x = lim n→∞ P + (φ n (g n ) + φ n−1 (g n−1 ) + · · · + φ(g 1 ) + g 0 ).
There exists a finite set N ⊂ Z(G) such that an equality lim n→∞ P + (φ n (g n ) + φ n−1 (g n−1 ) + · · · + φ(g 1 )
+ g 0 ) = lim n→∞ P + (φ n (g ′ n ) + φ n−1 (g ′ n−1 ) + · · · + φ(g ′ 1 ) + g ′ 0 )
holds for g i , g ′ i ∈ S, i ≥ 1, and g 0 , g ′ 0 ∈ Z(G) if and only if there exists a sequence h n ∈ N such that φ n (g ′ n ) + φ n−1 (g ′ n−1 ) + · · ·+ φ(g ′ 1 ) + g ′ 0 = φ n (h n + g n ) + φ n−1 (g n−1 ) + · · ·+ φ(g 1 ) + g 0 for all n big enough.
Proof. The sets K + g cover R d for g ∈ Z(G) = Z d , and the group Z(G) is φinvariant, hence for every x ∈ E + and n ≥ 0 there exists h n ∈ Z(G) such that
x ∈ φ n (K) + h n . We have then φ n−1 (K) + h n−1 ∩ φ n (K) + h n = ∅, hence φ −1 (K) ∩ K + φ −n (h n ) − φ −n (h n−1 ) = ∅ which implies that φ −n (h n ) − φ −n (h n−1 ) = g n ∈ S,
i.e., h n = φ n (g n ) + h n−1 . It follows that there exists a sequence g i ∈ S such that h n = φ n (g n ) + φ n−1 (g n−1 ) + · · · + φ(g 1 ) + h 0 .
Note that since x ∈ φ n (K) + h n , we have P + (h n ) − x < Cλ n for some constant C. It follows that x = lim n→∞ P + (h n ).
Note that the set of all limits lim n→∞ P + (φ n (g n ) + φ n−1 (g n−1 ) + · · · + φ(g 1 )) for all choices of g i ∈ S is a bounded subset T + of E + .
Suppose that lim n→∞ P + (φ n (g n ) + φ n−1 (g n−1 ) + · · · + φ(g 1 ) + h) = lim n→∞ P + (φ n (g ′ n ) + φ n−1 (g ′ n−1 ) + · · · + φ(g ′ 1 ) + h ′ ) for g i , g ′ i ∈ S and h, h ′ ∈ Z(G). Then for every n ≥ 0 we have P + (φ n (g n )+φ n−1 (g n−1 )+· · ·+φ(g 1 )+h)−P + (φ n (g ′ n )+φ n−1 (g ′ n−1 )+· · ·+φ(g ′ 1 )+h ′ ) = P + (φ n+1 (g ′ n+1 )) + P + (φ n+2 (g ′ n+2 )) + · · · − P + (φ n+1 (g n+1 )) + P + (φ n+2 (g n+2 )) + · · · ∈ φ n (T + − T + ).
It follows that
P + ((g n + φ −1 (g n−1 ) + · · · + φ −(n−1) (g 1 ) + φ −n (h))− (g ′ n + φ −1 (g ′ n−1 ) + · · · + φ −(n−1) (g ′ 1 ) + φ −n (h ′ ))) ∈ T + − T + . Since φ −1 is contracting on E − ,
there exists a compact set T − ⊂ E − such that for any h and any sequence g i ∈ S we have P − (g n + φ −1 (g n−1 ) + · · · + φ −(n−1) (g 1 ) + φ −n (h)) ∈ T − for all n big enough.
It follows that for all n big enough the difference
(g n + φ −1 (g n−1 ) + · · · + φ −(n−1) (g 1 ) + φ −n (h))− (g ′ n + φ −1 (g ′ n−1 ) + · · · + φ −(n−1) (g ′ 1 ) + φ −n (h ′ )) belongs to a bounded set T = (T + − T + ) ⊕ (T − − T − ), hence we can take N = T ∩ Z d . Fix a stable plaque W + = W + (x 0 ) of M. The group G acts on W + by x → [g(x), x], since G preserves the direct product structure of M.
Let v ∈ R d , and denote v + = P + (v) and v − = P − (v). Using Proposition 6.6, find a sequence g i ∈ S, i ≥ 1, and g 0 ∈ Z(G) such that v + = lim n→∞ P + (φ n (g n ) + · · · + φ(g 1 ) + g 0 ),
and define for x ∈ W + (8) v + (x) = lim n→∞ [(φ n (g n ) + · · · + φ(g 1 ) + g 0 )(x), x].
We also define for x ∈ W − , where W − is an unstable plaque:
(9) v − (x) = lim n→∞ [x, (φ −n (g n ) + · · · + φ −1 (g 1 ) + g 0 )(x)],
where g i ∈ S, for i ≥ 1, and g 0 ∈ Z(G) are such that
v − = lim n→∞ P − (φ −n (g n ) + · · · + φ −1 (g 1 ) + g 0 ).
(Replacing in Proposition 6.6 φ by φ −1 and E + , P + by E − , P − , we see that such a sequence g n exists.)
Proposition 6.7. The limit (8) exists and depends only on v + and x. The limit (9) exists and depends only on v − and x.
Proof. Follows directly from (7), Propositions 6.3 and 6.6.
Theorem 6.8. The limits (8) and (9) define continuous actions of E + and E − on W + = W + (x 0 ) and W − = W − (x 0 ), respectively. Their direct sum is a continuous action of R d on M. This action satisfies the following conditions:
(1) it is free and proper;
(2) its restriction onto Z(G) = Z d < R d coincides with the original action of
Z(G) on M; (3) it preserves the direct product structure, i.e., v([x, y]) = [v(x), v(y)] for all v ∈ R d and x, y ∈ M; (4) F (v(x)) = φ(v)(F (x)) for all x ∈ M and v ∈ R d ; (5) the action commutes with G, i.e., v(g(x)) = g(v(x)) for all x ∈ M, g ∈ G,
and v ∈ R d ;
(6) if g(x) = v(x) for g ∈ G and v ∈ R d , then g = v ∈ Z(G).
Proof. The fact that (8) and (9) define actions follows directly from the fact that the limits do not depend on S and the choice of the sequences g i .
Let us prove that the action is continuous. It is enough to prove that the action of E + on W + is continuous. We have to show that for every v 1 ∈ E + , x ∈ W + , and ǫ > 0 there exists δ > 0 such that if v 2 ∈ E + and y ∈ W + are such that v 1 − v 2 < δ and d + (x, y) < δ, then d + (v 1 (x), v 2 (y)) < ǫ.
Take an arbitrary ǫ > 0. For every n there exists δ 1 (n) and a sequence g 0 ∈ H, g i ∈ S, i ≥ 1, such that v 1 = lim m→∞ P + (g 0 +φ(g 1 )+· · ·+φ m (g m )) and all points v 2 in the δ 1 (n)-neighborhood of v 1 can be represented as limits v 2 = lim m→∞ P + (h 0 + φ(h 1 ) + · · · + φ m (h m )) for h 0 ∈ Z(G), and h i ∈ S, i ≥ 1, such that h i = g i for i = 0, 1, . . . , n (see the proof of Proposition 6.6 and use Lebesgue's covering lemma). There exists δ 2 (n) such that if y ∈ W + is such that d + (x, y) < δ 2 (n), then d + ([(g 0 + φ(g 1 ) + · · · + φ n (g n ))(x), x], [(g 0 + φ(g 1 ) + · · · + φ n (g n ))(y), y]) < ǫ/2, since the function y → [(g 0 + φ(g 1 ) + · · · + φ n (g n ))(y), y] is continuous. There exist constants C > 0 and λ ∈ (0, 1) such that d + (u(z), [(g 0 + φ(g 1 ) + · · · + φ n (g n ))(z), z]) < Cλ n for all z ∈ W + and u ∈ E + such that u = lim m→∞ P + (g 0 + φ(g 1 ) + · · · + φ m (g m )) for g 0 ∈ Z(G) and g i ∈ S for i ≥ 1.
Take n ≥ − log(ǫ/4C) log λ . Then for all v 2 ∈ E + and y ∈ W + such that v 1 − v 2 < δ 1 (n) and d + (x, y) < δ 2 (n) we have
d + (v 1 (x), v 2 (y)) ≤ d + (v 1 (x), [(g 0 + φ(g 1 ) + · · · + φ n (g n )(x), x])+ d + ([(g 0 + φ(g 1 ) + · · · + φ n (g n )(x), x], [(g 0 + φ(g 1 ) + · · · + φ n (g n )(y), y])+ d + (v 2 (y), [(g 0 + φ(g 1 ) + · · · + φ n (g n )(y), y]) ≤ ǫ/4 + ǫ/2 + ǫ/4 = ǫ.
Which shows that the action of E + on W + is continuous.
The same arguments (using Proposition 6.3 and inequalities (7)) as in the proof of the criterion of equality of two limits in Proposition 6.6 show that an equality lim n→∞ [(φ n (g n ) + · · · φ(g 1 )
+ g 0 )(x), x] = lim n→∞ [(φ n (g ′ n ) + · · · φ(g ′ 1 ) + g ′ 0 )(x), x]
for g i , g ′ i ∈ S, i ≥ 1, and g 0 , g ′ 0 ∈ Z(G) is equivalent to the equality lim n→∞ P + (φ n (g n ) + · · · φ(g 1 ) + g 0 ) = lim n→∞ P + (φ n (g ′ n ) + · · · φ(g ′ 1 ) + g ′ 0 ).
This (and a similar statement for P − and the action on W − ) shows that the action of R d is free. Let us show that the action is proper. Let B ⊂ M be a compact set. We have to show that the set {v ∈ R d : v(B) ∩ B = ∅} is compact. It is closed, since the action is continuous.
Denote K = {(x i ) d i=1 ∈ R d : |x i | ≤ 1}. Then for every v ∈ R d there exists h ∈ Z(G) = Z d such that v − h ∈ K. The set K(B) = {v(x) : x ∈ B
, v ∈ K} is compact, since the action is continuous and the sets B and K are compact. The action of G on M is proper, hence the set A of elements h ∈ Z(G) such that
h(K(B)) ∩ B = ∅ is finite. Suppose that x ∈ B and v ∈ R d are such that v(x) ∈ B. There exists h ∈ Z(G) such that v − h ∈ K. Then v(x) = (h + v − h)(x) ∈ h(K(B)) ∩ B, hence h ∈ A, so that v ∈ K + A.
But the set K + A is compact, which proves that the action of R d on M is proper.
The proof of statements (2)-(5) is straightforward, using the fact that the action does not depend on the choice of S.
Let us prove the last statement. Suppose that g(x) = v(x) for g ∈ G and v ∈ R d . Then g leaves invariant the orbit R d (x) of x. Let G 1 be the group of all elements leaving the R d (x) invariant. The action of Z(G) on R d (x) is co-compact, the action of G on M is proper, hence the index of Z(G) in G 1 is finite, i.e., the image of G 1 in G/Z(G) is finite. But G/Z(G) is torsion free (see [Rob96,5.2.19]). Consequently, G 1 = Z(G). Since the action of R d on R d (x) is free, this implies that g = v ∈ Z(G).
Proposition 6.9. If G is abelian, then the action of R d on M is transitive. Proof. It is enough to show that for every point x ∈ W + there exists a sequence g i ∈ S, i ≥ 1, and an element g 0 ∈ G such that
(10) x = lim n→∞ [(φ n (g n ) + · · · + φ(g 1 ) + g 0 )(x 0 ), x 0 ].
The action of G on M is co-compact, hence there exists a relatively compact open rectangle R ⊂ M containing x 0 and such that h∈G h(R) = M.
Then for every x ∈ W + and every n ≥ 0 there exists h n ∈ G such that x ∈ h n (F n (R)). Assume that S is big enough so that it contains all elements h ∈ G such that h(R) ∩ (F (C) ∪ F −1 (C)) = ∅. Then the same arguments as in the proof of Proposition 6.6 show that there exists a sequence g i ∈ S, i ≥ 1, and an element g 0 ∈ G for which (6.6) holds.
Theorem 6. 10. Let (X , f ) be a locally connected and connected Smale space which has a splitting with a free abelian group of deck transformations G ∼ = Z d . Let φ be the automorphism of G induced by a lift of f . Then (X , f ) is topologically conjugate to the hyperbolic automorphism of the torus R d /Z d induced by φ. In particular, (X , f ) has a fixed point. Proof. By Proposition 6.9, the action of R d on M defined in Theorem 6.8 is transitive.
Fix a basepoint x 0 ∈ M, define ρ 0 : R d −→ M by v → v(x 0 ). The map ρ 0 is a homeomorphism, since it is continuous, bijective, and proper.
Denote v 0 = ρ −1 0 (F (x 0 )), i.e., v 0 ∈ R d is such that v 0 (x 0 ) = F (x 0 ). Then the map φ 0 = ρ −1 0 F ρ 0 : R d −→ R d satisfies φ 0 (v) = ρ −1 0 (F (v(x 0 )) = ρ −1 0 (φ(v)(F (x 0 ))) = ρ −1 0 ((φ(v) + v 0 )(x 0 )) = φ(v) + v 0 . The linear operator 1 − φ is invertible, since φ is hyperbolic. Therefore, there exists w 0 ∈ R d such that w 0 − φ(w 0 ) = v 0 , i.e., φ 0 (w 0 ) = φ(w 0 ) + v 0 = w 0 . Define then ρ 1 (v) = ρ 0 (v + w 0 ). We have then F (ρ 1 (v)) = F (ρ 0 (v + w 0 )) = ρ 0 (φ 0 (v + w 0 )) = ρ 0 (φ(v + w 0 ) + v 0 ) = ρ 0 (φ(v) + φ(w 0 ) + v 0 ) = ρ 0 (φ(v) + w 0 ) = ρ 1 (φ(v)).
The statement of the theorem follows now directly from Theorem 6.8.
Let us go back to the case when G is torsion free nilpotent.
Proposition 6.11. The action of R d on M is uniformly locally Lipschitz, i.e., there exist ǫ > 0 and C > 1 such that for every v ∈ R d and all x, y ∈ M such that d M (x, y) < ǫ we have d M (v(x), v(y)) ≤ Cd M (x, y).
Note that it follows that
C −1 d M (x, y) ≤ d M (v(x)
, v(y)) for all x, y ∈ M such that d M (x, y) ≤ C −1 ǫ.
Proof. By Theorem 6.8, G maps R d -orbits to R d -orbits.
Let
K = {(x i ) d i=1 ∈ R d : |x i | ≤ 1}
, and let R ⊂ M be a relatively compact rectangle such that π(R) = X . Let ǫ > 0 and C > 1 be such that
(11) C −1 d M (x, y) ≤ d + (x, [y, x]) + d − (x, [x, y]) ≤ Cd M (x, y)
for all x, y ∈ M such that d M (x, y) < ǫ, see Subsection 5.2, where the metric d M is defined. We also assume that ǫ is sufficiently small so that for all x, y ∈ M such that d M (x, y) < ǫ there exists g ∈ G such that g(x), g(y) ∈ R.
Let δ is such that d M (v(x), v(y)) < ǫ for all v ∈ K and all x, y ∈ R such that d M (x, y) < δ. It exists, since the action of R d is continuous, and the set K and the closure of R are compact. For all x, y ∈ M such that d M (x, y) < δ and all v ∈ R d there exists g ∈ G and h ∈ Z(G) such that g(x), g(y) ∈ R, and v + h ∈ K. Then d M (g(x), g(y)) < δ, hence
d M (v(x), v(y)) = d M (g −1 · v · g(x), g −1 · v · g(y)) = d M ((h + v)(g(x)), (h + v)(g(y))) < ǫ.
We have shown that for all x, y ∈ M such that d M (x, y) < δ and all v ∈ R d we have d M (v(x), v(y)) < ǫ.
Let x, y ∈ M be such that d M (x, y) < ǫ. Let n be biggest positive integer such that d + (F −n (x), F −n ([y, x])) < C −1 δ. Then n is equal, up to an additive constant, to − log d + (x, [y, x])/α + , where α + is the exponent of d + .
We have then d
+ (v(F −n (x)), v(F −n ([y, x])) < ǫ for all v ∈ R d .
Applying F n , and using the fact that φ is an automorphism of R d , we get that d + (u(x), u([y, x])) ≤ C 1 e −nα+ ≤ C 2 d + (x, [y, x]) for all u ∈ R d , where C 1 , C 2 are constant (not depending on x, y).
In the same way we prove that d − (u(x), u([x, y])) < C 3 d − (x, [x, y]) for all u ∈ R d , if d M (x, y) < ǫ. It follows then from (11) that there exist ǫ 1 > 0 and C 4 > 0 such that if x, y ∈ M are such that d M (x, y) < ǫ 1 , then d M (u(x), u(y)) < C 4 d M (x, y) for all u ∈ R d .
Let M be the quotient of M by the R d -action defined in Theorem 6.8. We denote for x ∈ M by x the R d -orbit of x. Since G maps R d -orbits to R d -orbits, the action of G on M induces a well-defined action of G on M. Denote G = G/Z(G), it is a torsion-free finitely generated nilpotent group of nilpotency class one less than the class of G. By Theorem 6.8, Z(G) is equal to the kernel of the action of G on M, and the action of G on M is free.
The action of R d on M descends to a free action of R d /Z d on X , whose orbits are the images of the R d -orbits under the map π : M −→ X . Let π : M −→ X be the corresponding map induced by π : M −→ X .
Denote, for x, y ∈ M δ(x, y) = inf{x 1 ∈ x, y 1 ∈ y : d M (x 1 , y 1 )}.
Lemma 6.12. There exist ǫ > 0, C > 1 and a G-invariant metric d on M such that C −1 δ(x, y) ≤ d(x, y) ≤ Cδ(x, y) for all x, y such that δ(x, y) < ǫ.
Proof. The function δ is G-invariant, since the metric d M is G-invariant. Note that δ(x, y) > 0 for all x = y, since the quotient map π : (M, d M ) −→ (X , d) is a local isometry, the images of x and y in X are compact, hence distance between any two points of π(x) and π(y) are bounded from below.
Let ǫ be as in Proposition 6.11. Define d(x, y) as infimum of δ(x i , x i+1 ) over all sequences x = x 0 , . . . , x n = y such that δ(x i , x i+1 ) < ǫ. Note that by Proposition 6.11 there exists C > 1 such that δ(x, y) ≤ C δ(x i , x i+1 ), hence
δ(x, y) ≤ Cd(x, y)
for all x, y ∈ M. We also have
d(x, y) ≤ δ(x, y)
for all x, y such that δ(x, y) < ǫ.
Proposition 6.13. The topology defined by the metric d coincides with the quotient topology on M. The map π : M −→ X is uniformly locally bi-Lipschitz with respect to d and the metric on X coming from the Hausdorff distance between compact subsets of X .
Proof. By Proposition 6.11 and definition of d, there exist C > 1 and ǫ > 0 such that if d(x, y) < ǫ, then for every y ∈ y there exists x ∈ x such that d M (x, y) < Cǫ. Suppose that U ⊂ M is open with respect to d. Let U ⊂ M be the preimage of U . Then for every x ∈ U there exists ǫ > 0 such that the ǫ-neighborhood of x (with respect to d) is contained in U . Let y be such that d(x, y) < C −1 ǫ. Then for every y ∈ y there exists x ∈ x such that d M (x, y) < ǫ. It follows that the ǫ-neighborhood of the set x contains the set y. It follows that U is open in M.
Suppose that U ⊂ M is an R d -invariant open subset of M. Then for every x ∈ U there exists ǫ > 0 such that the ǫ-neighborhood of x is contained in U . Suppose that y ⊂ U is such that d(x, y) < C −1 ǫ/2. Then there exists y ∈ y such that d M (x, y) ≤ ǫ. Then y ∈ U , hence y ⊂ U , since U is R d -invariant. We have shown that every set that is open in the quotient topology is also open with respect to d.
The statement about the Hausdorff distance also follows directly from the definition of d and Proposition 6.11 (and the fact that π : M −→ X is a local isometry).
Note that v(x) = [v + (x), v − (x)],
where v + = P + (v) and v − = P − (v). It follows that for any v, u ∈ R d and x, y ∈ M we have
[v(x), u(y)] = [v + (x), u − (y)] = [(v + + u − )(x), (v + + u − )(y)] = (v + + u − )([x, y]),
i.e., the value of [x, y] depends only on x and y. It follows that the function [x, y] = [x, y] is well defined and satisfies the equalities (1) and (2) of Definition 2.1. We will prove that it is continuous in the next proposition.
We also have a well defined homeomorphism F (x) = F (x), by condition (4) of Theorem 6.8.
Lemma 6.14. The metric d agrees with the local product structure on M. In particular, the map [·, ·] is continuous. Proof. We know that the metric d M on M agrees with the local product structure on M, since it is locally isometric to the standard metric on X . Let x, y be points of M such that δ(x, y) is small. Let us prove that for all x and y that are close enough to each other. (Here F 1 ≍ F 2 means that there exists a constant C > 1 such that
C −1 F 1 ≤ F 2 ≤ CF 1 .)
There exist points x ∈ x and y ∈ y such that d M (x, y) ≤ 2δ(x, y). Since d agrees with the local product structure on M, we have
d M (x, y) ≍ d M (x, [x, y]) + d M (y, [x, y]). We have δ(x, [x, y]) ≤ d M (x, [x, y]) and δ(y, [x, y]) ≤ d M (y, [x, y]), hence there exists a constant C 1 > 1 such that δ(x, y) ≥ C −1 1 (δ(x, [x, y]) + δ(y, [x, y]))
. On the other hand, since δ is equivalent to a metric (see Lemma 6.12), there exists C 2 > 1 such that δ(x, y) ≤ C 2 (δ(x, [x, y]) + δ(y, [x, y])), by the triangle inequality. This proves (12).
Lemma 6. 15. There exists C > 1 and ǫ > 0 such that if x, y ∈ M are such that [x, y] = x (i.e., x and y belong to the same stable plaque of M) and δ(x, y) < ǫ, then there exist x ∈ x and y ∈ y such that [x, y] = x and d M (x, y) ≤ Cδ(x, y).
Proof. Let x, y belong to one stable plaque of M. There exist ǫ > 0 and C > 1 (not depending on
x, y) such that if d M ([x, g(x)], [y, h(y)]) < ǫ for g, h ∈ G, then d M ([x, g(x)], [y, h(y)]) ≥ C −1 (d M ([x, g(x)], [y, g(y)]) + d M ([y, g(y)], [y, h(y)])) ≥ C −1 d M ([x, g(x)], [y, g(y)]),
see Figure 3. It follows that
(13) inf{d M (v(x), u(y)) : v, u ∈ E − } ≥ C −1 inf{d M (v(x), v(y)) : v ∈ E − },
if the left hand side of the inequality is less than ǫ.
We always can find x 1 ∈ x and y 1 ∈ y such that d M (x 1 , y 1 ) ≤ 2δ(x, y). Then [x 1 , y 1 ] = v(x 1 ) for some v ∈ R d .
Note that [x 1 , y 1 ] and x 1 belong to the same unstable plaque of M, hence v ∈ E − . Then [v(x 1 ), y 1 ] = [x 1 , y 1 ] = v(x 1 ), i.e., the points v(x 1 ) and y 1 belong to the same stable plaque.
It follows from (13) that
inf{d M (uv(x 1 ), u(y 1 )) : u ∈ E − } ≤ C inf{d M (uv(x 1 ), w(y 1 )) : u, w ∈ E − } ≤ Cd M (x 1 , y 1 ) ≤ 2Cδ(x, y).
It follows that there exists a pair of points x ′ ∈ x, y ′ ∈ y such that x ′ and y ′ belong to the same stable plaque, and d M (x ′ , y ′ ) ≤ 2Cδ(x, y).
Let x, y ∈ M be such that δ(x, y) is small, and [x, y] = x. Let x 2 ∈ M be a point close to x. Using Lemma 6.15, find x ∈ x and y ∈ y such that [x, y] = x and d M (x, y) ≤ C 1 δ(x, y). By Proposition 6.11, there exists x 2 ∈ x 2 such that d M (x, x 2 ) ≤ C 2 δ(x, x 2 ). We conclude from this, and from the fact that d M agrees with the local product structure on M, that there exists a constant C 3 > 1 such that if d M (x, y) and δ(x, x 2 ) are small enough, we have
d M ([x, x 2 ], [y, x 2 ]) ≤ C 3 d M (x, y). Consequently, δ([x, x 2 ], [y, x 2 ]) ≤ d M ([x, x 2 ], [y, x 2 ]) ≤ C 3 d M (x, y) ≤ C 3 Cδ(x, y).
It follows that the maps
W + (x 1 ) −→ W + (x 2 ) : x → [x,
x 2 ] are locally Lipschitz with respect to the metric d. Since inverses of these maps are also maps of the same form, they are in fact locally bi-Lipschitz. This shows that the local product structure on M agrees with d.
Proposition 6.16. The dynamical system (X , f ), where f is the map induced by F is a connected and locally connected Smale space. The quotient map M −→ X is a splitting with the group of deck transformations G.
Proof. It follows from Theorem 6.8 that f : X −→ X is a well defined homeomorphism. The plaques of the direct product structure of M are continuous images of plaques of M, hence they are connected. The map F is a lift of f .
The map π : M −→ X is uniformly locally bi-Lipschitz with respect to d and the Hausdorff distance on X , by Proposition 6. 13. It follows that the Hausdorff distance agrees with the quotient topology on X and that π is a covering.
It follows from Proposition 6.13, Lemma 6.14, and the fact that G preserves the direct product structure on M, that the image under π of the local product structure on M is a well defined local product structure on X .
Suppose that x, y ∈ M are such that [x, y] = x (i.e., x and y belong to the same stable plaque of M). By Lemma 6.15, there exist x ∈ x and y ∈ y belonging to the same stable plaque of M. Note also that it follows from Lemmas 6.12 and 6.14 that there exists a constant C > 1, not depending on x and y, such that we can find x, y satisfying
C −1 d(x, y) ≤ d M (x, y) ≤ Cd(x, y), provided that d(x, y) is small enough.
Then d M (F n (x), F n (y)) ≤ Cλ n d M (x, y) for some fixed C > 1 and λ ∈ (0, 1). It follows that there exists a constant C 2 > 1 such that for any two points x, y ∈ M such that [x, y] = x, and d(x, y) is small enough we have d(F n (x), F n (y)) ≤ C 2 λ n d(x, y) for all n. Analogous statement about the unstable plaques of M is proved in the same way. The map π : M −→ X is locally bi-Lipschitz with respect to d and the Hausdorff distance on X . It follows that the images of the stable and unstable plaques of M are stable and unstable leaves of (X , f ).
Suppose that x, y ∈ M are such that π(x) = π(y), i.e., the R d -orbits x and y are mapped to the same set in X . Then there exist x ∈ x and y ∈ y such that π(x) = π(y), i.e., there exists g ∈ G such that g(x) = y. Then g(x) = y. Since the action of G on M is free, we conclude that G is the group of deck transformations of the splitting π : M −→ X .
Proposition 6.17. If a connected and locally connected Smale space (X , f ) has a splitting with a nilpotent torsion free group of deck transformations, then f has a fixed point. Proof. We argue by induction on the nilpotency class. We know that the statement is true for abelian groups of deck transformations, see Theorem 6.10. The Smale space (X , f ) is a locally connected Smale space with the group of deck transformations G of lower class. Therefore, by the inductive hypothesis, f has a fixed point. Its preimage in X is an f -invariant torus T ⊂ X equal to an orbit of the R d /Z d -action. It follows from the definition of the action of R d on M that T is locally closed with respect to the local product operation [·, ·], hence (T, f ) is a Smale space, and f restricted to this torus is a hyperbolic automorphism, hence it has a fixed point (see Theorem 6.10).
If G is a torsion-free finitely generated nilpotent group, then there exists a unique simply connected nilpotent Lie group L such that G is isomorphic to a co-compact lattice in L, see [Mal49]. Moreover, every automorphism of G is uniquely extended to L. The Lie group L is called the Malcev completion of G.
Let G be a finitely generated torsion free nilpotent group, and let φ be its automorphism. We say that φ : G −→ G is hyperbolic if its unique extension φ : L −→ L to the Malcev completion is hyperbolic, i.e., if the automorphism Dφ of the Lie algebra of L has no eigenvalues on the unit circle.
Proposition 6. 18. Let (X , f ) be a connected and locally connected Smale space with a splitting π : M −→ X with a torsion free nilpotent group of deck transformations G. Let φ be an automorphism of G induced by a lift F of f which has a fixed point in M. Then φ is hyperbolic.
Proof. Let us prove our proposition by induction on the nilpotency class of G. It is true for abelian groups, by Proposition 6.5.
Suppose that we have proved the proposition for all nilpotent groups of class n. Suppose that nilpotency class of G is n + 1. By Proposition 6.16 and the inductive hypothesis, the automorphism of G induced by φ is hyperbolic.
Let x 0 ∈ M be the fixed point of F . Then x 0 ∈ M is a fixed point of F . Image of x 0 in X is an f -invariant torus T , such that (T, f ) is topologically conjugate to a hyperbolic automorphism of the torus.
The map π : x 0 −→ T is its splitting with the group of deck transformations equal to Z(G) = Z d . It follows then from Proposition 6.5 that the restriction of φ onto Z(G) is hyperbolic.
We see that restriction of φ : L −→ L onto Z(L) and the automorphism induced by φ on L/Z(L) both are hyperbolic, hence φ itself is hyperbolic. The space X is a quotient of the Cantor set under a finite-to-one map (see [Bow70,Fri87]) with an upper bound on the cardinality of its fibers. It follows then from Hurewicz formula [Kur61] that X has finite topological dimension.
By a theorem of Alexandroff [Ale29], X is homeomorphic to an inverse limit of simplicial complexes, which are nerves of finite open coverings of X . We can make the elements of the coverings sufficiently small, so that they can be lifted to a G-invariant covering of M. It follows that M is an inverse limit of a sequence of simplicial complexes with G-actions and G-equivariant maps between them. In particular, there exists a G-equivariant map A from M to a simplicial complex ∆ with a G action on it. Since L is homeomorphic to R n , there exists a G-equivariant map B : ∆ −→ L. Composition h 0 = B • A is then a G-equivariant map from M to L.
Let us show now that there exists a G-equivariant map h : M −→ L such that φ • h = h • F . We will use the arguments of [Fra70, Theorem 2.2], which we repeat here for sake of completeness, and since our setting is slightly different.
Consider the space Q of all continuous maps γ : X −→ L such that γ(π(x 0 )) = 1 with topology of uniform convergence on X . It is a nilpotent group (of the same class as L) with respect to pointwise multiplication. Define Φ 0 (γ) = φ −1 • γ • f . Then Φ 0 is a continuous automorphism of the group Q.
Let L be the Lie algebra of L, and let exp : L −→ L be the exponential map. It is a diffeomorphism, since L is simply connected and nilpotent. Let Q be the Banach space of continuous maps X −→ L mapping π(x 0 ) to zero. Then Log : γ → exp −1 •γ is a homeomorphism of Q with Q.
Let T 0 : Q −→ Q be defined by T 0 (γ) = Φ 0 (γ)γ −1 . Let us show that T 0 is a homeomorphism. We show at first that it is a local homeomorphism at the identity (i.e., the constant map x → 1), using the homeomorphism Log : Q −→ Q and computing the derivative of T = Log
•T 0 • Log −1 . Denote Φ = Log •Φ 0 • Log −1 .
We have exp •dφ = φ • exp, where dφ is the derivative of φ : L −→ L at the identity. It follows that dφ −1 • exp −1 = exp −1 •φ −1 , and for every γ ∈ Q we have
Φ(γ) = Log •Φ 0 • Log −1 (γ) = exp −1 •φ −1 • exp •γ • f = dφ −1 • exp −1 • exp •γ • f = dφ −1 • γ • f.
It follows that Φ : Q −→ Q is linear.
hence Φ 0 (γ −1 2 γ 1 ) = γ −1 2 γ 1 . But then γ 1 = γ 2 , as the identity is the only fixed point of Φ 0 .
We have proved that T 0 : Q −→ Q is a homeomorphism. Let h 0 : M −→ L be any G-equivariant map. Consider the map
γ(x) → (φ −1 • h 0 • F (x)) −1 · h 0 (x).
It is easy to see that for every g ∈ G we have γ(g(x)) = γ(x), i.e., γ is constant on G-orbits, hence it descends to a continuous map γ : X −→ L, which is an element of Q. There exists γ ′ ∈ Q such that T (γ ′ ) = γ. Then
F 0 (γ ′ )(γ ′ ) −1 = γ, which means that γ ′ (f (x)) = φ(γ(x)) · φ(γ ′ (x)).
Then the map h(x) = h 0 (x) · γ ′ (π(x)) is G-equivariant, and
h(F (x)) = h 0 (F (x)) · γ ′ (f (π(x))) = h 0 (F (x)) · φ(γ(x)) · φ(γ ′ (π(x))) = h 0 (F (x))h 0 (F (x)) −1 φ(h 0 (x))φ(γ ′ (π(x))) = φ(h 0 (x)γ ′ (π(x))) = φ(h(x)),
which finishes the proof of the proposition.
Theorem 5.11 shows now that (X , f ) and (G\L, f L ) are topologically conjugate.
Let us finish the proof of Theorem 6.1. Let (X , f ) be a connected and locally connected Smale space, and let π : M −→ X be its splitting with a virtually nilpotent group of deck transformations G. Let F be a lift of f to M. Let φ be the automorphism induced by F on G.
Every finitely generated virtually nilpotent group G contains a torsion free nilpotent subgroup G 0 of finite index (see [KM79,17.2.2]). For every g ∈ G and n ∈ Z the subgroup g −1 φ n (G 0 )g has the same index in G as G 0 . There exists only a finite number of subgroups of given index in a finitely generated group. Taking then intersection of all subgroups of the form g −1 φ n (G 0 )g for n ∈ Z and g ∈ G, we get a normal φ-invariant torsion free nilpotent subgroup G 1 of finite index in G. It will be finitely generated as a finite index subgroup of a finitely generated group.
Then G 1 \M together with the map f 1 induced by F is a Smale space. It is a finite covering of X , and its group of deck transformations is G 1 . Then, by Proposition 6.17, f 1 has a fixed point, hence we may assume that F has a fixed point x 0 . We assume then that φ : G −→ G is given by φ(g)(x 0 ) = F (g(x 0 )).
Let Consider the action of G on L obtained by conjugating by Φ the action of G on M. The action of its subgroup G 1 ≤ G will coincide then with the natural action of G 1 ≤ L on L by left multiplication.
The Smale space (X , f ) is then topologically conjugate to the homeomorphism induced by φ on G\L.
Proposition 6. 21. The group G acts on L by affine transformations.
Proof. The action of G 1 ≤ G on L coincides with the natural left action of G 1 on L as a subgroup of L. The action of G on G 1 by conjugation can be uniquely extended to an action of G on L by automorphism. Denote by α g (h) for g ∈ G and h ∈ L the image of h under the automorphism of L equal to the extension of the automorphism h → ghg −1 of G 1 .
Let g ∈ G. Then a g = g(1) = Φ −1 (g(x 0 )) is an element of L. Let us prove that the action of g on L is given by the formula
g(x) = α g (x) · a g .
Consider the map A g (x) = a −1 g α g (x)a g : L −→ L. Note that if h ∈ G 1 , then A hg = A g , since a hg = hg(1) = ha g and α hg (x) = hα g (x)h −1 . It follows that there is a finite number of possibilities for A g , since G 1 has finite index in G. Note also that φ(a g ) = φ(g(1)) = φ(g)(1) and φ(α g (x)) = α φ(g) (φ(x)), so that φ(A g (x)) = A φ(g) (φ(x)).
Suppose that x ∈ L + . Then φ n (x) → 1 as n → ∞. Since the set of possible maps of the form A φ n (g) is finite and they are continuous, we have φ n (A g (x)) = A φ n (g) (φ n (x)) → 1, i.e., A g (x) ∈ L + . Consequently, the maps A g preserve L + .
If x, y belong to one stable plaque, then x −1 y ∈ L + , hence
A g (x −1 y) = (α g (x)a g ) −1 (α g (y)a g ) ∈ L + .
Consequently, the affine map x → α g (x)a g preserves the stable plaques of L. It is proved in the same way that it preserves the unstable plaques, hence it preserves the local product structure. Let h ∈ G 1 . Then g(hL + ) = ghg −1 (g(L + )) = α g (h)a g L + , since g(L + ) is the stable plaque a g L + of the point a g = g(1). By the same argument, g(hL − ) = α g (h)a g L − for all h ∈ G 1 .
Let R ⊂ L be a relatively compact open rectangle such that G 1 \R = G 1 \L and 1 ∈ R. Then for every x ∈ L − and every n ∈ N there exists g n ∈ G 1 such that x ∈ φ n (g n R). Note that then distance from x to φ n (g n L + ) is exponentially decreasing with n. It follows that the union of the stable plaques of the form hL + for h ∈ G 1 is dense in L. Similarly, the union of the unstable plaques of the form hL − for h ∈ G 1 is also dense in L.
The actions of the maps x → g(x) and x → α g (x)a g on the stable and unstable plaques of the form hL ± for h ∈ G 1 coincide. Both maps are continuous on L and preserve the direct product structure, hence they are equal.
This finishes the proof of Theorem 6.1.
7.
Smale spaces with pinched spectrum 7.1. Splitting.
Definition 7.1. Let (X , f ) be a Smale space such that X is connected and locally connected. Let a 0 , a 1 be the stable lower and upper critical exponents, and let b 0 , b 1 be the unstable lower and upper critical exponents.
We say that the Smale space has pinched spectrum if
a 0 a 1 + b 0 b 1 > 1,
Theorem 7.1. A Smale space with pinched spectrum is splittable.
Proof. Choose numbers α 0 , α 1 , β 0 , β 1 such that 0 < α < a 0 ≤ a 1 < α 1 , 0 < β 0 < b 0 ≤ b 1 < β 1 , and α 0 α 1 + β 0 β 1 > 1.
Let d + and d − be metrics associated with the internal log-scales ℓ + and ℓ − on the stable and unstable leaves of exponents α 0 and β 0 , respectively. All distances inside the leaves will be measured using these metrics.
Let R be a finite covering of X by open rectangles. We assume (making the rectangles small enough) that the holonomies inside the rectangles R ∈ R are bi-Lipschitz. Then it follows from equalities (3) and (4) in Section 2 that the holonomies inside f k (R) for all R ∈ R and k ∈ Z are bi-Lipschitz with a common Lipschitz constant L > 1.
There exists ǫ > 0 such that for any x ∈ X there exists a rectangle R ∈ R such that the ǫ-neighborhood of x in W + (x) (with respect to d + ) and the ǫ-neighborhood of x in W − (x) (with respect to d − ) are contained in R. Then, for some constant c > 0 and for every k ∈ Z, x ∈ X , there exists R ∈ R such that the ce −α0kneighborhood of x in W + (x) is contained in f k (R), and the ce α0k -neighborhood of
x in W − (x) is contained in f k (R).
Choose x 0 ∈ X . Let us construct a splitting π x0 : W + (x 0 ) × W − (x 0 ) −→ X . Let x ∈ W + (x 0 ), and let n be a positive integer. Since α 1 is a stable upper exponent, there exists a sequence x 0 , x 1 , . . . , x m1 = x of points of W + (x 0 ) such that m 1 ≤ C 1 e α1(n−ℓ−(x0,x)) , and ℓ + (x i , x i+1 ) ≥ n, for some constant C 1 .
Passing to d + , we get that
m 1 ≤ C 2 d + (x 0 , x) α1/α0 e α1n , d + (x i , x i+1 ) ≤ C 3 e −α0n .
For every k ∈ Z there exist rectangles R i,1 ∈ R such that the ce −α0k -neighborhood of x i in W + (x i ) = W + (x 0 ) and the ce β0k -neighborhood of x i in W − (x i ) belong to f k (R i,1 ). If ce −α0k > C 3 e −α0n , then R i,1 contains x i−1 and x i+1 . The last inequality is equivalent to k ≤ n − r 1 for some constant r 1 ∈ Z. Choose k 1 = n − r 1 , and find a sequence of rectangles R i,1 satisfying the above conditions for k = k 1 .
Let y ∈ W − (x 0 ) be such that d − (x 0 , y) ≤ ce α0k1 = C 4 e α0n (where C 4 = ce −α0r1 ). Denote z 0 = y, z 1 = [x 1 , z 0 ] f k 1 (R0,1) , z 2 = [x 2 , z 1 ] f k 1 (R1,1) , e.t.c.. If all points z 0 , . . . , z m1 are defined, then we say that y can be continued to x, and denote π x0 (x, y) = z m1 . Note that x 0 can be continued to x and π x0 (x 0 , x) = x.
If y can be continued to x, then, in the above notation,
d + (z i , z i+1 ) ≤ Ld + (x i , x i+1 ) ≤ C 5 e −α0n ,hence d + (y, π x0 (x, y)) ≤ C 5 m 1 e −α0n ≤ C 6 d + (x 0 , x) α1/α0 e (α1−α0)n .
Let k 2 be such that (14) ce α0k2 ≥ C 6 d + (x 0 , x) α1/α0 e (α1−α0)n , so that for every point z ∈ X there exists a rectangle R ∈ R such that the C 6 d + (x 0 , x) α1/α0 e (α1−α0)n -neighborhood of z in W + (z) is contained in f −k2 (R), and the ce −β0k2 -neighborhood of z in W − (z) is also contained in f −k2 (R). Inequality (14) follows from an inequality
k 2 ≥ α 1 α 2 0 log d + (x 0 , x) + α 1 − α 0 α 0 n + s
for some constant s. Consequently, we can take
(15) k 2 = α 1 α 2 0 log d + (x 0 , x) + α 1 − α 0 α 0 n + s 1 ,
where s 1 > 0 is bounded from above. Let n 2 be such that ℓ − (z 1 , z 2 ) ≥ n 2 implies d − (z 1 , z 2 ) ≤ ce −β0k2 . There exists a constant r 2 (not depending on k 2 ) such that we can take n 2 = k 2 + r 2 . Let y ∈ W − (x 0 ). There exists a sequence y 0 = x 0 , y 1 , y 2 , . . . , y m2 = y such that
m 2 ≤ C 7 d − (y, x 0 ) β1/β0 e β1n2 , d − (y i , y i+1 ) ≤ ce −β0n2 ≤ ce −β0k2 .
Choose a sequence of rectangles R i,2 ∈ R such that the ce −β0n2 -neighborhood of y i in W − (y i ) = W − (x 0 ), and the ce α0n2 -neighborhood of y i in W + (y i ) are contained in f −n2 (R).
Suppose that y i can be continued to x. Then d + (y i , π x0 (x, y i )) ≤ ce α0k2 ≤ ce α0n2 , hence π x0 (x, y i ) ∈ f −n2 (R i,2 ). Define then a sequence z 0,i = y i+1 , z 1,i = [x 1 , z 0,i ] f k 1 (R0) , z 2,i = [x 2 , z 1,i ] f k 1 (R1) , e.t.c.. Each of the points z j,i will be defined, provided d − (z j−1,i , x j−1 ) ≤ ce β0k1 . We have an estimate
d − (z j−1,i , x j−1 ) ≤ C 8 (d − (x 0 , y 1 ) + d − (y 1 , y 2 ) + · · · + d − (y j−2 , y j−1 )) ≤ C 8 m 2 ce −β0k2 ≤ C 9 d − (y, x 0 ) β1/β0 e (β1−β0)k2 .
(We used that n 2 = k 2 + r 1 for some constant r 1 .)
It follows that y can be continued to x if
C 9 d − (y, x 0 ) β1/β0 e (β1−β0)k2 ≤ ce β0k1 , i.e., if (β 1 − β 0 )k 2 + β 1 β 0 log d − (y, x 0 ) + s 2 ≤ β 0 n for some constant s 2 .
Replacing k 2 by the value given in (15), we get that y can be continued to x if
(16) (β 1 − β 0 ) α 1 − α 0 α 0 n + α 1 α 2 0 log d + (x 0 , x) + β 1 β 0 log d − (y, x 0 ) + s 3 ≤ β 0 n for some constant s 3 . If (17) (β 1 − β 0 )(α 1 − α 0 ) α 0 < β 0 ,
then taking n big enough, we can guarantee that inequality (16) is satisfied. Inequality (17) is equivalent to
(β 1 − β 0 )(α 1 − α 0 ) < α 0 β 0 ,
i.e., to
β 1 α 1 < β 0 α 1 + β 1 α 0 ⇐⇒ α 0 α 1 + β 0 β 1 > 1.
It follows that if the Smale space has pinched spectrum, then every point y ∈ W − (x 0 ) can be continued to every x ∈ W + (x 0 ), and we can define π x0 : W + (x 0 ) × W − (x 0 ) using the rules described above.
Let us show that the map π x0 : W + (x 0 ) × W − (x 0 ) −→ X is well defined, i.e., does not depend on the choice of the rectangles R i,1 (we did not use the rectangles R i,2 in the definition of π x0 ).
It follows from the construction that the map y → π x0 (x, y) is equal to composition of holonomy maps of a sequence of rectangles R i,1 ∈ f k1 (R) for some positive k 1 .
We also showed that for every y, the germ of the map y → π x0 (x, y) is equal to a germ of a holonomy in a rectangle R j,1 ∈ f −n2 (R) for some positive n 2 .
Note also that given such a sequence R i,1 ∈ f k1 (R) we can find a sequence R ′ i ∈ f m (R) such that m is arbitrarily big and the map y → π x0 (x, y) defined by the original sequence R 1,i is a restriction of the maps defined by the new sequence R ′ i . Suppose that h i : W − (x 0 ) −→ W − (x) for i = 1, 2 are compositions of holonomies defined using two sequences R 1,i , and R ′ 1,i . We may assume that both sequences belong to f k1 (R) for some fixed k 1 . Let y belongs to the domain of both maps h i . We may assume (taking k 1 big enough) that x and y belong to connected components of the domains of h i . Then there exists a connected chain of rectangles R 2,i ∈ f −n2 (R), for some n 2 > 0, such that restrictions of h i onto the corresponding plaques of R 2,i are equal to holonomies in R 2,i . It follows then from h 1 (x 0 ) = h 2 (x 0 ) = x that h 1 (y) = h 2 (y). Consequently, π x0 is well defined. The map π x0 is obviously a local homeomorphism.
Note that if R ∈ R is such that x 0 ∈ R, then π x0 : P + (R, x 0 ) × P − (R, x 0 ) −→ R coincides with [·, ·] R .
Let (a, b) ∈ W + (x 0 ) × W − (x 0 ) be an arbitrary point, and let x 1 = π x0 (a, b). Then it follows from the definition of the maps π xi and their uniqueness that π x0 (x, y) = π x1 (π x0 (x, b), π x0 (a, y)).
It follows that π x0 is onto, since its range contains every rectangle R ∈ R intersecting it. It also follows that the map π x0 is a covering, since every point of W + (x 0 ) × W − (x 0 ) has a neighborhood mapped homeomorphically by π x0 to an element of R.
Another corollary of (18) is that π x0 homeomorphically maps plaques of W + (x 0 )× W − (x 0 ) to leaves of X , since π x1 maps the direct factors of W + (x 1 ) × W − (x 1 ) identically onto the leaves W + (x 1 ) and W − (x 1 ). This finishes the proof of the theorem.
Polynomial growth.
Theorem 7.2. Let (X , f ) be a Smale space with pinched spectrum. Then the group of deck transformations of the splitting of (X , f ) has polynomial growth.
Proof. Our proof essentially repeats the proof of the main theorem of [Bri78]. Let R be a finite covering of X by open connected rectangles. Let π : M −→ X be the splitting constructed in Theorem 7.1, where M = W + (x 0 ) × W − (x 0 ). Denote by R the union of the sets of connected components of π −1 (R) for R ∈ R.
Consider the graph Γ with the set of vertices identified with R in which two vertices are connected if the corresponding sets have non-empty intersection. It is easy to show (see the proof of Proposition 5.2) that the graph Ξ is quasi-isometric to the Cayley graph of the group G of deck transformations of π and has the same growth rate as G.
Let B(r) be the set of elements of R that are on distance at most r in Γ from a vertex R ∈ R such that (x 0 , x 0 ) ∈ R.
Let 0 < α 0 < a 0 ≤ a 1 < α 1 and 0 < β 0 < b 0 ≤ b 1 < β 1 such that α 0 /α 1 +β 0 /β 1 > 1. Denote by d + and d − the metrics of exponents α 0 and β 0 on the corresponding leaves of X and plaques of W + (x 0 ) × W − (x 0 ). (Every plaque of M is identified with a leaf of X by π.)
Take R ∈ B(r). Choose a sequence R 0 ∋ (x 0 , x 0 ), R 1 , . . . , R m = R of elements of R forming a chain in Γ of length m ≤ r.
We will denote by [·, ·] the direct product structure on W + (x 0 ) × W − (x 0 ). Let D − and D + be the suprema of the d − -and d + -diameters of the sets [y, ∪R i ] and [∪R i , y] for all y ∈ ∪R i (see Figure 5).
There exists constants C 1 , C 2 not depending on r, and a number n = n(r) such that |n − log D − /β 0 | < C 1 , and for every point y ∈ M there exists a rectangle V ∈ f n (R) such that the C 2 e −α0n -neighborhood of y in the stable plaque of y (with respect to d + ) and the set [y, ∪R i ] are both contained in V .
It follows that we can find a sequence of rectangles V i ∈ f n (R) of length at most C 3 e α1n such that V i ∩ V i+1 = ∅, the first rectangles in the sequence intersects contains a given point of R i−1 ∩ R i , while the last one contains a given point of R i ∩ R i+1 .
Figure 5. Growth estimation
Consequently, we can find a sequence of rectangles V 0 , . . . , V l ∈ f n (R) of length at most C 3 re α1n ≤ C 4 rD α1/β0 − such that V i ∩ V i+1 = ∅, x 0 ∈ V 0 , V m contains a point x ∈ R = R m , and for every V i there exists a point z i ∈ V i such that [z i , ∪R i ] ⊂ V i . Moreover, we may assume that the chain V i covers any given in advance three point y 1 , y 2 , y 3 ∈ ∪R i . It follows that the d + -distance from [y 1 , y 2 ] to [y 3 , y 2 ] is bounded from above by Note that α1 α0 + β1 β0 − α1β1 α0β0 = α1β1 α0β0 α0 α1 + β0 β1 − 1 > 0, hence D + ≤ C 8 r p+ for p + = 1 + α1−α0 In particular (taking C 10 = max(C 8 , C 9 ) and p = max(p + , p − )) we have that d − ([x 0 , x], x 0 ) and d + ([x, x 0 ], x 0 ) are less than C 10 r p for any x ∈ R∈B(r) R.
Let µ + and µ − be the SRB measures satisfying the conditions of Theorem 3. 6. Let µ be their direct product on M = W + (x 0 ) × W − (x 0 ). Since the measures µ + and µ − on the leaves of X are invariant under holonomies, G acts by measure preserving transformations on M. It follows that there exist positive constants A 1 and A 2 such that
A 1 |B(r)| ≤ µ R∈B(r) R ≤ A 2 |B(r)|.
By the proven above, the set R∈B(r) R is contained in the direct product of the balls of radius C 10 r p with center in x 0 in W + (x 0 ) and W − (x 0 ). By condition (1) of Theorem 3.6 volumes of these balls are bounded from above by C(C 10 r p ) η/α0 and C(C 10 r p ) η/β0 for some constant C. It follows that |B(r)| is bounded above by a polynomial in r.
By the Gromov's theorem on groups of polynomial growth [Gro81], G is virtually nilpotent. Theorem 6.1 now implies the following description of Smale spaces with pinched spectrum.
Theorem 7.3. Every connected and locally connected Smale space with pinched spectrum is topologically conjugate to an infra-nilmanifold automorphism.
Mather spectrum of Anosov diffeomorphisms
Let f : X −→ X be a diffeomorphism of a compact Riemann manifold X . It induces a linear operator f * on the Banach space of continuous vector fields by f * ( X)(x) = Df • X(f −1 (x)).
By a theorem of J. Mather [Mat68], f is an Anosov diffeomorphism if and only if the spectrum of f + does not intersect with the unit circle. It belongs then to the set {z ∈ C : λ 1 < |z| < λ 2 } ∪ {z ∈ C : µ 2 < |z| < µ 1 }, where 0 < λ 1 < λ 2 < 1 < µ 2 < µ 1 . The tangent bundle T X is decomposed then into a direct sum W s ⊕ W u such that there exists a constant C > 1 such that for all vectors v + ∈ W s , v − ∈ W u , and for every positive integer n we have C −1 λ n 1 v + ≤ Df n v + ≤ Cλ n 2 v + and C −1 µ n 2 v − ≤ Df n v − ≤ Cµ n 1 v − . For more detail, see [Mat68,Bri77,Bri78]. Every Anosov diffeomorphism is a Smale space, see [BS02, Proposition 5. 10.1]. Stable and unstable leaves of (X , f ) are manifolds, and the vectors of W s and W u are tangent to the stable and unstable leaves, respectively.
Proposition 8.1. Let λ 1 , λ 2 , µ 1 , µ 2 be as above. Then log µ 2 and log µ 1 are unstable lower and upper exponents of (X , f ), and − log λ 2 and − log λ 1 are stable lower and upper exponents of (X , f ).
Proof.
It is easy to show that for every fixed k 0 the metrics d k0 on leaves W + (x 0 ) and W − (x 0 ) are quasi-isometric to the restrictions of the Riemannian metric of X onto W + (x 0 ) and W − (x 0 ), with the quasi-isometry constants depending only on k 0 (and (X , f )). For every stable leaf W + of (X , f ) and all x, y ∈ W + , n ∈ Z, d 0 (f −n (x), f −n (y)) = d n (x, y).
It follows that there exist constants C 1 > 1, ∆ > 0 such that
C −1 1 d + (f −n (x)
, f −n (y)) − ∆ ≤ d n (x, y) ≤ C 1 d + (f −n (x), f −n (y)) + ∆, where d + is the Riemannian metrics on the stable leaves.
If γ is a curve in the stable leaf connecting f −n (x) to f −n (y), then f n (γ) is a curve connecting x to y, and length(f n (γ)) = d dt f n • γ(t) dt = Df n • d dt γ(t) dt, hence C −1 λ n 1 length(γ) ≤ length(f n (γ)) ≤ Cλ n 2 length(γ), and C −1 λ n 1 · d + (f −n (x), f −n (y)) ≤ d + (x, y) ≤ Cλ n 2 · d + (f −n (x), f −n (y)). It follows that C −1 1 C −1 d + (x, y) · λ −n 2 − ∆ ≤ d n (x, y) ≤ C 1 C d + (x, y) · λ −n 1 + ∆ for all stably equivalent x, y and all positive n. Then, by Propositions 3.5 and 4.11, − log λ 1 and − log λ 2 are upper and lower exponents.
The case of unstable leaves is proved in the same way. [Bri77,Bri78,Bri80,BM81] considers Anosov diffeomorphisms such that either (18) 1 + log µ 2 log µ 1 > log λ 1 log λ 2 or (19) 1 + log λ 2 log λ 1 > log µ 1 log µ 2 .
M. Brin considers in
Note that since log µ1 log µ2 and log λ1 log λ2 are both greater than one, each of the inequalities (18) and (19) imply (20) log λ 2 log λ 1 + log µ 2 log µ 1 > 1.
For instance, in the case of (18):
log λ 1 log λ 2 + log µ 1 log µ 2 > 1 + log µ 1 log µ 2 = log µ 1 log µ 2 log µ 2 log µ 1 + 1 > log µ 1 log µ 2 · log λ 1 log λ 2 .
Multiplying by log µ2 log µ1 · log λ2 log λ1 , we get (20). Note that if a 0 and a 1 are stable lower and upper critical exponents, and b 0 and b 1 are unstable lower and upper critical exponents of the Smale space, then − log λ 2 ≤ a 0 , and − log λ 1 ≥ a 1 , log µ 2 ≤ b 0 , and log µ 1 ≥ b 1 , so that log µ 2 log µ 1 ≤ a 0 a 1 , log λ 2 log λ 1 ≤ b 0 b 1 , and therefore a 0 a 1 + b 0 b 1 ≥ log µ 2 log µ 1 + log λ 2 log λ 1 .
hence k − 1 ≤ e ηn d(x, y), which implies that η is also an upper stable exponent. It follows that η is equal to stable lower and upper critical exponents, which shows that the Smale space has pinched spectrum.
The group G of deck transformations of a splitting of X acts on a stable leaf W + (x 0 ) ∼ = R by transformations x → [x, x 0 ]. This action preserves the measure µ + . Let us identify (W + (x 0 ), µ + ) and R with the Lebesgue measure by a measurepreserving homeomorphism. Since G acts by measure-preserving transformations, the corresponding action of G on R is by transformations of the form x → ±x + a for a ∈ R.
The action of G on W + (x 0 ) is free, since otherwise an unstable leaf is not mapped homeomorphically onto its image in X . But this implies that G acts on R by translations, hence is torsion-free abelian. Therefore, by Proposition 5.2 and Theorem 6.10, (X , f ) is topologically conjugate to a hyperbolic automorphism of the torus R d /Z d for some d.
4. 1 .
1Connectivity. The aim of this section is to prove the following description of locally connected Smale spaces.
Theorem 4 . 1 .
41Let (X , f ) be a Smale space. The following conditions are equivalent.
F : M −→ M be a lift of f .
Definition 5 . 3 .
53Let G be the group of deck transformations of the splitting π : M −→ X , and let F : M −→ M be a lift of f with a fixed point x 0 . A set Σ ⊂ G is a coarse stable (resp. unstable) plaque if the stable plaque W + (x 0 ) (resp. unstable plaque W − (x 0 )) and the set Σ(x 0 ) are on finite Hausdorff distance from each other.
Theorem 5 . 10 .
510Let (X , f ) be a connected and locally connected Smale space. Suppose that there exists a splitting π : M −→ X and a lift F : M −→ M of f with a fixed point x 0 . Let φ be the associated automorphism of G, and let Σ + and Σ − be coarse stable and unstable plaques of x 0 . Then (X , f ) is uniquely determined, up to topological conjugacy by the quadruple (G, φ, Σ + , Σ − ). Proof. The dynamical system (X , f ) is uniquely determined by the G-space M and the map F : M −→ M. The group G acts on the plaques W + and W − by the actions g : x → [g(x), x], g : x → [x, g(x)], respectively. The action of G on M ∼ = W + ×W − is reconstructed from these actions by the formula g(x) = [[g(x), x], [x, g(x)]]. Similarly, the map F : M −→ M is determined by the action of F on W + and W − , since F ([x, y]) = [F (x), F (y)]. Consequently, it is enough to show that the quadruple (G, φ, Σ + , Σ − ) uniquely determines the dynamical systems (W + , G), (W + , F ), (W − , G), and (W − , F ), up to topological conjugacy. Let us prove that the triple (G, φ, Σ + ) uniquely determines the dynamical systems (W + , G) and (W + , F ). The same proof will show that (G, φ, Σ − ) uniquely determines (W − , G) and (W − , F ). Let R ⊂ M be a relatively compact open rectangle such that x 0 ∈ R and g∈G g(R) = M.
Figure 2 .
2Figure 2.
Figure 2 .
2Let R ⊂ M be a compact rectangle such that π(R) = X . Let D be an upper bound on the value of d − (x, [x, y]
(
12) δ(x, y) ≍ δ(x, [x, y]) + δ(y, [x, y])
Figure 3 .
3Figure 3.
Theorem 6 . 19 .
619Let (X , f ) be a connected and locally connected Smale space with a splitting π : M −→ X with a torsion free nilpotent group of deck transformations G. Let φ be an automorphism of G induced by a lift of f . Let f L : G\L −→ G\L be the diffeomorphism induced by φ. Then (X , f ) and (G\L, f L ) are topologically conjugate. Proof. Let F be a lift of f to M with a fixed point x 0 . Extend the automorphism φ : G −→ G to an automorphism φ : L −→ L of the Lie group. Proposition 6.20. There exists a G-equivariant map h : M −→ L such that φ • h = h • F and h(x 0 ) = 1. Proof. Let us show at first that there exists a G-equivariant map h 0 : M −→ L.
L be the Malcev completion of G 1 . Extend φ : G 1 −→ G 1 to an automorphism φ : L −→ L. Then, by Theorem 6.19, there exists a homeomorphism Φ : L −→ M conjugating the actions of G 1 on M and L, and such thatΦ • F = φ • L. Note that Φ(1) = x 0 , Φ(L + ) = W + , and Φ(L − ) = W − , where L + = W + (1), L − = W − (1)are the stable and unstable plaques of the identity element of L, and W + = W + (x 0 ), W − = W − (x 0 ) are the stable and unstable plaques of x 0 . Note that L + = {g ∈ L : lim n→+∞ φ n (g) = 1} and L − = {g ∈ L : lim n→−∞ φ n (g) = 1} are closed subgroups of L (they are closed since they are plaques of a splitting). Stable plaques of L are the left cosets of L + ; unstable plaques of L are the left cosets of L − .
Figure 4 .
4Splitting
Acknowledgments. I started writing this paper during a visit to Institut Mittag-Leffler (Djursholm, Sweden) as a participant of the semester "Geometric and Analytic Aspects of Group Theory". I am very grateful to the Institute for excellent conditions for work and to the organizers of the semester for inviting me.This paper is based upon work supported by the National Science Foundation under grant DMS1006280.For every γ ∈ Q, we have T (γ) = Log •T 0 • Log −1 (γ) = exp −1 (T 0 (exp •γ)) = exp −1 (Φ 0 (exp •γ) · (exp •γ) −1 ) = exp −1 (exp •Φ(γ) · (exp •γ) −1 ) = Log(Log −1 (Φ(γ)) · Log −1 (γ) −1 ) = Log(Log −1 (Φ(γ)) · Log −1 (−γ)), since (Log −1 (γ(x))) −1 = (exp(γ(x))) −1 = exp(−γ(x)) for all x ∈ X .Let us compute the derivative of T at zero. If γ ∈ Q, then lim t→0 1 t T (tγ) = limby the Campbell-Hausdorf formula.Since dφ is hyperbolic, there exist a direct sum decomposition L = L + ⊕ L − and constants C > 0 and 0 < λ < 1 such that dφ n (v) ≤ Cλ n v for all n ≥ 0 and v ∈ L + , and dφ −n (v) ≤ Cλ n v for all n ≥ 0 and v ∈ L − . DefineThis shows that T is a local homeomorphism at zero. Consequently, T 0 is a local homeomorphism at the identity of Q. Let us show that T 0 is surjective. Let Z 1 (Q) = Z(Q) ⊂ Z 2 (Q) ⊂ · · · ⊂ Z n (Q) = Q be the upper central series of Q. Let us prove by induction on i that T 0 (Q) ⊃ Z i (Q). It is easy to see that T 0 (γ 1 γ 2 ) = T 0 (γ 1 )T 0 (γ 2 ) for all γ 1 , γ 2 ∈ Z(Q). Since T 0 is a local homeomorphism at the identity and Z(Q) is generated by any neighborhood of the identity (as any connected topological group, see[Pon66,Theorem 15,p. 76], this implies thatis the commutator, and has nothing to do with the direct product decomposition). By the inductive hypothesis, there exists γ 3 ∈ Q such that γ ′ = T 0 (γ 3 ). Thenis connected, hence generated by any neighborhood of the identity, it follows that Z i+1 (Q) ⊂ T (Q).Since Φ − I is invertible, the only fixed point of Φ is 0. Consequently, the only fixed point of Φ 0 is the unit of Q. If T (γ 1 ) = T (γ 2 ), then Φ 0 (γ 1 )γ −1Consequently, each of the conditions (18), (19) implies the inequalitywhich is the condition of Theorems 7.1 and 7.2. In particular, we conclude that Theorem 7.3 is a generalization of the results of[BM81].Co-dimension one Smale spacesWe say that a Smale space is of co-dimension one if its stable or unstable leaves are homeomorphic to R with respect to their intrinsic topology.It was proved by Franks [Fra70, Theorem 6.3] that every co-dimension one Anosov diffeomorphism is a linear Anosov diffeomorphism of a torus. Here we prove this statement for all locally connected and connected Smale spaces.Theorem 9.1. A locally connected and connected co-dimension one Smale space is topologically conjugate to a co-dimension one hyperbolic automorphism of a torus R d /Z d .Proof. Let us show that every co-dimension one locally connected and connected Smale space has pinched spectrum. In fact, we will show that the upper and lower critical exponents of the leaves homeomorphic to R coincide. This will obviously imply that a0 a1 + b0 b1 > 1. Consider the case when the stable leaves are homeomorphic to R. Let µ + be the measure on stable leaves described in Theorem 3.6.Let W + be a stable leaf, and let x, y ∈ W + . Define d + (x, y) = µ + ((x, y)), where (x, y) is the interval with endpoints x and y. Then d + (x, y) is a metric on W + . It is associated with the standard log-scale ℓ + , by Theorem 3.6.Let R be a finite covering of X by open rectangles such that their projections onto the stable direction are intervals. Note that for any two plaques of a rectangle R ∈ R are intervals of the same µ + -measure. Let M be the maximum of measures of the stable plaques of elements of R.Define d ′ n (x, y) for x, y ∈ W + to be the smallest length k of a chain I 0 , I 1 , . . . , I k of plaques of elements of R such that f n (I i ) ⊂ W + for every i = 0, . . . , k, x ∈ f n (I 0 ), y ∈ f n (I k ), and the intersection f n (I i ) ∩ f n (I i+1 ) is non-empty for every i = 0, . . . , k − 1. Note that then (x, y) ⊂ k i=0 f n (I i ). It is easy to show that d ′ n is uniformly quasi-isometric to the metrics d n defined by the standard log-scale on stable leaves. Let η be as in Theorem 3.6. Then for every such a chain we havehence k + 1 ≥ M −1 e ηn d(x, y), i.e., η is a lower stable exponent. It follows from the Lebesgue's covering lemma that there exists ǫ > 0 such that for every point z ∈ W + there exists a plaque of an element of R containing the ǫ-neighborhood of z. Let us order the points of W + , identifying it with R. Then for any x, y ∈ W + such that x < y we can find a sequence of plaques I 0 = (a 0 , b 0 ), I 1 = (a 1 , b 1 ), . . . , I k = (a k , b k ) such that x ∈ f n (I 0 ), y ∈ f n (I k ) \ f n (I k−1 ), and I i+1 contains the ǫ-neighborhood of b i . Then b i+1 > b i , and d(b i , b i+1 ) > ǫ, so that d(x, y) = d(x, b 0 ) + d(b 0 , b 1 ) + · · · + d(b k−1 , b k ) + d(b k , y) ≥ (k − 1)ǫe −ηn ,
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| []
|
[
"Randomized FIFO Mechanisms *",
"Randomized FIFO Mechanisms *"
]
| [
"Francisco Castro ",
"Hongyao Ma ",
"Hamid Nazerzadeh ",
"Chiwei Yan "
]
| []
| []
| We study the matching of jobs to workers in a queue, e.g. a ridesharing platform dispatching drivers to pick up riders at an airport. Under FIFO dispatching, the heterogeneity in trip earnings incentivizes drivers to cherry-pick, increasing riders' waiting time for a match and resulting in a loss of efficiency and reliability. We first present the direct FIFO mechanism, which offers lower-earning trips to drivers further down the queue. The option to skip the rest of the line incentivizes drivers to accept all dispatches, but the mechanism would be considered unfair since drivers closer to the head of the queue may have lower priority for trips to certain destinations. To avoid the use of unfair dispatch rules, we introduce a family of randomized FIFO mechanisms, which send declined trips gradually down the queue in a randomized manner. We prove that a randomized FIFO mechanism achieves the first best throughput and the second best revenue in equilibrium. Extensive counterfactual simulations using data from the City of Chicago demonstrate substantial improvements of revenue and throughput, highlighting the effectiveness of using waiting times to align incentives and reduce the variability in driver earnings. * The authors would like to thank | 10.1145/3490486.3538353 | [
"https://arxiv.org/pdf/2111.10706v1.pdf"
]
| 244,478,523 | 2111.10706 | 5bac30977b1a6c9985c2c96b5c8ff8afbd741fba |
Randomized FIFO Mechanisms *
November 23, 2021
Francisco Castro
Hongyao Ma
Hamid Nazerzadeh
Chiwei Yan
Randomized FIFO Mechanisms *
November 23, 2021
We study the matching of jobs to workers in a queue, e.g. a ridesharing platform dispatching drivers to pick up riders at an airport. Under FIFO dispatching, the heterogeneity in trip earnings incentivizes drivers to cherry-pick, increasing riders' waiting time for a match and resulting in a loss of efficiency and reliability. We first present the direct FIFO mechanism, which offers lower-earning trips to drivers further down the queue. The option to skip the rest of the line incentivizes drivers to accept all dispatches, but the mechanism would be considered unfair since drivers closer to the head of the queue may have lower priority for trips to certain destinations. To avoid the use of unfair dispatch rules, we introduce a family of randomized FIFO mechanisms, which send declined trips gradually down the queue in a randomized manner. We prove that a randomized FIFO mechanism achieves the first best throughput and the second best revenue in equilibrium. Extensive counterfactual simulations using data from the City of Chicago demonstrate substantial improvements of revenue and throughput, highlighting the effectiveness of using waiting times to align incentives and reduce the variability in driver earnings. * The authors would like to thank
Introduction
Matching marketplaces play an instrumental role in economic exchanges and the allocation of public and private resources. Over the past decade, the rise of online platforms connecting people with gig workers has also radically changed many aspects of our daily lives. To improve efficiency and reduce waiting times, platforms often aim to match rider or grocery delivery trips with the closest available drivers. When requests are concentrated in space, however, matching by proximity has unintended consequences. As an example, Amazon drivers have been reportedly hanging their smartphones in trees near Amazon delivery stations and Whole Foods stores, in order to appear even closer and gain higher priority for job offers. 1 A similar problem existed for Uber and Lyft at airports and event venues. 2 Matching riders to the closest drivers incentivizes drivers to get as close to the terminal or venue as possible, leading to traffic congestion. 3 Many ridesharing platforms now maintain virtual queues at airports for drivers who are waiting in designated areas, and dispatch drivers from the queue in a first-in-first-out (FIFO) manner. 4 This resolves the congestion issues and is also considered more fair by many since drivers who have waited the longest in the queue are now the first in line to receive trip offers. At major U.S. airports, however, a driver at the head of the queue will receive the next trip offer in a few seconds under FIFO dispatching, if she declines an offer from the platform (see Figure 12). As we shall see, this lowered cost of cherry-picking substantially exacerbates existing problems on incentive alignment. Figure 1 shows the average trip fare by destination Census Tract in Chicago, for trips originating from the O'Hare and Midway airports. A short trip from O'Hare to a nearby area pays an average of $10-$20, but a long trip can pay an average of $60. During busy hours, instead of accepting an average trip, drivers who are close to the head of the queue are better off declining most trip offers and waiting for only the highest earning trips. Riders, however, have finite patience, despite being willing to wait for some time for a match. When each driver decline takes an average of 10 seconds, 2 minutes had passed after a trip with low or moderate earnings (e.g. trips to downtown Chicago) was offered to and declined by the top 12 drivers in the queue. 5 At this point, it is very likely that the rider cancels her trip request, not knowing when a driver will be assigned, if at all.
What we have seen is that in the presence of heterogeneous earnings and finite rider patience, trips with moderate or low earnings never reach drivers in the queue who are willing to accept them. This undercuts the platforms' mission of providing reliable transportation for riders, and leads to low revenue and trip throughput for the platform. Moreover, fulfilling only the small number of high earning trips is also a poor outcome for the drivers, since many drivers who just dropped off a rider at the airport will have to relocate back to the city with an empty car, and those who do join the queue would need to wait for a very long time for a ride.
Simple fixes by limiting dispatching transparency or drivers' flexibility are not desirable-in recent years, ridesharing platforms are moving towards sharing trip destination and earnings estimation upfront, as well as providing drivers the options to accept or decline any trips without penalties. 6 Hiding information or imposing penalties are not fully effective either. For example, experienced drivers often call riders to ask about trip details when destinations are hidden before the pick-up [Cook et al., 2018]. Forcing drivers to accept every dispatch improves reliability in the short run, but also imposes a lottery (with possible outcomes ranging from $9 to over $60) on drivers who might have waited for two hours in line. Such high variance in earnings discourages future engagement, and leads to drivers' churning from the platform in the long run.
Recognizing the inefficiencies under FIFO dispatching, alternative mechanisms have been studied extensively in the literature. In particular, last-in-first-out (LIFO) dispatching is shown to be optimal in the presence of waiting costs, with or without heterogeneous rewards [Hassin, 1985, Su andZenios, 2004]. Intuitively, participants' losing (instead of gaining) priority over time substantially reduces the incentive to "wait for a better offer". However, LIFO dispatching is perceived as "blatantly unfair" by many Zenios, 2004, Breinbjerg et al., 2016]. Moreover, as discussed by Hassin [1985] and Su and Zenios [2004], LIFO dispatching is easy to manipulate since participants may rejoin the queue at the end to (re)gain priority. This renders LIFO unsuitable and ineffective for ridesharing platforms, as drivers have the option to go offline and online again to rejoin the end of the virtual queue at any time.
Ideally, platforms may properly price trips by destination and eliminate drivers' incentives to cherry-pick. In recent work, Ma et al. [2019] propose the spatio-temporal pricing mechanism, which is welfare optimal, incentive aligned, and guarantees that drivers at the same origin are indifferent towards trips to all destinations. This remains an idealized target for our current setting, but is hard to achieve in practice. Consider, again, the O'Hare example as shown in Figure 1a. Tripling the fares of the short trips to match the earnings from the long trips is suboptimal. On the other hand, the platform is unable to decrease driver payouts below some pre-determined per-minute and per-mile rates, thus earnings from long trips cannot be effectively reduced either.
Our Results
We study the dispatch of trips to drivers who are waiting in a virtual queue, where some trips are necessarily more lucrative than the others due to operational constraints. Without the power to adjust trip prices, the mechanisms we design use drivers' waiting times in the queue to align incentives, improve reliability and efficiency, and reduce the variability in drivers' total payoffs. The model. We study a continuous time, non-atomic model, with one origin (e.g. the airport) and stationary arrival rates of riders and drivers. Riders request trips to a number of destinations with heterogeneous earnings for drivers. Upon the arrival of each rider, or after a rider's trip request is declined, the platform offers the rider's trip to a driver in the queue. Riders are willing to wait for some time for a match, but have finite patience and will cancel their requests after a certain number of declines from drivers. Drivers' waiting in the queue (as opposed to driving elsewhere in the city, for example) is costly for both the drivers and the platform. Drivers are strategic, aiming to optimize their total payoff, i.e. the earnings from trips minus the waiting costs they incur.
We study mechanisms that are fully transparent and flexible (see Footnote 6). At any point in time, drivers know about the supply, demand, the length of the queue and their positions in their queue. When offered trip requests, drivers are provided trip destinations and earnings upfront so that they can decide whether to accept based on this information. Moreover, drivers are not penalized for any actions they take, and have the flexibility to (i) decline any number of trip dispatches without losing their positions in the queue, (ii) rejoin the virtual queue at the tail at any point of time, and (iii) decide to not join the queue upon arrival, or leave the queue at any point of time to perhaps relocate back to the city without a rider.
Main results. To optimize trip throughput and the platform's net revenue (i.e. total earnings from completed trips, minus the opportunity costs the platform incurs due to drivers' waiting in the queue), the first best outcome has no driver in the queue, and dispatches all drivers upon arrival to destinations in decreasing order of earnings. However, under the status quo strict FIFO dispatching where trips are dispatched to each and every driver starting from the head of the queue, drivers close to the head of the queue are incentivized to cherry-pick and wait for higher-earning trips. We analyze the equilibrium outcome under strict FIFO and show that with finite rider patience, most trips except for the highest earning ones become unfulfilled. Drivers' excessive waiting in the queue further reduces drivers' total payoffs as well as the net revenue of the platform.
Recognizing that the moderate and low earning trips never reach drivers in the queue who are willing to accept them, we first present the direct FIFO mechanism, which offers lower-earning trips directly to drivers further down the queue. We prove that accepting all dispatches forms a subgame perfect equilibrium among drivers, and that the equilibrium outcome achieves the first best trip throughput, and the second best net revenue (i.e. the highest steady state net revenue achievable by any flexible and transparent mechanism). The direct FIFO mechanism, however, would be considered unfair in practice since a driver may have lower priority for trips to many destinations than drivers further down the queue, even when all drivers are non-strategic and accept every dispatch. Consider the Chicago Midway airport (Figure 1b) as an example. A driver close enough to the head of the queue will no longer receive any trip back to downtown Chicago, since direct FIFO skips drivers at the head of the queue when dispatching lower and moderate earning trips, and all high-earning trips the driver may receive will be heading to the suburbs.
To achieve optimal throughput and revenue without the use of an unfair dispatch rule, we introduce a family of randomized FIFO mechanisms. A randomized FIFO mechanism is specified by a set of "bins" in the queue (e.g., the top 10 positions, the 10 th to 20 th positions, and so on). Each trip request is first offered to a driver in the first bin uniformly at random. After each decline, the mechanism then offers the trip to a random driver in the next bin. By sending trips gradually down the queue in this randomized manner, the randomized FIFO mechanisms appropriately align incentives using waiting times, achieving the first best throughput and second best net revenue: the option to skip the rest of the line incentivizes drivers further down the queue to accept trips with lower earnings; randomizing each dispatch among a small group of drivers increases each individual driver's waiting time for the next dispatch, thereby allowing the mechanism to prioritize drivers closer to the head of the queue for trips to every destination without creating incentives for excessive cherry-picking.
Extensive counterfactual simulations using data from the City of Chicago suggest that in comparison to strict FIFO dispatching, the randomized FIFO mechanism achieves substantial improvements in revenue, throughput, and driver earnings. Moreover, the variance in drivers' total payoffs is small, and diminishes rapidly as riders' patience increases-with higher rider patience, the mechanism can more effectively match higher-earning trips with drivers who have incurred higher waiting costs in the queue. This demonstrates the desirable balance achieved by the randomized FIFO mechanisms between efficiency, reliability, fairness, and the variability in driver earnings, and highlights the effectiveness of using waiting times in queue to align incentives and to reduce earning inequity when the flexibility to set prices is limited due to operational constraints.
Related Work
Ridesharing platforms. The literature on pricing and matching in ridesharing platforms is rapidly growing. Castillo et al. [2017] and establish the importance of dynamic pricing in maintaining the spatial density of open driver supply, which reduces waiting times and improves operational efficiency. In the presence of spatial imbalance and temporal variation of supply and demand, Bimpikis et al. [2019] and Besbes et al. [2020] study revenue-optimal pricing; Ma et al. [2019] propose origin-destination based pricing that is appropriately smooth in space and time, achieving welfare optimality and incentive compatibility; Garg and Nazerzadeh [2020] show that additive instead of multiplicative "surge" pricing is more incentive aligned for drivers when prices need to be origin-based only. Considering the online arrival of supply and demand and their distribution in space, Kanoria and Qian [2020], Qin et al. [2020] andÖzkan and Ward [2020] study dynamic matching policies that dispatch drivers from areas with relatively abundant supply, and Ashlagi et al. [2019], Dickerson et al. [2018] and Aouad and Saritaç [2020] focus on the online matching between riders and drivers and the pooling of shared rides. In this work, we focus on a single origin where the optimal destination-based pricing is infeasible due to operation constraints such that some trips are necessarily more lucrative than the others. This leads to the need of using drivers' waiting times to align incentives and to reduce the variability in driver earnings.
The operation of ridesharing platforms is also studied using queueing-theoretic models. Banerjee et al. [2015] compare optimal dynamic and static pricing policies; Banerjee et al. [2018] propose state-dependent dispatching policies to minimize unfulfilled demand; Afeche et al. [2018] study the impact of admission control on platform revenue and driver income; Besbes et al. [2019] show that in comparison to traditional service settings, higher capacity is needed when spatial density of available supply affects operational efficiency; Castro et al. [2020] study practical dispatching policies when drivers have heterogeneous compatibility with trips. These works use queueingtheoretic frameworks to analyze the availability of driver supply, but study settings where drivers are spread out in space, and do not consider cherry-picking by drivers. In contrast, we focus on the matching of trips to drivers who are waiting in a virtual queue, addressing the problem of dispatching heterogeneous trips to drivers who have incurred different waiting costs in a way that is reliable, efficient and fair.
Various empirical studies analyze the Uber platform as a two-sided marketplace, focusing on the labor market of Uber drivers [Hall et al., 2017], the longer-term labor market equilibration [Hall and Krueger, 2016], the value of flexible work arrangements [Chen et al., 2019], learning-by-doing and the gender earnings gap [Cook et al., 2018], and the surplus of consumers [Cohen et al., 2016]. In regard to the dynamic "surge" pricing, Hall et al. [2015], Chen and Sheldon [2015], and Lu et al. [2018] demonstrate its effectiveness in improving reliability and efficiency, increasing driver supply during high-demand times, as well as incentivizing drivers to relocate to higher demand areas. In contrast, we use data from ridesharing platforms (including Uber and Lyft, made public by the City of Chicago) to estimate the heterogeneity in driver earnings by trip destination. We also demonstrate via counterfactual simulations the inefficiencies of FIFO dispatching when drivers are strategic, as well as the substantial improvements achieved by our proposed mechanisms.
Queueing mechanisms. The allocation of resources or jobs to participants waiting in a queue has been studied extensively in the literature. Naor [1969] first demonstrates the negative externalities from waiting: when agents make self-interested decisions on whether to join a FIFO queue, in equilibrium more agents line up in the queue in comparison to the socially optimal outcome. When monetary transfers are allowed, Naor shows that the optimal outcome can be achieved by levying an entrance toll, and a large body of subsequent work has studied how to align incentives and improve system efficiency in various settings (see Hassin [2016] for a comprehensive review). In many practical settings including ours, however, the use of monetary incentives is restricted due to regulatory or business constraints.
Without the use of monetary transfers, Hassin [1985] shows that the last-in-first-out (LIFO) queueing discipline achieves the socially optimal outcome in equilibrium, since when the agent who has waited the longest in the queue decides whether to leave, she imposes no externality on any current or future agents. With homogeneous agents who prefer items of higher quality (i.e. when all patients prefer kidneys from younger and healthier donors in the context of kidney transplantation), Su and Zenios [2004] demonstrate the excessive organ wastage resulting from patients' cherry-picking under FIFO, and proves that LIFO dispatching optimizes organ utilization. These works highlight the important role of the queueing discipline in shaping participants' strategic considerations. As is discussed in these papers, however, LIFO is practically infeasible since the dispatch rule (i) would be perceived as unfair, and (ii) can be easily manipulated by re-joining the queue. In this work, we propose practical mechanisms that allow drivers to decline dispatches and to re-join the queue at any point of time. Moreover, we model the fact that riders' finite patience limits the number of times a trip can be dispatched, and prove that no transparent and flexible mechanism can achieve a better outcome than ours even when assuming infinite rider patience.
On the flip side, Che and Tercieux [2021] establish the optimality of FIFO when the planner has full flexibility to (i) prevent participants from joining the queue and remove participants from the queue, and (ii) design the information provided to the participants. The objective is to optimize a weighted sum of the participants' utility and the service provider's profit. Intuitively, when the planner has the power to ensure that the queue is not too long, FIFO dispatching is the most effective since it provides the strongest incentive for participants to join and to stay in the queue. Su and Zenios [2006] and Ashlagi et al.
[2020] study settings where an agent's value for an item depends on the type of the item and the private type of the agent. Su and Zenios [2006] design disjoint queue mechanisms that optimize either efficiency or equity (i.e. the minimum utility across all agent types). Assuming that the value for a match is supermodular in the types of the agent and the item, Ashlagi et al. [2020] establishes that a monotone disjoint queue mechanism is welfareoptimal. In both settings, agents cannot decline the allocated items. 7 Therefore, the mix of items dispatched to each queue effectively determines a lottery over items, and the waiting times in the different queues function as prices and incentivize an agent to choose the lottery intended for her type. In contrast, instead of eliciting private information, we focus on improving reliability without using penalties or hiding information. Our mechanism effectively dispatches every trip according to "a sequence of lotteries over positions in the queue", aligning incentives using (i) the option to skip the rest of the line and (ii) the additional cost of cherry-picking introduced by randomization.
Existing work also compare FIFO and randomized allocation rules in various settings. Assuming an overloaded queue with fixed length, Bloch and Cantala [2017] show that agents in the queue prefer FIFO, but randomizing offers among all agents in the queue reduces waste, thus improves turnover and benefits agents who are not yet in the queue. Also assuming an overloaded queue, Leshno [2019] focuses on inefficiencies arising from the "mismatch", i.e. agents accepting their less preferred item since the wait for the more preferred item is too long. In a buffer queue for agents who have declined a less preferred item, randomizing offers reduces the variability of the expected waiting time for the more preferred item and reduces mismatches compared to FIFO. When agents have heterogeneous preferences over affordable housing developments, Arnosti and Shi [2020] prove that "individual lotteries" (one for each development) achieves the same outcome as a "wait-list without choice", both compelling agents to accept poor matches. More choices (via e.g., wait-list with choice) leads to better matching, but the authors also establish a trade-off between matching and targeting agents with worse outside options. In all three settings, the randomization is among all agents in the queue. In contrast, our proposed mechanisms randomize each dispatch among drivers from a small segment in the queue, which increases the costs of cherry-picking without introducing excessive variability in drivers' total payoffs.
In this work, we focus on settings where participants have the flexibility to decline dispatches without losing their positions in the queue. Schummer [2021] analyze the impact of limiting this "deferral right" for various settings, where participants are risk averse or discount the future.
Preliminaries
We study a continuous time model, with one origin (e.g. an airport) where trips are dispatched to drivers who are waiting in a queue. L = {1, 2, . . . , } denotes the set of ∈ Z >0 discrete trip types (e.g. trips to different destinations). Rider demand and driver supply are non-atomic and are stationary over time. For each location i ∈ L, µ i > 0 denotes the arrival rate of riders requesting trips to location i (i.e. the mass of riders arriving per unit of time). Upon arrival, riders' trip requests need to be dispatched to the drivers. All riders have a patience level of P ∈ Z >0 , meaning that a rider may be willing to wait for a while for a driver to accept her trip request, but she will cancel her request and leave after the P th time that her trip is declined by the drivers. Each driver can drive any rider to her destination, and riders do not have preferences over drivers.
Let λ > 0 be the arrival rate of drivers. Upon arrival, the driver may decide whether to join the queue. The net earnings of a trip to each location i ∈ L is w i , meaning that a driver who completes a rider trip to location i gets a payoff of w i from the trip, and the payoff of a driver who does not join the queue or leaves the queue without a rider is normalized to be zero. For each unit of time a driver spends waiting in the queue, the driver incurs an opportunity cost of c > 0, and the platform incurs an opportunity cost of c p ∈ [0, c]. 89 Drivers are strategic, aim to optimize their earnings from trips minus their waiting costs, and do not have preferences over riders or destinations.
An informal timeline of a dispatching mechanism is as follows (see Section 3 for the formal definition). Upon the arrival of each rider, the mechanism may dispatch the rider's trip request to a driver in the queue. If the driver accepts the dispatch, she leaves the queue to pick up the rider. Otherwise, the trip may be dispatched again, until (i) some driver accepts the trip, or (ii) the rider cancels her request when her patience runs out (after the trip is declined for P times), or (iii) the mechanism decides to not dispatch the trip again. 10 We consider a setting where the platform has complete information about demand, supply, opportunity costs, and the earnings from trips to different destinations. We assume drivers have the same information, and that this is common knowledge amongst the drivers. We study mechanisms that are fully transparent and flexible. At any point in time, all drivers know the total length of the queue and their positions in the queue. When offered trip dispatches, drivers are provided trip destinations and earnings upfront, so that they can decide whether to accept a dispatch based on this information. Moreover, drivers are not penalized for actions they take, and have the options to (i) decline dispatches they do not want to accept without losing their position in the queue, (ii) rejoin the virtual queue at the tail at any point of time, and (iii) decide to not join the queue upon arrival, or leave the queue at any point of time to perhaps relocate back to the city without a rider.
A platform's trip throughput is the mass of trips completed per unit of time by drivers in the queue. A platform's net revenue is the sum of the net earnings from trips made by drivers per unit of time, minus the opportunity cost the platform incurs due to drivers' waiting in the queue (this opportunity cost models the platform's loss of revenue elsewhere in the city, due to driver supply being tied-up in the queue). When drivers are non-strategic and accept all dispatches from the platform, we refer to the highest achievable trip throughput and net revenue as the first best.
For simplicity of notation, we assume the destinations are ordered such that w 1 > w 2 > · · · > w ≥ 0. 11 With stationary and infinitesimal demand and supply, a platform does not need a nonzero driver queue. In steady state, a platform that aims to optimize its net revenue should keep no driver in the queue, but dispatch drivers upon their arrival to destinations in decreasing order of w i until either all drivers are dispatched or all riders are picked-up. Denote the lowest-earning trip that is (partially) completed as
i * = max i ∈ L λ > i−1 j=1 µ j .
(1)
Proposition 1 (The first best). The steady state first best outcome has zero drivers in the queue. Upon arrival, drivers are dispatched to pick up arriving riders in decreasing order of w i . The remaining drivers (if any) are suggested to leave without joining the queue. The first best trip throughput is
T FB = min λ, i∈L µ i ,(2)
10 It takes some time for trip dispatches to be accepted or declined by the drivers (see Footnote 5). Drivers in the queue will be moving forward during the time a trip is repeatedly dispatched, but this does not affect our results since (i) the dispatch rules and driver strategies we study depend on the positions in the queue instead of the identities of individual drivers, and (ii) the optimality results we establish focus on the equilibrium outcome in steady state.
11 Combining destinations with the same net earnings does not affect the equilibrium outcome of any mechanism we study. For drivers who are free to decline dispatches based on trip destinations, no trip with wi < 0 will be accepted since completing one such trip is worse than declining the dispatch and leave the queue immediately without a rider. and the first best net revenue is
R FB = i * −1 i=1 w i µ i + w i * min λ − i * −1 j=1 µ i , µ i * .
(3)
Strict FIFO Dispatching
The FIFO queue discipline is considered fair by most, and is the default discipline in many everyday situations [Larson, 1987, Breinbjerg et al., 2016, Platz and Østerdal, 2017. We show that when drivers have the flexibility to decline undesired trips, offering each trip to every driver incentivizes excessive cherry-picking and leads to poor outcomes for the riders, drivers, and the platform. To avoid ambiguity, we refer to this mechanism as strict FIFO dispatching. We start by analyzing the equilibrium outcome under strict FIFO dispatching. Consider a rider request for a trip to location 1. Under strict FIFO, the trip will be accepted by the driver at the head of the queue, since a trip to location 1 has the highest earnings among all trips the driver may receive in the queue. Moreover, the (infinitesimal) driver at the head of the queue will be willing to accept only trips to location 1, since she is the first in line to receive all incoming trip dispatches, thus she does not have to wait any time for a trip dispatch to location 1.
Similar reasoning holds for drivers who are very close to the head of the queue, and a driver is willing to accept a trip to location 2 only if the additional waiting cost for a trip to location 1 outweighs the earnings gap w 1 − w 2 . Let τ 1,2 be the maximum additional time a driver is willing to wait for a trip to location 1, in comparison to immediately taking a trip to location 2. We know
τ 1,2 c = w 1 − w 2 ⇒ τ 1,2 = (w 1 − w 2 )/c.(4)
By Little's Law, the first position in the queue where the driver is willing to accept a location 2 trip is n 2 µ 1 τ 1,2 = µ 1 (w 1 − w 2 )/c, since the waiting time from this position to the head of the queue is τ 1,2 when all drivers ahead of this position only accept trips to location 1. For a driver at position n 2 , her continuation payoff (i.e. net earnings from the trip the driver accepts minus the waiting costs the driver incurs from this point of time onward) is w 2 , regardless of whether the driver accepted a trip to location 2, or if the driver continued to wait for a trip to location 1.
Similarly, in comparison to accepting a trip to location i + 1, a driver is willing to wait an additional τ i,i+1 = (w i − w i+1 )/c units of time for a trip to location i. We can compute the first positions in the queue where drivers are willing to accept trips to each location i ∈ L, assuming that riders have infinite patience and will not cancel their trip requests regardless of how many times their trips have been declined by the drivers (see Figure 2).
Lemma 1 (informal). Assume that riders have infinite patience. Under strict FIFO dispatching, it is an equilibrium for a driver to accept trip dispatches to each location i ∈ L only if the driver is at position q ≥ n i in the queue, where n 1 0 and
n i i−1 j=1 w j − w j+1 c j k=1 µ k , ∀i ≥ 2.(5)
We provide in Appendix A.1 the formal statement and the proof of the equilibrium outcome under strict FIFO dispatching. Briefly, we prove by induction that assuming infinite rider patience, for each location i ∈ L, a driver at position n i in the queue gets a continuation payoff of w i regardless of whether she accepted a trip to location i or not. Drivers at positions earlier than n i in the queue are, however, better off waiting for trips with higher earnings.
Riders
Head of the queue n 1 τ 1,2 periods n 2 τ 2,3 periods n 3 . . . When riders have a finite patience level P , however, trip requests to locations i ∈ L with n i > P will not reach a driver who is willing to accept this trip before the rider's patience runs out. As a result, trips to these destinations become unfulfilled, leading to poor efficiency and reliability. The following example demonstrates that drivers' excessive waiting in the queue further reduces drivers' total payoffs as well as the net revenue of the platform.
Example 1. Consider an airport queue, where riders request trips to three destinations L = {1, 2, 3}. The arrival rate of riders to each destination is µ 1 = 1, µ 2 = 6, and µ 3 = 3, and the net earnings from these trips are w 1 = 75, w 2 = 25, w 3 = 15, respectively. Intuitively, trips to location 1 represent the rare but high-earning long trips from the airport. Location 2 can be considered as the downtown area with high trip volumes and medium earnings, and think about location 3 trips as short rides to the hotels and towns surrounding the airport with low earnings.
Drivers arrive at a rate of λ = 5 per unit of time, and the opportunity costs for the drivers and the platform are c = c p = 1/3. Considering each unit of time as one minute, this corresponds to a scenario where a driver driving for the platform elsewhere in the city will make $20 per hour. Riders have a patience level of P = 12. When it takes an average of 10 seconds for each driver to decline a dispatch, this corresponds to the riders' being willing to wait for two minutes for a match before canceling their trip requests.
The first best. The first best outcome accepts all trips to location 1, and dispatches the remaining 4 units of drivers to trips to location 2. No driver waits in the line. The first best trip throughput is T FB = 5, and the first best net revenue is R FB = w 1 +4w 2 = 175. This outcome can be implemented, for example, by forcing each driver to always accept the first trip dispatch she receives. This, however, introduces a high variance in drivers' total payoffs (net earnings from trip minus waiting costs): the average total payoff of a driver who arrived at the queue is 35, and the variance is 400.
Strict FIFO dispatching. Under strict FIFO dispatching, when drivers have the flexibility to decide which trips to take, the driver at the head of the queue is only willing to accept trips to location 1. A driver with a location 2 trip in hand is willing to wait an additional τ 1,2 = (w 1 − w 2 )/c = 150 minutes for a trip to location 1. Therefore, the first position in the queue where the driver is willing to go to location 2 is n 2 = τ 1,2 µ 1 = 150. With a patience level of 12, riders requesting trips to location 2 will cancel their trip requests after their requests are declined by the 12 th driver in the queue. Location 3 trips are similarly unfulfilled, thus strict FIFO dispatching achieves a trip throughput of only T strict = 1 per minute.
The remaining 4 units of drivers will need to leave the queue without a rider trip in steady state. The drivers, however, will not leave if the payoff from joining the queue at the tail is better than that from relocating without a rider. Drivers are willing to wait for w 1 /c = 225 minutes for a trip to location 1, thus the steady state queue length will be µ 1 w 1 /c = 225 by Little's Law. In equilibrium, drivers get a payoff of zero regardless of whether they joined the queue or left without a rider. The large number of drivers waiting in the queue is also very costly for the platform, which achieves in this example a net revenue of zero: R strict = w 1 µ 1 − c p (µ 1 w 1 /c) = 0.
Strict FIFO dispatching is fair in the sense that drivers who are closer to the head of the queue have higher priority for trips to every destination. However, as we have seen in the above example, dispatching each trip to each and every driver in the queue leads to poor reliability for the riders, low trip throughput and net revenue for the platform, and zero earnings for the drivers despite their strategizing for better earnings. In the next section, we will see that by deprioritizing drivers at the head of the queue for trips to certain destinations (thereby violating what is typically perceived as fair dispatching), we are able to substantially improve the outcome for the riders, drivers, and the platform, even without the power to adjust trip prices.
The Direct FIFO Mechanism
In this section, we introduce the direct FIFO mechanism. The mechanism is based largely on FIFO dispatching, but sends lower-earning trips starting from positions further down the queue where drivers are willing to accept the dispatches for the option to skip the rest of the line. Accepting all trips forms a subgame perfect equilibrium among drivers, and the mechanism achieves the highest possible revenue and throughput under any mechanism that is flexible and transparent.
A Dispatching Mechanism
We first formally define a dispatching mechanism. Let Q ≥ 0 denote the length of the queue, and let q ∈ [0, Q] be a particular position in the queue. q = 0 and q = Q are the head and the tail of the queue, respectively, i.e. positions where the drivers have waited the longest and the shortest time in line. Let h denote the past dispatching history of a particular rider's trip request. This represents the positions in the queue to which the trip was offered (if any). Finally, we use φ to denote the decision to not dispatch a rider's trip request to any driver.
Definition 1 (Dispatching mechanism). Given the queue length Q, the past dispatching history h of a trip, and the trip's destination, a dispatching mechanism determines a probability distribution over [0, Q] ∪ {φ}. Upon the arrival of a rider, or after a rider's trip is declined by some driver, the mechanism either (i) dispatches the trip to a driver at some position q ∈ [0, Q] in the queue, or (ii) decides to not dispatch the trip (which we denote as φ).
The queue length Q represents the state of the queue. The dispatching mechanisms we study make dispatch decisions for each trip based on the state and the past dispatch history of this particular trip, but not on other factors such as how the state had evolved over time, or what actions the drivers had taken in the past. 12 Similarly, we focus on driver strategies that depend on the queue length and a driver's position in the queue, and we denote a strategy as a tuple σ = (α, β, γ). For any queue length Q ≥ 0, and at any position q ∈ [0, Q] in the queue, (i) α(q, Q, i) ∈ [0, 1] for each location i ∈ L is the probability with which the driver at position q in the queue accepts the trip dispatch if she is dispatched a trip to location i, (ii) β(q, Q) ∈ [0, 1] determines the probability with which the driver at position q in the queue re-joins the queue at the tail (by going offline and online again, for example), and (iii) γ(q, Q) ∈ [0, 1] is the probability for the driver at q to leave the queue without a rider.
Let U (q, Q, σ, σ ) denote the random variable representing the continuation payoff of the driver at position q in the queue, when the current length of the queue is Q, when this driver adopts strategy σ, and when all other drivers employ strategy σ (including those drivers who will arrive in the future). This includes the net earnings from the trip the driver may complete in the future, minus the total opportunity cost she incurs from this point of time onward waiting in the queue. Denote π(q, Q, σ, σ ) E [U (q, Q, σ, σ )] as the driver's expected continuation payoff from position q onward, where the expectation is taken over randomness in both the mechanism's decisions and the strategies of drivers. π(Q, Q, σ, σ ) thus represents the expected payoff of a driver with strategy σ, who joins the queue when the queue length is Q, and when all other drivers employ strategy σ .
We define the following properties.
Definition 2 (Subgame-perfect equilibrium). A strategy σ * forms a subgame perfect equilibrium (SPE) among drivers under a mechanism if for any economy and any feasible strategy σ,
π(q, Q, σ * , σ * ) ≥ π(q, Q, σ, σ * ), ∀Q ≥ 0, ∀q ∈ [0, Q].(6)
Definition 3 (Individual rationality). A mechanism is individually rational in SPE if under a strategy σ * that forms an SPE among drivers, for any economy,
π(q, Q, σ * , σ * ) ≥ 0, ∀Q ≥ 0, ∀q ∈ [0, Q].(7)
Definition 4 (Envy-freeness). A mechanism is envy-free in SPE if under a strategy σ * that forms an SPE among drivers, for any economy,
π(q 1 , Q, σ * , σ * ) ≥ π(q 2 , Q, σ * , σ * ), ∀Q ≥ 0, ∀q 1 , q 2 ∈ [0, Q] s.t. q 1 ≤ q 2 .(8)
Intuitively, under a mechanism that is individually rational and envy-free in SPE, a driver anywhere in the queue always gets non-negative continuation payoff, and does not envy the expected continuation earnings of any driver who is further down the queue.
Given a mechanism M and a strategy σ * that forms an SPE under M, let Q * denote the length of the queue under σ * in steady state. This is the case if the number of drivers joining the queue per unit of time is equal to the number of drivers dispatched from the queue when (i) the length of the queue is Q * and (ii) all drivers adopt strategy σ * . Moreover, let z i (σ * ) denote the fraction of trips to location i ∈ L that are completed in steady state when all drivers adopt σ * .
The trip throughput of mechanism M is the amount of trips completed per unit of time under σ * in steady state:
T M (σ * ) i∈L z i (σ * )µ i .(9)
The net revenue achieved by mechanism M is the total net earnings all drivers made from trips per unit of time under σ * in steady state, minus the total opportunity costs the platform incurs due to drivers' waiting in the queue:
R M (σ * ) i∈L z i (σ * )µ i w i − Q * c p .(10)
When c p = c, the net revenue of the platform is R M (σ * ) = i∈L z i (σ * )µ i w i − Q * c, i.e. the total net payoffs to all drivers who arrive at the queue.
The objective of a mechanism is to optimize trip throughput and net revenue achieved in equilibrium in steady state. 13 We say a mechanism is optimal if in equilibrium in steady state (i) the mechanism achieves the first best trip throughput, and (ii) the mechanism achieves the second best net revenue i.e., the highest steady sate equilibrium net revenue that is achievable by any dispatching mechanism that is flexible and transparent, provides trip information to drivers upfront, and does not penalize drivers for any actions they take. 14 3.2 Optimality of Direct FIFO Definition 5 (Direct FIFO). Under the direct FIFO mechanism, trips to each location i ∈ L are dispatched in a FIFO manner to drivers starting from position n i (as defined in (5)) in the queue, when the length of the queue is Q ≥ n i . When Q < n i , trips to location i are not dispatched.
Under the direct FIFO mechanism, the highest earning trips to location 1 are dispatched to the head of the queue, where the driver have waited for the longest time (thus have incurred the highest waiting costs). For a trip to location i > 1, the mechanism skips drivers close to the head of the queue who will be unwilling to accept, and dispatches the trip starting from the n th i positionthe first position in the queue where the driver is willing to accept a trip to location i under strict FIFO dispatching assuming infinite rider patience. The following theorem proves that this option to "skip the rest of the line" incentivizes drivers to accept all dispatches they receive.
Theorem 1 (Incentive compatibility of direct FIFO). It is a subgame-perfect equilibrium for drivers to accept all dispatches from the direct FIFO mechanism, and to join the queue if and only if the length of the queue is at mostQ
n + w c i∈L µ i .(11)
Moreover, the equilibrium outcome is individually rational and envy-free.
The proof is via induction on queue positions, and is provided in Appendix A.2. Intuitively, this is a "direct implementation" of the equilibrium outcome under strict FIFO dispatching when riders have infinite patience (see Lemma 1). Trips are dispatched starting from the positions in the queue where the drivers are indifferent towards accepting the trip or continuing to wait, and the equilibrium payoff from joining the queue is non-negative when the queue length is at mostQ.
When there are more drivers than needed to complete all trips to location 1, the direct FIFO mechanism does not achieve the first best net revenue-drivers are willing to spend time waiting for trips with higher earnings, leading to a queue of non-zero length and lowering the net revenue of the platform. This kind of "strategic waiting" is, however, not avoidable. We prove in the following theorem that the outcome under direct FIFO achieves the second best net revenue, i.e. the highest equilibrium net revenue achievable in steady state by any dispatching mechanism that provides trip destinations upfront and does not penalize drivers for declining dispatches.
13 When the platform takes as commission a fixed fraction of the earnings made by the drivers (from the queue as well as from driving elsewhere in the city), the problem of maximizing a platform's total commission is equivalent to that of maximizing the net revenue as defined in (10).
14 As we shall see later in this section, a platform may not be able to achieve the first best net revenue in certain settings, despite achieving the first best trip throughput. This is the case when a mechanism completes the same set of trips as those under the first best outcome, but drivers' strategically waiting for higher earning trips leads to a non-zero equilibrium queue length, thus increasing opportunity cost and reducing the net revenue of the platform.
Riders
Head of the queue n 1 n 2 n 3 . . . n i * −1 Tail of the queue n i * (a) The under-supplied scenario, with λ ≤ i∈L µ i .
Riders
Head of the queue n 1 n 2 n 3 . . . n
Tail of the queuē Q (b) The over-supplied scenario, with λ > i∈L µ i . Theorem 2 (Optimality of direct FIFO). For every economy, the direct FIFO mechanism achieves in SPE the first best trip throughput. Moreover, the equilibrium outcome achieves the first best net revenue when c p = 0, and the second best net revenue when c p ∈ (0, c].
We prove this theorem in Appendix A.2. Briefly, we first show that the steady state equilibrium outcome under direct FIFO is as illustrated in Figure 3. When λ ≤ i∈L µ i , i * as defined in (1) is the lowest-earning trip that is (partially) completed in equilibrium. Drivers will line up for trips to locations j < i * (which have higher earnings than w i * ), but the equilibrium queue length is Q * = n i * and there is no wait for a trip to location i * . See Figure 3a. Every driver gets a total payoff of w i * regardless of which trip they take, and all trips that are completed under the first-best outcome are completed. When λ > i∈L µ i , the queue is over-supplied such that all trips are accepted and completed, and the equilibrium queue length is Q * =Q (see Figure 3b). At this point, the drivers are indifferent between joining the queue and leaving, and all drivers get a zero payoff.
Given that all trips completed under the first best outcome are completed, direct FIFO achieves the first best trip throughput, and also the first best net revenue if c p = 0 (i.e. when the platform does not incur any opportunity cost due to drivers' waiting in the queue). To prove that the direct FIFO mechanism achieves the second best net revenue when c p > 0, we first establish that no mechanism can achieve in equilibrium a strictly higher total payoff for all drivers combined. This implies that if the same set of trips are completed, reducing the equilibrium queue length in comparison to that under direct FIFO (thereby reducing the total opportunity costs for the drivers as well as the platform) is not possible. We then prove that completing a different set of trips in return for a shorter queue cannot be an improvement.
We now revisit the economy analyzed in Example 1 in Section 2.
Example 1 (Continued). Consider the economy in Example 1, for which strict FIFO dispatching achieves trip throughput T strict = 1 and net revenue R strict = 0. Under direct FIFO, trips to each location will be dispatched to drivers in the queue starting at positions n 1 = 0, n 2 = 150, and n 3 = 360, respectively. With λ = 5, µ 1 = 1 and µ 2 = 6, the lowest earning trip accepted in equilibrium will be trips to location i * = 2, and the steady state queue length is Q * = n 2 = 150. Upon arrival at the tail of the queue at n 2 , one unit of driver moves on to wait for trips to location 1, and the remaining 4 units of drivers immediately accept trips to location 2 and leave. In equilibrium, all drivers get the same total payoff of w 2 = 25. The platform achieves a trip throughput of T direct = 5 and a net revenue of R direct = 1 · w 1 + 4 · w 2 − Q * c p = 125 per unit of time. Since c = c p this net revenue is also the total payoff for all drivers combined.
In comparison to strict FIFO dispatching, the direct FIFO mechanism substantially improves driver earnings, trip throughput, and the net revenue of the platform. The mechanism is, however, not fair because even when all drivers are non-strategic and accept all dispatches from the platform, a driver closer to the head of the queue may still receive trips to certain destinations at a lower rate than drivers further down the queue. Take the Midway airport as an example. A driver who has waited long enough in the queue will never receive a trip back to downtown Chicago againas we can see from Figure 1b, all high-earning trips direct FIFO dispatches to her will be heading to the suburbs. This renders the direct FIFO mechanism ill-suited for practice.
The Randomized FIFO Mechanism
In this section, we introduce a family of randomized FIFO mechanisms, which achieve optimal equilibrium throughput and revenue without using unfair dispatch rules-when drivers are straightforward and accept all dispatches, a driver closer to the head of the queue receives trip dispatches to every destination at a (weakly) higher rate than any driver further down the queue.
To demonstrate the effectiveness of randomization for aligning incentives, we first analyze the steady state Nash equilibrium under random dispatching, where every trip request is simply dispatched to drivers in the queue uniformly at random. 15
Definition 6 (Nash equilibrium in steady state). A strategy σ * forms a Nash equilibrium among drivers in steady state under a mechanism if there exists a queue length Q * ≥ 0 such that (i) for any feasible strategy σ and any position in the queue q ∈ [0, Q * ], π(q, Q * , σ * , σ * ) ≥ π(q, Q * , σ, σ * ),
(ii) when all drivers adopt strategy σ * , the steady state queue length is Q * .
Proposition 2 (Optimality of random dispatching). In Nash equilibrium in steady sate, dispatching every trip to all drivers in the queue uniformly at random achieves the first best trip throughput and the second best net revenue. When c p = 0, the equilibrium net revenue is also the first best.
See Appendix A.3 for the proof of this result. Briefly, we show that under random dispatching, every driver in the queue is willing to accept a trip to location i * (the lowest earning trip accepted in equilibrium under direct FIFO) despite the fact that the drivers may still receive higher-earning trips later. This is different from the outcome under strict FIFO, because in comparison to the driver at the head of the queue under strict FIFO, a driver who declines a dispatch under random dispatching will need to wait for a much longer time to receive her next dispatch. This additional waiting time introduced by randomization increases drivers' costs of cherrypicking, and allows random dispatching to align incentives without deprioritizing drivers at the head of the queue when dispatching trips to any location. Naively randomizing among all drivers in the queue, however, introduces substantial uncertainty in drivers' waiting times. This contributes to the variability in drivers' total payoffs, on top of the variability in the net earnings from trips. This is in stark contrast to direct FIFO, which matches lower-earning trips with drivers who have waited less time in the queue, thereby reducing the variation in drivers' total payoffs.
Riders
Head of the queuē Figure 4: Illustration of a randomized FIFO mechanism.
b (1)b(1)b(2)b(2) . . .b (m)b(m)
The randomized FIFO mechanisms we now introduce make proper use of drivers' waiting times in the queue in both ways. By gradually sending declined trips down the queue in a randomized manner, a randomized FIFO mechanism aligns incentives, and also guarantees that drivers in earlier segments in the queue (who have incurred higher waiting costs) will take trips with higher earnings.
Definition 7 (Randomized FIFO). A randomized FIFO mechanism is specified by m ≥ 1 bins in the queue ([b (1) ,b (1) ], [b (2) ,b (2) ], . . . , [b (m) ,b (m) ])
. For the k th time a trip is dispatched, the mechanism dispatches the trip to a driver in the k th bin [b (k) ,b (k) ] uniformly at random. See Figure 4 for an illustration. Intuitively, all rider requests are first dispatched to drivers in the first bin [b (1) ,b (1) ] uniformly at random. If a dispatch is declined, the mechanism will then dispatch the trip to a random driver in the next bin.
With a rider patience level of P , each trip may be dispatched a maximum of P times before the rider cancels her request. Recall that i * as defined in (1) is the lowest-earning trip that is (partially) completed under the first best outcome. Given any economy, let (
L (1) , L (2) , . . . , L (m) ) for some m ≤ min{i * , P } be an ordered partition of the top i * destinations {1, 2, . . . , i * } ⊆ L, i.e. (i) (collectively exhaustive) m k=1 L (k) = {1, 2, . . . , i * }, and for each k = 1, . . . , m, L (k) = ∅, (ii) (mutually exclusive) for all k 1 , k 2 ≤ m s.t. k 1 = k 2 , L (k 1 ) ∩ L (k 2 ) = ∅,
(iii) (monotone) for all k 1 , k 2 ≤ m s.t. k 1 < k 2 , we have i < j for all i ∈ L (k 1 ) and all j ∈ L (k 2 ) . Condition (iii) requires that trips in an earlier partition have higher earnings than those in a later partition. Given an economy and any ordered partition (L (1) , . . . , L (m) ) of the top i * destinations, we construct a corresponding set of m bins in the queue ([b (1)
,b (1) ], . . . , [b (m) ,b (m) ]) as follows: b (k) i∈∪ k <k L (k ) w i − min i ∈L (k) {w i } µ i /c,(13)b (k) i∈∪ k ≤k L (k ) w i − min i ∈L (k) {w i } µ i /c.(14)
In Lemma 4 in Appendix A.4 we show thatb (1) = 0,b (k) ≥b (k) ≥ 0 for each k ≤ m, and b (k+1) ≥b (k) for all k ≤ m − 1. This guarantees that the bins start from the head of the queue, are well defined, and do not overlap with each other.
We now present the main result of this paper, that the family of randomized FIFO mechanisms constructed in this way achieves the optimal steady state outcome in Nash equilibrium.
Theorem 3 (Optimality of randomized FIFO). For any economy and any ordered partition of the top i * destinations (L (1) , . . . , L (m) ) with m ≤ min{i * , P }, a randomized FIFO mechanism corresponding to (13) and (14) achieves the first best trip throughput and the second best net revenue in Nash equilibrium in steady state. When c p = 0, the net revenue is also the first best.
We provide the proof of this theorem in Appendix A.4. Briefly, let Q * denote the steady state equilibrium queue length under the direct FIFO mechanism. We first show that under a randomized FIFO mechanism, it is a Nash equilibrium in steady state that (i) all drivers in the k th bin in the queue accept all and only trips in the top k partitions ∪ k k =1 L (k ) (with potential randomization over trips to location i * ), (ii) after joining the queue, no driver leaves the queue without a rider trip, or rejoins the queue at its tail, (iii) drivers join the queue with probability min{1, i∈L µ i /λ} upon arrival, and (iv) the length of the queue remains constant at Q * . Under this equilibrium outcome, all trips that are completed under the first best are also completed, so that the randomized FIFO and direct FIFO mechanisms complete the same set of trips in steady sate. The queue lengths being the same then implies that randomized FIFO also achieves the optimal net revenue, given the optimality of the direct FIFO mechanism we have established in Theorem 2.
More formally, let π * (q) π(q, Q * , σ * , σ * ) be the continuation payoff of a driver at position q ∈ [0, Q * ] in the queue, when the queue length is Q * and when all drivers adopt the equilibrium strategy described above (which we denote as σ * ). We show that π * (q) is non-negative, piece-wise linear, and monotonically non-increasing in q. Moreover, π * (q) = min i∈L (k)
{w i } for all q ∈ [b (k) ,b (k) ]
for each k ≤ m, i.e. the continuation earning of any driver in the k th bin is equal to the net earning of the lowest earning trip in the k th partition. This allows us to prove by induction on k that σ * forms a Nash equilibrium. The non-negativity and monotonicity of π * (q) imply that the randomized FIFO mechanism is individually rational and envy free in Nash equilibrium in steady state.
We now demonstrate via the following example that the randomized FIFO mechanism substantially reduces the variability in drivers' earnings in comparison to dispatching every request to all drivers in the queue uniformly at random.
Example 2. Consider an economy with three destinations L = {1, 2, 3}, rider arrival rates µ 1 = 1, µ 2 = 6, µ 3 = 3, and net earnings from trips w 1 = 75, w 2 = 25, w 3 = 15. The opportunity costs per minute are c = c p = 1/3, and drivers arrive at a rate of λ = 8 per minute. Moreover, assume for simplicity that riders have a patience level of P = 2, i.e. each trip can be dispatched only twice before riders cancel their requests.
Strict FIFO dispatching. Trips to locations 2 and 3 cannot reach drivers in the queue who are willing to accept them. T strict = 1 as a result. Moreover, R strict = 0 since queue is long enough that drivers get a payoff of zero regardless of whether they had joined the queue, and when c p = c the platform's net revenue is equal to the total payoff of all drivers.
Random dispatching. When the length of the queue is Q * = 360, it is an equilibrium for drivers to accept all trips to locations 1 and 2, and randomize on trips to location 3. In steady state, all location 1 and 2 trips, and a third of location 3 trips are completed. The throughput is T rand = 8, and the platform achieves a net revenue of R rand = 120 per minute. The average waiting time in the queue is Q * /T rand = 45 minutes, and the drivers get an average total payoff of 15. However, due to the high level of variability in (i) a driver's waiting time for a trip, and (ii) the net earnings from the trip a driver may accept, the variance of drivers' total payoffs is 500 (see Appendix B.4 for the computation of the equilibrium outcome and earning variance).
Randomized FIFO. Consider a randomized FIFO mechanism corresponding to the ordered partition (L (1) , L (2) ) = ({1}, {2, 3}). The corresponding bins are given byb (1) =b (1) = 0,b (2) = 180, and b (2) = 360. All trips are first sent to drivers in [b (1) ,b (1) ] = {0}, i.e. the head of the queue. In equilibrium, drivers at the head of the queue accept only trips to location 1.The remaining trips to locations 2 and 3 will then be randomly dispatched to drivers at positions 180 to 360 in the queue.
In equilibrium, the length of the queue is Q * = 360. Compared to random dispatching, the randomized FIFO mechanism achieves the same trip throughput, net revenue, average driver wait-ing time, and average driver payoff. In contrast, the variance of the total payoffs of the drivers is reduced from 500 to 75 (see Appendix B.5 for more details). By matching higher earning trips with drivers in earlier bins who have incurred a higher waiting cost, the randomized FIFO mechanism is able to substantially reduces the variability in drivers' total payoffs.
A higher patience level of riders increases the number of times a trip can be dispatched before the rider cancels her request. This allows the randomized FIFO mechanisms to use a larger number of bins and better match higher-earning trips with drivers who have waited longer in the queue. When riders are sufficiently patient, the randomized FIFO mechanism is able to achieve zero variance in drivers' total payoffs. Consider an economy where riders' patience level is higher than the number of trip types completed in equilibrium, i.e. when P ≥ i * . The randomized FIFO mechanism corresponding to m = i * partitions has a single trip in each partition, and offers a trip to the driver at position n k in the queue if it is the k th time that the trip is dispatched. In equilibrium, trips to each location k ≤ i * are accepted by the drivers at n k , and the equilibrium outcome is the same as that under direct FIFO, where all drivers have the same total payoff.
Discussion
In real-life systems with the richness of a ridesharing platform, there typically exist multiple notions of fairness. In the context of the airport queues, a platform could be perceived as not treating drivers fairly if some drivers receive much more lucrative trips than others after waiting a similar amount of time, or if drivers who arrived later in time have higher priority for trips to certain destinations.
Under the randomized FIFO mechanisms, the small variance in drivers' total payoffs and the envy-freeness of the equilibrium outcome (which guarantees that no driver would want to swap positions with any other driver who joined the queue later in time) can both be considered as fairness properties for the equilibrium outcome (see Avi-Itzhak and Levy [2004], Platz and Østerdal [2017], and Wierman [2011]). The direct FIFO mechanism achieves zero earning variance in theory, but severely violates what is typically required of a fair dispatch rule since even when all drivers are straightforward and accept every dispatch, drivers closer to the head of the queue may still receive trips to certain destinations at a lower rate than drivers further down the queue.
Under a randomized FIFO mechanism, trips are only dispatched to drivers closest to the head of the queue when all drivers are straightforward. With strategic drivers, in equilibrium, it is possible for drivers in later bins to receive certain low-earning trips at a higher rate than drivers in an earlier bin, after these trips are first dispatched to and declined by drivers in the earlier bins. 16 As we have seen from the analysis of strict FIFO dispatching, improving efficiency and average driver earnings does require that lower-earning trips be quickly dispatched to drivers further down the queue who are willing to accept them, before riders' patience runs out. 17
In addition to the optimality of revenue and throughput, the proof of Theorem 2 also establishes that no mechanism can achieve a better total payoff for drivers, if drivers are provided trip details upfront as well as the flexibility to freely decline any dispatches. It is tempting to think that a mechanism that imposes penalties could easily achieve a better outcome. However, the same proof also implies that even if a mechanism is allowed to move drivers to the tail of the queue for 16 There may also be segments in the queue where the drivers do not receive any dispatches under the randomized FIFO mechanisms. It is possible forb (k−1) <b (k) , meaning that a driver who have just moved past the k th bin may need to wait for some time before she reaches the k − 1 th bin, and in the mean time will not receive any dispatches. The existence of such segments in between bins is to guarantee that a driver who is about to reachb (k) is not getting a continuation payoff that is too high such that the driver will decline the lower-earning trips in L (k) .
17 The trade-off between efficiency and fairness has been discussed in various contexts Zenios, 2004, 2006]. In our setting, the very limited rider patience leads to substantially higher efficiency loss under strict FIFO dispatching.
declining trip dispatches, no such mechanism can achieve a better throughput, revenue, or driver payoff. 18 Intuitively, all trips that are accepted under the first best outcome are also accepted under randomized FIFO. The only inefficiency arises because drivers strategically wait for better trips, which leads to a higher-than-necessary amount of driver supply being "tied-up" at the queue in equilibrium. Reducing the length of the queue lowers the opportunity cost for the platform, but also decreases drivers' cost of being moved to the tail of the queue at the same time. This renders such penalties less effective. All things considered, the randomized FIFO mechanisms achieve a desirable balance between flexibility, efficiency, fairness and variability in drivers' earnings.
Simulation Results
In this section, we present counterfactual simulation results for the Chicago O'Hare International Airport. As we vary the level of driver supply or rider patience, we compare various mechanisms and benchmarks in the equilibrium net revenue, trip throughput, queue length, and drivers' average waiting time, average earnings, and earning variance. Additional simulations for O'Hare and for the Chicago Midway International Airport are provided in Appendix D.
To estimate the distribution of trips and the net earnings from trips by destination, we make use of trip-level data from ridesharing platforms (including Uber and Lyft) made public by the City of Chicago. 19 The dataset provides the fare for each trip (rounded to the nearest $2.50), the origin and destination of each trip on the Census Tract level, as well as the timestamps at the beginning and the end of each trip (rounded to the nearest 15 minutes). There are a total of 801 census tracts within the City of Chicago, which we consider as the set of destinations. 20 From November 1, 2018 to March 11, 2020, there are a total of 4.53 million trips originating from O'Hare (see Figure 11 in Appendix D.1). The number of trips by destination census tract is as shown in Figure 5a, and the average trip fare by destination is shown in Figure 1a in Section 1. Without driver identifiers, we are unable to estimate the average hourly earnings of a driver in Chicago. We assume throughout this section that the opportunity cost of a driver is c = 1/3, representing the scenario that an average driver driving in the city makes $20 per hour. 21 Combining the average fare, average trip duration (see Figure 13a), and the opportunity cost, we estimate the net earnings 18 Su and Zenios [2004] suggest that a mechanism could impose penalties such that patients who decline an organ offer would expect a decrease in their priority position. In today's ridesharing platforms, Uber and Lyft drivers may lose their positions in line and move back to the tail of the queue after declining (multiple) trip dispatches.
See https://help.lyft.com/hc/en-us/articles/115012922787-Receiving-Airport-FIFO-pickuprequests and https://www.uber.com/us/en/drive/san-francisco/airports/san-francisco-international/, accessed 09/27/2020. Such penalties can improve the outcome under strict FIFO dispatching.
19 https://data.cityofchicago.org/Transportation/Transportation-Network-Providers-Trips/m6dm-c72p, accessed December 12, 2020. This dataset contains all trips in Chicago from November 2018 onward, but we use data up to mid March of 2020, before the COVID-19 pandemic substantially changed the dynamics of the market. During this time of consideration, drivers in Chicago do not have trip destinations up front. 20 https://data.cityofchicago.org/Facilities-Geographic-Boundaries/Boundaries-Census-Tracts-2010/ 5jrd-6zik, accessed September 14, 2020. The destination census tracts are not available for trips ending outside of the City of Chicago, and may also be hidden due to privacy considerations when trips are sparse. Overall, 42.6% of trips originating from O'Hare do not have a destination census tract, thus we cannot take these trips into consideration for our simulations. We do not expect any qualitative change in the simulation results if trips to all destinations are included. In fact, incorporating the long trips to the suburbs with very high earnings (which are currently missing) will likely lead to a worse outcome under strict FIFO dispatching, since these trips provide strong incentives for drivers at the head of the queue to wait and cherry-pick. 21 The simulation results are not sensitive to the choice of c. A proposal from Uber in 2019 (see https://p2a.co/ H9gttWA, accessed September 14, 2020) discussed ensuring drivers are paid an average of $21 per hour while on trip, the earnings per hour online could be currently slightly lower, depending on the average utilization level. by trip destination as shown in Figure 5b. 22 Throughout this section, we fix the total arrival rate of riders at i∈L µ i = 12 per minute. This is roughly equal to the rate of completed trips during early evening hours on weekdays (see Figure 12 in Appendix D.1 for the average number of completed trips by hour-of-week). We assume that the platform's opportunity cost of drivers' time is c p = c = 1/3 per minute, which corresponds to the scenario where the gap between the first best and the second best net revenue (which is achieved by the mechanisms we propose) is the largest. Finally, the randomized FIFO mechanism we evaluate in this section corresponds to an ordered partition of the set of completed trips into at most P subsets, each containing (approximately) the same number of destinations.
Varying Driver Supply. We first compare the different mechanisms and benchmarks as the arrival rate of drivers λ varies from zero to twenty percent over the total rider arrival rate. We fix the rider patience level at P = 12, representing the scenario where each driver decline takes 10 seconds on average, and where riders are willing to wait for 2 minutes for a match. Figure 6 presents the steady state net revenue, trip throughput, and queue length achieved in equilibrium.
When the arrival rate of drivers is very low, the outcome under direct FIFO, randomized FIFO, strict FIFO and random dispatching coincide, and all mechanisms achieve a net revenue very close to that under the first best outcome. This is because all drivers are able to accept trips with high earnings, and do not spend much time lining up in the queue. As the arrival rate of drivers increases, the length of the queue increases, and so does the gap between the first best and the second best net revenue (which is achieved by direct FIFO, randomized FIFO, and random dispatching).
In contrast to the other mechanisms, the trip throughput under strict FIFO dispatching quickly plateaus despite the increasing driver supply, since rider requests for lower earning trips cannot reach drivers in the queue who are willing to accept them. These trips become unfulfilled, and at the same time, some drivers will have to deadhead back to the city without a rider. As a result, the net revenue under strict FIFO (which is equal to the total payoff of all drivers combined when c p = c) drops to zero-drivers will continue to join the queue until the queue is so long that joining is no better than leaving without a rider, thus in equilibrium all drivers get a zero total payoff. Once the queue is over-supplied, i.e. when the driver arrival rate exceeds the total rider demand, the net revenue under all mechanisms drop to zero. This is inevitable, since no driver is willing to leave the airport without a rider as long as joining the queue and wait leads to a strictly positive payoff, but some driver has to deadhead in steady state. Nevertheless, we can see from Figure 7a that the average waiting time under randomized FIFO is still shorter than that under strict FIFO dispatching despite the longer queue length, since the trip throughout is substantially higher.
In Figures 7b and 7c, we compare the average payoff (i.e. the net earnings from trips minus the waiting costs) of all drivers who arrived at the airport, and also the standard deviation of drivers' payoffs. As expected, random dispatching introduces substantial uncertainty in drivers' payoffs. In contrast, by matching higher-earning trips with drivers who have waited longer in the queue, the randomized FIFO mechanism achieves a much smaller variance in drivers' payoffs, in comparison to random dispatching as well as the first best outcome.
Varying Rider Patience. Fixing the arrival rate of drivers at λ = 10, we compare the equilibrium, steady state outcomes under different mechanisms when riders' patience level increases from P = 1 to P = 120. Figure 8 shows the net revenue, trip throughput, and the length of the queue, and Figure 9 shows drivers' average waiting times in queue, drivers' average payoff after arriving at the queue, and the standard deviation of drivers' payoffs. The equilibrium outcomes under the direct FIFO mechanism and random dispatching are not affected by riders' patience level. Both mechanisms achieve the first best trip throughput, a high net revenue for the platform, and a low waiting time for the drivers. The randomized FIFO mechanism achieves the same throughput, revenue, and average driver waiting time. Moreover, we see from Figure 9c that (i) the variance in drivers' total payoffs is substantially lower than that under random dispatching, and (ii) this variance diminishes rapidly as riders' patience level increases. Intuitively, riders' patience level P determines the number of times a trip can be dispatched, hence the number of bins a randomized FIFO mechanism may employ. As P increases, the mechanism is able to better match trips with higher earnings with drivers who have waiting longer in the queue.
Strict FIFO dispatching, on the other hand, performs poorly. As the patience level increases, trips to more destinations can reach drivers in the queue who are willing to accept them, thus the throughput increases. The net revenue and the average driver payoff remain at zero, however, because drivers continue to join the queue until the payoff from joining is no better than that from leaving without a rider. Once P exceeded 100, strict FIFO is finally able to dispatch all drivers that arrive at the airport, achieving the second best net revenue. This level of rider patience is not practical, however, since even when each driver decline takes only 10 seconds, P > 100 requires that riders wait for over fifteen minutes to get matched to a driver.
Conclusion
We study the dispatching of trips to drivers in a queue, where some trips are necessarily more lucrative than the others due to operational constraints. We propose a family of randomized FIFO mechanisms, which send declined trips gradually down the queue in a randomized manner, and achieve in equilibrium the highest possible revenue and throughput under any mechanism that is transparent and flexible. Extensive counterfactual simulations demonstrate substantial improvements of throughput and revenue in comparison to the status quo strict FIFO dispatching, highlighting the effectiveness of using drivers' waiting times in the queue to align incentives, improve efficiency and reliability, and reduce the variability in driver earnings.
From a technical perspective, our setting generalizes existing work in the literature by modeling rider impatience and endogenizing drivers' decisions to join, leave, or re-join the platform. The randomized FIFO mechanisms we propose are also appealing for practice since drivers are provided trip destination and earnings information upfront, as well as the flexibility to freely accept or decline any dispatches. Even when a mechanism is allowed to impose penalties such that drivers would lose their position in the queue after declining a dispatch (i.e., moving back to the tail of the queue), no such mechanism can achieve a higher better throughput, revenue, or total driver earnings. All things considered, the randomized FIFO mechanism achieves a desirable balance between efficiency, flexibility, fairness and variability in driver earnings. Appendix A provides proofs omitted from the body of the paper. We derive the equilibrium outcome under various mechanisms and benchmarks in Appendix B. Additional examples and discussions are provided in Appendix C, and we include in Appendix D detailed description of the data from the City of Chicago as well as additional simulation results.
References
A Proofs
A.1 Equilibrium Outcome Under Strict FIFO
Before formally stating and proving the equilibrium outcome under strict FIFO dispatching, we first provide the following result on necessary conditions of best-response strategies. Recall from Section 3.1 that given a mechanism, π(q, Q, σ, σ ) denotes the expected continuation payoff (net earnings from trip minus waiting costs) of a driver at position q ≥ 0 in the queue, when the length of the queue is Q ≥ q, when this driver adopts strategy σ, and when every other driver employs strategy σ . Moreover, under a strategy σ = (α, β, γ), α(q, Q, i), β(q, Q) and γ(q, Q) denote the probability for a driver to (i) accept a trip to location i ∈ L, (ii) re-joins the queue at the tail, and (iii) leave the queue without a rider, when the length of the queue is Q ≥ 0 and when the driver is at some position q ∈ [0, Q].
Lemma 2. Fix a strategy σ * adopted by the rest of the drivers. σ = (α, β, γ) is a best-response strategy only if for any queue length Q ≥ 0 and at any position in the queue q ≤ Q, (i) the driver accepts (or declines) with probability one trips for which the net earnings is strictly above (or below) the continuation payoff, i.e. for all i ∈ L, w i > π(q, Q, σ, σ * ) ⇒ α(q, Q, i) = 1, and w i < π(q, Q, σ, σ * ) ⇒ α(q, Q, i) = 0, (ii) the driver rejoins at the tail of the queue with probability one (or zero) if the continuation payoff at the tail of the queue is strictly higher (or lower), i.e. π(q, Q, σ, σ * ) < π(Q, Q, σ, σ * ) ⇒ β(q, Q) = 1 and π(q, Q, σ, σ * ) > π(Q, Q, σ, σ * ) ⇒ β(q, Q) = 0, and (iii) the driver leaves the queue without a rider trip with probability one (or zero) if the continuation payoff is strictly negative (or positive), i.e. π(q, Q, σ, σ * ) < 0 ⇒ γ(q, Q) = 1 and π(q, Q, σ, σ * ) > 0 ⇒ γ(q, Q) = 0.
When the length of the queue is Q, a driver at location q who is dispatched a trip to location i ∈ L faces the decision of whether to accept the trip and get a continuation payoff of w i , or to decline the trip and remain in the queue. The continuation payoff from remaining in the queue given strategy σ is π(q, Q, σ, σ * ), thus a best response must satisfy condition (i) in Lemma 2. Similarly, it is easy to see that a violation of either condition (ii) or (iii) leads to a useful deviation that improves the driver's payoff, contradicting the assumption that σ is a best-response strategy.
Condition (i) also implies that an optimal acceptance strategy α must have a cut-off structure, such that for any Q ≥ 0 and any q ∈ [0, Q], α(q, Q, i) > 0 for location i ∈ L implies α(q, Q, j) = 1 for all j < i, since trips to these destinations have higher net earnings.
Recall that n 1 0, and observe that for each i ≥ 2, n i as defined in (5) can be rewritten as:
n i i−1 j=1 w j − w j+1 c j k=1 µ k = i−1 j=1 w j − w i c µ j .(15)
Similarly,Q as defined in (11) can be rewritten as:
Q n + w c i=1 µ i = i∈L w i µ i /c,(16)
and it is straightforward to see that n i ≤Q for all i ∈ L. We now formally state and prove Lemma 1, on the equilibrium outcome of strict FIFO dispatching when riders have infinite patience and never cancel their trip requests.
Lemma 1 (SPE under strict FIFO with infinite rider patience). Assume that riders are infinitely patient. Under strict FIFO dispatching, it is a subgame-perfect equilibrium for drivers to: 23
• accept trips to each location i ∈ L if and only if the driver is at position q ≥ n i in the queue, and
• join the queue if and only if the length of the queue is weakly belowQ, and never leave the queue or move to the tail after joining.
Proof. Let σ * = (α * , β * , γ * ) be the strategy specified by the lemma, i.e. for any queue length Q ≥ 0 and any position in the queue q ∈ [0, Q],
α * (q, Q, i) =1{q ≥ n i }, β * (q, Q) =0, γ * (q, Q) =1{q >Q}.
Here, 1{·} is the indicator function. γ * (q, Q) = 1{q >Q} means that the driver will leave (or not join) the queue if and only if the driver's position is (or will be) later thanQ. What we need to show is that starting from any queue length Q ≥ 0, assuming that the rest of the drivers all adopt strategy σ * , it is a best response for a driver at any position q ∈ [0, Q] in the queue to also employ strategy σ * . We prove this by induction on (segments of) positions in the queue, starting from the head of the queue.
The base case. First, consider the driver at the head of the queue (i.e. at position q = 0). The (infinitesimal) driver does not have to wait any time for a dispatch to location 1. The continuation payoff for the driver at q = 0 under σ * is therefore π(0, Q, σ * , σ * ) = w 1 .
This is the highest net earnings a driver may get from any trip, thus no other strategy may achieve a higher payoff, and σ * is a best response for the driver at the head of the queue.
The induction step. Now assume that for some i ≥ 2, it is a best response for drivers at positions q ≤ n i−1 in the queue to employ strategy σ * , and that a driver's optimal continuation payoff starting from position q = n i−1 onward is π(n i−1 , Q, σ * , σ * ) = w i−1 for any Q ≥ n i−1 . We prove the induction step by showing that:
(i) σ * is a best response for a driver at any position q ∈ (n i−1 , n i ] in the queue, and (ii) the optimal continuation payoff from position n i onward is π(n i , Q, σ * , σ * ) = w i .
23 Note that the strategy prescribed by this lemma is a particular SPE. There exist other strategies that may form an SPE among the drivers, depending on how drivers break ties between alternatives with equal continuation payoffs.
We first compute the continuation payoff for drivers at positions (n i−1 , n i ] in the queue, assuming that all drivers adopt strategy σ * . First, consider a driver at some position q ∈ (n i−1 , n i ). Under σ * , the driver accepts trips to locations j < i, does not leave the queue without a rider trip, or move to the tail of the queue. When all drivers adopt σ * , trips to locations 1 through i − 1 are accepted by drivers at or ahead of n i−1 , thus the driver at q will not accept any trip she receives from the platform. By Little's Law, a driver at q will wait (q − n i−1 )/ j≤i−1 µ j units of time before she reaches position n i−1 in the queue. By the induction assumption, the driver gets an optimal continuation payoff of w i−1 starting from n i−1 . As a result, for any q ∈ (n i−1 , n i ),
π(q, Q, σ * , σ * ) = π(n i−1 , Q, σ * , σ * ) − c(q − n i−1 )/ j≤i−1 µ j = w i−1 − c(q − n i−1 )/ j≤i−1 µ j .
Now consider a driver at position q = n i in the queue. If the driver is dispatched and accepts a trip to location i, she gets w i . If not, the driver moves forward in the queue and her continuation payoff is again
lim q→n i − π(q, Q, σ * , σ * ) = w i−1 − c(n i − n i−1 )/ j≤i−1 µ j = w i .
Combining the two cases, we know
π(q, Q, σ * , σ * ) = w i−1 − c(q − n i−1 )/ j≤i−1 µ j , ∀q ∈ (n i−1 , n i ],(18)
and we have π(q, Q, σ * , σ * ) > w i when q < n i and π(n i , Q, σ * , σ * ) = w i .
We now prove that σ * is a best response for drivers at (n i−1 , n i ] in the queue. Assume towards a contradiction, that there exists some strategy σ, and some q ∈ (n i−1 , n i ], such that π(q, Q, σ, σ * ) > π(q, Q, σ * , σ * ) for some Q ≥ q. Note that the driver does not get dispatched any trip with net earnings higher than w i until the driver reaches position n i−1 in the queue. Consider the following two scenarios:
• If the driver left the queue (with or without a rider) under σ before she reaches n i−1 , the driver's payoff is upper-bounded by w i ≤ π(q, Q, σ, σ * ).
• When the driver did reach n i−1 under σ, her optimal continuation payoff from n i−1 onward is w i−1 given the induction assumption. Moreover, the driver will incur a waiting cost of at least c(q − n i−1 )/ j≤i−1 µ j before reaching n i−1 thus driver's continuation payoff starting from q is again upper bounded by π(q, Q, σ * , σ * ).
Combining the two cases, we know that π(q, Q, σ, σ * ) > π(q, Q, σ * , σ * ) is not achievable for any q ∈ (n i−1 , n i ] under any strategy σ, thus σ * is a best response. This completes the proof of the induction step.
End of the queue. What we have proved by induction is that σ * is a best response for any driver at positions q ∈ [0, n ] in the queue, and that the optimal continuation payoff from n onward under any strategy is w . Now consider drivers at positions q ∈ (n ,Q] in the queue. Following σ * implies waiting until reaching n in the queue, thus the continuation payoff is:
π(q, Q, σ * , σ * ) = w − c(q − n )/ j≤ µ j ≥ w − c(Q − n )/ j≤ µ j = 0, ∀q ∈ (n ,Q].(19)
An argument very similar to the proof of the induction step shows that regardless of whether a driver reached n in the queue or not, achieving continuation payoff higher than π(q, Q, σ * , σ * ) is not possible and σ * is a best response. Moreover, π(Q,Q, σ * , σ * ) = 0 holds, thus it is a best response to leave (or not join) the queue when at position q >Q (or when the length of the queue is longer thanQ). This completes the proof of this lemma.
A.2 Incentive Compatibility and Optimality of Direct FIFO
In this section, we provide proofs for the incentive compatibility and optimality of the direct FIFO mechanism.
Theorem 1 (Incentive compatibility of direct FIFO). It is a subgame-perfect equilibrium for drivers to accept all dispatches from the direct FIFO mechanism, and to join the queue if and only if the length of the queue is at mostQ
n + w c i∈L µ i .(11)
Moreover, the equilibrium outcome is individually rational and envy-free.
Proof. Let σ * denote the strategy of (i) always accepting trip dispatches from the direct FIFO mechanism, (ii) join the queue if and only if the length of the queue is Q ≤Q, and (iii) once in the queue, never leave the queue without a rider or move to the tail of the queue.
To establish the incentive compatibility of the direct FIFO mechanism, we need to show that starting from any queue length Q ≥ 0, and assuming the rest of the drivers all adopt strategy σ * , it is a best response for a driver to also employ strategy σ * . This can be established using a very similar (and in fact, slightly simpler) argument as in the proof of Lemma 1. We do not repeat the same arguments here but refer the readers to Appendix A.1, where we established the SPE under strict FIFO (assuming infinite rider patience) by induction on segments of the queue.
It is also straightforward to show that the equilibrium continuation payoff under direct FIFO is also identical to that under strict FIFO dispatching where riders are infinitely patient. Combining equations (17), (18) and (19), we have the following expression for the equilibrium continuation payoff of a driver at position q ∈ [0, Q] in the queue for any queue length Q ≥ 0 under the direct FIFO mechanism:
π(q, Q, σ * , σ * ) = w 1 , if q = 0, w i−1 − c(q − n i−1 )/ j≤i−1 µ j , if q ∈ (n i−1 , n i ], ∀i ≥ 2, w − c(q − n )/ j∈L µ j , if q ∈ (n ,Q], 0, if q >Q.(20)
For any q ≤Q, π(q, Q, σ * , σ * ) is non-negative, continuous, and monotonically decreasing in q, thus the direct FIFO mechanism is individually rational and envy-free (i.e., no drivers envies the other drivers in positions behind her in the queue). Also observe that there is no randomness at all in a driver's continuation payoff starting from any position in the queue, since at each point n i , the driver gets precisely w i regardless of whether the driver accepted a trip to location i and left, or if the driver moved forward in the queue. As a result, the individual rationality and envy-freeness properties also hold ex post. This completes the proof of the theorem.
Theorem 2 (Optimality of direct FIFO). For every economy, the direct FIFO mechanism achieves in SPE the first best trip throughput. Moreover, the equilibrium outcome achieves the first best net revenue when c p = 0, and the second best net revenue when c p ∈ (0, c].
Proof. Let σ * denote the equilibrium strategy of accepting all dispatches from the direct FIFO mechanism, and joining the queue if and only if the length of the queue is at mostQ (see Theorem 1). We prove the optimality of the direct FIFO mechanism with the following three steps:
Step 1. Establish the steady state outcome when all drivers adopt strategy σ * , and prove that the same set of trips that are completed under the first best outcome are also completed under direct FIFO.
Step 2. Show that in equilibrium, no transparent and flexible mechanism is able to achieve a higher total payoff than that under direct FIFO for all drivers who arrive at the queue.
Step 3. Complete the proof that no mechanism is able to achieve a better net revenue.
We start from Step 1.
Step 1. We first establish the steady state equilibrium outcome under the direct FIFO mechanism. There are two cases, depending on whether the platform is over or under-supplied.
Step 1.1: λ > i∈L µ i . We first show that in the over-supplied case, when all drivers adopt strategy σ * , Q * =Q is a steady-state queue length. To prove this, first observe that with Q * =Q ≥ n i for all i ∈ L, all rider trips are accepted. The rate at which drivers are dispatched from the queue is i∈L µ i < λ, thus drivers effectively join the queue with probability i∈L µ i /λ and the length of the queue remains constant at Q * =Q. Observe that the total payoff achieved by all drivers who arrive at the queue is zero, because a driver gets a zero payoff regardless of whether she join the queue upon arrival, or left immediately without joining.
We also show that Q * =Q is the unique steady queue length, by proving that starting from any queue length Q =Q, the length of the queue will converge toQ within a finite amount of time. First, we know from (20) that the equilibrium continuation payoff of a driver at any position q <Q in the queue is strictly positive. If the length of the queue Q is strictly shorter thanQ, a driver strictly prefers to join the queue upon arrival, and drivers join the queue at a rate of λ under σ * . This cannot be the steady state outcome, since the rate at which drivers are dispatched from the queue is at most i∈L µ i < λ, and even lower when Q < n . As a result, the queue length will grow at a rate of at least λ − i∈L µ i , whenever Q <Q. Moreover, any queue length Q >Q cannot be an steady sate either, since (20) implies that the drivers at positions q >Q have strictly negative continuation payoffs, thus will leave the queue immediately.
Step 1.2: λ ≤ i∈L µ i . Recall that when a platform is not over-supplied, i * ∈ L as defined in (1) denotes the lowest-earning (i.e. highest index) trip that is (partially) completed under the first best outcome, when the λ units of drivers are dispatched to destinations in decreasing order of w i .
We first show that Q * = n i * is a steady state equilibrium queue length. When the length of the queue is n i * , all trips to locations i < i * will be dispatched and accepted by drivers in the queue.
i<i * µ i out of the λ drivers move forward in the queue upon arrival, and the remaining λ − i<i * µ i drivers leave the queue immediately with trips to location i * that are dispatched to the tail of the queue Q * = n i * . In this way, rate at which drivers join the queue is the same as the rate at which drivers are dispatched from the queue, and the length of the queue remains at n i * .
Observe that the set of trips completed in steady state under direct FIFO is the same as those completed under the first best outcome. Moreover, a driver gets a payoff of w i * regardless of whether the driver accepted a trip to location i * immediately after arrival. As a result, the total payoff of all drivers is λw i * per unit of time.
We also show that Q * = n i * is the unique steady state queue length for all non-degenerate economies, meaning that λ = i j=1 µ j for any i ∈ L. Consider the following two scenarios:
• When the length of the queue is Q < n i * , trips to locations j ≥ i * are not dispatched under the direct FIFO mechanism. The excess drivers, however, will still join the queue under σ * (at Q <Q, the payoff from joining is strictly positive). As a result, the length of the queue will grow at a rate at least λ − j<i * µ > 0j, as long as it is strictly below n i * .
• When Q > n i * , all trips to locations j ≤ i * are dispatched and accepted under direct FIFO. As a result, the length of the queue will decrease at a rate of j≤i * µ j − λ when λ < j≤i * µ j , until it reaches Q = n i * . In the degenerate case where λ = j≤i * µ j , any queue length between n i * and n i * +1 may be a steady state queue length, and we break ties in favor of shorter queues under the direct FIFO mechanism.
Combining the two settings in Step 1.1 and 1.2, we know that the same set of trips that are completed under the first best outcome are also completed under direct FIFO. This implies that the direct FIFO mechanism achieves in equilibrium the first best steady state trip throughput of T direct = min{ i∈L µ i , λ}. Moreover, when c p = 0, the outcome under direct FIFO also achieves the first best revenue, since the total net earnings from trips is the same as that under the first best, and drivers' lining up in the queue is not costly for the platform.
Step 2. We now prove that it is not possible to improve the total payoff of all drivers who had arrived at the queue, when drivers have access to trip destinations upfront and have the option to decline trips and to re-join the queue at the tail at any point of time. Again we discuss the under-supplied and the over-supplied cases separately.
Step 2.1: λ > i∈L µ i . We need to prove that in equilibrium, under any mechanism that is transparent and flexible, drivers cannot get a strictly positive average payoff after arriving at the queue. To show this, consider a mechanism M that is flexible and transparent. It cannot be a steady state equilibrium under M for every driver to leave the queue with a rider trip. As a result, some driver must willingly leave without a rider, and the net payoff of such drivers is non-positive.
Assume towards a contradiction that M achieves a strictly positive average driver payoff, and let σ and Q denote the equilibrium strategy under M, and the steady state queue length under M, respectively. The expected continuation payoff of a driver who joined the queue at the tail must be strictly positive: π M (Q , Q , σ , σ ) > 0. This is because the drivers who did not join the queue upon arrival (if any) have zero net earnings thus if π M (Q , Q , σ , σ ) ≤ 0, the average payoff of all drivers who arrived at the queue will be non-positive. π M (Q , Q , σ , σ ) > 0, however, contradicts the assumption that the outcome forms an equilibrium. In this case, no driver will be willing to leave the queue without a rider trip, since it is a useful deviation to join the queue at the tail and get a strictly positive payoff.
Step 2.2: λ ≤ i∈L µ i . As we've shown in Step 1.2, in this case drivers have an average payoff of w i * after arriving at the queue, where i * is the lowest earning trip that is (partially) completed in equilibrium. What we need to prove is that under any mechanism M that does not penalize drivers for declining dispatches or rejoining the queue at the tail, the average payoff of a driver who arrived at the queue cannot exceed w i * .
First, by definition of i * , it cannot be a steady state equilibrium under M for every driver who arrive at the virtual queue to leave the queue with a rider trip to a location j < i * . As a result, some driver must leave with a trip to some location j ≥ i * , or leave without a rider. In both cases, the driver's continuation payoff after accepting a trip or leaving the queue is upper bounded by w i * . This cannot form an equilibrium when π M (Q , Q , σ , σ ) > w i * (since a driver is better off re-joining the queue at the tail instead, therefore π M (Q , Q , σ , σ ) ≤ w i * must hold.
This completes the proof of Step 2.
Step 3. We now prove that no mechanism can achieve a higher net revenue in equilibrium than that under the direct FIFO mechanism. The case of c p = 0 was already discussed in Step 1. The case where c p = c is also straightforward, since in this case the net revenue of the platform is equal to the total net payoff of all drivers combined (see discussions in Section 3), thus
Step 2 implies that no mechanism can achieve a higher net revenue.
What is left to prove is the case where c p ∈ (0, c). Consider an alternative mechanism M, and let {μ i } i∈L be the rate at which mechanism M completes trips to each destination in equilibrium in steady state. We are going to prove that the net revenue under M is optimized when the outcome under M is the same as that under direct FIFO, and we again discuss the over and under-supplied cases separately.
Step 3.1: λ > i∈L µ i . Given Step 2, drivers get a total payoff of zero under M. Assuming that the equilibrium queue length is Q * M , we have:
i∈Lμ i w i − cQ * M = 0.(21)
The platform, however, may still get a non-zero net revenue
R M = i∈Lμ i w i − c p Q * M = (c − c p )Q * M ≥ 0,
and it is straightforward to see that R M is optimized when Q * M is the maximized. With (21), Q * M = i∈Lμ i w i /c is maximized whenμ i = µ i for all i ∈ L. This is the same outcome as that under the direct FIFO mechanism, thus no mechanism can achieve a better net revenue.
Step 3.2: λ ≤ i∈L µ i . In this case, drivers get an average payoff of w i * under direct FIFO, and the equilibrium queue length is Q * direct = n i * . Let T M i∈Lμ i denote the trip throughput under mechanism M, and let u * M be the average equilibrium payoff of drivers achieved under M. Consider the following two cases:
• T M < λ, in which case not all drivers receive rider trips in equilibrium under M. An argument very similar to that in Step 2 shows that in this case, the average payoff of a driver who joined the queue upon arrival must be zero, thus u * M = 0. Similar to the over-suppllied setting, we have i∈Lμ i w i − cQ * M = 0, which implies
R M = i∈Lμ i w i − c p Q * M = (c − c p )Q * M .
R M is again optimized when Q * M is the longest. For any fixed throughput T M = i∈Lμ i < λ, the queue length Q * M = i∈Lμ i w i /c is maximized when the T M units of drivers are dispatched to trips in decreasing order of w i , and this implies that the net revenue R M = (c − c p )Q * M is upper bounded by:
R M ≤(c − c p ) i<i * µ i w i /c + (λ − i<i * µ i )w i * /c = i<i * µ i w i + (λ − i<i * µ i )w i * − c p c i<i * µ i w i + (λ − i<i * µ i )w i *
This is weakly below the net revenue under direct FIFO, which can be written as:
R direct = i<i * µ i w i + (λ − i<i * µ i )w i * − c p n i * = i<i * µ i w i + (λ − i<i * µ i )w i * − c p c i<i * µ i (w i − w i * ) + (λ − i<i * µ i )(w i * − w i * ) .
• Consider now the case where T M = λ. Drivers' getting an average payoff of u * M implies:
i∈Lμ i w i − cQ * M = λu * M .(22)
For each i ∈ L, denote ∆ i (w i − w i * )/c. The equilibrium queue length can be written as:
Q * M = 1 c i∈Lμ i w i − i∈Lμ i u * M = i∈Lμ i (∆ i + (w i * − u * M )/c) .(23)
The net revenue under M is therefore of the form:
R M = i∈Lμ i w i − c p Q * M = i∈Lμ i (w i − c p ∆ i ) − λ(w i * − u * M )c p /c.(24)
For the first term in (24)
, w i − c p ∆ i = w i − (w i − w i * )c p /c = w i (1 − c p /c) + w i * c p /c is higher for smaller i with higher w i . The second term −λ(w i * − u * M )c p /c is non-positive given
Step 2, therefore achieves its maximum when u * M = w i * . Putting the two parts together, we know that R M is optimized when whenμ i is maximized for smallest i ∈ L first (until we have i∈Lμ i = λ), in which case the average payoff achieves u * M = w i * . This is, again, the same outcome as that under direct FIFO.
This completes the proof of Step 3, and concludes the proof of the optimality of direct FIFO.
A.3 Optimality of Random Dispatching
Before proving the optimality of random dispatching, we first provide the following lemma on the best response strategy of a driver in a stationary environment.
Lemma 3. Consider a driver in a stationary environment, where she receives trip offers to each location i ∈ L at a rate of η i ≥ 0. The highest achievable net payoff from any feasible strategy is max {max j∈L ρ j , 0}, where
ρ j j i=1 w i η i − c j i=1 η i .(25)
Moreover, j * is a maximizer of ρ j if and only if ρ j * ≤ w j * and ρ j * ≥ w j * +1 .
Proof. Lemma 2 implies that any best response strategy on acceptance in this setting must have a cutoff structure, meaning that if the driver accepts a trip to some location j ∈ L with non-zero probability, then she must accept any trip to locations i < j with probability 1. Moreover, the driver will decide to leave the queue only if the expected continuation payoff from the optimal acceptance strategy is non-positive. We now show that the highest achievable net payoff under any best-response strategy is this stationary environment is max{max j∈L ρ j , 0}. Consider for now a deterministic strategy such that the driver stays in the queue, and accepts all trips to locations 1 through j if offered. We denote this strategy as σ j . The average net earnings the driver gets from the an average trip she accepts is j i=1 w i η i / j i=1 η i , and in expectation, the driver will wait 1/ j i=1 η i units of time to receive a trip dispatch she will accept. Therefore, the expected net payoff (i.e. the net earnings from trip a driver accepts minus her expected waiting cost) under strategy σ j is
j i=1 w i η i j i=1 η i − c j i=1 η i = j i=1 w i η i − c j i=1 η i = ρ j .
Among all deterministic strategies such that the driver does not leave, the highest achievable net payoff is therefore max j∈L ρ j .
The cutoff structure proved by Lemma 2 also implies that the only potentially useful randomization in a driver's acceptance strategy is on the lowest earning trip that is accepted. Consider a strategy where the driver accepts all trips to locations 1 through j − 1, but accepts location j trips with probability θ ∈ [0, 1]. The expected net payoff in this setting is:
j−1 i=1 w i η i + w j θη j − c j−1 i=1 η i + θη j = ρ j−1 j−1 i=1 η i + w j θη j j−1 i=1 η i + θη j .
This is a weighted average of ρ j−1 and w j , thus can be optimized at θ = 0 (or θ = 1) if ρ j−1 ≥ w j (or if ρ j−1 ≤ w j ). Therefore, for a driver who does not choose to immediately leave the queue, the highest achievable net payoff can be achieved by a deterministic acceptance strategy, and the optimal payoff under any acceptance strategy is equal to max j∈L ρ j . When this payoff is negative, the driver is better off leaving the queue instead of waiting for any trip dispatches. As a result, a driver's highest possible payoff a driver may achieve in this stationary environment is max {max j∈L ρ j , 0}.
What is left to show is that j * is a maximizer of ρ j if and only ρ j * ≤ w j * and ρ j * ≥ w j * +1 . To prove this, first observe that for any j > 1, ρ j is a weighted average of ρ j−1 and w j :
ρ j = ρ j−1 j−1 i=1 η i + w j η j j i=1 η i .(26)
This implies (i) when ρ j ≥ ρ j−1 , it must be the case that w j ≥ ρ j ≥ ρ j−1 , and (ii) ρ j ≥ ρ j+1 ⇒ ρ j ≥ w j+1 . Therefore, if j * is a maximizer of ρ j , we must have ρ j * ≥ ρ j * −1 ⇒ w j * ≥ ρ j * , and
ρ j * ≥ ρ j * +1 ⇒ ρ j * ≥ w j * +1 .
On the other hand, if ρ j * ≤ w j * and ρ j * ≥ w j * +1 both hold, we now prove that j * must be a maximizer of ρ j . Denoteĵ ∈ L as the first location for which ρ j > w j+1 , i.e. j min{j ∈ L | ρ j > w j+1 }.
We first claim that ρ j must be monotonically non-decreasing when j ≤ĵ, i.e. for all j <ĵ, ρ j ≤ ρ j+1 . This is because for any j <ĵ, ρ j ≤ w j+1 holds by definition ofĵ, thus by (26) we have ρ j ≤ ρ j+1 . Moreover, given (26) and the fact that w j is monotonically decreasing, we can prove by a simple induction (ρĵ > wĵ +1 ⇒ ρĵ > ρĵ +1 > wĵ +1 > wĵ +2 and so on) that (i) ρ j must be monotonically decreasing for all j ≥ĵ, i.e. ∀j ≥ĵ, ρ j ≥ ρ j+1 , and (ii) ρ j > w j+1 for all j ≥ĵ. Combining the two cases, we know thatĵ is a maximizer of ρ j .
For j * , we know from (26) that ρ j * ≤ w j * ⇒ w j * ≥ ρ j * ≥ ρ j * −1 , therefore j * − 1 <ĵ. Given ρ j * ≥ w j * +1 , consider the two possible scenarios.
• If ρ j * > w j * +1 , we must have j * ≥ĵ, thus j * =ĵ holds and j * is a maximizer of ρ j .
• If ρ j * = w j * +1 , we have j * <ĵ. Moreover, (26) implies ρ j * +1 = ρ j * = w j * +1 > w j * +2 , which means j * +1 ≥ĵ. As a result, j * =ĵ−1, and j * is still a maximizer of ρ j because ρ j * = ρ j * +1 = ρĵ.
This completes the proof of this lemma.
With Lemma 3 at hand, we now prove the result on the equilibrium outcome under a mechanism that dispatches every trip request to all drivers in the queue, uniformly at random.
Proposition 2 (Optimality of random dispatching). In Nash equilibrium in steady sate, dispatching every trip to all drivers in the queue uniformly at random achieves the first best trip throughput and the second best net revenue. When c p = 0, the equilibrium net revenue is also the first best.
Proof. We prove this result by showing that the equilibrium outcome under random dispatching has the same queue length Q * as that under direct FIFO, and that the same set of trips that are completed under direct FIFO is also completed under random dispatching. Theorem 2 then implies the same optimality results for random dispatching.
We discuss the over-supplied and under-supplied settings separately.
Case 1: λ > i∈L µ i . When the platform is over-supplied, we have proved in Theorem 2 that all rider trips are completed under direct FIFO, and that the equilibrium queue length is Q * =Q (as defined in (11)). We now prove that under random dispatching, when the queue length isQ, it is a Nash equilibrium for drivers to (i) join the queue with probability i∈L µ i /λ upon arrival, (ii) accept all trip dispatches while in the queue, and (iii) never move to the tail of the queue or leave the queue after joining. More formally, we prove that the strategy σ * = (α * , β * , γ * ) defined as follows forms a Nash equilibrium among the drivers when the queue length isQ:
α * (q,Q, i) = 1, ∀i ∈ L, ∀q ∈ [0,Q], β * (q,Q) = 0, ∀q ∈ [0,Q], γ * (q,Q) = 0, if q <Q 1 − i∈L µ i /λ, if q =Q.
When all drivers adopt strategy σ * , the length of the queue remains atQ, since the numbers of drivers who join the queue and who are dispatched from the queue are both i∈L µ i per unit of time. All rider trips are completed, implying the same steady state revenue and trip throughput as those under direct FIFO.
We now prove that σ * forms a Nash equilibrium among the drivers under random dispatching when the queue length isQ. First, observe that when the queue length isQ and when the rest of the driver adopts σ * , (i) each rider trip is dispatched once since the probability of declines is zero, and (ii) a driver's position in the queue has no impact on the rate at which she receives dispatches to each destination. This is therefore a stationary setting we have analyzed in Lemma 3. For a driver anywhere in the queue, the rate at which she receives dispatches to each location i ∈ L is:
η i = µ i /Q.
Recall from (16) thatQ can be rewritten asQ = i=1 w i µ i /c. Therefore, the expected payoff ρ j from accepting only trips to locations 1 through j (as defined in (25)) is of the form:
ρ j = j i=1 w i η i − c j i=1 η i = j i=1 w i µ i − cQ j i=1 µ i = j i=1 w i µ i − i=1 w i µ i j i=1 µ i .
This implies ρ = 0, and ρ j < 0 for all j < . By Lemma 3, we know that the best acceptance strategy is to accept all dispatches, which is aligned with α * . This also implies that when all drivers adopt σ * , the continuation payoff of a driver anywhere in the queue is π(q,Q, σ * , σ * ) = ρ = 0.
The drivers' being indifferent towards being in the queue and leaving the queue means that there is no useful deviation from joining the queue with probability i∈L µ i /λ (hence the probability of not joining the queue is γ * (Q,Q) = 1 − i∈L µ i /λ). Moreover, re-joining the queue at the tail is not useful since a driver's position in the queue has no impact on the rate at which the driver receives trip dispatches. This completes the proof that σ * forms a Nash equilibrium among the drivers, thus concludes the discussion for Case 1, the over-supplied setting.
Case 2: λ ≤ i∈L µ i . In the case without excess drivers, i * as defined in (1) denotes the index of the lowest-earning trip that is (partially) completed in equilibrium under the direct FIFO mechanism and the first best. We know from Theorem 2 that the equilibrium queue length under direct FIFO is Q * = n i * , and the drivers complete all trips to locations j < i * , and in each unit of time the drivers also complete λ − i * −1 i=1 µ i out of the µ i * trips to location i * . We now prove that random dispatching achieves the same equilibrium outcome (queue length and set of trips completed). Fix the length of the queue at Q * = n i * , and consider the strategy σ * = (α * , β * , γ * ) such that for all q ∈ [0,Q],
α * (q,Q, i) = 1, if i < i * , 1 − 1 − (λ − i * −1 i=1 µ i )/µ i * 1/P , if i = i * , 0, if i < i * ,(28)β * (q,Q) = 0,(29)
γ * (q,Q) = 0.
For simplicity of notation, let
θ i * 1 − 1 − (λ − i * −1 i=1 µ i )/µ i * 1/P
. When every driver adopts strategy σ * , each trip to locations i < i * is dispatched once, the trip to location i * is dispatched
P k=1 (1 − θ i * ) (k−1) θ i * k + (1 − θ i * ) P P = (1 − (1 − θ i * ) P )/θ i * times,
and each trip to locations i > i * is dispatched P times. Given the queue length Q * = n i * , the rate at which a driver anywhere in the queue receives trip dispatches to each location is:
η i = µ i /n i * , if i < i * , µ i (1 − (1 − θ i * ) P )/(θ i * n i * ), if i = i * , µ i P/n i * , if i > i * .
As we observed in (15),
n i * = i * −1 i=1 (w i − w i * )µ i /c.
For each j < i * , the expected payoff from accepting only the top j trips can be written as:
ρ j = j i=1 w i µ i − cn i * j i=1 µ i = j i=1 w i µ i − i * −1 i=1 (w i − w i * )µ i j i=1 µ i .
This implies that:
ρ i * −1 = i * −1 i=1 w i µ i − i * −1 i=1 (w i − w i * )µ i i * −1 i=1 µ i = w i * .
We know from (26) that ρ i * = w i * must hold as well since ρ i * is a weighted average of ρ i * −1 and w i * .Moreover, since w i is strictly decreasing in i, we have ρ i * −1 < w i * −1 . Applying Lemma 3, we know that the highest possible expected payoff a driver may receive in this stationary setting is w i * , and this can be achieved by accepting all trips to location i < i * , and accepting trips to location i * with any probability in [0, 1]. α * is therefore an optimal acceptance strategy. It is also straightforward to see that no strategy that involves not joining the queue the queue, and moving to the tail of the queue, or leave the queue without a rider trip, could achieve a higher expected payoff than w i * , thus σ * forms a Nash equilibrium when the queue length is Q * = n i * .
What is left to prove is that the length of the queue remains at Q * = n i * when all drivers adopt σ * . To show this, we prove that the rate at which drivers are dispatched from the queue is equal to λ, the rate at which drivers join the queue. First, all trips to locations i ≤ i * − 1 are accepted, so we only need to prove that λ − i * −1 i=1 µ i drivers accept trips to location i * per unit of time. For trips to location i * , each time a trip is dispatched, it is not accepted with probability
(1 − (λ − i * −1 i=1 µ i )/µ i * ) 1/P .
Thus the probability for the trip to be unfulfilled after P dispatches is
1 − (λ − i * −1 i=1 µ i )/µ i * .
This implies that the probability for a trip to location i * to be completed
is (λ − i * −1 i=1 µ i )/µ i * , so that λ − i * −1 i=1 µ i drivers
A.4 Optimality of Randomized FIFO
In this section, we prove the optimality of the randomized FIFO mechanisms. We first provide the following lemma, which shows that the bins constructed as in (13) and (14) given any ordered partition are well-defined and not overlapping.
b (k) −b (k) = 1 c i∈∪ k ≤k L (k ) (w i − min i ∈L (k) {w i })µ i − 1 c i∈∪ k <k L (k ) w i − min i ∈L (k) {w i } µ i = 1 c i∈L (k) w i − min i ∈L (k) {w i } µ i ≥ 0.
Note that when L (k) contains a single location,b (k) =b (k) holds, meaning that for the k th time each trip is dispatched, the trip will be offered to the driver at position q =b (k) in the queue. This completes the proof of part (i).
For part (ii), first observe that for any k > 1, min i∈L (k−1) {w i } > min i∈L (k) {w i }, since the partition is ordered thus w i > w j for all i ∈ L (k−1) and all j ∈ L (k) . As a result,
b (k−1) = 1 c i∈∪ k <k L (k ) w i − min i ∈L (k−1) {w i } µ i < 1 c i∈∪ k <k L (k ) w i − min i ∈L (k) {w i } µ i =b (k) .
This completes the proof of this lemma.
We now prove the main result of our paper on the optimality of randomized FIFO.
Theorem 3 (Optimality of randomized FIFO). For any economy and any ordered partition of the top i * destinations (L (1) , . . . , L (m) ) with m ≤ min{i * , P }, a randomized FIFO mechanism corresponding to (13) and (14) achieves the first best trip throughput and the second best net revenue in Nash equilibrium in steady state. When c p = 0, the net revenue is also the first best.
Proof. We first show that given a randomized FIFO mechanism corresponding to an ordered partition of the top i * locations, under the Nash equilibrium in steady state, (i) the length of the queue is equal to the equilibrium queue length under the direct FIFO mechanism, and (ii) the same set of trips completed under direct FIFO are also completed. Theorem 2 then implies that the equilibrium outcome under randomized FIFO is optimal. We also establish individual rationality and envy-freeness under randomized FIFO by showing that a driver's continuation payoff as a function of the driver's position in the queue is non-negative and monotonically non-increasing.
Recall that i * (defined in (1)) is the index of the lowest-earning trip that is (partially) completed in equilibrium under direct FIFO. We discuss the following cases: Case 1. The total number of bins m = min{i * , P } = 1, in which case all trips are dispatched to drivers in the first bin. There are again two scenarios:
Case 1.1 i * = 1, and in which case only trips to location 1 are (partially) completed under the direct FIFO mechanism. Case 1.2 i * > 1, but P = 1, meaning that riders are impatient, and will cancel their trip request after any driver decline.
Case 2. The number of bins m = min{i * , P } > 1 , in which case trips may be dispatched multiple times, and we establish the equilibrium result by induction.
Case 1.1: m = i * = 1. In this case, there is a single partition under randomized FIFO: L (1) = {1}, and we haveb (1) =b (1) = 0. As a result, all trips are dispatched (only once) to the driver at the head of the queue. Recall that no driver will decline a trip to location 1, since there is no other trip with better earnings that the driver would like to wait for. Consider two cases:
• When λ ≤ µ 1 , it is straightforward to verify that (i) the length of the queue is zero, and (ii) all drivers accept a trip to location 1 immediately upon arrival, forms a Nash equilibrium among drivers in steady state. This is the same outcome as that under direct FIFO.
• When λ > µ 1 but i * = 1, the number of locations must be = 1, and all trips are accepted at the head of the queue. The equilibrium outcome is again the same as that under direct FIFO, where it is also the case that all trips are dispatched to and accepted by the driver at the head of the queue. In steady state, drivers join the queue with probability µ 1 /λ, all trips are completed, and the length of the queue is Q * = µ 1 w 1 /c =Q (at which point a driver is indifferent toward joining the queue and leaving without a rider trip).
Combining the two cases, we know that when i * = 1, randomized FIFO achieves the same optimal outcome achieved by direct FIFO in equilibrium. Every driver gets a payoff of w 1 when λ ≤ µ 1 , and when λ > µ 1 , the continuation payoff decreases linearly in the driver's position in the queue at takes value zero atQ. The equilibrium outcome is, therefore, individually rational and envy-free.
Case 1.2: i * > 1, m = P = 1. With P = 1, riders cancel their trip requests after a single driver decline, and all trips are dispatched by the randomized FIFO mechanism to the first (and only) bin of drivers uniformly at random. L (1) = {1, . . . , i * , and the first bin is given byb (1) = 0 and b (1) = n i * > 0. There are two cases, depending on whether the queue is under or over-supplied.
Case 1.2.1: λ ≤ i∈L µ i . Assume that the length of the queue is Q * = n i * . Under the randomized FIFO mechanism, all trips are randomly dispatched to drivers in [b (1) ,b (1) ] = [0, n i * ], i.e. all drivers in the queue. This is the same scenario as the under-supplied setting under random dispatching, which we analyzed in Case 2 in the proof of Proposition 2. It is straightforward to verify that the same strategy (specified by (28), (29) and (30)) forms a Nash equilibrium in steady state under randomized FIFO, and the equilibrium queue length remains constant at Q * = n i * . We refer the readers to the proof of Proposition 2, and do not repeat the same arguments here. Drivers at any position q ∈ [0, Q * ] in the queue has the same continuation payoff w i * ≥ 0, thus the equilibrium outcome is individually rational and envy-free.
Case 1.2.2: λ > i∈L µ i . When the queue is over-supplied, all trips are completed under direct FIFO, and i * = . The randomized FIFO mechanism dispatches each trip once (since P = 1) to drivers in [b (1) ,b (1) ] = [0, n ] uniformly at random. 24 Consider the strategy σ * = (α * , β * , γ * ):
α * (q,Q, i) = 1, ∀i ∈ L, ∀q ∈ [0,Q] β * (q,Q) = 0, ∀q ∈ [0,Q] γ * (q,Q) = 0, if q <Q, 1 − i∈L µ i /λ, if q =Q.
Here, γ * (Q,Q) = 1 − i∈L µ i /λ means that the drivers join the queue at the tail q =Q with probability i∈L µ i /λ. It is clear that when σ * is adopted by all drivers, the length of the queue remains atQ. We now prove that with Q * =Q, it is a Nash equilibrium for all drivers to adopt strategy σ * . In other words, when the length of the queue isQ and when σ * is adopted by the rest of the drivers, σ * is a best response strategy for a driver at any q ∈ [0,Q] Let us first consider a driver at some position q ∈ [0, n ] in the queue. If the driver does not leave the queue or move to the tail of the queue, this again is a stationary environment analyzed in Lemma 3. When every other driver adopts σ * , every trip is dispatched only once, thus the rate at which a driver receives trip dispatches to each location i ∈ L is (15)), a driver's expected payoff from accepting only the top j trips ρ j can therefore be rewritten as:
η i = µ i /n . Since n = −1 i=1 (w i − w )µ i /c = i=1 (w i − w )µ i /c (seeρ j = j i=1 w i µ i − cn j i=1 µ i = j i=1 w i µ i − i=1 (w i − w )µ i j i=1 µ i = i=j+1 (w − w i )µ i + w j i=1 µ i j i=1 µ i = w + i=j+1 (w − w i )µ i j i=1 µ i . 24
The equilibrium queue length is Q * =Q >b (1) , as a result, drivers at positions q ∈ (b (1) ,Q] do not receive any dispatches under randomized FIFO. This is, therefore, a different setting from the over-supplied setting under random dispatching (Case 1 of Proposition 2), where trips are dispatched to all drivers in the queue at random.
Since w − w i ≤ 0 for all i ∈ L, ρ j is maximized at j = , and we also have ρ = w . Lemma 3 implies that σ * , i.e. accepting all trips, is the optimal acceptance strategy for a driver at q ∈ [0,b (1) ]. Under σ * , the continuation payoff is of the form:
π(q,Q, σ * , σ * ) = ρ = w , ∀q ∈ [0, n ].(31)
Since w ≥ 0, there is no incentive for a driver to leave the queue without a rider trip, hence there is no useful deviation from γ(q,Q) = 0. To see why moving to the tail of the queue is not a useful deviation either, observe that a driver will not receive any trip dispatch until she moves back to positionb (1) = n in the queue. Once the driver moves here (after incurring a non-negative waiting cost), the driver is in the exact same position as she is before, achieving an optimal payoff of w when accepting all trips from the platform. This completes the proof that σ * is a best response for a driver at some q ∈ [0,b (1) ] in the queue, when σ * is adopted by the rest of the drivers. Now consider any driver at some position q ∈ (n ,Q] in the queue. The driver will not receive any trip dispatches until she reaches position n in the queue, thus there is no useful deviation from the acceptance strategy σ * . Fromb (1) onward, the driver gets an optimal continuation payoff of w as we have shown above. As a result, the driver's continuation payoff under σ * is of the form:
π(q,Q, σ * , σ * ) = w − c(q − n )/ i∈L µ i , ∀q ∈ (b (1) ,Q].(32)
We can verify that π(q,Q, σ * , σ * ) > 0 for all q <Q and that π(q,Q, σ * , σ * ) = 0. Leaving the queue without a rider (and get zero) is therefore not a useful deviation. Moreover, atQ the drivers are indifferent towards being in the queue or leaving without a rider, thus a randomized joining decision is a best response. Individual rationality and envy-freeness both hold, since π(q,Q, σ * , σ * ) is non-negative and monotonically non-increasing for all q ∈Q. This completes Case 1.2.
Case 2: m > 1. In this case we prove by induction on k (the index of bins, starting from the first bin) that in Nash equilibrium given the steady state queue length Q * , drivers in the k th bin accept all dispatches for trips in the first k partitions ∪ k k =1 L (k ) , and decline all lower earning trips in ∪ k >k L (k ) . We then establish individual rationality and envy-freeness by checking that the continuation payoff is non-negative and monotonically non-increasing.
Before analyzing the base case of the induction, we first provide some notations. Denote j (k) ∈ L andj (k) ∈ L as the lowest and highest indices of trips in the k th partition L (k) :
j (k) min{i ∈ L|i ∈ L (k) },(33)j (k) max{i ∈ L|i ∈ L (k) }.(34)
We know that a trip to location j (k) (orj (k) ) is the highest (or lowest) paying trip in L (k) . Let Q * denote the equilibrium queue length under direct FIFO, i.e. Q * = n i * when λ ≤ i∈L µ j , and Q * =Q when λ > i∈L µ j . Let σ * = (α * , β * , γ * ) be a strategy given by:
• accepting all trips in the top k partitions while in the k th bin in the queue, and randomize only on trips to location i * while in the last bin:
α * (q, Q * , i) = 1{i ∈ ∪ k k =1 L (k ) }, if q ∈ [b (k) ,b (k) ] for some k ≤ m, and i = i * , min{(λ − i<i * µ i ), µ i * }/µ i * , if q ∈ [b (m) ,b (m) ] and i = i * ,(35)
• never move to the tail of the queue:
β * (q, Q * ) = 0, ∀q ∈ [0, Q * ],(36)
• never leave the queue without a trip after joining the queue, and join the queue with probability min{ i∈L µ i /λ, 1}, i.e.
γ * (q, Q * ) = 0, if q < Q * , 1 − min{ i∈L µ i /λ, 1}, if q = Q * .(37)
We now prove by induction that σ * forms a Nash equilibrium among the drivers in steady state with queue length Q * .
Step 1: the base case with k = 1. We first prove that when the length of the queue is Q * and when every other driver adopts strategy σ * , it is a best response for any driver in the first bin [b (1) ,b (1) ] to also adopt strategy σ * . The first bin consists of the topb (1) drivers at the head of the queue. When L (1) = {1}, b (1) =b (1) = 0, thus all trips are first dispatched to the driver at the head of the queue. In this case, it is clear that accepting only trips to location 1 is the best response for a driver at q = 0, and this is aligned with σ * . Now consider the case where |L (1) | > 1, such thatb (1) > 0. With m > 1, the equilibrium length of the queue Q * is aboveb (1) , thus the first bin is "full". For a driver at any position q ∈ [0,b (1) ], the rate at which she receives dispatches to each location i ∈ L is
η (1) i = µ i /b (1) .
Note that the rates {η
(1) i } i∈L are independent to both the strategies adopted by the rest of the drivers in the first bin, and also the strategies employed by all drivers later in the queue.
We first prove that if a driver does not leave the queue or move to the tail of the queue, then there is no useful deviation from α * (q, Q * , i) = 1{i ∈ L (1) }, i.e. accepting all trips in L (1) . This is a stationary setting that we have analyzed in Lemma 3. Given (14), we know thatb (1) is of the form:
b (1) = 1 c i∈L (1) (w i − min i ∈L (1) {w i })µ i .
The utility for a driver in the first bin from accepting only the topj (1) trips (as defined in (25)) can therefore be written as:
ρ (1) j (1) = i∈L (1) w i η (1) i − c i∈L (1) η (1) i = i∈L (1) w i µ i − cb (1) i∈L (1) µ (1) i = i∈L (1) w i − i∈L (1) w i − min i ∈L (1) {w i } µ i i∈L (1) µ (1) i = min i∈L (1) {w i }.
This implies ρ
(1) j (1) ≤ w i for all i ∈ L (1) , and ρ (1) j (1) > w i for all i / ∈ L (1) (recall that the partitions are ordered). Lemma 3 then implies that an optimal acceptance strategy is to accept all trips to locations 1 throughj (1) , and this is aligned with σ * . Lemma 3 also implies that the continuation payoff of any driver in the first bin given strategy σ * is: π(q, Q * , σ * , σ * ) = min
i∈L (1) {w i }, ∀q ∈ [0,b (1) ].(38)
Since min i∈L (1) {w i } ≥ 0, deviating from γ * (q, Q * ) = 0 and leaving the is not a useful deviation. Moreover, by moving to the tail of the queue, a driver will not receive any trip with net earnings higher than min i∈L (1) {w i }, if the driver does not move all the way back to the first bin again. Once a driver is back to the first bin (after incurring some non-negative waiting costs), the driver is in the exact same situation as before moving to the tail, receiving trips again at rates {η (1) i } i∈L . Deviating from β * (q, Q * ) = 0 is therefore not a useful strategy either. This implies that σ * is a best response for drivers in the first bin, and completes the proof of the base case with k = 1.
Step 2: induction step for 1 < k < m. Assume that when the length of the queue is Q * , and when every other driver adopts strategy σ * , it is a best response for a driver at any position q ∈ [0,b (k−1) ] to adopt strategy σ * . We now prove in this induction step, that it is also a best response for any driver at positions q ∈ (b (k−1) ,b (k) ] to adopt strategy σ * .
We take the following steps in proving this result:
Step 2.1 Under strategy σ * , the equilibrium continuation payoff π(q, Q * , σ * , σ * ) is linearly decreas-
ing in q when q ∈ [b (k−1) ,b (k) ], and constant for q ∈ [b (k) ,b (k) ]: π(q, Q * , σ * , σ * ) = min i∈L (k−1) {w i } − c q −b (k−1) i∈∪ k−1 k =1 L (k ) µ i , if q ∈ [b (k−1) ,b (k) ], min i∈L (k) {w i }, if q ∈ [b (k) ,b (k) ].(39)
Step 2.2 Under any feasible strategy σ = (α, β, γ) such that the driver does not leave the queue or move to the tail of the queue (i.e. if β(q, Q * ) = γ(q, Q * ) = 0 for all q ∈ [b (k−1) ,b (k) ]), the continuation payoff cannot exceed that under σ * :
π(q, Q * , σ, σ * ) ≤ π(q, Q * , σ * , σ * ), ∀q ∈ [b (k−1) ,b (k) ].
Step 2.3 σ * is a best response for drivers at positions q ∈ [b (k−1) ,b (k) ] in the queue.
Step 2.1 implies that for any driver at some q ∈ [b (k) ,b (k) ], it cannot be a useful deviation from σ * to accept any trip in later bins ∪ k >k L (k ) since the driver gets π(q, Q * , σ * , σ * ) ≥ min i∈L (k) w i > max i∈∪ k >k L (k ) w i from following strategy σ * . Moreover, all best-response strategies must have γ(q, Q * ) = 0 for all q ∈ [b (k−1) ,b (k) ], because min i∈L (k) {w i } > 0 thus leaving the queue and getting zero cannot be a useful deviation.
With
Step 2.2, we know that across all feasible strategies where the driver does not move to the tail of the queue, σ * is a best strategy for drivers at any q ∈ [b (k−1) ,b (k) ]. With 2.1 and 2.2, we prove the final step, that even when we consider all feasible strategies where people may move to the tail of the queue, there is still no strategy that results in a higher payoff than σ * .
We start from Step 2.1.
Step 2.1. We prove (39) in this step. First, we show that under σ * , the continuation payoff of drivers satisfy π(b (k−1) , Q * , σ * , σ * ) = min i∈L (k−1) w i , and π(b (k) , Q * , σ * , σ * ) = π(b (k) , Q * , σ * , σ * ) = min i∈L (k) w i . For simplicity of notation, denote the continuation payoff under strategy σ * given the equilibrium queue length Q * as:
π * (q) π(q, Q * , σ * , σ * ).
Moreover, denote the total trip volume and average net earnings for a given subset of partitions as:
s k 1 :k 2 i∈∪ k 2 k =k 1 L (k ) µ i ,(41)w k 1 :k 2 i∈∪ k 2 k =k 1 L (k ) w i µ i i∈∪ k 2 k =k 1 L (k ) µ i = i∈∪ k 2 k =k 1 L (k ) w i µ i s k 1 :k 2 .(42)
Consider now a driver who had just reached the position positionb (k−1) in the queue. Under σ * , a driver at [0,b (k−1) ] in the queue only accept trips in the top k − 1 partitions ∪ k ≤k−1 L (k ) . When all drivers adopt the same strategy σ * , a priori there is no difference in the waiting times or earnings from trips of any driver who are atb (k−1) in the queue. The average net earnings a driver at q =b (k−1) will get from the trip she will accept in the future is thereforew 1:k−1 . By Little's Law, the average amount of time the driver spends waiting in the queue isb (k−1) /s 1:k−1 . Thus the average continuation payoff for the driver at q =b (k) is:
π * (b (k−1) ) =w 1:k−1 − cb (k−1) /s 1:k−1 .
Givenb (k) as defined in (14), we know:
π * (b (k−1) ) =w 1:k−1 − i∈∪ k <k L (k ) µ i w i − min i ∈L (k−1) {w i } s 1:k−1 = min i∈L (k−1) {w i }.(43)
Similarly, by reasoning about the net earnings an average driver gets from an average trip, and the average waiting cost a driver incurs, we can show that π(b (k) , Q * , σ * , σ * ) = π(b (k) , Q * , σ * , σ * ) = min i∈L (k) w i . Under σ * , drivers at some position q ∈ (b (k−1) ,b (k) ) will wait for (q −b (k−1) )/s 1:k−1 units of time before reachingb (k−1) in the queue, therefore her continuation payoff is of the form:
π * (q) = min i∈L (k−1) {w i } − c q −b (k−1) s 1:k−1 , if q ∈ [b (k−1) ,b (k) ).(44)
It is straightforward to verify that π * is left continuous atb (k) :
lim q→b (k) − π * (q) = min i∈L (k−1) {w i } − c b (k) −b (k−1) s 1:k−1 = min i∈L (k) {w i }.
What is left to prove for Step 2.1 is that π * (q) remains constant where q ∈ [b (k) ,b (k) ]. This is trivial when |L (k) | = 1, in which caseb (k) =b (k) . Therefore, we now consider the case where |L (k) | > 1 such thatb (k) >b (k) . When all drivers adopt strategy σ * , all trips in the first k − 1 partitions ∪ k−1 k =1 L (k ) are accepted before reaching the k th bin. For a driver in the k th bin, the rate at which she receives trip dispatches to each location i ∈ L is therefore:
η (k) i = 0, if i ∈ ∪ k−1 k =1 L (k ) µ i /(b (k) −b (k) ), if i ∈ ∪ k ≥k L (k )(45)
Note that the rates {η (k)
i } i∈L are independent to the strategies taken by drivers in later bins of the queue. With a slight abuse of notation, let
η (k) i∈L (k) η (k) i
be the total rate at which drivers in the k th bin receives trips in L (k) .
Fix an arbitrary point in time and call it time t = 0, and consider a driver who is at position b (k) at time t = 0. Let g(t) be the position of the driver in the queue, if the driver has not yet accepted a trip and leave the queue. We know g(0) =b (k) . Before the driver reaches positionb (k) in the queue, we know that in the next dt units of time, when every other driver adopts strategy σ * , there are s 1:k−1 dt drivers who are dispatched from queue positions earlier thanb (k) , and there are dts k:k (g(t) −b (k) )/(b (k) −b (k) ) drivers who are dispatched from the k th bin, ahead of this driver. As a result, the time derivative of the driver's queue position satisfies
dg(t) dt = −s 1:k−1 − s k:k g(t) −b (k) b (k) −b (k) ,(46)
i.e. the driver moves forward in the queue at a rate of s 1:k−1 + s k:k (g(t) −b (k) )/(b (k) −b (k) ) positions per unit of time. Since s 1:k−1 > 0, we know that the driver will reachb (k) within finite time. π * (g(t)) denotes continuation payoff of this driver as a function of time. For a driver at some position g(t) ∈ (b (k) ,b (k) ] at time t, the probability for the driver to be dispatched a trip she will accept under σ * in the next dt units of time is η (k) dt. If the driver is not dispatched, she moves forward in the queue to position g(t + dt) after incurring a cost of cdt. If the driver is dispatched, she takes a trip with an average net earnings ofw k:k after incurring a waiting cost in the order of cO(dt). Therefore, the driver's continuation payoff as a function of time t can be written as:
π * (g(t)) = (1 − η (k) dt)(π * (g(t + dt)) − cdt) + η (k) dt(w k:k − cO(dt))(47)
Reorganizing (47), and taking the limit as dt → 0, we have dπ * (g(t)) dt = c + η (k) (π * (g(t)) −w k:k ) = η (k) π * (g(t)) − w k:k − c/η (k) ,
and this implies
π * (g(t)) =w k:k − c/η (k) + Ce η (k) t ,(49)
where C is some constant. Given that g(0) =b (k) and π(b (k) ) = min i∈L (k) {w i }, we have:
π * (g(0)) = min i∈L (k) {w i } =w k:k − c/η (k) + C.
Given (13) and (14), the size of the k th bin is:
b (k) −b (k) = 1 c i∈L (k) (w i − min i ∈L (k) {w i })µ i .(50)
w k:k − c/η (k) therefore satisfies
w k:k − c/η (k) = i∈L (k) w i µ i − c(b (k) −b (k) ) s k:k = min i∈L (k) {w i }.
As a result, C = 0 must hold, meaning that π * (q) remains constant with respect to t for all t such that g(t) ≤b (k) and g(t) ≥b (k) . This completes the proof that π * (q) =w k:k −c/η (k) = min i∈L (k) {w i } for all q ∈ (b (k) ,b (k) ].
This completes the proof of Step 2.1. What we know from this step and Lemma 2 is that for any driver at some q ∈ [b (k) ,b (k) ], it cannot be a useful deviation from σ * to accept any trip in later bins ∪ k >k L (k ) since the driver gets π(q, Q * , σ * , σ * ) ≥ min i∈L (k) w i > max i∈∪ k >k L (k ) w i from following strategy σ * . Moreover, all best-response strategies must have γ(q, Q * ) = 0 for all q ∈ [b (k−1) ,b (k) ], because min i∈L (k) {w i } ≥ 0 thus leaving the queue and getting zero cannot be a useful deviation.
Step 2.2. We now prove that under any feasible strategy σ = (α, β, γ) such that the driver does not leave the queue or move to the tail of the queue (i.e. if β(q, Q * ) = γ(q, Q * ) = 0 for all
q ∈ [b (k−1) ,b (k) ]), for any position q ∈ [b (k−1) ,b (k) ], π(q, Q * , σ, σ * ) ≤ π(q, Q * , σ * , σ * ).(51)
First, (51) is straightforward to establish for q ∈ [b (k−1) ,b (k) ), since for a driver who does not leave or move to the tail of the queue, the driver will wait in line until she reaches positionb (k−1) in the queue, and this is aligned with σ * . Once a driver is atb (k−1) , the best strategy moving forward is σ * (by induction assumption). As a result, it is impossible to achieve a better continuation payoff than that under σ * . Now consider q ∈ [b (k) ,b (k) ], and there are two cases depending on whether |L (k) | = 1. When |L (k) | = 1, Lemma 4 implies thatb (k) =b (k) , thus all trips in ∪ k ≥k L (k ) are dispatched to the driver at positionb (k) in the queue. Under σ * , a driver at q =b (k) gets a continuation payoff of w i where i ∈ L (k) is the only trip in the k th partition, regardless of whether the driver accepts a trip or moved forward in the queue. An argument very similar to the proof of the induction step of Lemma 1 shows no alternative strategy may achieve a higher continuation payoff.
What is left to study in the case where |L (k) | > 1 andb (k) −b (k) > 0. Assume towards a contradiction that there exists some q ∈ (b (k) ,b (k) ] such that π(q, Q * , σ, σ * ) > π(q, Q * , σ * , σ * ) = min i∈L (k) w i . We introduce the following notation:
• If the driver did not accept any trip dispatches under σ before reaching positionb (k) in the queue, denote the time it takes for the driver to move from q tob (k) as κ(q).
• Denote the probability for the driver to be dispatched a trip she is willing to accept under σ before the driver reachesb (k) (i.e. within the next κ(q) units of time, given strategy σ) as ξ(κ(q)).
• Conditioning on a driver's receiving a trip within the κ(q) units of time while following strategy σ, let ω(κ(q)) be the driver's expected payoff, which includes both the net earnings from the trip the driver accepts and the waiting cost the driver incurs.
The driver's continuation payoff at position q under strategy σ can therefore be written as:
π(q, Q * , σ, σ * ) =ξ(κ(q))ω(κ(q)) + (1 − ξ(κ(q))) π(b (k) , Q * , σ, σ * ) − cκ(q) .(52)
π(b (k) , Q * , σ, σ * ) shows up in the second term because if a driver did not accept a dispatch before time κ(q) had passed (starting from the time of her being at position q), the driver has now reached b (k) . We have just argued that this continuation payoff is bounded by π(b (k) , Q * , σ, σ * ) ≤ π * (b (k) ) = min i∈L (k) w i . When ξ(κ(q)) = 0, π(q, Q * , σ, σ * ) ≤ min i∈L (k) w i trivially holds. When ξ(κ(q)) > 0, combining (52) and the assumption that π(q, Q * , σ, σ * ) > min i∈L (k) w i , we get ω(κ(q)) > min i∈L (k) {w i } + (1 − ξ(κ(q))) · cκ(q)/ξ(κ(q)).
Observe that in the first κ(q) units of time, the driver receives trip dispatches at rates {η (k) i } i∈L . Now consider a stationary setting that we analyzed in Lemma 3, where a driver always receives trip dispatches at rates {η (k) i } i∈L . 25 If the driver employs strategy σ (restricted to the first κ(q) units of time starting from position q in the queue) in this stationary setting, the driver's expected utility, which we denote asπ(σ), can be written as: π(σ) =ξ(κ(q))ω(κ(q)) + (1 − ξ(κ(q)))(π(σ) − cκ(q)).
Intuitively, if the driver gets dispatched in the first κ(q) units of time given strategy σ, she gets an expected payoff of ω(κ(q)), and this happens with probability ξ(κ(q)). If the driver is not dispatched in the first κ(q) units of time, the driver's continuation payoff starting from that point of time onward is againπ(σ). Reorganizing this expression, and applying (53), we get: ξ(κ(q))π(σ) =ξ(κ(q))ω(κ(q)) − (1 − ξ(κ(q)))cκ(q) >ξ(κ(q)) min i∈L (k)
{w i } + (1 − ξ(κ(q))) · cκ(q) − (1 − ξ(κ(q)))cκ(q) =ξ(κ(q)) min i∈L (k) {w i }.
This impliesπ(σ) > min i∈L (k) {w i }, meaning that there exists a strategy for a driver to get a continuation payoff strictly above min i∈L (k) {w i } in the stationary setting where the driver always receives trip dispatches at rate {η (k) i } i∈L given by (45). We now prove that this is not possible, and as a result we have a contradiction.
Given (45) and (50), for a driver who receives trip dispatches at rates {η (k) i } i∈L , the expected utility from accepting top j trips (as defined in (25)) for some j ≥ j (k) can be rewritten as:
ρ (k) j = i≤j w i η (k) i − c i≤j η (k) i = j i=j (k) w i µ i − c(b (k) −b (k) ) j i=j (k) µ i (54) = j i=j (k) w i µ i − i∈L (k) (w i − min i ∈L (k) {w i })µ i j i=j (k) µ i .(55)
Recall that j (k) andj (k) are the lower-index and highest-index trips in L (k) , respectively. When j ≤j (k) ,
ρ (k) j = min i ∈L (k) {w i } − j (k) i=j+1 (w i − min i ∈L (k) {w i })µ i j i=j (k) µ i ≤ min i ∈L (k) {w i }.
When j ≥j (k) , we also have:
ρ (k) j = j i=j (k+1) w i µ i + min i ∈L (k) {w i } i∈L (k) µ i j i=j (k) µ i ≤ min i ∈L (k) {w i }.
As a result, ρ
j is optimized at j =j (k) , taking value min i∈L (k) {w i }. Lemma 3 then implies that a driver in such a stationary setting cannot achieve a utility strictly higher than min i∈L (k) {w i }. This completes the proof of Step 2.2.
Step 2.3. We now complete the induction step by proving that σ * is a best response for drivers at any position q ∈ [b (k−1) ,b (k) ] in the queue. We prove this by contradiction. Assume that there exists a strategy σ such that π(q, Q * , σ, σ * ) > π * (q) for someq ∈ [b (k−1) ,b (k) ]. We show a contradiction with the following steps.
(i) We first argue that it is without loss of generality to restrict our analysis to strategies such that the driver does not leave the queue, i.e. σ = (α, β, γ) for which γ(q, Q * ) = 0, ∀q ∈ [b (k−1) ,b (k) ]. This is because if we have a strategy σ where the driver leaves with a non-zero probability at some q ∈ [b (k−1) ,b (k) ], we may construct an alternative strategy where instead of leaving the queue at q, the driver stays in the queue and follows σ * from then on.
Step 2.1 implies that this is an improvement, since the driver gets a continuation payoff of π * (q) ≥ min i∈L (k) {w i } > 0 instead of 0. Thus we get an alternative strategy that improves over σ * , and also satisfies γ(q, Q * ) = 0 for all q ∈ [b (k−1) ,b (k) ]. This contradicts Step 2.2.
(ii) We now prove that the continuation payoff at the tail of the queue under σ must satisfy
π(Q * , Q * , σ, σ * ) > min i∈L (k) {w i }. First, there must exist someq ∈ [b (k−1) ,b (k) ] such that β(q, Q * ) > 0. Otherwise, given (i), σ is a strategy such that γ(q, Q * ) = β(q, Q * ) = 0 for all q ∈ [b (k−1) ,b (k) ]
, and we have proved in Step 2.2 that among all such strategies, σ * is a best response. Assuming now towards a contradiction, that π(Q * , Q * , σ, σ * ) ≤ min i∈L (k) {w i }. π * (q) ≥ min i∈L (k) {w i } (from Step 2.1) implies that it is a (weak) improvement if instead of moving to the tail of the queue, the driver remains in the queue and adopts σ * from then on. This, again, is a strategy with γ(q, Q * ) = β(q, Q * ) = 0 for all q ∈ [b (k−1) ,b (k) ], thereby contradicting Step 2.2.
(iii) With π(Q * , Q * , σ, σ * ) > min i∈L (k) {w i }, we claim that π(b (k) , Q * , σ, σ * ) > π(Q * , Q * , σ, σ * ) > min i∈L (k) {w i }.(56)
First, observe that a driver at the tail of the queue q = Q * will not receive any trip with net earnings weakly above min i∈L (k) {w i } until the driver reaches positionb (k) in the queue. In the scenarios where the driver is dispatched under σ before reachingb (k) , the driver's payoff is strictly below min i∈L (k) {w i }. π(Q * , Q * , σ, σ * ) is a weighted average of (I) the payoff the driver gets from being dispatched before reachingb (k) , and (II) the continuation payoff after reachinḡ b (k) π(b (k) , Q * , σ, σ * ), minus the waiting cost a driver incurs before reachingb (k) . Therefore, we must have π(b (k) , Q * , σ, σ * ) > π(Q * , Q * , σ, σ * ) in order for π(Q * , Q * , σ, σ * ) > min i∈L (k) {w i } to hold.
(iv) It is without loss of generality to assume that there existsq ∈ [b (k−1) ,b (k) ] such that β(q, Q * ) = 1, i.e. the driver always moves back to the tail of the queue atq. First, observe that π(q, Q * , σ, σ * ) ≤ π(Q * , Q * , σ, σ * ) must hold for someq ∈ [b (k−1) ,b (k) ]-otherwise, reducing β(q, Q * ) to zero for all q ∈ [b (k−1) ,b (k) ] is a weak improvement, again contradicting Step 2.2. Now, increasing β(q, Q * ) to 1 for one suchq ∈ [b (k−1) ,b (k) ] will be a weak improvement over σ, thus in this way, we've constructed a strategy that achieves a better continuation payoff than σ * at some point, withq ∈ [b (k−1) ,b (k) ] for some β(q, Q * ) = 1.
(v) Now consider an alternative setting, where the driver follow strategy σ, except that the driver always moves back tob (k) instead of the tail of the queue, whenever the driver is prescribed to move to the tail of the queue under σ. 26 Denote this new strategy as σ . (56) implies that this will be an improvement, such that the continuation payoff under this new setting, which we denote asπ, must also satisfyπ(b (k) , Q * , σ , σ * ) > min i∈L (k)
{w i }.
This is, however, not possible. Observe that in this alternative setting, under σ , the driver at b (k) will either accept a trip and leave the queue before reachingq ∈ [b (k−1) ,b (k) ], or move back tob (k) before reachingq or atq. As a result, the driver will either be receiving trip dispatches at rates {η (k)
i } i∈L defined in (45) (when the driver is at some position in [b (k) ,b (k) ]), or not receive any trip dispatches at all (when the driver is in [b (k−1) ,b (k) )). The driver's payoff is therefore upper bounded by the scenario where she is in a stationary setting, always receiving trip dispatches at rates {η (k) i } i∈L , but we have proved in Step 2.2 that the highest achievable expected payoff in this setting is min i∈L (k) {w i }.
This is a contradiction, and concludes the proof of the induction step, that σ * is a best response for a driver at any position [b (k−1) ,b (k) ] in the queue.
Step 3: The last bin k = m and beyond. Given Steps 1 and 2, we know that σ * is a best response for a driver at any position q ≤b (m−1) in the queue. What is left to prove that σ * is also a best response for any driver at q ∈ (b (m−1) , Q * ] in the queue.
First, with the same arguments as in Step 2.1, we can show that π(q, Q * , σ * , σ * ) = min
i∈L (m−1) {w i } − c(q −b (m−1) )/s 1:m−1 , if q ∈ [b (m−1) ,b (m) ],(57)
and that
π(b (m) , Q * , σ * , σ * ) = π(b (m) , Q * , σ * , σ * ) = min i∈L (m) {w i } = w i * .(58)
In the case where |L (m) | = 1, i.e. when L (m) = {i * }, Lemma 4 implies thatb (m) =b (m) = n i * . Under σ * , all trips to locations j < i * are accepted by drivers in the top m − 1 bins, and all trips to locations j ≥ i * are dispatched (for the last time) to drivers at position n i * in the queue, where drivers accept only trips to location i * . The equilibrium queue length is Q * = n i * when λ ≤ i∈L µ i . When λ > i∈L µ i , the equilibrium queue length is Q * =Q, and π * (q) decreases linearly in q when q ≥ n i * = n , with π * (Q) = 0 at the tail of the queue. Using the same arguments as those in the proof of Lemma 1, we can show that when the length of the queue is Q * and when the rest of the drivers adopt σ * , it is also a best response for a driver at any q ∈ [b (m−1) , Q * ] to adopt σ * . We do not repeat the same reasoning here.
What is left to analyze is the case where |L (m) | > 1, in which caseb (m) −b (m) > 0. When all drivers adopt strategy σ * , for any driver in the last bin [b (m) ,b (m) ], the rates at which the driver receives trip dispatches to each location are given by
η (m) i = 0, if i ∈ ∪ m−1 k =1 L (k ) , µ i /(b (m) −b (m) ), if i ∈ L (m) .(59)
Applying Lemma 3 in the same way as we did in Step 2.2 above, we can show that for a driver in a stationary setting, where the driver always receives trips to all locations at rates {η (m) i } i∈L , the highest expected payoff a driver may get is min i∈L (m) {w i } = w i * .
We prove that under σ * , π * (q) = w i * holds for all q ∈ [b (m) ,b (m) ]. Denoteμ i * min{µ i * , λ − j<i * µ j }. Under σ * , a driver in the last bin accepts trip dispatches to location i * with probability μ i * /µ i * (see (35)). As a result, for a driver at position q ∈ [b (m) ,b (m) ] in the queue, under σ * , the total rate at which the driver receives and accepts trip dispatches is
η (m) i∈L (m) , i<i * η (m) i +μ i * /(b (m) −b (m) ).
Consider now a driver who is at positionb (m) at time t = 0, and denote the driver's position as a function of time t as g(t). The same argument as in the proof of Step 2.1 implies that for all t such that g(t) ∈ [b (m) ,b (m) ] the derivative of the continuation payoff π * (g(t)) with respect to t is of the form:
dπ * (g(t)) dt =η (m) π * (g(t)) − i∈L (m) ,i<i * w i µ i + w i * μ i * i∈L (m) ,i<i * µ i +μ i * − c/η (k) .
It is straightforward to verify that
i∈L (m) ,i<i * w i µ i + w i * μ i * i∈L (m) ,i<i * µ i +μ i * − c/η (m) = w i * ,(60)
thus solving dπ * (g(t)) dq =η (m) (π * (g(t)) − w i * ) with boundary condition π * (g(t)) = w i * , we know that π * (g(t)) = w i * holds for all t such that
g(t) ∈ [b (m) ,b (m) ].
Regarding the tail of the queue: in the under-supplied scenario where λ ≤ i∈L µ i , the equilibrium queue length is Q * =b (m) = n i * , thus there is no more driver in the queue beyondb (m) . In the over-supplied scenario with λ > i∈L µ i , i * = andb (m) = n , and we have:
π * (q) = w − c(q − n ) i∈L µ i , ∀q ∈ [n ,Q].(61)
Assume that the queue length is Q * and the rest of the drivers adopt strategy σ * . To prove that σ * is a best-response for drivers in [b (m−1) ,b (m) ] i.e. π * (q) ≥ π(q, Q * , σ, σ * ) for all q ∈ [b (m−1) ,b (m) ] and any feasible strategies σ, we use arguments very similar to those in Steps 2.2 and 2.3, and therefore do not repeat the details here. Intuitively, if σ * is not a best response, we are able to construct a strategy under which a driver gets a payoff strictly higher than min i∈L (m) {w i } = w i * in the stationary setting where the river always receives trips to all locations at rates {η (m) i } i∈L . This is not possible, as we have discussed above.
This establishes that the highest continuation payoff a driver at q =b (m) may get under any strategy is π * (b (m) ). To show that σ * is also a best response for any driver at q ∈ [b (m) ,Q] in the setting with λ > i∈L µ i , the same arguments used in the proof of Lemma 1 applies, thus we again refer the readers to Appendix A.1.
This completes the proof for Case 2, m > 1, and establishes that when the length of the queue is Q * , strategy σ * forms a Nash equilibrium in steady state among the drivers. As we have discussed earlier, this equilibrium outcome is optimal for trip throughput and the platform's net revenue, since it has the same queue length and completes the same set of trips as the equilibrium outcome under direct FIFO (which is optimal -see Theorem 2). Combining the three steps of Case 2, we also see that the continuation payoff π * (q) is non-negative and monotonically non-increasing in q. As a result, the equilibrium outcome of Case 2 is also individually rational and envy-free.
This completes the proof of this theorem.
B Equilibrium Outcome Under Various Mechanisms
In this section, we derive the steady state equilibrium outcome under various benchmarks and mechanisms that we discussed in this paper. For each mechanism, (the first best, strict FIFO, direct FIFO, random dispatching, and randomized FIFO), we compute the equilibrium trip throughput, net revenue, average driver payoff, length of the queue, the minimum and maximum waiting times in the queue, and the variance in drivers' total payoffs.
Recall that i * as defined in (1) is the lowest earning trip that is (partially) completed under the first best outcome, and thatμ i * min{µ i * , λ − i * −1 i=1 µ j } denotes the amount of type i * jobs fulfilled per unit of time in steady state. Moreover, for any i ∈ L, τ i,i+1 denotes the amount of time a driver is willing to wait for a trip to location i, assuming that the driver has the option to immediately accept a trip to location i + 1:
τ i,i+1 = (w i − w i+1 )/c.
B.1 The First Best
We first derive the steady state first best outcome as discussed in Section 2, which dispatches available drivers (upon arrival) to destinations in decreasing order of w i , until either all drivers are dispatched, or all riders are picked up.
• Trip throughput: T FB = min{λ, i∈L µ i }.
• Net revenue:
R FB = i * −1 i=1 w i µ i + w i * μ i * .
• The average payoff of each driver who arrived at the queue: u FB = R FB /λ, since no driver incurs any waiting cost, and R FB is equal to the total payoff achieved by all drivers per unit of time.
• The first best outcome maintains an empty driver queue, therefore the queue length, the minimum and maximum waiting time of any driver, and the average driver waiting time are all zero.
• Since drivers are either dispatched a random trip or asked to leave the queue without waiting any time in the queue, the variance in drivers' payoffs can be computed as follows:
-When the platform is under-supplied, i.e. λ ≤ i∈L µ i , every driver gets dispatched a trip to some location i ≤ i * . The variance in drivers' payoffs is therefore the variance of the net earnings of the completed trips:
Var (U FB ) = 1 λ i * −1 i=1 µ i (w i − u FB ) 2 +μ i * (w i * − u FB ) 2 .
-When the platform is over-supplied, there are µ j drivers per unit of time each getting a payoff of w j , and the rest of the drivers all get zero since they leave the airport a rider trip. Therefore,
Var (U FB ) = 1 λ i∈L µ i (w i − u FB ) 2 + (λ − i∈L µ i )(0 − u FB ) 2 .
B.2 Equilibrium Outcome under Direct FIFO
As we have proved in the paper, the direct FIFO mechanism has the same trip throughput as the first best throughput, and zero variance in drivers' earnings. Moreover:
• As we have shown in Theorem 2, the steady state equilibrium queue length is Q * direct =Q if λ > i∈L µ i , and Q * direct = n i * , otherwise.
• Net revenue:
-When λ > i∈L µ i , all trips are completed, thus
R direct = i∈L µ i w i − Q * direct c p . -When λ ≤ i∈L µ i , we have R direct = i * −1 i=1 µ i w i +μ i * w i * − Q * direct c p .
• The average driver payoff in equilibrium is u * direct = 0 if λ > i∈L µ i , and u * direct = w i * otherwise.
• Regarding the maximum, minimum, and average waiting times in queue:
-When λ ≤ i∈L µ i , the minimum waiting time for a trip (in this case, a trip to location i * ) would be zero. The maximum waiting time (which would be for a trip to location 1) is
i * −1 i=1 τ i,i+1 = (w 1 − w − i * )/c.
The average waiting time for a driver who arrived at the queue is Q * direct /λ, and the average waiting time for a driver who joined the queue is the same. -When the queue is λ > i∈L µ i , the minimum amount of time the driver needs to wait in the queue for a trip (which would be for a trip to location ) is w /c. The maximum waiting time is w 1 /c. The average waiting time for a driver who arrived at the airport is Q * direct /λ, and the average waiting time for a driver who joined the queue Q * direct / i∈L µ i .
B.3 Equilibrium Outcome under Strict FIFO Dispatching
Under strict FIFO, when P ≥ n i * , all trips that are completed under direct FIFO will be able to reach a driver who is willing to accept them under strict FIFO. As a result, the equilibrium outcome will be identical to that under direct FIFO. When P < n i * , some trips that are completed under direct FIFO will not be completed under strict FIFO, thus there exists excess drivers who need to leave the queue without a rider trip. Let i * (P ) be the lowest earning trip with n i ≤ P , the equilibrium outcome is as follows:
• Trip throughput:
T strict = i * (P ) i=1 µ i .
• The average payoff of drivers is thereby also u * strict = 0, since drivers will join the queue until when they are indifferent between joining the queue and leaving without a rider.
• Drivers will be willing to wait w i * (P ) /c units of time for a trip to location i * (P ), thus the total length of the queue would be Q * strict = n i * (P ) + T strict w i * (P ) /c, which is equal to
i * (P ) i=1 µ i w i /c. • Net revenue: R strict = i * (P ) i=1 µ i w i − Q * strict c p .
• w i * (P ) /c is the minimum amount of time a driver has to wait for any trip, and the maximum waiting time (which would be for a trip to location 1) is w 1 /c.
• On average, the total amount of time spent by all drivers on waiting is Q * units of time, per unit of time. Therefore, the average waiting time for a driver who joined the virtual queue is Q * /T strict , and the average waiting time for a driver who arrived at the origin is Q * /λ.
• Every driver gets a zero net payoff, thus the variance in drivers' earnings is also zero.
B.4 Equilibrium Outcome under Random Dispatching
As we have proved in Proposition 2, random dispatching achieves the same equilibrium trip throughput, net revenue, and queue length as those under the direct FIFO mechanism. As a result, drivers also have the same average payoff and average waiting time. Given the fact that dispatching is random, theoretically drivers might not have to wait any time for a trip dispatch, and there is also no upper bound on a driver's waiting time in the queue.
What is left to compute is the variance in drivers' total payoff. We discuss the over-supplied and the under-supplied settings separately.
Over-supplied. With λ > i∈L µ i , the average net earnings from a completed trip isw i∈L w i µ i i∈L µ i . The average waiting time of a driver who joined the queue isw/c, since in equilibrium drivers are indifferent towards whether to join the queue.
Note that (i) a driver's waiting time in the queue is independent to the driver's net earnings from the trip she accepts, and (ii) whether a driver gets dispatched in memoryless, thus a driver's waiting time is exponentially distributed, with meanw/c. The variance in drivers' waiting times is therefore (w/c) 2 , thus the variance in drivers waiting costs isw 2 .
In steady state, i∈L µ i drivers join the queue per unit of time. The rest of the drivers do not join the queue thus get 0, which is equal to the average payoff of all drivers. The total variance in the payoff of a driver who arrived at the airport is therefore:
w 2 + i∈L (w i −w) 2 µ i i∈L µ i i∈L µ i /λ.
Under-supplied. Consider now the case when the queue is not over-supplied, and the lowestearning trips that's completed is i * . In equilibrium,μ i * units of trips to location i * are completed in each unit of time. The average net earnings of the trip completed by each driver
w = i * i=1 w i µ i +μ i * w i * λ,
and the variance in drivers' net earnings from trips is
i * −1 i=1 (w i −w) 2 µ i + (w i * − w i * ) 2μ i * λ.
The average waiting time for a driver is Q * /T = (w − w i * )/c, and the distribution of waiting times is exponential. Therefore, the variance in drivers waiting costs is (cQ * /T ) 2 = (w − w i * ) 2 , and the total variance in drivers' payoff is:
(w − w i * ) 2 + i * −1 i=1 (w i −w) 2 µ i λ.
B.5 Equilibrium Outcome under Randomized FIFO
As we have proved in Theorem 3, the randomized FIFO mechanisms achieve the same equilibrium trip throughput, net revenue, and queue length as those under the direct FIFO mechanism. As a result, drivers also have the same average payoff and average waiting time in the queue. With randomization in dispatching, it is generally possible for drivers to wait zero or infinite units of time for a dispatch, although there exist special cases where the minimum and maximum waiting times in the queue are non-zero or finite.
We now derive the minimum and maximum waiting times, and the variance in drivers' payoffs. We discuss the same set of cases as analyzed in the proof of Theorem 3 in Appendix A.4. Case 1.1: m = i * = 1. As we have proved in Appendix A.4, the outcome under randomized FIFO in this case is identical to that under the direct FIFO mechanism.
Case 1.2: i * > 1, m = P = 1. As we've discussed in Appendix A.4, the outcome under randomized FIFO in this case will be identical to that under random dispatching, when the queue is not over supplied, i.e. when λ ≤ i∈L µ i .
What is left to discuss in the over-supplied case with λ > i∈L µ i . In this case, every trip is randomly dispatched to drivers at positions [0, n ] in the queue, and no driver declines any dispatches in equilibrium. With this randomization, the maximum waiting time for a driver in the queue can be infinite, but the minimum time a driver has to wait for a trip would be (Q−n )/ i∈L µ i = w /c, since a driver does not receive any dispatch until she has moved from the tail of the queue Q * =Q to position n in the queue.
For a driver who joined the queue, the variance in her earnings from the trip she completes is i∈L (w i −w) 2 µ i i∈L µ i , wherew = i∈L w i µ i i∈L µ i is the average net earnings from a rider trip. Once a driver had reached n in the queue, the additional time she has to wait for a trip is exponentially distributed with mean (w − w )/c. As a result, for a driver who joined the queue, the variance in her waiting cost is (w − w ) 2 , and the overall variance of all drivers who arrived at the queue is:
(w − w ) 2 + i∈L (w i −w) 2 µ i i∈L µ i i∈L µ i /λ.
Case 2: m > 1. We now consider the case where there are multiple bins under randomized FIFO.
No driver receives any dispatch until the driver reachesb (m) . Sinceb (m) = n i * , the minimum amount of time any driver has to wait for a trip is equal to (Q * rand − n i * )/T rand , which is equal to 0 if the queue is not over-supplied, and is equal to (Q − n )/ i∈L µ i otherwise. What is left to compute is drivers' maximum waiting time in queue, and the variance in drivers' total payoffs.
Recall that U * = U (Q * , Q * , σ * , σ * ) is the random variable representing the payoff of all drivers who arrived at the queue, and that u * E [U * ] = π * (Q * ). From Theorem 3, the average payoff of all drivers who arrived at the queue is the same as that under direct FIFO, i.e. u * = w i * when the queue is not oversupplied, and u * = 0, otherwise.
Let U (k) represent the (equilibrium, steady state) total payoff of a driver who is dispatched from the k th bin, and let U (0) = 0 denote the payoff of drivers who did not join the queue (if any). We know that
• U * takes value U (0) with probably ψ (0) max{λ − i∈L µ i , 0}/λ.
• U * takes value U (k) with probability ψ (k) s k:k /λ = i∈L (k) µ i /λ for each k = 1, . . . , m − 1, and • U * takes value U (m) with probability ψ (m) min{λ − s 1:m−1 , s m:m }/λ. The overall variance in drivers' total payoff Var (U * ) can be written as:
E (U * − u * ) 2 = m k=0 ψ (k) E U (k) − u * 2 = m k=0 ψ (k) Var U (k) + E U (k) − u * 2 .
E U (0) = Var U (0) = 0. We now compute E U (k) and Var U (k) for each k ≥ 1.
The last bin: k = m. When |L (m) | = 1, L (m) = {i * } andb (m) =b (m) = n i * . In this case, there is no variance in the payoffs of all drivers who are dispatched from the last bin: Var U (m) = 0. Moreover, drivers dispatched from the last bin gets the average payoff of all drivers: E U (m) = u * . Now consider the setting where |L (m) | > 1. Recall thatμ i * min{µ i * , λ − j<i * µ j } is the unit of location i * trip completed per unit of time in steady state. The rate at which drivers are dispatched from the last bin is:s m:m i∈L (m) , i<i * µ i +μ i * and the average net earnings from a trip accepted by a driver dispatched from the last bin is:
w (m) i∈L (m) , i<i * w i µ i + w i * μ i * s m:m .
U (m) is equal to the earning from the trip the driver (who is dispatched from the last bin) completes, minus the total waiting costs the driver incurred. The expected net earnings from trip isw (m) . For the expected waiting cost, note that once a driver reachedb (m) , the driver's additional waiting time for a dispatch should be exponentially distributed, truncated at the time the driver reachesb (m) . The parameter of this exponential distribution isη (m) s m:m /(b (m) −b (m) ), and by the time the driver reachesb (m) ,s m:m out of the T rand = min{λ, i∈L µ i } drivers who reached positionb (m) in the queue are dispatched. Denote ζ (m) s m:m /T rand as the fraction of drivers who are dispatched in the last bin (out of all of the drivers who joined the queue), and denote ∆ (m) as the time it takes for a driver to move fromb (m) tob (m) in the queue if the driver is not dispatched before reachingb (m) , we know:
1 − e −∆ (m)η(m) = ζ (m) ⇒ ∆ (m) = − log(1 − ζ (m) )/η (m) .(62)
Denote ν (m) ∆ (m) + (Q * −b (m) )/T rand , we know that ν (m) is the time it takes for a driver to move from the tail of the queue to the lower bound of the last binb (m) , if the driver is not dispatched before reachingb (m) . For a trip for a driver who is dispatched from the last bin, the average waiting time the driver spends in the last bin is therefore:
∆ (m) 0 t ·η (m) e −η (m) t dt = 1 η (m) (1 + (1/ζ (m) − 1) log((1 − ζ (m) )).(63)
Let κ (m) be the random variable representing the total waiting time of a driver who is dispatched from the last bin, we know that E κ (m) = (Q * −b (m) )/T rand + 1 η (m) (1 + (1/ζ (m) − 1) log((1 − ζ (m) )), and with this we can compute E U (m) =w (m) − cE κ (m) .
What is left to compute is Var U (m) . For a driver who is dispatched from the last bin, the earning from the trip the driver receives is independent to the time at which the driver receives the trip. As a result, the variance Var U (m) should be the sum of the variance in trip earnings and the variance in the waiting costs. The former is equal to:
i∈L (m) , i<i * (w i −w (m) ) 2 µ i + (w i * −w (m) )μ i * s m:m ,
and the latter is of the form:
c 2 ∆ (m) 0 t − 1 η (m) (1 + (1/ζ (m) − 1) log((1 − ζ (m) )) 2η (m) e −η (m) t dt = c η (m) ζ (m) 2 (ζ (m) ) 2 + (−1 + ζ (m) ) log(1 − ζ (m) ) 2 .
A middle bin. Now consider each k = m − 1, m − 2, . . . , 2. When |L (k) | = 1, there is no variance in the payoffs of drivers who are dispatched from the k th bin: Var U (k) = 0. It takes the driver a total of ν (k+1) + (b (k+1) −b (k) )/s 1:k units of time to move from the tail of the queue tob (k) , and once the driver gets tob (k) , the driver receivesw k:k , which is equal to the net earnings from the only trip in L (k) . Therefore,
E U (k) =w k:k − c ν (k+1) + (b (k+1) −b (k) )/s 1:k .
For the case where |L (k) | > 1, recall that η (k) s k:k /(b (k) −b (k) ), and denote:
ζ (k) s k:k /s 1:k .
With the same argument as those for the last bin, the time it takes a driver to move fromb (k) tō b (k) (if the driver is not dispatched before reachingb (k) ) is
∆ (k) − log(1 − ζ (k) )/η (k) .
This implies that the total waiting time for a driver to reachb (k) (if the driver is not dispatched before then) is ν (k) ν (k+1) + (b (k+1) −b (k) )/s 1:k + ∆ (k) . The expected time a driver waits in the k th bin, if the driver is dispatched from the k th bin, is of the form:
∆ (k) 0 t · η (k) e −η (k) t dt = 1 η (k) (1 + (1/ζ (k) − 1) log((1 − ζ (k) )).
thus the expected payoff of a driver who is dispatched in the k th bin is:
E U (k) =w k:k − c ν (k+1) + (b (k+1) −b (k) )/s 1:k + 1 η (k) (1 + (1/ζ (k) − 1) log((1 − ζ (k) )) .
The variance Var U (k) is similarly consisted of two parts. The variance from the net earnings from a trip a driver accepts in the k th bin is i∈L (k) (w i −w (k) ) 2 µ i /s k:k , and the variance of drivers' waiting costs is:
c 2 ∆ (k) 0 t − 1 η (k) (1 + (1/ζ (k) − 1) log((1 − ζ (k) )) 2 η (k) e −η (k) t dt = c η (k) ζ (k) 2 (ζ (k) ) 2 + (−1 + ζ (k) ) log(1 − ζ (k) ) 2 .
The first bin. When |L (1) | = 1, Var U (1) = 0. The total waiting time for a driver to reach b (1) = 0 is ν (2) +b (2) /µ 1 , thus the expected payoff of a driver dispatched from the first bin is E U (1) = w 1 − c ν (2) +b (2) /µ 1 . Moreover, in this case, the maximum waiting time for any driver who have joined the queue is ν (2) +b (2) /µ 1 .
When |L (1) | > 1, the waiting time a driver may spend waiting in the queue is unbounded. Once a driver reachedb (1) , the driver's waiting time for a trip is exponentially distributed with parameter η (1) s 1:1 /(b (1) −b (1) ). The expected waiting time in the first bin is 1/η (1) , thus the expected payoff of a driver who is dispatched from the first bin is E U (1) =w 1:1 − c 1/η (1) + ν (2) + (b (2) −b (1) )/s 1:1 , and Var U (1) , the variance of the earnings of a driver who is dispatched from the first bin is:
Var U (1) = i∈L (1) (w i −w 1:1 ) 2 µ i /s 1:1 + (c/η (1) ) 2 .
C Additional Discussion and Examples
C.1 Net Earnings from Prices and Distances
For each location i ∈ L, the net earnings w i from a trip to location i represents the total payoff of a driver who completed a trip to location i, minus the total payoff to a driver who left the queue without a rider (in this way, the net earnings of a driver who left the queue without a rider is normalized to be zero). The net earnings incorporate payments from the immediate trip, as well as drivers' continuation earnings after arriving at different destinations (which are affected by market conditions at the destinations).
Assuming that drivers get the same continuation earnings from every destination onward, we now illustrate how net earnings of trips can be derived from the prices and distances of trips to different destinations. For each destination i ∈ L, let δ i > 0 denote the amount of time (e.g. minutes) it takes for a driver to complete a trip to destination i, which includes time it takes for a driver to pick up the rider. p i > 0 denotes the effective earnings rate from a trip to location i ∈ L, meaning that the total payment to a driver for a trip to location i is p i δ i . The earnings rates are induced by the time and distance rates the platform pays the drivers, and may vary across destinations due to differences in trip lengths and traffic conditions, etc. For drivers who decides to relocate without a rider and drive elsewhere, δ > 0 is the minimum relocation distance, i.e. the amount of time a driver needs to spend driving from the airport in order to start making an average earnings rate of c.
A driver who accepts a trip from the virtual queue to some location i ∈ L will make p i per unit of time for δ i periods, followed by making c per period after arriving at location i. A driver who relocates back to the city without a rider makes 0 for the first δ periods, and then starts to earn c per period. See Figure 10. The additional earnings from a trip to location i, relative to that from relocating without a rider (and then driving in the city), is the net earnings from this trip. For each location i ∈ L, the net earnings w i is of the form:
w i = δ i p i − (δ · 0 + (δ i − δ)c) = δ i (p i − c) + δc.(64)
C.2 Additional Examples
Example 3 shows that a randomized FIFO mechanism may not achieve the second best outcome, if some trips with lower earnings than w i * are included in the ordered partition of destinations.
Example 3 (Last bin w/ trips to location i ≥ i * ). Consider an economy with three destinations, where µ 1 = 1, w 1 = 100, µ 2 = 2, w 2 = 40, and µ 3 = 5, w 3 = 10. The arrival rate of drivers is λ = 2, and that the opportunity cost for drivers' time is c = 1. Under the first best outcome, one unit of trips to location 1 and one unit of trips to location 2 are completed per unit of time. With i * = 2, the average net payoff of drivers under the second best outcome would be w i * = w 2 = 40, and the equilibrium, steady state queue length is Q * = n i * = µ 1 (w 1 − w 2 )/c = 60. Assume that riders have a patience level of P = 2. The appropriate construction of randomized FIFO corresponds to the ordered partition L (1) = {1} and L (2) = {2}. Now consider a randomized FIFO mechanism associated with the ordered partition L (1) = {1} and L (2) = {2, 3}. Constructing the bins according to (13) and (14), we haveb (1) =b (1) = 0 and b (2) = 1 c µ 1 (w 1 − w 3 ) = 90.
Note thatb (2) is higher than the equilibrium queue length Q * under the second best outcome. We now show that the randomized FIFO mechanism constructed in this way will not achieve the second best as long as c p > 0. For a driver in the first bin, i.e. at the head of the queue, the driver is only willing to accept a trip to location 1. When the queue is shorter thanb (2) , trips to location 2 or 3 will not be dispatched again by the randomized FIFO mechanism after being rejected by the driver at the head of the queue, thus all but location 1 trips become unfulfilled. But when the queue is longer thanb (2) , we will not be achieving our second best outcome either, since the total waiting costs incurred by the drivers will be higher than that under the second best outcome, even when we are completing the same set of trips.
D Additional Simulations
We include in this section descriptions of the dataset made public by the City of Chicago, additional simulation results for O'Hare that are omitted from Section 5 of the paper, as well as simulation results for the Chicago Midway International airport.
D.1 Chicago O'Hare International Airport
D.1.1 Trip Volume by Day and Hour-of-Week
We first provide the volume of trips originating from Chicago O'Hare, and the average duration and earning rates by destination. Figure 11 shows the number of trips that originate from the O'Hare airport on each day, from November 1, 2018 to mid March, 2020. We can see strong weekly patterns, seasonality patterns (e.g. low trip volume during Christmas through New Year), and also the sharp decline in trip volume after the onset of the COVID-19 pandemic. The average number of trips originating from O'Hare during each hour-of-week is as shown in Figure 12. Here, the 0 th hour-of-week corresponds to midnight -1am on Mondays, and the 1 st hour-of-week corresponds to 1am -2am on Mondays, and so on. We can see that the number of trips originating from the airport peaks during early evenings, averaging around 12 trips per minute during the weekdays, and reaches a maximum of over 15 trips per minute on Thursday. Note that these are completed trips, thus the rider request rates are strictly higher. Figure 13 illustrates the average duration and the average earning rates (trip fare divided by trip duration) for trips ending in each census tract. We can see that longer trips take more time on average, and trips ending closer to major highways have better earnings rates.
D.1.2 Counterfactual Simulations
We now provide additional results for O'Hare that are omitted from the body of the paper.
Varying Driver Supply As the arrival rate of driver varies, Figure 14 presents the minimum and maximum waiting times for drivers who joined the queue in equilibrium in steady state. For strict FIFO, and direct FIFO, the minimum waiting time is the time a driver needs to wait in the queue for the lowest earning trip that is completed in equilibrium. Under randomized FIFO, the minimum waiting time is the time it takes for a driver to move from the tail of the queue to the last bin (i.e. positionb (m) in the queue). When the queue is not over-supplied, the minimum waiting times under direct FIFO and randomized FIFO are both zero. Under strict FIFO and direct FIFO, the maximum waiting time is the time a driver needs to wait in the queue for a trip to location 1, the highest earning trip. Under random dispatching or randomized FIFO with |L (1) | > 1, there is no upper bound on how long a driver may need to wait in the queue.
Varying Rider Patience Figure 15 presents the minimum and maximum waiting times for drivers who joined the queue as we vary the patience level of riders. Under strict FIFO, the minimum waiting time decreases very slowly as riders' patience level increases, despite the fact that the minimum waiting time for a trip under every other mechanism is zero.
D.2 Chicago Midway International Airport
In this section, we present simulation results for the Chicago Midway International airport. O'Hare. Figure 18 shows the total trip count by destination census tract, and the estimated net earnings by destination assuming that drivers' opportunity cost is c = 1/3 per minute. Varying Driver Supply We now compare the equilibrium, steady state outcome under various mechanisms and benchmarks, as we vary the arrival rate of drivers from 0 to 6 drivers per minute. We fix the total arrival rate of riders at i∈L µ i = 5, and the rider patience level at P = 12. See Figures 19, 20 and 21. The observations here are aligned with those for the O'Hare airport presented in Section 5. Varying Rider Patience Fixing the arrival rate of drivers at λ = 4, Figures 22, 23 and 24 compare the equilibrium outcome under various mechanisms and benchmarks as we vary the patience level of the riders. The observations are, again, fully aligned with those for the O'Hare airport presented in Section 5 of the paper.
Figure 1 :
1Average trip fare by destination Census Tract in Chicago, for trips originating from the O'Hare International Airport and the Midway International Airport. See Section 5 for more details.
Figure 2 :
2The equilibrium outcome under strict FIFO dispatching, assuming infinite rider patience.
Figure 3 :
3The steady-state equilibrium outcome under the direct FIFO mechanism.
Figure 5 :
5Trip volume and estimated net earnings (assuming c = 1/3) by destination Census Tract in Chicago, for trips originating from the Chicago O'Hare International Airport.
Figure 6 :
6Equilibrium net revenue, trip throughput, and length of the queue in steady state, as the arrival rate of drivers varies. Chicago O'Hare.
Figure 7 :
7Drivers' average waiting times, average payoff, and the standard deviation (SD) in drivers' payoff in equilibrium in steady state, as the arrival rate of drivers varies. Chicago O'Hare.
Figure 9 :
9Drivers' average waiting times, average payoff, and the standard deviation (SD) in drivers' payoff in equilibrium in steady state, as riders' patience level varies. Chicago O'Hare.
accept trips to location i * per unit of time. This completes the proof of the under-supplied case, and concludes the proof of this proposition.
Lemma 4 .
4For any ordered partition (L (1) , L (2) , . . . , L (m) ) of the top i * destinations {1, 2, . . . , i * }, the corresponding set of bins satisfies:(i) 0 ≤b (k) ≤b (k) for each k = 1, . . . , m, andb (k) =b (k) if |L (k) | = 1, (ii)b (k−1) <b (k) for all k = 2, 3, . . . , m.Proof. For part (i),b (1) ≥b (1) = 0 trivially holds. For all k = 2, 3, . . . , m, we havē
Figure 10 :
10The timeline of a trip to location i (above) and relocation without a rider (below).
Figure 11 :
11Total number of trips per day from the Chicago O'Hare International Airport.
Figure 12 :
12Average number of trips from the O'Hare International Airport, by hour-of-week. The gray stripes indicate the morning rush hours (7am -9am) and the green stripes indicate the evening rush hours (5pm -7pm).(a) Average trip duration.(b) Average fare per minute.
Figure 13 :
13Average trip duration (in minutes) and the average fare per minute by destination Census Tract in Chicago, for trips originating from the Chicago O'Hare International Airport.
Figures 16 and 17 plot the daily number of trips originating from Midway and the average number of of trips by hour-of-week. The weekly and seasonality patterns are similar to what we observed for
Figure 14 :
14The minimum and maximum waiting time for drivers who join the queue, in equilibrium in steady state, as the arrival rate of drivers varies. Chicago O'Hare.
Figure 15 :
15The minimum and maximum waiting time for drivers who join the queue, in equilibrium in steady state, as the patience level of the riders varies. Chicago O'Hare.
Figure 16 :
16Total number of trips per day from the Chicago Midway International Airport.
Figure 17 :
17Average number of trips from the Midway International Airport, by hour-of-week.(a) Trip count by destination.(b) Net earnings by destination.
Figure 18 :
18Trip volume and net earnings (assuming c = 1/3) by destination Census Tract in Chicago, for trips originating from the Chicago Midway International Airport.
Figure 19 :
19Equilibrium net revenue, trip throughput, and length of the queue in steady state, as the arrival rate of drivers varies. Chicago Midway.
Figure 20 :
20Drivers' average waiting times, total payoff, and the standard deviation (SD) in drivers' total payoff in equilibrium in steady state, as the arrival rate of drivers varies. Chicago Midway.
Figure 21 :
21The minimum and maximum waiting time for drivers who join the queue, in equilibrium in steady state, as the arrival rate of drivers varies. Chicago Midway.
Figure 22 :
22Equilibrium net revenue, trip throughput, and length of the queue in steady state, as the rider patience level varies. Chicago Midway.
Figure 23 :
23Drivers' average waiting times, total payoff, and the standard deviation (SD) in drivers' total payoff in equilibrium in steady state, as the rider patience level varies. Chicago Midway.
Figure 24 :
24The minimum and maximum waiting time for drivers who join the queue, in equilibrium in steady state, as the patience level of the riders varies. Chicago Midway.
Figure 8: Equilibrium net revenue, trip throughput, and length of the queue in steady state, as riders' patience level varies. Chicago O'Hare.0
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The specific policies and their implementations vary across companies and geographical regions. As an example, see https://www.uber.com/blog/california/keeping-you-in-the-drivers-seat-1/ (accessed 02/21/2021). This is in part due to regulatory requirements for categorizing drivers as independent contractors. For context, we cite an excerpt from California Proposition 22: "The network company does not require the app-based driver to accept any specific rideshare service or delivery service request as a condition of maintaining access to the network company's online-enabled application or platform."
The same optimal outcome in Ashlagi et al.[2020] can also be achieved by a FIFO queue that allows agents to decline undesired items, assuming that the items are infinitely patient, and that the mechanism does not have to reveal full information on the offered items to the agents.
The opportunity costs for drivers captures the value of their forgone outside options, which include, for example, the potential earnings a driver can make from driving elsewhere in the city for the same platform instead of waiting in the queue. Having driver supply tied-up in the queue is thereby potentially costly for the platform as well.9 Drivers who are waiting in the queue may not drive for the same platform at all times, and the market might be oversupplied already. As a result, the opportunity cost for the platform cp may be lower than that for the drivers.
When a mechanism is allowed to make dispatch decisions based on drivers' actions in the past, the mechanism may easily align incentives by no longer sending any trip offers to a driver who had declined a dispatch, for example.
For simplicity of analysis, we work in this section with Nash equilibrium in steady state because drivers' equilibrium strategy depends on the length of the queue when dispatches are randomized.
See Appendix C.1 for more details. Note that without driver identifiers, we are not able to appropriately estimate the continuation payoff of drivers after arriving at different destinations. As a result, the net earnings used in our simulations incorporate only payments from the immediate trip, effectively assuming that there is no heterogeneity in the continuation earnings from different locations onward.
In other words, we allow the driver to remain in the k th bin forever, instead of forcing her to move pastb(k) .
This is not allowed under randomized FIFO-we construct this scenario for the purpose of this proof only.
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| []
|
[
"Electromagnetic and strong isospin-breaking corrections to the muon g − 2 from Lattice QCD+QED",
"Electromagnetic and strong isospin-breaking corrections to the muon g − 2 from Lattice QCD+QED"
]
| [
"D Giusti \nDipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy\n",
"V Lubicz \nDipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy\n",
"G Martinelli \nDipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy\n",
"F Sanfilippo \nDipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy\n",
"S Simula \nDipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy\n"
]
| [
"Dipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy",
"Dipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy",
"Dipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy",
"Dipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy",
"Dipartimento di Matematica e Fisica\nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare\nUniversità degli Studi Roma Tre\nINFN\nSezione di Roma Tre\nVia della Vasca Navale 84, Università degli Studi di Roma \"La Sapienza\" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy"
]
| []
| We present a lattice calculation of the leading-order electromagnetic and strong isospinbreaking corrections to the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon. We employ the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with N f = 2 + 1 + 1 dynamical quarks at three values of the lattice spacing (a 0.062, 0.082, 0.089 fm) with pion | 10.1103/physrevd.99.114502 | [
"https://arxiv.org/pdf/1901.10462v1.pdf"
]
| 118,885,855 | 1901.10462 | ce06b1120b860a8e65a6d50c61bf9695af48bbca |
Electromagnetic and strong isospin-breaking corrections to the muon g − 2 from Lattice QCD+QED
28 Jan 2019
D Giusti
Dipartimento di Matematica e Fisica
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare
Università degli Studi Roma Tre
INFN
Sezione di Roma Tre
Via della Vasca Navale 84, Università degli Studi di Roma "La Sapienza" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy
V Lubicz
Dipartimento di Matematica e Fisica
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare
Università degli Studi Roma Tre
INFN
Sezione di Roma Tre
Via della Vasca Navale 84, Università degli Studi di Roma "La Sapienza" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy
G Martinelli
Dipartimento di Matematica e Fisica
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare
Università degli Studi Roma Tre
INFN
Sezione di Roma Tre
Via della Vasca Navale 84, Università degli Studi di Roma "La Sapienza" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy
F Sanfilippo
Dipartimento di Matematica e Fisica
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare
Università degli Studi Roma Tre
INFN
Sezione di Roma Tre
Via della Vasca Navale 84, Università degli Studi di Roma "La Sapienza" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy
S Simula
Dipartimento di Matematica e Fisica
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare
Università degli Studi Roma Tre
INFN
Sezione di Roma Tre
Via della Vasca Navale 84, Università degli Studi di Roma "La Sapienza" and INFN, Sezione di Roma, Piazzale Aldo Moro 5, Sezione di Roma Tre, Via della Vasca Navale 84, II-00146, 00185, 00146Rome, Roma, RomeItaly, Italy, Italy
Electromagnetic and strong isospin-breaking corrections to the muon g − 2 from Lattice QCD+QED
28 Jan 20192
We present a lattice calculation of the leading-order electromagnetic and strong isospinbreaking corrections to the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon. We employ the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with N f = 2 + 1 + 1 dynamical quarks at three values of the lattice spacing (a 0.062, 0.082, 0.089 fm) with pion
The muon anomalous magnetic moment a µ ≡ (g − 2)/2 is one of the most precisely determined quantities in particle physics. It is experimentally known with an accuracy of 0.54 ppm [1] (BNL E821) and the current precision of the Standard Model (SM) prediction is at the level of 0.4 ppm [2].
The discrepancy between the experimental value, a exp µ , and the SM prediction, a SM µ , corresponds to 3.7 standard deviations, namely a exp µ − a SM µ = 27.1 (7.3) · 10 −10 [3]. Since the above tension may be an exciting indication of new physics (NP) beyond the SM, an intense research program is currently underway in order to achieve a significant improvement of the uncertainties. The forthcoming g − 2 experiments at Fermilab (E989) [4] and J-PARC (E34) [5] aim at reducing the experimental uncertainty by a factor of four, down to 0.14 ppm, making the comparison of the experimental value a exp µ with the theoretical prediction a SM µ one of the most stringent tests of the SM in the quest of NP effects. On the theoretical side, the main uncertainty on a SM µ comes from hadronic contributions, related to the hadronic vacuum polarization (HVP) and light-by-light terms [6,7]. With the planned reduction of the experimental error, the uncertainty of the hadronic corrections will soon become the main limitation of this SM test.
The theoretical predictions for the hadronic contribution a HVP µ have been traditionally obtained from experimental data using dispersion relations for relating the HVP function to the experimental cross section data for e + e − annihilation into hadrons [8,9]. An alternative approach, proposed in Refs. [10][11][12], is to compute a HVP µ in Lattice QCD from the Euclidean correlation function of two electromagnetic (em) currents. In this respect an impressive progress in the lattice determinations of a HVP µ , which at leading order in α em is a quantity of order O(α 2 em ), has been achieved in the last few years [13][14][15][16][17][18][19][20][21][22][23][24].
With the increasing precision of the lattice calculations, it becomes necessary to include strong and em isospin-breaking (IB) corrections, which contribute to the HVP at order O(α 2 em (m d − m u )/Λ QCD ) and O(α 3 em ), respectively. In Ref. [20] a lattice calculation of the IB corrections to the HVP contributions due to strange and charm quarks, δa HVP µ (s) and δa HVP µ (c) 1 , was carried out using the RM123 approach [25,26], which is based on the expansion of the path integral in powers of the mass difference (m d − m u ) and of the em coupling α em . The quenched QED (qQED) approximation, which neglects the effects of sea-quark charges, was adopted and quark disconnected contractions were not included because of the large statistical fluctuations of the corresponding signals. The dominant source of uncertainty in the results of Ref. [20] was related to the em corrections to the renormalization constant (RC) of the local vector current, computed through the axial Ward-Takahashi identity derived in the QCD+QED theory.
In this work we present our determination of the IB corrections to the HVP contribution due to the light u-and d-quarks, δa HVP µ (ud), using the same methods and lattice setup adopted in Ref. [20] in the case of the strange and charm contributions. A preliminary result for δa HVP µ (ud) was presented in Ref. [27]. Thanks to a recent nonperturbative evaluation of QCD+QED effects on the RCs of bilinear operators performed in Refs. [28,29] we can update the determinations of δa HVP µ (s) and δa HVP µ (c) made in Ref. [20], obtaining a drastic improvement of the uncertainty by a factor of ≈ 4 and ≈ 6, respectively.
Within the qQED approximation and neglecting quark-disconnected diagrams the main results of the present study are:
δa HVP µ (ud) = 7.2 (1.1) stat+f it (1.4) input (1.3) chir (1.2) FVE (0.5) a 2 · 10 −10 = 7.2 (2.5) · 10 −10 ,(1)δa HVP µ (s) = −0.0219 (25) stat+f it (17) input (1) chir (3) FVE (1) a 2 · 10 −10 = −0.0219 (30) · 10 −10 ,(2)δa HVP µ (c) = −0.0344 (13) stat+f it (16) input (1) chir (1) FVE (1) a 2 · 10 −10 = −0.0344 (21) · 10 −10 ,(3)
where the errors come from (statistics + fitting procedure), input parameters, chiral extrapolation, finite-volume and discretization effects. Thus, we confirm that the em corrections δa HVP µ (s) and
δa HVP µ (c) turn out to be negligible with respect to the current uncertainties of the corresponding lowest-order terms a HVP µ (s) = 53.1 (2.5) · 10 −10 and a HVP µ (c) = 14.75 (0.56) · 10 −10 determined in Ref. [20]. In the case of the u-and d-quarks our finding (1) corresponds to about 1.2% of the lowest-order value a HVP µ (ud) = 619 (17.8) · 10 −10 obtained recently in Ref. [24].
Recent calculations of the IB corrections to the HVP are: δa HVP µ (ud) = 9.0 (4.5) · 10 −10 from FNAL/HPQCD/MILC [21], which includes only strong IB effects, and δa HVP µ (ud) = 9.5 (10.2) · 10 −10 from RBC/UKQCD [23], which includes also one disconnected QED diagram. In Ref. [22] the BMW collaboration has estimated the value δa HVP µ (ud) = 7.8 (5.1) · 10 −10 from results of the dispersive analysis of e + e − data [7]. In the case of the strange contribution δa HVP µ (s) RBC/UKQCD has recently obtained the result δa HVP µ (s) = −0.0149 (32) · 10 −10 [23], which confirms the smallness of such contribution though it differs slightly from our finding (2).
Summing up the three contributions (1)-(3) and adding a further ≈ 15% uncertainty related to the qQED approximation and to the neglect of quark-disconnected diagrams (see Section III), we get δa HVP µ (udsc) = 7.1 (2.6) (1.2) qQED+disc · 10 −10 = 7.1 (2.9) · 10 −10 ,
which represents the most accurate determination of the IB corrections to a HVP µ to date.
The paper is organized as follows. In section II we describe the evaluation of the em and strong IB corrections to the light-quark HVP contribution at order O(α 2 em (m d − m u )/Λ QCD ) and O(α 3 em ) using the RM123 approach [25,26]. Details of the lattice simulations are collected in the Appendix A. In section III we describe the extrapolation to the physical pion mass and to the continuum and infinite volume limits. Finally, section IV contains our conclusions and outlooks for future developments.
II. ISOSPIN-BREAKING CORRECTIONS IN THE RM123 APPROACH
We adopt the time-momentum representation for the evaluation of the HVP contribution a HVP µ to the muon (g − 2), namely [30] a HVP
µ = 4α 2 em ∞ 0 dt K µ (t)V (t) ,(5)
where the kernel function K µ (t) is given by
K µ (t) = 4 m 2 µ ∞ 0 dω 1 √ 4 + ω 2 √ 4 + ω 2 − ω √ 4 + ω 2 + ω 2 cos(ωm µ t) − 1 ω 2 + 1 2 m 2 µ t 2(6)
with m µ being the muon mass. In Eq. (5) the quantity V (t) is the vector current-current Euclidean correlator defined as
V (t) ≡ 1 3 i=1,2,3 d x J i ( x, t)J i (0) ,(7)
where
J µ (x) ≡ f =u,d,s,c,... J f µ (x) = f =u,d,s,c,... q f ψ f (x)γ µ ψ f (x) (8)
is the em current operator with q f being the electric charge of the quark with flavor f in units of the electron charge e, while ... means the average of the T -product over gluon and quark fields.
We will limit ourselves to the HVP contribution of the light u and d quarks, indicated by
For sake of simplicity we drop the suffix (conn), but it is understood that in the following we refer always to quark-connected contractions only.
In the RM123 method of Refs. [25,26] the vector correlator V (t) is expanded into a lowestorder contribution V ud (t), evaluated in isospin symmetric QCD (i.e. m u = m d and α em = 0), and a correction δV ud (t) computed at leading order in the small parameters (m d − m u )/Λ QCD and α em :
V (t) = V ud (t) + δV ud (t) + . . . ,(10)
where the ellipses stand for higher order terms in (m d − m u )/Λ QCD and α em .
The separation between the isosymmetric QCD and the IB contributions, V ud (t) and δV ud (t), is prescription dependent. In this work we follow Ref. [20] and we impose the matching condition in which the renormalized coupling and quark masses in the full theory, α s and m f , and in isosym-
metric QCD, α(0)
s and m
(0) f , coincide in the MS scheme at a scale of 2 GeV. Such a prescription is known as the GRS one [31].
The calculation of the IB correlator δV ud (t) requires the evaluation of the self-energy, exchange, tadpole, pseudoscalar and scalar insertion diagrams depicted in Fig. 1. More specifically one has
(a) (b) (c) (d) (e)δV ud (t) ≡ δV J (t) + δV T (t) + δV P S (t) + δV S (t) + δV SIB (t) + δV Z A (t) (11) 1 2 ψ f (y)(iτ 3 γ 5 − γ ν )U ν (y)ψ f (y + aν) + ψ f (y + aν)(iτ 3 γ 5 + γ ν )U † ν (y)ψ f (y) .Z QCD+QED m = Z m (1 − 4πα em Z m ) + O(α m em α n s ) , (m > 1, n ≥ 0)(19)
where Z m is the mass RC in QCD only and the product Z m Z m encodes the corrections at first order in α em . The quantity Z m can be written as
Z m = Z (1) m · Z f act m ,(20)
where Z
(1) m is the pure QED contribution at leading order in α em , given in the MS scheme at the renormalization scale µ by [32,33]
Z (1) m = q 2 f 16π 2 [6 ln(aµ) − 22.5954] ,(21)
while Z f act m accounts for the corrections of order O(α n s ) with n ≥ 1 to Eq. (20). It represents the QCD correction to the "naive factorization" approximation Z m = Z
u = m (0) d = m (0)
ud being the renormalized light-quark mass in isosymmetric QCD.
In the numerical evaluation of the photon propagator the photon zero-mode has been removed according to the QED L prescription [34], i.e. the photon field A µ satisfies A µ (k 0 , k = 0) ≡ 0 for all k 0 .
In this work we make use of the same isosymmetric QCD gauge ensembles used in Ref. [20], i.e. those generated by the European Twisted Mass Collaboration (ETMC) with N f = 2 + 1 + 1 dynamical quarks, which include in the sea, besides two light mass-degenerate quarks, also the strange and the charm quarks with masses close to their physical values [35,36]. For earlier investigations of finite volume effects (FVEs) the ETMC produced three dedicated ensembles, A40.20, A40.24 and A40.32 (see Appendix A for details), which share the same light-quark mass and lattice spacing and differ only in the lattice size L. To improve such an investigation a further gauge ensemble, A40.40, has been generated at a larger value of the lattice size L.
For our maximally twisted-mass setup δm crit f has been determined in Ref. [37], while 1/Z m = Z P , where Z P is the RC of the pseudoscalar density evaluated in Ref. [38]. The coefficient Z f act m has been recently computed in Ref. [28] in a non-perturbative framework within the RI -MOM scheme [39].
Within the qQED approximation, which treats the dynamical quarks as electrically neutral particles, the correlator δV J (t) corresponds to the sum of the diagrams (1a)-(1b), while the correlators δV T (t) and δV P S (t) represent the contributions of the diagrams (1c) and (1d), respectively. The diagram (1e) contributes to both δV S (t) and δV SIB (t).
In our numerical simulations we have adopted the following local version of the vector current:
J µ (x) = Z A q f ψ f (x)γ µ ψ f (x) ,(22)
where ψ f and ψ f represent two quarks with the same mass, charge and flavor, but regularized with opposite values of the Wilson r-parameter (i.e. r f = −r f ). Being at maximal twist the current (22) renormalizes multiplicatively with the RC Z A of the axial current. By construction the local current (22) does not generate quark-disconnected diagrams.
As discussed in Ref. [20], the properties of the kernel function K µ (t), given by Eq. (6), guarantee that the contact terms, generated in the HVP tensor by a local vector current, do not contribute to both a HV P µ and its IB correction.
Since we have adopted the renormalized vector current (22), the contribution δV Z A (t), appearing in Eq. (11), takes into account the em corrections to the RC Z A in QCD+QED, namely
Z QCD+QED A = Z A (1 + 4πα em Z A ) + O(α m em α n s ) , (m > 1, n ≥ 0)(23)
where Z A is the RC of the axial current in pure QCD (determined in Ref. [38]), while the product Z A Z A encodes the corrections at first order in α em . The quantity Z A can be written as
Z A = Z (1) A · Z f act A ,(24)
where Z
(1)
A is the pure QED correction at leading order in α em , given by [32,33] Z (1)
A = −15.7963 q 2 f 16π 2 ,(25)
and Z f act A takes into account QCD corrections of order O(α n s ) with n ≥ 1 to Eq. (24). In this work we make use of the non-perturbative determination obtained in Ref. [28] within the RI -MOM scheme, which improves significantly the value Z f act A = 0.9 (1) obtained through the axial Ward-Takahashi identity in Ref. [20]. The values adopted for the coefficients Z f act m and Z f act A are collected in Table IV of Appendix A.
Thus, the IB term δV Z A (t) is simply given by
δV Z A (t) ≡ −0.2001 4πα em (q 4 /q 2 ) Z f act A V ud (t) ,(26)
where
q 4 /q 2 ≡ f =u,d q 4 f / f =u,d q 2 f = 17/45 and V ud (t)
is the lowest-order contribution of the light-quarks to the vector correlator, calculated for our lattice setup in Ref. [24].
To sum up, the IB corrections δV ud (t) can be written as the sum of two (prescription dependent) contributions as
δV ud (t) = δV QED (t) + δV SIB (t) ,(27)
where
δV QED (t) = δV J (t) + δV T (t) + δV P S (t) + δV S (t) + δV Z A (t)(28)
and δV SIB (t) is given by Eq. (16).
Within the qQED approximation, where the shift δm crit f is proportional to q 2 f [37], and neglecting quark-disconnected diagrams the QED correlator δV QED (t) is proportional to
q 4 ≡ f =u,d q 4 f = 17/81. Instead, the SIB correlator δV SIB (t) is proportional to f =u,d q 2 f (m (0) f −m f ) = (1/6) (m d − m u ).
Using as inputs the experimental charged-and neutral-kaon masses the value m d − m u = 2.38 (18) MeV was determined in Ref. [37] at the physical point in the MS(2 GeV) scheme. Such a value is adopted in Eq. (16) for all gauge ensembles.
In Fig. 2 we show the dependence of both δV QED (t) and δV SIB (t) on the time distance t in the case of the ETMC gauge ensemble D20.48 (see Appendix A).
III. RESULTS
A convenient procedure [18,20,24] consists in splitting Eq. (5) into two contributions corresponding to 0 ≤ t ≤ T data and t > T data , respectively. In the first contribution the vector correlator is numerically evaluated on the lattice, while for the second contribution an analytic representation is required. If T data is large enough that the ground-state contribution is dominant for t > T data and smaller than T /2 in order to avoid backward signals, the IB corrections δa HVP µ (ud) can be written as
δa HVP µ (ud) ≡ δa HVP µ (<) + δa HVP µ (>)(29)
with δa HVP
µ (<) = 4α 2 em T data t=0 K µ (t) δV ud (t) ,(30)δa HVP µ (>) = 4α 2 em ∞ t=T data +a K µ (t) δ Z ud V 2M ud V e −M ud V t = 4α 2 em ∞ t=T data +a K µ (t) Z ud V 2M ud V e −M ud V t δZ ud V Z ud V − δM ud V M ud V (1 + M ud V t) ,(31)
where M ud V is the ground-state mass of the lowest-order correlator V ud (t) and Z ud V is the squared matrix element of the vector current between the ground-state |V and the vacuum:
Z ud V ≡ (1/3) i=x,y,z f =u,d q 2 f | 0|ψ f (0)γ i ψ f (0)|V | 2 . In Ref.
[24] the ground-state masses M ud V and the matrix elements Z ud V have been determined using appropriate time intervals t min ≤ t ≤ t max for each value of β and of the lattice volume for the ETMC ensembles adopted in this work. For the reader's convenience the values chosen for t min and t max in Ref. [24] are shown in Table I I. Values of t min and t max adopted in Ref. [24] to extract the ground-state signal from the light-quark vector correlator V ud (t) for each value of β and of the lattice volume V /a 4 for the ETMC gauge ensembles adopted in this work (see Table III of Appendix A).
In Eq. (31) the quantities δM ud V and δZ ud V can be extracted respectively from the "slope" and the "intercept" of the ratio δV ud (t)/V ud (t) at large time distances (see Refs. [20,25,26,37]
), namely δV ud (t) V ud (t) − −−−−−−−−−−− → t>>a,(T −t)>>a δZ ud V Z ud V + δM ud V M ud V f ud (t)(32)
where
f ud (t) ≡ M ud V T 2 − t e −M ud V t − e −M ud V (T −t) e −M ud V t + e −M ud V (T −t) − 1 − M ud V T 2 ≈ − 1 + M ud V t(33)
is almost a linear function of the Euclidean time t. This procedure is shown in Fig. 3 in the case of the gauge ensemble D20.48.
The time dependencies of the integrand functions K µ (t) δV QED (t) and K µ (t) δV SIB (t) are shown in Fig. 4 in the case of the ETMC gauge ensemble B55.32 (see Appendix A). After summation over the time distance t, the SIB contribution dominates over the QED one.
The results for the separate contributions δa HVP Table I), together with the uncertainty (at 1σ level) of a linear fit applied to the data. (see Table I for the values of t min and t max , and it will not be given separately in the final error budget.
T data (t min + 2a) (t min + t max )/2 (t max − 2a) (T /2 − 4a)
We have considered also the ratio of the IB correction δa HVP For the separate QED and SIB contributions the FVEs differ qualitatively and quantitatively.
In the case of the QED data a power-law behavior in terms of the inverse lattice size 1/L is expected. According to the general findings of Ref. [40] the universal, structure-indepedent FVEs are expected to vanish, since they depend on the global charge of the meson states appearing in the spectral decomposition of the vector correlator, while the structure-dependent (SD) FVEs start at order O(1/L 2 ). Moreover, using the effective field theory approach of Ref. [41] one may argue that in the case of mesons with vanishing charge radius (as the ones appearing in the vector correlator) the SD FVEs start at order O(1/L 3 ) (see also Ref. [20]). In the case of the SIB correlator (16), since a fixed value m d −m u = 2.38 (18) MeV [37] is adopted for all gauge ensembles, an exponential dependence in terms of the quantity M π L is expected [42].
In Fig. 6 the data for the QED and SIB contributions to the ratio δa HVP Since the SIB data dominate over the QED ones, the FVEs for the ratio δa HVP For the combined extrapolations to the physical pion mass and to the continuum and infinite-2 We remind the reader that the lowest-order term a HVP µ (ud) has nonnegligible FVEs, which are exponentially suppressed in terms of MπL [42] (see Fig. 9 of Ref. [24]). volume limits we adopt the following fit ansatz:
δa HVP µ (ud) a HVP µ (ud) = δ 0 1 + δ 1 m ud + δ 1l m ud ln(m ud ) + δ 2 m 2 ud + D a 2 + δ F V E ,(34)
where the FVE term is estimated by using alternatively one of the fitting functions
δ F V E = F e −M L or δ F V E = F n M 2 16π 2 f 2 0 e −M L (M L) n (n = 1 2 , 1, 3 2 , 2)(35)
with B 0 and f 0 being the leading-order low-energy constants of ChPT and M 2 ≡ 2B 0 m ud . For the chiral extrapolation we consider either a quadratic (δ 1l = 0 and δ 2 = 0) or a logarithmic (δ 1l = 0 and δ 2 = 0) dependence. Half of the difference of the corresponding results extrapolated to the physical pion mass is used to estimate the systematic uncertainty due to the chiral extrapolation.
Discretization effects are estimated by including (D = 0) or excluding (D = 0) the term proportional to a 2 in Eq. (34). The free parameters to be determined by the fitting procedure are δ 0 , δ 1 , δ 1l (or δ 2 ), D and F (or F n ).
In our combined fit (34) the values of the free parameters are determined by a χ 2 -minimization procedure adopting an uncorrelated χ 2 . The uncertainties on the fitting parameters do not depend on the χ 2 -value, because they are obtained by using the bootstrap samplings of Ref. [38]. This guarantees that all the correlations among the lattice data points and among the fitting parameters are properly taken into account. The quality of our fitting procedure is illustrated in Fig. 7.
where the errors come in the order from (statistics + fitting procedure), input parameters of the eight branches of the quark mass analysis of Ref. [38], chiral extrapolation, finite-volume and discretization effects. In Eq. (36) the uncertainty in the square brackets corresponds to the sum in quadrature of the statistical and systematic errors.
Using the leading-order result a HVP µ (ud) = 619.0 (17.8) · 10 −10 from Ref. [24], we obtain our determination of the leading-order IB corrections to a HVP
The above results show that the IB correction (37) is dominated by the strong SU (2)-breaking term, which corresponds roughly to ≈ 85% of δa HVP µ (ud).
Our determination (37), obtained with N f = 2 + 1 + 1 dynamical flavors of sea quarks, agrees within the errors with and is more precise than both the phenomenological estimate δa HVP µ (ud) = 7.8 (5.1) · 10 −10 , obtained by the BMW collaboration [22] using results of the dispersive analysis of e + e − data [7], and the lattice determination δa HVP µ (ud) = 9.5 (10.2) · 10 −10 , obtained by the RBC/UKQCD collaboration [23] at N f = 2+1, which includes also one disconnected QED diagram.
Recently, adopting N f = 1 + 1 + 1 + 1 simulations, the FNAL/HPQCD/MILC collaboration has found for the SIB contribution the value δa HVP µ (ud) (SIB) = 9.0 (4.5) · 10 −10 [21].
Thanks to the recent nonperturbative evaluation of QCD+QED effects on the RCs of bilinear operators performed in Refs. [28,29] we can update the determinations of the strange δa HVP [20]. Recently [23] in the case of the strange contribution the RBC/UKQCD collaboration has found the result δa HVP µ (s) = −0.0149 (32)·10 −10 , which deviates from our finding (40) by ≈ 1.6 standard deviations.
The sum of our three results (37), (40) and (41) Recently, in Ref. [23] one QED disconnected diagram has been calculated in the case of the u-and d-quark contribution and found to be of the same order of the corresponding QED connected term.
Thus, we estimate that the uncertainty related to the qQED approximation and to the neglect of quark-disconnected diagrams is approximately equal to our QED contribution (38), obtaining δa HVP µ (udsc) = 7.1 (2.6) (1.2) qQED+disc · 10 −10 = 7.1 (2.9) · 10 −10 ,
which agrees within the errors with the recent determinations based on dispersive analyses of the experimental cross section data for e + e − annihilation into hadrons (see Ref.
[3] and references therein).
IV. CONCLUSIONS
We have presented a lattice calculation of the isospin-breaking corrections to the HVP contribution of light quarks to the anomalous magnetic moment of the muon at order O[α 2 em (m d − m u )/Λ QCD ] in the light-quark mass difference and O(α 3 em ) in the em coupling. We have employed the gauge configurations generated by ETMC with N f = 2+1+1 dynamical quarks at three values of the lattice spacing (a 0.062 − 0.089 fm) with pion masses in the range M π 210 − 450 MeV and with strange and charm quark masses tuned at their physical values determined in Ref. [38].
The calculation of the IB corrections has been carried out adopting the RM123 approach of Refs. [25,26], which is based on the expansion of the lattice path-integral in powers of the small parameters (m d − m u )/Λ QCD and α em , which are both of the order of O(1%).
In this work we have taken into account only connected diagrams in which each quark flavor contributes separately. The leading-order em contributions to the renormalization constant of the local version of the lattice vector current, adopted in this work, have been evaluated using a recent nonperturbative calculation performed within the RI -MOM scheme in Refs. [28,29] . Thanks to that we have updated also the determinations of the strange δa HVP µ (s) and charm δa HVP µ (c) IB contributions made in Ref. [20], obtaining a drastic improvement of the uncertainties.
Within the qQED approximation and neglecting quark-disconnected diagrams the main results of the present study are:
δa HVP µ (ud) = 7.2 (2.5) · 10 −10 ,(44)δa HVP µ (s) = −0.0219 (30) · 10 −10 ,(45)
δa HVP µ (c) = −0.0344 (21) · 10 −10 ,
Summing up the three contributions (44)- (46) and adding a further ≈ 15% uncertainty related to the qQED approximation and to the neglect of quark-disconnected diagrams, we get δa HVP µ (udsc) = 7.1 (2.6) (1.2) qQED+disc · 10 −10 = 7.1 (2.9) · 10 −10 ,
which represents the most accurate determination of the IB corrections to a HVP µ to date.
New QCD simulations with N f = 2+1+1 dynamical quarks close to the physical pion point [43] and the evaluation of quark-disconnected diagrams are in progress. and of the product M π L for the 16 ETMC gauge ensembles with N f = 2 + 1 + 1 dynamical quarks used in this contribution (see Ref. [38]) and for the gauge ensemble, A40.40 added to improve the investigation of FVEs. The bare twisted masses µ σ and µ δ describe the strange and charm sea doublet according to Ref. [51].
The central values and errors of the pion mass are evaluated using the bootstrap events of the eight branches of the analysis of Ref. [38]. The valence quarks in the pion are regularized with opposite values of the Wilson r-parameter in order to guarantee that discretization effects on the pion mass are of order O(a 2 µ ud Λ QCD ).
randomly. In the case of the light-quark contribution we have used 160 stochastic sources (diagonal in the spin variable and dense in the color one) per each gauge configuration.
masses between 210 and 450 MeV. The results are obtained adopting the RM123 approach in the quenched-QED approximation, which neglects the charges of the sea quarks. Quark disconnected diagrams are not included. After the extrapolations to the physical pion mass and to the continuum and infinite-volume limits the contributions of the light, strange and charm quarks are respectively equal to δa HVP µ (ud) = 7.2 (2.5) · 10 −10 , δa HVP µ (s) = −0.0219 (30) · 10 −10 and δa HVP µ (c) = −0.0344 (21) · 10 −10 . At leading order in α em and (m d − m u )/Λ QCD we obtain δa HVP µ (udsc) = 7.1 (2.9) · 10 −10 , which is currently the most accurate determination of the isospin-breaking corrections to a HVP µ . arXiv:1901.10462v1 [hep-lat] 28 Jan 2019 I. INTRODUCTION
FIG. 1 .
1Fermionic connected diagrams contributing to the IB corrections to a HVP µ (ud): self-energy (a), exchange (b), tadpole (c), pseudoscalar (d) and scalar (e) insertions. Solid lines represent light-quark propagators in isosymmetric QCD.
. (14) δm crit f is the em shift of the critical mass for the quark flavor f . In Eq. (15) the quantity Z m is related to the em corrections to the mass RC in QCD+QED, Z QCD+QED m , as
m
(i.e. Z f act m = 1) adopted in Ref. [20]. Finally, Eq. (16) corresponds to the strong IB (SIB) effect (in the GRS prescription) with m (0)
FIG. 2 .
2Left panel: time dependence of the IB contribution δV QED (t) [see Eq. (28)] in lattice units in the case of the gauge ensemble D20.48 (see Appendix A). Right panel: the same as in the left panel, but for the SIB term δV SIB (t) [see Eq. (16)]. The simulated pion mass is M π 260 MeV and the lattice spacing is equal to a 0.06 fm.
(
ud), are obtained adopting four choices of T data , namely: T data = (t min +2a), (t min +t max )/2, (t max − 2a) and (T /2 − 4a). These results are collected inTable IIfor some of the ETMC gauge ensembles. We find that the separation between δa HVPµ (<) and δa HVP µ (>) depends on the specific value of T data , as it should be, but their sum δa HVP µ (ud) is independent of the specific choice of the FIG. 3. Ratios δV QED (t)/V ud (t) (left panel) and δV SIB (t)/V ud (t) (right panel) in the case of the gauge ensemble D20.48 versus the time distance t. The shaded areas correspond to the time interval where the ground-state is dominant (see
FIG. 4 .
4Time dependence of the integrand functions K µ (t) δV QED (t) (left panel) and K µ (t) δV SIB (t) (right panel) for the u and d-quark contributions to the IB corrections δa ud µ [see Eq. (30)] in the case of the ETMC gauge ensemble B55.32. The simulated pion mass is M π 375 MeV and the lattice spacing is equal to a 0.082 fm. In the left panel the labels "self ", "exch", "T+PS", "S', "Z A " indicate the QED contributions of the diagrams (1a), (1b), (1c)+(1d), (1e) and the one generated by the QED corrections to the RC Z A of the local vector current [see Eq. (26)]. value of T data within the statistical uncertainties. Note that for T data = t max − 2a the contribution δa HVP µ (>), which depends on the identification of the ground-state signal, is still a significant fraction of the total value δa HVP µ (ud), as it was already observed in the case of the lowest-order term a HVP µ (ud) in Ref. [24]. All four choices of T data are employed in the various branches of our bootstrap analysis. The corresponding systematics is sub-dominant with respect to the other sources of uncertainties and ensemble A80.24
II. Results for the contributions δa HVP µ (<), δa HVP µ (>) and their sum δa HVP µ (ud), in units of 10 −10 , obtained adopting in Eqs. (30-31) four different choices of T data , namely: T data = (t min + 2a), (t min + t max )/2, (t max − 2a) and (T /2 − 4a) for the ETMC gauge ensembles A80.24, A50.32, B55.32 and D30.48
), which was evaluated in Ref.[24] for the same gauge ensembles. The attractive feature of the ratio δa HVPµ (ud)/a HVP µ (ud) is to be less sensitive to some of the systematics effects, in particular to the uncertainties of the scale setting. The data for δa HVP µ (ud) and the ratio δa HVP µ (ud)/a HVP µ (ud) are shown respectively in the left and right panels of Fig. 5. It can be seen that discretization effects play a minor role, while FVEs are more relevant.
FIG. 5 .
5Results for δa HVP µ (ud) (left panel) and the ratio δa HVP µ (ud)/a HVP µ (ud) (right panel) versus the renormalized average u/d mass m ud (in the MS(2 GeV) scheme). Errors are the quadrature of the statistical uncertainties and of the error generated by the uncertainties of the input parameters of the quark mass analysis of Ref. [38] (see Appendix A).
the case of the four ensembles A40.XX, which share common values of the light-quark mass and of the lattice spacing, but differ in the lattice size L. It can be seen that the theoretical expectations for the FVEs are consistent with the lattice data for both the QED and SIB contributions 2 .
to be mainly exponentially suppressed in M π L.
FIG. 6 .
6Results for the ratio δa HVP µ (ud)/a HVP µ (ud) versus the quantity M π L in the case of the four ensembles A40.XX, which share common values of the light-quark mass and of the lattice spacing, but differ in the lattice size L. The empty (full) markers correspond to the SIB (QED) contribution. The solid line is a fit of the SIB data using the phenomenological Ansatz A + Be −MπL . The dashed and dotted lines correspond to a fitting function of the form A + B/L n with n = 3 (dashed line) and n = 6 (dotted line) both applied to the QED data.
FIG. 7 .
7Results for the ratio δa HVP µ (ud)/a HVP µ (ud) versus the renormalized average u/d mass m ud in the MS(2 GeV) scheme. The empty markers correspond to the raw data, while the full ones represent the lattice data corrected by the FVEs obtained in the fitting procedure (34) with δ 1l = 0 and δ 2 = 0. The solid lines correspond to the results of the combined fit (34) obtained in the infinite-volume limit at each value of the lattice spacing. The black asterisk represents the value of the ratio δa HVP µ (ud)/a HVP µ (ud) extrapolated to the physical pion mass, corresponding to m phys ud (MS(2 GeV) = 3.70 (17) MeV and to the continuum limit, while the red area indicates the corresponding uncertainty as a function of m ud at the level of one standard deviation. Errors are statistical only. At the physical pion mass and in the continuum and infinite-volume limits we get δa HVP µ (ud) a HVP µ (ud) = 0.0117 (18) stat+f it (22) input (21) chir (19) FVE (8) a 2 [41] ,
stat+f it (1.4) input (1.3) chir (1.2) FVE (0.5) a 2 · 10 −10 = 7.2 (2.5) · 10 −10 ,(37)which comes (within the GRS prescription) from the sum of the QED contribution
) contributions to the IB effects made in Ref.[20]. We getδa HVP µ (s) = −0.0219 (25) stat+f it (17) input (1) chir (3) FVE (1) a 2 · 10 −10 = −0.0219 (30) · 10 −10 ,(40)δa HVP µ (c) = −0.0344 (13) stat+f it (16) input (1) chir (1) FVE (1) a 2 · 10 −10 = −0.0344 (21) · 10 −10(41)to be compared with δa HVPµ (s) = −0.018 (11) · 10 −10 and δa HVP µ (c) = −0.030 (13) · 10 −10 given in Ref. [20]. The updated results confirm that the em corrections δa HVP µ (s) and δa HVP µ (c) are negligible with respect to the current uncertainties of the corresponding lowest-order terms a HVP µ
yields the contribution of quark-connected diagrams to δa HVP µ within the qQED approximation, namely δa HVP µ (udsc)| conn = 7.1 (2.6) · 10 −10 .
which represents the most accurate determination of the IB corrections to a HVP µ to date. Using the recent ETMC determinations of the lowest-order contributions of light, strange and charm quarks, a HVP µ (ud) = 619.0 (17.8) · 10 −10 , a HVP µ (s) = 53.1 (2.5) · 10 −10 and a HVP µ (c) = 14.75 (0.56) · 10 −10[20,24], and an estimate of the lowest-order quark-disconnected diagrams, a HVP µ (disc) = −12 (4) · 10 −10 , obtained using the results of Refs.[22] and[23], our finding (42) for the IB corrections leads to an HVP contribution to the muon (g − 2) equal to a HVP µ = 682 (19) · 10 −10 ,
ACKNOWLEDGMENTS 267985 ".
267985We gratefully acknowledge the CPU time provided by CINECA under the initiative INFN-LQCD123 on the Marconi KNL and SKL systems at CINECA (Italy). We thank S. Bacchio and B. Kostrezwa for their help in setting up the interface and the parameters for the DDαAMG library [44] used to evaluate the quark propagators. We thank B. Kostrezwa for his help in the HMC simulations used to produce the A40.40 gauge ensemble with the tmLQCD software package [45-47]. G. M., V.L. and S.S. thank MIUR (Italy) for partial support under Contract No. PRIN 2015P5SBHT. G. M. thanks the partial support from ERC Ideas Advanced Grant No. DaMeSyFlaValues of the simulated-quark bare masses (in lattice units), of the pion mass (in units of MeV)
.β
V /a 4 t min /a t max /a
1.90 40 3 × 80
12
22
32 3 × 64
12
22
24 3 × 48
12
20
20 3 × 48
12
20
1.95 32 3 × 64
13
22
24 3 × 48
13
20
2.10 48 3 × 96
18
30
TABLE
TABLE
Table III of
IIIAppendix A for the ETMC gauge ensembles).
Finally, the values adopted the coefficients Z f act m [see Eq. (20)] and Z f act A [see Eq. (24)] are collected in Table IV. [1] G. W. Bennett et al. [Muon g-2 Collaboration], Phys. Rev. D 73 (2006) 072003 [hep-ex/0602035]. [2] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) 030001. [3] A. Keshavarzi et al., Phys. Rev. D 97 (2018) 114025 [arXiv:1802.02995 [hep-ph]].TABLE IV. Values adopted for the coefficients Z f act m [see Eq. (20)] and Z f act A [see Eq. (24)] for the M1 and M2 renormalization methods of Ref.[38] at the three values of β. In Ref.[20] a common value Z f act estimated through the axial Ward-Takahashi identity derived in the presence of QED effects, was adopted at all values of β.β
Z f act
m
(M1)
Z f act
A
(M1)
Z f act
m
(M2)
Z f act
A
(M2)
1.90
1.652 (35)
0.860 (17)
1.596 (11)
0.981 (11)
1.95
1.527 (35)
0.872 (12)
1.556 (12)
0.979 (8)
2.10
1.453 (14)
0.909 (7)
1.437 (5)
0.958 (4)
A
= 0.9 (1),
In the strange and charm sectors the strong IB corrections are absent at leading order in (m d − mu).
Appendix A: Simulation detailsThe ETMC gauge ensembles used in this work are the same adopted in Ref.[38]to determine the up-, down-, strange-and charm-quark masses in isospin symmetric QCD. We employ the Iwasaki action[48]for gluons and the Wilson Twisted Mass Action[49][50][51]for sea quarks. Working at maximal twist our setup guarantees an automatic O(a)-improvement[50,52].We consider three values of the inverse bare lattice coupling β and different lattice volumes, as shown inTable IIIWe made use of the bootstrap samplings elaborated for the input parameters of the quark mass analysis of Ref.[38]. There, eight branches of the analysis were adopted differing in:• the continuum extrapolation adopting for the scale parameter either the Sommer parameter r 0 or the mass of a fictitious pseudoscalar meson made up of strange(charm)-like quarks;• the chiral extrapolation performed with fitting functions chosen to be either a polynomial expansion or a Chiral Perturbation Theory (ChPT) Ansatz in the light-quark mass;• the choice between the methods M1 and M2, which differ by O(a 2 ) effects, used to determine the mass RC Z m = 1/Z P in the RI -MOM scheme.Throughout this work the renormalized average u/d quark mass m ud is given in the MS scheme at a renormalization scale equal to 2 GeV. We recall that, in the GRS prescription we have chosen, the renormalized average u/d quark mass in isosymmetric QCD m The statistical accuracy of the meson correlator is based on the use of the so-called "oneend" stochastic method[53], which includes spatial stochastic sources at a single time slice chosen
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| []
|
[
"Feynman-path analysis of Hardy's paradox: measurements and the uncertainty principle",
"Feynman-path analysis of Hardy's paradox: measurements and the uncertainty principle"
]
| [
"D Sokolovski \nSchool of Mathematics and Physics\nQueen's University of Belfast\nBT7 1NNBelfatsUnited Kingdom\n",
"I Puerto Giménez \nDepartamento de Física Fundamental II\nUniversidad de La Laguna\nLa Laguna38204\n\nS/C de Tenerife\nSpain\n",
"R Sala Mayato \nDepartamento de Física Fundamental II\nUniversidad de La Laguna\nLa Laguna38204\n\nS/C de Tenerife\nSpain\n"
]
| [
"School of Mathematics and Physics\nQueen's University of Belfast\nBT7 1NNBelfatsUnited Kingdom",
"Departamento de Física Fundamental II\nUniversidad de La Laguna\nLa Laguna38204",
"S/C de Tenerife\nSpain",
"Departamento de Física Fundamental II\nUniversidad de La Laguna\nLa Laguna38204",
"S/C de Tenerife\nSpain"
]
| []
| Hardy's paradox is analysed within Feynman's formulation of quantum mechanics. A transition amplitude is represented as a sum over virtual paths which different intermediate measurements convert into different sets of real pathways. Contradictory statements emerge when applying to the same statistical ensemble. The "strange" weak values result is also investigated in this context. | 10.1016/j.physleta.2008.02.079 | [
"https://arxiv.org/pdf/0903.4795v1.pdf"
]
| 17,257,698 | 0903.4795 | 7b3ae3d80c03eb835ec283d0f02f086863cc4025 |
Feynman-path analysis of Hardy's paradox: measurements and the uncertainty principle
27 Mar 2009
D Sokolovski
School of Mathematics and Physics
Queen's University of Belfast
BT7 1NNBelfatsUnited Kingdom
I Puerto Giménez
Departamento de Física Fundamental II
Universidad de La Laguna
La Laguna38204
S/C de Tenerife
Spain
R Sala Mayato
Departamento de Física Fundamental II
Universidad de La Laguna
La Laguna38204
S/C de Tenerife
Spain
Feynman-path analysis of Hardy's paradox: measurements and the uncertainty principle
27 Mar 2009PACS: 0365Ca, 0365Ta
Hardy's paradox is analysed within Feynman's formulation of quantum mechanics. A transition amplitude is represented as a sum over virtual paths which different intermediate measurements convert into different sets of real pathways. Contradictory statements emerge when applying to the same statistical ensemble. The "strange" weak values result is also investigated in this context.
Introduction
The Hardy's paradox first introduced in [1] continues to attract attention in the literature [2,3,4,5,6,7,8]. The paradox consists in that, in a two-particle interferometer set up, detection of particle(s) in different arms yields results obviously incompatible with each other. One way to resolve the paradox is by noting that the conflicting results refer not to the same but to different physical situations thereby avoiding counterfactual reasoning -in the words of Ref. [4] "talking about the values of non-measured attributes". The legitimacy of counterfactual statements was explored in [3,4] within the framework of formal logic. A different resolution involving inaccurate or weak quantum measurements was proposed in [5]. The purpose of this paper is to analyse the Hardy's paradox, counterfactual statements and the weak measurements in terms of virtual (Feynman) paths and the uncertainty principle [9,10]. In Feynman's quantum mechanics a transition probability amplitude is found by adding amplitudes for all interfering paths which, together, form a single indivisible pathway connecting the initial and final states of the system. The same paths can be either interfering or exclusive alternatives, depending on whether or not the system interacts with other (e.g. meter) degree's of freedom. In addition, Feynman's uncertainty principle states that any determination of the path taken must destroy the interference between the alternatives [9].
The simplest illustration of the dangers of counterfactual reasoning is Young's double-slit experiment. One observes the probabilities with which electron starting in its initial state (source) reaches a variety 1 E-mail address: [email protected] (D. Sokolovski); [email protected] (R. Sala Mayato) 1 of final states (points on the screen) via single pathway comprising both slits number 1 and 2 so that an interference pattern is produced. An accurate observation of the slit chosen by an electron produces a system in which each final state can be reached via two real pathways (one through each slit) travelled with probabilities P 1 and P 2 . It is a different system: the interference pattern is destroyed, and the probabilities to arrive in the final state do not agree with the unobserved ones. Answering the "which way?" question by attributing the probabilities P 1 and P 2 to unobserved system would constitute a counterfactual statement which is, obviously, wrong. Below we will show that the Hardy's set up is equivalent to a three-slit Young experiment, and that, in a similar way, the contradicting statements at the centre of the "paradox" refer to different sets of real pathways produced by different intermediate measurements and not to a single statistical ensemble.
The authors of [5] have come to the defence of counterfactual reasoning suggesting to resolve the paradox with the help of weak measurements [11,12,13,14] performed by a meter whose interaction with the system is so weak, that the interference between different paths is not destroyed. There are two distinct issues associated with the weak measurements. The answer to the first (and easier) question of whether they can be performed in practice is yes [6,7,8,15]. The second (and, to our knowledge not yet fully answered) question concerns the interpretation of the weak results. Weak measurements are often seen "as an extension to the standard von Neumann model of measurements" [8] which allows one to refer to some of the outcomes as "strange and surprising" [5]. The problem is captured by the following simple example: consider applying a weak measurement to determine the slit number in the above double-slit experiment. For a particle arriving near a minimum of the interference pattern one can obtain, say, a number 10 [16] (a similar unusual value has been obtained in the optical realisation of the experiment in [15]). This is not a valid (hence "surprising") slit number as there are only two holes drilled in the screen [17]. Possible interpretations are that (i) the weak measurement reveals some new information about how the particle traverses the screen or, (ii) that in the weak limit the meter is not working properly. Here we will follow Ref. [16] in adopting the second view and turn to the uncertainty principle for an explanation. If the measured result is wrong, what is the correct answer to the "which slit?" question? The uncertainty principle tells us that for an unobserved electron the paths through the first and the second holes form (and this is the only paradox of quantum mechanics [10]) a single indivisible pathway so that the correct answer does not exist. Thus a weak meter must either fail or the uncertainty principle would be proven wrong. The meter does fail by producing an answer which appears to have nothing to do with the original question. Mathematically, a weak value is an improper average obtained with an alternating non-probabilistic amplitude distribution [16] and as such is not tied to its support (in this case, slit numbers 1 and 2), and serves mostly to demonstrate that the particle cannot be seen as passing through a slit with any particular probability. We will argue further that the same can be said about the "resolution" of the Hardy's paradox, which just as the above example relies on the surprising value −1 of the pair occupation number.
The rest of the paper is organised as follows: in Section 2 we briefly describe measurements as a way of converting interfering virtual paths into exclusive real ones. In Section 3 we discuss the three-box case of Aharonov et al [14] and its relation to the uncertainty principle. In Section 4 we identify the virtual paths for the Hardy's set up. In Section 5 we analyse real pathways produced from these by different measurements. In Section 6 we consider differences between measurements with and without post-selection. In Section 7 we discuss weak measurements for the Hardy's scheme and show that a different choice of the final state can lead to extremely large "anomalous" weak occupation numbers. Secttion 8 contains our conclusions.
Quantum measurements and virtual paths
Consider a quantum system with a zero HamiltonianĤ = 0 in a N-dimesional Hilbert space. The system is prepared (pre-selected) in some state |i and, at a later time T , observed (post-selected) in a final state | f . It is convenient to choose an othonormal basis {|n }, n = 1, 2..N corresponding to the "position" operator [18]n ≡ ∑ N n=1 |n n n|. In general, the transition amplitude can be written as a sum over all virtual paths n(t) which take the values 1, 2...N at any given time t [19,20]. SinceĤ = 0, there are only N constant paths n(t) = 1, 2...N and the path decomposition of the transition amplitudes takes a simple form
f |i = N ∑ n=1 f |n n|i ≡ ∑ {n} Φ{n} ,(1)
where Φ{n} = f |n n|i is the amplitude for the n-th path. With these notations the problem becomes equivalent to a N-slit Young experiment with N discrete final destinations (positions on the screen) given by the chosen final state | f and any N − 1 orthonormal states spanning the Hilbert subspace orthogonal to | f . The virtual paths provide a convenient way to describe an intermediate von Neumann measurement at 0 < t < T of any operator F(n) which commutes with the positionn,
F(n) ≡ N ∑ n=1 |n F(n) n| ,(2)
where F(n) is an arbitrary function. For an accurate measurement the initial pointer position f is known exactly, its initial state is the delta-function δ( f ) and the probability amplitude Φ( f ) to register a meter reading f is given by
Φ( f ) = ∑ {n} δ( f − F(n))Φ{n} .(3)
It is readily seen that if F(n) has M < N distinct eigenvalues F 1 , F 2 ,... The transition probabilities are, in general, altered by the measurement,
P f ←i ≡ M ∑ j=1 | N ∑ n=1 δ(F(n) − F j )Φ{n}| 2 = | f |i | 2 .(4)
Note that if all eigenvalues of F(n) are the same, no new real pathways are produced, nothing is measured and no perturbation is incurred. Note also that Eq. (4) is just an example of the Feynman's rule for assigning probabilities [10] which prescribes adding amplitudes for the interfering, and probabilities for the exclusive alternatives.
Another parameter which affects both the obtained information and the incurred perturbation is the accuracy of the measurement ∆ f . If the meter's sharp initial state is replaced with a distributed one, δ( f ) → G( f ), where G( f ) can be chosen a Gaussian with a width ∆ f , the amplitude to obtain a reading f is given by
Ψ( f ) = Z d f ′ G( f − f ′ )Φ( f ′ ) = ∑ n G( f − F(n))Φ{n} .(5)
In general, an inaccurate measurement produces a continuum of real pathways labelled by the variable f . The same path {n} contributes to many pathways, its contribution to Ψ( f ) being G( f − F(n))Φ{n}. The perturbation decreases with the increase of the uncertainty ∆ f and becomes negligible for ∆ f >> δ f where δ f is the difference between the largest and the smallest eigenvalue of F(n). Application of the meter under these conditions leads to so-called weak measurements first proposed in [11,12,13] and recently discussed in [16]. We will return to weak measurements in Section 7.
The uncertainty principle and the three-box example
The uncertainty principle states that [9]: "Any determination of the alternative taken by a process capable of following more that one alternative destroys the interference between alternatives". An illustration of what the principle may mean in practical circumstances is provided by the three-box case by Aharonov et al [13,14,21,22,23]. Consider a three-state system with a zero Hamiltonian pre-and post-selected in states such that the amplitudes in (3) are (β < 1 is real)
Φ{1} = β, Φ{2} = −β and Φ{3} = −β(6)
so that for an unobserved system we have
f |i = ∑ {n} Φ{n} = β .(7)
One sees that there is a cancellation between the paths but cannot decide whether it is paths {1} and {2} or {1} and {3} that make each other redundant. All three paths must therefore be treated as a single indivisile pathway, denoted {1 + 2 + 3}, travelled with the probability P f ←i
{1+2+3} = |Φ{1} + Φ{2} + Φ{3}| 2 = β 2 .
If one decides to accurately measure the projector on |2 , whose matrix form iŝ
P 2 ≡ diag(0, 1, 0) ,(8)
in order to see that the particle does indeed follow the path {2}, the meter will read 1 confirming the assumption and the particle will be detected in | f with the same probability P f ←i = β 2 . Perhaps surprisingly, measurement of the projectorP 3 ≡ diag(0, 0, 1) (9)
will confirm that particle always travels along the path {3}. Assuming that these results refer to the same system (ensemble) prompts a somewhat paradoxical conclusion that the particle is in several places simultaneously [13,14]. This "paradox" disappears [23] once one notices that measurement ofP 2 creates a new network in which a final state can be reached via two real pathways, {2}, and a coherent superposition of the two remaing ones, {1 + 3}. In a similar way, measureament ofP 3 fabricates yet another different statistical ensemble with two real pathways, {3} and {1 + 2}. What is true for one ensemble is not true for the other even though both are produced by observations made on the same system. Equally, neither measurement reveals what "actually" happens in the unobserved system where the "which way?" information is lost, in accordance with the uncertainty principle, to quantum interference.
Virtual paths in the Hardy's setup
The Hardy's set up shown in Fig. 1 has been discussed by many authors [1,2,5,6,7] and a brief description will suffice here. An electron (e−) and a positron (e+) are injected into their respective Mach-Zehnder interferometers, each equipped with two detectors, labelled C−, D−, C+ and D+. So long as two interferometers are independent, the outcome of the experiment consists in two detectors clicking in coincidence.
There are, therefore, four possible final outcomes. Each particle has a choice of one of the two arms of the corresponding interferometer, so that there are four virtual paths to reach each outcome. The system is, therefore, equivalent to the four-slit Young diffraction experiment, with a minor distinction that there are only four discrete final states (positions on the screen). Hardy changes this arrangement by allowing two interferometer arms, labelled 2− and 2+ overlap so that if both particles are injected simultaneously and choose to travel along the overlapping arms, annihilation follows with certainty. Further, each interferometer is tuned in such a way that neither D− nor D+ click if only an electron or a positron is injected. Now the outcome of the experiment consist in two of the four detectors clicking in coincidence or in a photon, γ, produced in annihilation. There are, therefore, five possible final states. Let the orthogonal state |1 − (|2 + ) and |2 − (|1 + ) correspond to the electron (positron) travelling via non-overlapping (overlapping) arms and vice versa of the corresponding interferometer, respectively. Similarly, the states |1 − |1 + and |2 − |2 + correspond to both particle travelling via non-overlapping and overlapping arms. The four paths can be labelled as follows:
{1} via |1 − |1 + {2} via |1 − |2 + {3} via |2 − |1 + {4} via |2 − |2 + .(10)
Just past the point P in Fig.1, the state of the electron-positron pair is
|i = (|1 + |1 − + |1 + |2 − + |2 + |1 − + |γ )/2 ,(11)
where |γ is the state of the photon produced in annihilation. Of the five final states, the four corresponding to electron and positron arriving at the detectors (D−, D+), (C−, D+), (D−,C+) and (C−,C+), respectively, are given by Figure 2 shows the network of paths connecting the initial and final states of the Hardy's system. There is only one path which connects the initial state with |γ , while each of the final states (12)(13)(14)(15)
| f = (|1 − − |2 − )(|1 + − |2 + )/2 (12) |g = (|1 − + |2 − )(|1 + − |2 + )/2 (13) |h = (|1 − − |2 − )(|1 + + |2 + )/2 (14) | j = (|1 − + |2 − )(|1 + + |2 + )/2 .(15)Ψ z←i {1} = z|1 − |1 + 1 + | 1 − |i , z = f , g, h, j, γ .(16)
Probability amplitudes for the paths {2} and {3} can be found in asimilar manner and are given in Table.1. Paths connecting different final states (see Fig.2) are mutually exclusive and cannot interfere. If no additional observations are made on the system, virtual paths ending in the same final state are interfering alternatives and must be treated as a single real pathway. Coherence between such paths can be destroyed, e.g., by an intermediate accurate measurement of an operator. As discussed in Section 2, each such measurement would produce, depending on the multiplicity of the operator's eigenvalues, a number of additional real pathways. In the next Section we will consider these measurements in more detail.
Accurate measurements and real pathways in Hardy's set up. Counterfactuals
Following [5] we wish to perform intermediate measurements of "pair occupation" operators which establish whether the electron and the positron are propagating along specified arms of their respective interferometers. In the basis consisting of the states in the r.h.s. of Eq. (10) these projectors take the form
N(1 − |1+) = diag(1, 0, 0, 0) (17) N(1 − |2+) = diag(0, 1, 0, 0)(18)N(2 − |1+) = diag(0, 0, 1, 0) (19) N(2 − |2+) = diag(0, 0, 0, 1) .(20)
We will also require single particle occupation operators which establish whether the electron (positron) travels along the specified arm, while the position of the other member of the pair remains indeterminate,
N(1−) =N(1 − |1+) +N(1 − |2+) = diag(1, 1, 0, 0) = 1 −N(2−)(21)N(1+) =N(1 − |1+) +N(2 − |1+) = diag(1, 0, 1, 0) = 1 −N(2+) .(22)
Consider now accurate measurements of these operators for the system post-selected in | f (electron and positron are detected in D− and D+, respectively Fig. 4. The probabilities for these pathways are found by adding, where appropriate, the corresponding amplitudes in Table 1 and squaring the moduli,
P f ←i {1} = |Φ f ←i {1} | 2 = 1/16, P f ←i {2+3} = |Φ f ←i {2} + Φ f ←i {3} | 2 = 1/4 .(23)
Repeating the calculation for the caseN(1 − |2+) is measured yields
P f ←i {1} = |Φ f ←i {2} | 2 = 1/16, P f ←i {1+3} = |Φ f ←i {1} + Φ f ←i {3} | 2 = 0 ,(24)
suggesting that if D+ and D− click (which happens with a probability of 1/16) (I) the electron and the positron always travel along the non-overlapping and overlapping arms, respectively. Similarly, forN(2 − |1+) one finds
P f ←i {3} = 1/16, P f ←i {1+2} = 0 ,(25)
and might conclude, in contradiction to the above, that (II) the electron and the positron always travel along the overlapping and non-overlapping arms, respectively. For the single particle operators (21)(22) real pathways and corresponding probabilities can be constructed in a similar manner and are given in Table 2. Thus, measuringN(1−) we find that (III) the electron always travels along the overlapping arm. MeasuringN(1+) reveals that (IV) the positron always travels along the overlapping arm. The italicised statements (I),(II),(III) and (IV ), if referred to the same system, would imply that "and electron and a positron in some way manage to "be" and "not be" at the same time at the same location" [5]. As in Section 3 the "paradoxical" nature of the Hardy's example is removed once one notices that the above statements refer to different networks of classical (real) pathways produced from the same parent unobserved system. That these networks are indeed different is seen already from the fact that the transition probabilities to arrive in the final states |g , |h and | j in Table 2 are different for each choice of the measured quantity. Note that for the final state | f the transition probability remains unchanged, but only due to the special choice of the system's parameters. Thus, only the statement (I) applies under the conditionN(1 − |2+) is measured, while statements (II), (III) and (IV ) refer to unmeasured attributes and should be discarded. The approach to resolving quantum interference "paradoxes" based on avoiding counterfactual reasoning is by no means new [3,4,5]. We note however, that the path analysis with its notion of converting interfering virtual paths into exclusive real ones, provides a helpful insight into the argument.
"Which way?" probabilities without post-selection. The sum and the product rules
In the above P f ←i
{1} = |Φ f ←i
{1} | 2 gave the probability for the pair to travel along the non-overlapping arms provided the detectors D+ and D− click in coincidence. Alternatively, we may choose to record the frequency with which the route is travelled regardless of which detectors click, or just switch the detectors off completely. Bearing in mind that the paths connecting different final states cannot interfere we find the corresponding probability to be
P all←i {1} = |Φ f ←i {1} | 2 + |Φ g←i {1} | 2 + |Φ h←i {1} | 2 + |Φ j←i {1} | 2 = i|N(1 − |1+)|i .(26)
The operator average in the r.h.s. of (26) is the standard [24] expression for the probability of an outcome for a system in the state |i , which we have derived using (16), completeness of the states (12) remains the same for all choices of orthogonal final sates | f , |g , |h and | j , and only depends on the inital state |i . Also, from Table 2 one notes that (we apologise for the somewhat awkward sentence about to follow) the state-to-state probability for the electron to travel along the non-overlapping arm ifN(1−) is measured does not equal the sum of probabilites for the electron and positron to travel along non-overlapping arms ifN(1 − |1+) is measured, and that for the electron to travel along non-overlapping and the positron along the overlapping arms, respectively, provided we measureN(1 − |2+), e.g.,
P f ←i {1+2} = 0 = P f ←i {1} + P f ←i {2} = 1/8 ,(27)P all←i {1+2} ≡ i|N(1−)|i = i|N(1 − |1+) +N(1 − |2+)|i = P all←i {1} + P all←i {2} .(28)
A closely related subject is the "failure" of the product rule for post-selected systems [2,13,25]. As was shown in the previous Section, a measurement ofN(2−), conditioned on detectors D+ and D− clicking at the same time, shows that the electron always travels along its overlapping arm. MeasuringN(2+) shows that the same can be said about the positron as well (see Table 2). Table 1 shows that a measurement of N(2 − |2+) gives P f ←i {4} = |Φ f ←i | 2 = 0 and reveals, as it should, that the two particles cannot travel the overlapping paths simultaneously as they would annihilate and never reach the detectors. Morover, from Eqs. (17) to (22) it is clear thatN(2 − |2+) =N(2−)N(2+). Again, assuming that all three results refer to the same statistical ensemble leads to a contradiction, as the joint probability of two certain events must also equal one. Hence inapplicability of the product rule gives another evidence that with post-selection a different ensemble is produced with each choice of the measured quantity, even though all three operators commute [23]. It is a simple matter to verify that without post-selection conditions i|Â|i = 1 and i|B|i = 1 would force i|ÂB|i = 1 for any two commuting operators andB.
One of the purposes of this Section is to stress the fundamental nature of the Feynman's rule for assigning probabilities [10] which we have used first for constructing P z←i {1} , z = f , g, h, j and then in the derivation of Eq. (26). On the other hand, someone who chooses the operator average in Eq. (26) as a starting point for defining quantum probabilities might find generalisation to pre-and post-selected ensembles more difficult, and the apparent break down of the sum and the product rules an unexpected property of post-selection.
Weak measurements in Hardy's set up
In Ref. [5] the authors argued against discarding counterfactual statements on the ground that they can be -to some extent-verified simultaneously provided the accuracy of the measurements is so low that the system remains essentially unperturbed. This can be achieved by making the meter state G( f ) in (5) very broad,
G( f ) → α −1/4 G( f /α), α → ∞(29)
8 so that the mean meter reading f takes the form [13,16] (
G( f ) = G(− f )) f ≡ Z f |Ψ( f )| 2 d f / Z |Ψ( f )| 2 d f ≈ Re N ∑ n=1 F(n)Φ{n}/ N ∑ n=1 Φ{n} ≡F ,(30)
whereF is the weak value of the operator F(n). Since there are no apriori restrictions on the phases of (in general complex valued) amplitudes Φ{n},F is an improper non-probabilistic average [16] with the obvious properties1 = 1 ,
F 1 + F 2 =F 1 +F 2 ,(31)andF = F(m) i f Φ{n} = Φ{m}δ nm ,(32)
i.e., in the absence of interference, when a single pathway connects initial and final states of the system. In the three-box case of Section 3 one findsP 2 = 1 for the projectorP 2 , just as it would be if an accurate measurement was conducted. Similarly, for the projectorP 3 one hasP 3 = 1, which suggests that, since weak measurements do not perturb, the two expectation values, normally observed in two different experiments, can be obtained simultaneously.
With the | f ← |i transition in the Hardy's set up being equivalent to the three-box case we therefore have: Here we adopt a different view. Clearly,N(1 − |1+) = −1 is not a valid pair occupational number for the non-overlapping arms of the interferometers, and as long as it is an essential part of the reasoning, we rather doubt the whole "resolution" of the above paradox. It is easy to see how this anomalous value arises. Measurement ofN(1 − |1+) = diag(1, 0, 0, 0) creates two pathways, {1} with the probability amplitude 1/4, labelled 1, and {2 + 3} with the amplitude −1/2, labelled 0 in Fig. 4. In the weak limit (29), the two pathways interfere and we have, in fact, a double-slit experiment, where we try to determine the chosen slit without destroying the interference pattern on the screen, which the uncertainty principle forbids [9,10]. A weak meter complies with the uncertainty principle by yielding a value which may lie anywhere on the real axis, and not just between 0 and 1. The mathematical reason for this is that the averaging in the r.h.s. of Eq. (30) is done with an improper alternating distribution [16], and the negative occupation number just manifests the failure to sensibly answer the "which way?" question.
N (1 − |2+) =N(2 − |1+) = 1 .(34)
This can be illustrated further by considering a more extreme case with the final state (12) replaced by (we assume that we can do that)
| f = (|1 − |1 + − |1 − |2 + − ε|2 − |1 − + ε|2 − |2 + )/2 1/2 (1 + ε) 1/2 ,(35)
where ε is a parameter, so that for ε = 1 we recover the original transition in the Hardy's set up. Note that N(2 − |1+) creates a single pathway {3}, since the amplitude for {1 + 2} vanishes, whereas forN(1 − |1+)
andN(1 − |2+) there exist two pathways with non-zero amplitudes. The weak pair occupation numbers in (30) now areN
(1 − |1+) = −1/ε N(1 − |2+) = 1/ε N(2 − |1+) = 1 .(36)
Similarly, for the single particle occupation numbers we find
N(1+) = 1 − 1/ε N(2−) = 1.(37)
Now let ε → 0 so that the transition becomes very improbable. For ε = 10 −6 the last of Eqs. (36) suggests that, in defiance of the uncertainty principle, we have established that in a system not perturbed by observations electron and positron always travel along the overlapping and non-overlapping arms, respectively. One must, however, add that there are also a million electron-positron pairs travelling along the non-overlapping and overlapping arms, and also minus one million pairs in the non-overlapping arms. We might try to retain only the results where a weak value is obtained under the non-interference condition (33) but then we would have to discard also the valueN(1 − |1+) = −1 needed to balance the books in the original example of Ref. [5]. A more realistic view [16] is that the unusual properties of the weak values signal a failure of our measurement procedure in the case when it is expected to fail. This failure occurs in a consistent way and can easily be observed, e.g., in an optical experiment [15]. What one observes, however, are the properties of a meter in a regime where it ceases to be a proper meter, not to be confused with the attributes of the measured system which, quantum mechanics tells us, simply do not exist. For example, treatingN(1+) as the actual charge in the non-overlapping positron arm [5] would mean that with no effort whatsoever (although not very often) we may create an arbitrary large charge, perhaps exceeding the largest one allowed by the relativistic quantum mechanics [26].
Conclusions and discussion
In summary, Feynman path approach offers a convenient language for describing some controversial aspects of quantum measurement theory. A quantum system can be seen as arriving in a given final state via a set of virtual (Feynman) paths defined for a particular "position operator"n, all of whose eigenvalues are non-degenerate. With no other measurements performed, the paths form one indivisible (real) pathway. Intermediate von Neumann measurements of operators which commute withn produce, by combining Feynman paths into classes and destroying coherence between them, a network of real pathways connecting the initial and final states of the system. The probabilities which can now be assigned to the pathways define a classical statistical ensemble which one observes in an experiment. The following statement can be regarded as a corollary to the Feynman's uncertainty principle [9]: each set of intermediate measurements produces a different network of real pathways, in such a way that in general no properties of a network A can be inferred from a network B, and, similarly, no properties of the unobserved system can be inferred from either of the networks. It is readily seen that an attempt to ascribe properties of different networks (e.g., the four italicised statements of Section 5) to a single (e.g., unobserved) system may lead to contradiction. Such attempts constitute counterfactual reasoning which, as has been noticed before [3,4] ought to be avoided. For the Hardy's set up we find, for example, that (I) electron and positron always travel along the non-overlapping and overlapping arms, respectively, (path {2} in (10) (10)) if {3} is decohered, while coherence between Feynman paths {1} and {2} is not destroyed, which presents no contradiction with the qualifications added.
We argue further that weak measurements do not provide a sufficient justification for counterfactual reasoning. It is important to go through the statements made in [5] in greater detail. It is true that where only one path contributes to a transition, a weak value would coincide with that obtained in an ideal strong measurement -Eq. (5) shows that that is the case for a meter of arbitrary accuracy. One cannot, however, consistently avoid the "strange" anomalous weak values where there is more than one interfering paths (cf. the exceptionally large occupation numbers discussed in Section 6). Reference [5] tells us that "strangeness is not a problem; consistency is the real issue". This is not quite so: one is lead to believe that a weak measurement is, in some sense, a valid extension of the accurate von Neumann measurement. This notion largely accounts for one's interest in the subject as well as for one's surprise when a "strange" reading, such as a slit number 10 in the two-slit case, is produced. An analogy with a purely classical meter employed to measure the slit number for a classical particle is helpful: the meter malfunctions and produces readings of, say, 10 if the slit 1 is used and 15 if the slit 2 is used. The link between a reading and the measured attribute of the system (slit number 1 or 2) is lost and, unless one knows how to re-calibrate the meter, the measurement becomes a study of the meter's rather than the system's properties. The peculiarity of the quantum situation is that a correct suitable answer to the question just does not exist under the weak conditions and a weak meter cannot be re-calibrated. In consequence, a weak measurement ceases to be a valid extension of the von Neumann procedure, and its result becomes a manifestation of the failure to find a suitable value of the system's intrinsic attribute which, quantum mechanics tells us, is unavailable. Accordingly, the meter, which can no longer be a good one, behaves as if the slit number were 10 or as if there were 10 6 electron-positron pairs in a situation where it can be verified independently that only two holes have been drilled in the screen and only two particles were injected into the system. Many authors [5,13,15] have emphasised the "surprising" aspect of such anomalous weak values. It would, however, been far more surprising had weak measurements consistently produced "suitable" results consistent, for example, with unobserved particle passing through each slit with a certain probability. Existence of strange anomalous values is a proof of the contrary and can be seen as a direct consequence of the uncertainty principle [16]. That not all weak values are strange and that the strange ones usually occur where there is destructive interference between virtual paths does not alter this conclusion. Finally, it is not surprising that a weak meter and the weak values obey their own self-consistent logic [5]. This would be the case for any degree of freedom whose interaction with another system is described by a reasonable coupling. This logic does not, however, extend to (non-existing) intrinsic properties of the measured system, as is required if one wishes to make an argument in favor of counterfactual reasoning. For this reason, weak measurements do not extend the limit set by the uncertainty principle on what can be learned about a quantum system. It seems appropriate to conclude with Feynman's own warning against such an extension [27]: "Do not keep saying to yourself, if you can possibly avoid it, "But how can it be like that?" because you will get "down the drain", into a blind alley from which nobody has yet escaped." (10) which connect the states |i and | f . The path {4} has a zero amplitude due to annihilation, and is shown by a dashed line. An accurate measurement of one of the operators listed above creates two real pathways, each comprising the paths joined by the vertical lines. The nunmbers 1 or 0 next to the lines are the eigenvalues of the measured operator. Figure 5: Table 2. Transition probabilities for the final states in Fig.2 if intermediate measurements in Fig.4 are performed. Also shown are the probabilities for the real pathways created by measurements for the final state | f in (12). In all cases the annihilation occurs with probability of 1/4.
F M with the multiplicities m 1 , m 2 , ...m M , respectively, N virtual paths are divided into M exclusive classes. Each such class can be seen to form a real indivisible pathway to which one can assign a probability P f ←i {m} , with which a reading f = F m would occur if the measurement is performed.
even thoughN(1−) =N(1 − |1+) +N(1 − |2+). This should not come as a surprise since, as discussed above, measurements ofN(1−),N(1 − |1+) andN(1 − |2+) create three different statistical ensembles and Eq. (27) is nothing more than an indication of this fact. An additional "sum rule" is obtained only if the individual "which way?" probabilities are summed over all final states
This would have confirmed that two scenarios (i) electron in the non-overlapping, positron the overlapping arms and (ii) electron in the overlapping, positron the non-overlapping arms take place at the same time, had not the weak value of the third pair occupation operatorN(1 − |1+) = 1 −N(1 − |2+) −N(2 − |1+) turned out to be −1 as required by Eqs. (31) and (32). The authors of Ref. [5] see in this "remarkable way" in which quantum mechanics solves the paradox of having more than one electron-positron pair involved.
Figure 1 :Figure 2 :
12The Hardy's set up of mirrors and beam splitters. An electron (e−) and a positron (e+) are simultaneously injected into the system. Their simultaneous presence in the overlapping arms 2− and 2+, respectively, leads to certain annihilation. Five final states and twelve paths connecting them with the initial state |i .
Figure 3 :Figure 4 :
34Table 1. Probability amplitudes for the virtual paths shown inFig. 2. Virtual paths in
can be reached via each of the pathways {1}, {2} and {3}. For example, the probability amplitude to reach a final state |z via pathway {1} is given by
{2 + 3}, is formed by interfering virtual paths {2} and {3} as is shown in). There are only three contributing paths, {1}, {2} and
{3} with the amplitudes 1/4, −1/4 and −1/4, so that the problem becomes equivalent to the three-box case
of Section 3. IfN(1 − |1+) is measured accurately, path {1} becomes a real pathway, while the second real
pathway,
{1}clearly varies with the choice of the final state |z , P all←i-(15) and
the fact thatN(1 − |1+) 2 =N(1 − |1+). Note that summation over all final outcomes has given P all←i
{1}
properties not possessed by the state-to-state probabilities P z←i
{1} , z = f , g, h, j. For example, while P z←i
{1}
) if {2} is decohered, while coherence between Feynman paths {1} and {3} is not destroyed, (II) electron and positron always travel along the overlapping and non-overlapping arms, respectively, (path {3} in
AcknowledgemntsR. Sala Mayato is greatful to acknowledge Ministerio de Educación y Ciencia, Plan Nacional under grants No. FIS2004-05687 and No. FIS2007-64018, and Consejería de Educación, Cultura y Deporte, Gobierno de Canarias under grant PI2004-025. I. Puerto is greatful to acknowledge Ministerio de Educación y Ciencia under grant AP-2004-0143. D. Sokolovski is greatful to acknowledge Universidad de La Laguna for partial support under project "Ayudas para la estancia de profesores e investigadores no vinculados a la Universidad de La Laguna".
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Unlike the value of a spin component, the number of slits (arms) can established independently by purely classical methods. Thus any measurement result which appears to imply the existence of more than two slits or arms. While conceptually identical, the double-slit (Mach-Zehnder interferometer) example provides somewhat stronger case against over-interpretation of weak values. would require an explanation of the apparent contradictionThe original paper on weak measureaments was [11] entitled "How can a measurement of a spin 1/2 give a result 100?". While conceptually identical, the double-slit (Mach-Zehnder interferometer) example provides somewhat stronger case against over-interpretation of weak values. Unlike the value of a spin component, the number of slits (arms) can established independently by purely classical methods. Thus any measurement result which appears to imply the existence of more than two slits or arms would require an explanation of the apparent contradiction.
Compare with the position operator for a particle in one spacial dimension,x = R |x x x|dx. Compare with the position operator for a particle in one spacial dimension,x = R |x x x|dx.
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The character of physical law. R P Feynman, Modern LibraryR. P. Feynman, The character of physical law, (Modern Library, 1994).
| []
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[
"HAAGERUP APPROXIMATION PROPERTY FOR QUANTUM REFLECTION GROUPS",
"HAAGERUP APPROXIMATION PROPERTY FOR QUANTUM REFLECTION GROUPS"
]
| [
"\nFRANÇ OIS LEMEUX\n\n"
]
| [
"FRANÇ OIS LEMEUX\n"
]
| []
| In this paper we prove that the duals of the quantum reflection groups H s+ N have the Haagerup property for all N ≥ 4 and s ∈ [1, ∞). We use the canonical arrow π : C(H s+ N ) → C(S + N ) onto the quantum permutation groups, and we describe how the characters of C(H s+ N ) behave with respect to this morphism π thanks to the description of the fusion rules binding irreducible corepresentations of C(H s+ N ) ([4]). This allows us to construct states on the central C * -algebra C(H s+ N ) 0 generated by the characters of C(H s+ N ) and to use a fundamental theorem proved by M.Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group G of Kac type ([6]). | null | [
"https://arxiv.org/pdf/1303.2151v2.pdf"
]
| 117,354,761 | 1303.2151 | 10395f2c2ffb33b028a5b914849d14f910cf7173 |
HAAGERUP APPROXIMATION PROPERTY FOR QUANTUM REFLECTION GROUPS
5 Sep 2013
FRANÇ OIS LEMEUX
HAAGERUP APPROXIMATION PROPERTY FOR QUANTUM REFLECTION GROUPS
5 Sep 2013
In this paper we prove that the duals of the quantum reflection groups H s+ N have the Haagerup property for all N ≥ 4 and s ∈ [1, ∞). We use the canonical arrow π : C(H s+ N ) → C(S + N ) onto the quantum permutation groups, and we describe how the characters of C(H s+ N ) behave with respect to this morphism π thanks to the description of the fusion rules binding irreducible corepresentations of C(H s+ N ) ([4]). This allows us to construct states on the central C * -algebra C(H s+ N ) 0 generated by the characters of C(H s+ N ) and to use a fundamental theorem proved by M.Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group G of Kac type ([6]).
[13]
) but which are not amenable. Thus, one says that the Haagerup property is a weak form of amenability. This property is also known as a "strong negation" of Kazhdan's property (T ): the only (classical) discrete groups with both properties are finite. Another weak form of amenability is the weak amenability, see below for examples in the quantum setting. One can find more examples and a more complete approach to the problems and questions related to the Haagerup property, also called "a-Tamenability", in [9].
The Haagerup property has many interests in various fields of mathematics such as geometry of groups or functional analysis. We can mention e.g. groups with wall space structures (see [10] and [8]) as illustrations of the interest in the Haagerup property with respect to the theory of geometry of groups. In functional analysis, the Haagerup property appears e.g. in questions related to the Baum-Connes conjecture (see [14]) or in Popa's deformation/rigidity techniques (see [16]).
In [6], a natural definition of the Haagerup property for compact quantum groups G of Kac type is proposed: G has the Haagerup approximation property if and only if its associated (finite) von Neumann algebra L ∞ (G) has the Haagerup property (see also below, Definition 1.1). We use this definition with the slight modification: the dual G of G has the Haagerup property if L ∞ (G) has the Haagerup property, so that this definition is closer to the classical case where G is a classical discrete 2010 Mathematics Subject Classification. Primary : 46L54, 16T20. Secondary : 46L65, 20G42. 1 group. The author of [12] proposes another definition for the Haagerup property of discrete quantum groups: G has the Haagerup property if there exists a net (a i ) in c 0 ( G) which converges to 1 pointwise and such that the associated multipliers m ai are unital and completely positive. These approaches are equivalent in the unimodular case.
In [6] and [7], the author shows that the duals of the compact quantum groups O + N , U + N and S + N , introduced by Wang (see [19] and [20]) have the Haagerup property. In fact, in [7], it is proved that any trace-preserving quantum automorphism group of a finite dimensional C * -algebra has the Haagerup property. In [12], using some block decompositions and Brannan's proof of the fact that O + N has the Haagerup property (precisely that some completely positive multipliers can be found), the author proves that O + N is weakly amenable (in fact, it is also proved in [12] that the U + N ⊂ Z * O + N is weakly amenable, and an argument of monoidal equivalence allows to prove, in particular, that S + N is weakly amenable too). In [11], a definition of property (T ) for discrete quantum groups and some classical properties for discrete groups are generalized, for instance: discrete quantum groups with property (T ) are finitely generated and unimodular.
The aim of this paper is to prove that the duals of quantum reflection groups H s+ N , introduced in [1], have the Haagerup property. It is a natural generalization of the case s = 1 treated in [7] (since H 1+ N = S + N ). However, this generalization is not immediate: as a matter of fact, the sub C * -algebra generated by the characters is not commutative so that the strategy used in [6] and [7] does not work anymore. However, a fundamental tool of the proof of the main result of our paper is [6, Theorem 3.7].
What motivates our paper is also the fact that quantum reflection groups are free wreath products between Z s and S + N (see [5] and Theorem 1.17 below) and the result proved in our paper naturally leads to the following question: is it true that if Γ is a discrete group which has the Haagerup property then Γ ≀ w S + N has the Haagerup property ? One can notice the similarity with the result in [10] concerning (classical) wreath products of discrete groups: If Γ, Γ ′ are countable discrete groups with the Haagerup property then Γ ≀ Γ ′ also has the Haagerup property. This similarity is however formal: in our paper we are considering (free) wreath products of groups whose duals have the Haagerup property.
Our proof of the fact H s+ N have the Haagerup property relies on the knowledge of the fusion rules of the associated compact quantum group H s+ N , determined in [4]. Indeed, there is no general result about fusion rules for free wreath products of compact quantum groups yet.
The rest of the paper is organized as follows. In the section 1, we recall the definition of the Haagerup property for compact quantum groups of Kac type and we give the result of Brannan concerning the construction of normal, unital, completely positive and trace-preserving maps on L ∞ (G) (see Theorem 1.2). We also give a positive answer to a question asked in [22], in the discrete and Kac setting case, concerning symmetric tensors with respect to the coproduct. Then we collect some results on Tchebyshev polynomials: some are already mentioned and used in [6], but we give suitably adapted statements and proofs for our purpose. Thereafter, we recall the definition of quantum reflection groups H s+ N , and we describe their irreducible corepresentations and the fusion rules binding them. We also recall that at s = 1, we get the quantum permutation groups S + N . In section 2, we identify the images of the irreducible characters of C(H s+ N ) by the canonical morphism onto C(S + N ). In the section 3, we prove that the duals of the quantum reflection groups H s+ N have the Haagerup approximation property for all N ≥ 4.
Preliminaries
Let us first fix some notations. One can refer to [6], [18], [15] and [24] for more details. In this paper, G = (C(G), ∆) will denote a compact quantum group, where C(G) is a full Woronowicz C * -algebra. Furthermore, every compact quantum group G considered in this paper is of Kac type (or equivalentely, its dual G is unimodular) that is: the unique Haar state h on C(G), is tracial. (We recall that
L ∞ (G) is defined by L ∞ (G) = C r (G) ′′ = π h (C(G)) ′′ , where (L 2 (G), π h )
is the GNS construction associated to h). One essential tool to construct nets of normal, unital, completely positive and trace preserving maps (we will say NUCP trace preserving maps) is the next theorem proved in [6]. We will denote by Irr(G) the set indexing the equivalence classes of irreducible corepresentations of a compact quantum group G and by P ol(G) the linear space spanned by the matrix coefficients of such corepresentations u α , α ∈ Irr(G). If α ∈ Irr(G), let L 2 α (G) ⊂ L 2 (G) be the subspace spanned by the GNS images of matrix coefficients u α ij , i, j ∈ {1, . . . , d α } of the irreducible unitary corepresentation u α (d α = dim(u α ij )) and p α : L 2 (G) → L 2 α (G) be the associated orthogonal projection. Then L 2 (G) = l 2 − α∈Irr(G) L 2 α (G). We denote by C(G) 0 ⊂ C(G) the C * -algebra generated by the irreducible characters χ α = dα i=1 u α ii of a compact quantum group G and χ α the character of the associated conjugate corepresentation u α .
T ψ = α∈Irr(G) ψ(χ α ) d α p α
is a unital contraction on L 2 (G) and the restriction of T ψ to L ∞ (G) defines a NUCP h-preserving map still denoted T ψ .
The averaging methods used to prove this theorem allow us to answer, in a restricted setting, a question asked in [22].
Let G = (C(G), ∆) be a compact quantum group. Then consider the C *subalgebra C(G) central := {a ∈ C(G) : ∆(a) = Σ • ∆(a)} i.e. the C * -subalgebra of the symmetric tensors in C(G) ⊗ C(G) with respect to ∆ (Σ denotes the usual flip map Σ :
C(G) ⊗ C(G) → C(G) ⊗ C(G), a ⊗ b → b ⊗ a).
In [22], the author also defines P ol(G) central := {a ∈ P ol(G) : ∆(a) = Σ • ∆(a)}. We recall the question asked by Woronowicz (see [22] thereafter Proposition 5.11):
Question 1.3. Is P ol(G) central dense in C(G) central (for the norm of C(G)) ?
Then the answer is yes, at least in the Kac and discrete setting. We simply denote by ||.|| the norm on C(G). It is clear, and proved in [22], that P ol
(G) central = span{χ α : α ∈ Irr(G)} where χ α = dα i=1 u α
ii denotes the character of an irreducible finite dimentional corepresentation (u α ij ). So the problem reduces to prove that C(G) central ⊂ span ||.|| {χ α : α ∈ Irr(G)}, the other inclusion being clear.
Theorem 1.4. Let G r = (C(G r ), ∆ r ) be a compact quantum group of Kac type with faithful Haar state. Then P ol(G r ) central ||.|| = C(G r ) central .
Proof. We first note that ∆ r preserves the trace in the sense that
(h ⊗ h) • ∆ r = h.
As a result the Hilbertian adjoint ∆ * r , of the L 2 -extension of ∆ r , is well-defined and we have ||∆ * r (x)|| ≤ ||x|| for x ∈ C(G r ) ⊗ C(G r ) with respect to the operator norms (note that this is particular to the tracial situation). Since ∆ * r clearly maps the
subspace P ol(G r ) ⊗ P ol(G r ) of L 2 (G r ) ⊗ L 2 (G r ) to P ol(G r ), it also restricts to a contractive map from C(G r ) ⊗ C(G r ) to C(G r ), still denoted ∆ * r . Now we put E = ∆ * r • Σ • ∆ r : C(G r ) → C(G r ). We have ||E|| ≤ 1, and for a ∈ C(G r ) central , E(a) = ∆ * r • ∆ r (a) = a so that C(G r ) central ⊂ E(C(G r )).
But, on the other hand, for any matrix coefficient of a finite dimensional unitary corepresentations (u α ij ), we have
E(u α ij ) = ∆ * r • Σ • ∆ r (u α ij ) = ∆ * r • Σ k u α ik ⊗ u α kj = ∆ * r k u α kj ⊗ u α ik .
We compute ∆ * r k u α kj ⊗ u α ik using the duality pairing induced by the inner product coming from the Haar state h: let β ∈ Irr(G r ), then for all 1 ≤ p, q ≤ d β :
u β pq , ∆ * r k u α kj ⊗ u α ik h = l,k u β pl ⊗ u β lq , u α kj ⊗ u α ik h = δ αβ δ ij δ pq d 2 α = u β pq , δ ij d α χ α h . Then summarizing, we have E(u α ij ) = δij dα χ α ∈ Pol(G r ) central , ||E|| = 1 and E| P ol(Gr ) central = id. Thus we obtain a conditional expectation E : C(G r ) → P ol(G r ) central ||.|| = span ||.|| {χ α : α ∈ Irr(G r )}. But we have C(G r ) central ⊂ E(C(G r )
) and the result follows.
Notation 1.5. We will denote by P ol(G) 0 and C(G) 0 the central * -algebras and C * -algebras generated by the irreducible characters of a compact quantum group G.
Tchebyshev polynomials.
Definition 1.6. We define a family of polynomials (A t ) t∈N as follows:
A 0 = 1, A 1 = X and for all t ≥ 1 (1.1) A 1 A t = A t+1 + A t−1 .
We call them the dilated Tchebyshev polynomials of second kind.
We will use the following results on Tchebyshev polynomials A t . The second one is based upon a result proved in [6,Proposition 4.4], but suitably adapted to our purpose.
Proposition 1.7. for all t, s ≥ 1 we have: A t A s = A t+s + A t−1 A s−1
Proof. This result is easily proved by induction on t ≥ 1.
Proposition 1.8. Let N ≥ 2. For all x ∈ (2, N ), there exists a constant c ∈ (0, 1) such that for all integers t ≥ 1 we have 0 < A t (x) A t (N ) ≤ x N ct .
Proof. First, we follow the proof of [6,Proposition 4.4] and introduce the function
q(x) = x+ √ x 2 −4 2
, for x > 2. Then an induction and the recursion formula (1.1) for the polynomials A t show that for all t ≥ 0, we have
A t (x) = q(x) t+1 − q(x) −t−1 q(x) − q(x) −1
Then using the same tricks as in [6], we get that for all fixed x ∈ (2, N ) and all t ≥ 1
A t (x) A t (N ) = q(x) t+1 − q(x) −t−1 q(N ) t+1 − q(N ) −t−1 q(N ) − q(N ) −1 q(x) − q(x) −1 = x N t 1 + 1 − 4 x 2 1 + 1 − 4 N 2 t 1 − q(x) −2t−2 1 − q(N ) −2N −2 1 − q(N ) −2 1 − q(x) −2 . Now notice that the factor 1−q(x) −2t−2 1−q(N ) −2N −2 is less than 1 because q is increasing. Furthermore, we have 1 + 1 − 4 x 2 1 + 1 − 4 N 2 t 1 − q(N ) −2 1 − q(x) −2 −→ t→∞ 0
since the last factor does not depend on t and
1+ 1− 4 x 2 1+ 1− 4 N 2 < 1. Hence, there ex- ists t 0 such that A t (x) A t (N ) ≤ x N t for all t ≥ t 0 . It remains to show that there exists c ∈ (0, 1) such that A t (x) A t (N ) ≤ x N ct0 for all t = 1, . . . , t 0 − 1, since for all 0 < t < t 0 , x N ct0 ≤ x N ct .
To prove that such a c exists, we notice that max At(x) At(N ) : t = 1, . . . , t 0 − 1 := D < 1 since the Tchebyshev polynomials are increasing on (2, +∞). Hence, it is clear that we can find c > 0 such that
x N ct0 ≥ D.
Remark 1.9.
(1) In [6, Proposition 6.4], the exponent is better (there is no
constant c) but there is a constant multiplying x N t .
Our version allows an easy proof of Proposition 3.3 below.
(2) The previous proposition gives information on the behavior of the dilated Tchebyshev polynomials on (2, +∞): the quotient
A t (x) A t (N )
has an exponential decay with respect to t ≥ 1. We will also need some informations on this quotient when x ∈ (0, 2) and N = 2. That is the aim of the next paragraph.
The polynomials A t are linked to the Tchebyshev polynomials of second kind U t by the following formula: ∀t ∈ N, x ∈ [0, 1], A t (2x) = U t (x). Indeed, we recall (see [17] for more details) that the Tchebyshev polynomials of second kind U t are defined for all x ∈ [−1, 1] by
(1.2) U t (x) = sin((t + 1) arccos(x)) √ 1 − x 2 = sin((t + 1)θ) sin(θ)
, with x = cos(θ).
In particular, U 0 = 1, U 1 (x) = 2x and for all t ∈ N * , U t (1) = t + 1. Then one can check that for all t ∈ N and x ∈ [0, 1]: 2xU t (x) = U t+1 (x) + U t−1 (x). Proposition 1. 10. Let x ∈ (0, 2). Then for any integer t ≥ 1
A t (x) A t (2) = 1 t + 1 sin((t + 1)θ) sin(θ)
, with x = cos(θ).
In particular, there exists a positive constant D < 1 such that ∀t ≥ 1,
A t (x) A t (2) ≤ D and A t (x) A t (2) −→ 0 as t → ∞.
Proof.
First, by what we recalled above, we can write
A t (x) A t (2) = U t ( x 2 ) U t (1) = U t ( x 2 ) t + 1 .
Thus, if x = 2 cos(θ), we have by the relation (1.2) above
A t (x) A t (2) = 1 t + 1 sin((t + 1)θ) sin(θ) −→ t→∞ 0.
On the other hand, on [0, 1], the polynomials U t , t ≥ 1 have t + 1 as a maximum, only attained in 1. Then, it is clear that for all t ≥ 1 and x ∈ (0, 2):
0 < A t (x) A t (2) = U t ( x 2 ) t + 1 < 1.
So the existence of the announced constant D is clear.
Quantum reflection groups.
In this subsection, we recall the definition of the quantum reflection groups H s+ N and the particular case of the quantum permutation groups S + N . We also recall that C(H s+ N ) is the free wreath product of two quantum permutation algebras. In the end of this subsection, we recall the description of the irreducible corepresentations of C(H s+ N ) together with the fusion rules binding them.
(1) U = (U ij ) is unitary, (2) t U = (U ji ) is unitary, (3) p ij = U ij U * ij is a projection, (4) U s ij = p ij together with the coproduct ∆ : C(H s+ N ) → C(H s+ N ) ⊗ C(H s+ N ) given by ∆(U ij ) = k U ik ⊗ U kj . Remark 1.12.
(1) For s = 1 we get the quantum permutation group S + N . The definition of S + N thus may be summed up as follows (see also [20]
): S + N is the pair (C(S + N ), ∆) where (a) C(S + N ) is the universal C * -algebra generated by N 2 elements v ij such that the matrix v = (v ij ) is unitary and v ij = v * ij = v 2 ij (i.e. v is a magic unitary). (b)
The coproduct is given by the usual relations making of v a corepresentation (the fundamental one) of C(S + N ). (2) For s = 2, we find the hyperoctahedral quantum group, i.e. the easy quantum group H + N studied e.g. in [21]. (3) There is a morphism C(H s+ N ) → C(S + N ) of compact quantum groups: one only has to check that the generators v ij of C(S + N ), satisfy the relations described in Definition 1.11, which is clear. Notation 1.13. We will denote by π : C(H s+ N ) → C(S + N ) the canonical arrow mentioned in the remark above.
Here are the results concerning the irreducible corepresentations of C(S + N ): Theorem 1.14. [3, Theorem 4.1] There is a maximal family v (t) t∈N of pairwise inequivalent irreducible finite dimensional unitary representations of S + N such that:
(1) v (0) = 1 and v is equivalent to 1 ⊕ v (1) . (2) The conjugate of any v (t) is equivalent to itself that is v (t) ≃ v (t) , ∀t ∈ N.
(3) The fusion rules are the same as for SO (3):
v (s) ⊗ v (t) ≃ 2 min(s,t) k=0 v (s+t−k)
We denote by
χ k = d k i=1 v (k)
ii the character associated to v (k) . We will need the following proposition, proved in [7]: Proposition 1.15. Let χ be the character associated to the fundamental corepresentation v of C(S + N ). Then, χ * = χ and there is a * -isomorphism C * (χ) = C(S +
N ) 0 = C * (χ t : t ∈ N) ≃ C([0, N ]) identifying χ t to the polynomial defined Π t by Π 0 = 1, Π 1 = X − 1 and ∀t ≥ 1, Π 1 Π t = Π t+1 + Π t + Π t−1 . Remark 1.16.
(1) The recursion formula defining the polynomials Π t is the one satisfied by the irreducible characters χ t .
(2) The polynomials A t and Π t are linked by the formula:
Π t (x) = A 2t ( √ x).
Before describing the fusion rules of C(H s+ N ), we recall that these compact quantum groups are free wreath products:
-C(H s+ N ) ≃ C(Z s ) * w C(S + N ) = C * (Z * N s ) * C(S + N )/ < [z i , v ij ] = 0 > where z i is the generator of the i-th copy Z s in the free product Z * N s . -In particular C(H s+ 2 ) ≃ C(Z s ) * w C(Z 2 ), C(H s+ 3 ) ≃ C(Z s ) * w C(S 3 ).(1) U k = (U k ij ) for any k > 0. (2) U k = U k+s for any k ∈ Z.
(3) U k = U −k for any k ∈ Z.
= U 0 ⊖ 1 (where U 0 = (U ij U * ij )
). The proof of the first three assertions follows from the definitions of corepresentations of compact quantum groups and of the definition of C(H s+ N ). The proof of the last three assertions is based upon Woronowicz's Tannaka-Krein duality (see [23]) and methods inspired by [2], [3] and [1]. Now, we can give the description of the fusion rules:
(1.3) vaz i ⊗ z j aw = vaz i+j aw ⊕ δ i+j,0 (v ⊗ w)
Then the irreducible corepresentations r α of C(H s+ N ) can be indexed by the elements α of the submonoid S generated by the elements az i a, i = 0, . . . , s − 1, with involution and fusion rules above.
Remark 1.21.
(1) S is composed of elements a L1 z J1 . . . z JK−1 a LK with -J i , L i > 0 integers.
-L 1 , L K odd integers and all the L i 's, i ∈ {2, . . . , K − 1} even integers.
-Except if K = 1, then L K is an even integer.
(2) With this description, we can identify the basic corepresentations introduced above: the corepresentation r a 2 is the corepresentation ρ 0 = (U ij U * ij ) ⊖ 1 and for t = 0, r az t a is the corepresentation ρ t = (U t ij ). (3) In Proposition 2.1, we will use the suggestive notation
vaz i+j aw = (vaz i ⊗ z j aw) ⊖ δ i+j,0 (v ⊗ w),
which simply means that we have the relation (1.3) in the monoid S.
(4) If α = a L1 z J1 . . . z JK−1 a LK ∈ S, then the conjugate corepresentation of r α is indexed by α = a LK z −JK−1 . . . z −J1 a L1
We end this subsection by the following proposition which summarizes the results above:
Proposition 1.22. The canonical morphism π : C(H s+ N ) → C(S + N ) maps all the corepresentations U t , t ∈ Z onto the fundamental corepresentation v of C(S + N )
; in other words, it maps all ρ t = r az t a , t = 0 onto v and ρ 0 = r a 2 onto v (1) .
Characters of quantum reflection groups and quantum permutation groups
As announced in the introduction, we find the images of the irreducible characters of C(H S+ N ) under the canonical morphism π : C(H s+ N ) → C(S + N ). Proposition 2.1. Let χ α be the character of an irreducible corepresentation r α of C(H s+ N ). Write α = a l1 z j1 . . . z j k−1 a l k . Then, identifying C(S + N ) 0 with C([0, N ]), the image of χ α , say P α , satisfies:
P α (X 2 ) = π(χ α )(X 2 ) = k i=1 A li (X).
Proof. We shall prove this proposition by induction on the even integer k i=1 l i using the description of the fusion rules given by Theorem 1.20, the recursion formula satisfied by the Tchebyshev polynomials, Proposition 1.7 and Proposition 1.22.
Let HR(λ) be the following statement: π(χ α )(X 2 ) = k i=1 A li (X) for any α = a l1 z j1 . . . z j k−1 a l k such that 2 ≤ i l i ≤ λ.
Let us begin by studying simple examples (and initializing the induction).
Consider the element aza. Then, the irreducible corepresentation r aza (written ρ 1 in Notation 1.19) is sent by π onto v = 1 ⊕ v (1) by Proposition 1.22. Thus, in term of characters, we obtain by Proposition 1.15
π(χ aza )(X) = 1 + (X − 1) = X = A 1 (X) i.e. P aza (X 2 ) = X 2 = A 1 (X)A 1 (X).
Actually, this holds for all elements α = az j a, j ∈ {1, . . . , s − 1} (since every irreducible corepresentation r az j a is sent by π onto 1 ⊕ v (1) , as is r aza ). Consider the element a 2 . Then, the irreducible corepresentation r a 2 (written ρ 0 in Notation 1.19) is sent by π onto v (1) . Thus π(χ a 2 )(X) = X − 1. i.e. P a 2 (X 2 ) = X 2 − 1 = A 2 (X).
To prove HR(2) one has to show that π(χ a 2 )(X 2 ) = A 2 (X) and π(χ az j a )(X 2 ) = A 1 A 1 (X) for all j ∈ {1, . . . , s − 1}, what we have just done above. Now assume HR(λ) holds: π(χ β )(X 2 ) = k i=1 A li (X) for any β = a l1 z j1 . . . z j k−1 a l k such that 2 ≤ i l i ≤ λ. We now show HR(λ + 2).
Let α = a L1 z J1 . . . a LK , with i L i = λ + 2. In order to use HR(λ), we must "break" α using the fusion rules as in the examples above. Then, essentially, one has to distinguish the cases L K = 1, L K = 3 and L K ≥ 5 (in the case L K ≥ 5 we can "break α at a LK " but in the other cases we must use a LK−1 or a LK−2 if they exist, that is if there are enough factors a Li ). So first, we deal with two special cases below, in order to have "enough" factors a L in α in the sequel. We use the fusion rules described in Theorem 1.20 (and the notations described after, see Remark 1.21).
-If K = 1 i.e. L K = λ + 2, J i = 0 ∀i, write:
α = a λ+2 = (a λ ⊗ a 2 ) ⊖ (a λ−1 ⊗ a) = (a λ ⊗ a 2 ) ⊖ a λ ⊖ a λ−2 .
Then using the hypothesis of induction and Proposition 1.7, we get
π(χ α )(X 2 ) = A λ A 2 (X) − A λ (X) − A λ−2 (X) = A λ A 2 (X) − (A λ (X) + A λ−2 (X)) = A λ A 2 (X) − A λ−1 A 1 (X) = A λ+2 (X).
(Notice that if λ = 2 one has λ − 2 = 0 and a 4 = (a 2 ⊗ a 2 ) ⊖ (a ⊗ a) = (a 2 ⊗ a 2 ) ⊖ a 2 ⊖ 1 so that the result we want to prove then is still true.) -If K = 2, J := J 1 = 0, write α = a L1 z J a L2 . We have L 1 + L 2 = λ + 2 ≥ 4 and L 1 , L 2 are odd hence L 1 or L 2 ≥ 3, say L 1 ≥ 3. Write
a L1 z J a L2 = (a 2 ⊗ a L1−2 z J a L2 ) ⊖ (a ⊗ a L1−3 z J a L2 ).
If L 1 = 3 then the tensor product a ⊗ a L1−3 z J a L2 is equal to az J a L2 hence α = a 3 z J a L2 satisfies
π(χ α )(X 2 ) = A 2 A 1 A L2 (X) − A 1 A L2 (X) = A 3 (X)A L2 (X).
If L 1 > 3 (i.e. L 1 ≥ 5), then the tensor product a ⊗ a L1−3 z J a L2 is equal to a L1−2 z J a L2 ⊕ a L1−4 z J a L2 . We get
π(χ α )(X 2 ) = A 2 A L1−2 A L2 (X) − A L1−2 A L2 (X) − A L1−4 A L2 (X) = A L1 (X)A L2 (X).
-From now on, we suppose that there are more than three factors a Li in α i.e. K ≥ 3. We will have to distinguish three cases: L K = 1, L K = 3 and L K ≥ 5.
If 5 ≤ L K < i L i , write L K = m K + 2. Then we have m K ≥ 3, so a L1 z J1 . . . a LK = a L1 z J1 . . . a mK +2 = (a L1 z J1 . . . a mK ⊗ a 2 ) ⊖ (a L1 z J1 . . . a mK −1 ⊗ a) = (a L1 z J1 . . . a mK ⊗ a 2 ) ⊖ a L1 z J1 . . . a mK ⊖ a L1 z J1 . . . a mK −2 .
Then
π(χ α )(X 2 ) = A L1 . . . A LK−1 A m k A 2 (X) − A L1 . . . A mK (X) − A L1 . . . A mK −2 (X) = A L1 . . . A LK−1 A LK (X).
If m K = 1, i.e. L K = 3, we proceed in the same way using
a L1 z J1 . . . z JK−1 a 3 = (a L1 z J1 . . . a ⊗ a 2 ) ⊖ a L1 z J1 . . . z JK−1 a.
To conclude the induction, one has to deal with the case L K = 1. We have to distinguish the following cases:
If L K−1 ≥ 4. We have a L1 z J1 . . . a LK−1 z JK−1 a = (a L1 z J1 . . . a LK−1−1 ⊗ az JK−1 a) ⊖ (a L1 z J1 . . . a LK−1−2 ⊗ z JK−1 a) = (a L1 z J1 . . . a LK−1−1 ⊗ az JK−1 a) ⊖ a L1 z J1 . . . a LK−1−2 z JK−1 a. Then π(χ α )(X 2 ) = A L1 . . . A LK−1−1 A 1 A 1 (X) − A L1 . . . A LK−1−2 A 1 (X) = A L1 . . . A LK−1 A 1 (X).
If L K−1 = 2 and J K−1 + J K−2 = 0 mod s, we can proceed in the same way using
a L1 z J1 . . .a LK−2 z JK−2 a 2 z JK−1 a = (a L1 z J1 . . . a LK−2 z JK−2 a ⊗ az JK−1 a) ⊖ a L1 z J1 . . . a LK−2+1 ⊖ a L1 z J1 . . . a LK−2−1 .
The last case to deal with is L K−1 = 2 and J K−1 + J K−2 = 0 mod s, and again we can conclude thanks to a L1 z J1 . . . z JK−2 a 2 z JK−1 a = (a L1 . . . a LK−2 z JK−2 a ⊗ az JK−1 a) ⊖ a L1 z J1 . . . a LK−2 z JK−2+JK−1 a.
As a corollary, we can get the result also proved in [4] (see Theorem 9.3):
Corollary 2.2. Let r α be an irreducible corepresentation of C(H s+ N ) with α = a l1 z j1 . . . a l k . Then dim(r α ) = k i=1 A li ( √ N ).
Proof. We have dim(r α ) = ǫ C(H s+ N ) (χ α ) = ǫ C(S + N ) • π(χ α ) since π is a morphism of Hopf algebras. But the counit on C(S + N ) 0 is given by the evaluation in N . Indeed, an immediate corollary of Theorem 1.14 and Proposition 1.15, is ǫ(Π t ) = Π t (N ) for all polynomials Π t , which form a basis of R[X]. Now by the previous proposition
π(χ α )(x) = k i=1 A li ( √ x), then ǫ C(S + N ) • π(χ α ) = k i=1 A li ( √ N ).
Haagerup property for quantum reflection groups
In this section we show that duals of the quantum reflection groups C(H s+ N ) = C(Z s ) * w C(S + N ), s ≥ 1 have the Haagerup property for N ≥ 4. We still denote by π the canonical surjection π : C(H s+ N ) → C(S + N ) and by ψ x = ev x the states on C(S + N ) 0 ≃ C([0, N ]) used to show that C(S + N ) have the Haagerup property (see [7]). Essentially, we are going to use both morphisms π, ψ x in this way: we can define states φ x composing these maps, ψ x • π, where π sends characters of C(H s+ N ) on characters of C(S + N ). Thus, we obtain states on the central algebra C(H s+ N ) 0 and, after checking that these states have some decreasing properties, we can use the Theorem 1.2 and conclude.
x ∈ [0, N ], φ x = ψ x • π is a state on C(H s+ N ) 0 . Proof.
One just has to note that π is Hopf * -homomorphism and hence sends C(H s+ N ) 0 to C(S + N ) 0 . Then ψ x • π is indeed a functional on C(H s+ N ) 0 . The rest is clear.
Notation 3.2. We introduce a proper function on the monoid S (see Theorem 1.20). Let L be defined by
L(α) = kα i=1 l i for α = a l1 z j1 . . . a l kα . Notice that for all R > 0 the set B R = α = a l1 z j1 . . . a l kα : L(α) = kα i=1 l i ≤ R ⊂ S is finite.
Thus we get that a net (f α ) α∈S belongs to c 0 (S) ⇐⇒ ∀ǫ > 0 ∃R > 0 : ∀α ∈ S, (L(α) > R ⇒ |f α | < ǫ). We say that a net (f α ) α converges to 0 as α → ∞ if (f α ) α ∈ c 0 (S).
C α (x) := φ x (χ α ) dim(r α ) = ψ x • π(χ α )(X) dim(r α ) = kα i=1 A li ( √ x) A li ( √ N ) .
Moreover C α (x) converges to 0 as α → ∞ for all x ∈ [4, N ).
Proof. Let α = a l1 z j1 . . . a l kα . We obtain the first assertion using Proposition 2.1 and Corollary 2.2:
π(χ α )(X) = A l1 . . . A l kα ( √ X), d α := dim(r α ) = kα i=1 A li ( √ N ).
By Proposition 1.8, for any fixed x ∈ (4, N ), there exists a constant 0 < c < 1
such that A l ( √ x) A l ( √ N ) ≤ √ x √ N cl for all l ≥ 1. Then C α (x) = φ x (χ α ) dim(r α ) = kα i=1 A li ( √ x) A li ( √ N ) ≤ x N c 2 i li = x N c 2 L(α) −→ α→∞ 0.
Proposition 3.4. (Case N = 4) Let χ α be an irreducible character of C(H s+ 4 ) associated to the irreducible corepresentation r α with α = a l1 z j1 a l2 . . . a l kα . Then for all x ∈ [0, 4]
C α (x) := φ x (χ α ) dim(r α ) = ψ x • π(χ α )(X) dim(r α ) = kα i=1 A li ( √ x) A li (2) .
Moreover C α (x) converges to 0 as α → ∞ for all x ∈ (0, 4).
Proof. The proof of the first assertion is similar to the one of the previous proposition. For the second assertion, we use Proposition 1.10. We recall that we proved in that proposition that there exists a constant D < 1 such that for all x ∈ (0, 4) and all l ≥ 1
(3.1) A l ( √ x) A l (2) ≤ D.
Let ǫ > 0 and x ∈ (0, 4). We want to prove that,
(3.2) kα i=1 A li ( √ x)
A li (2) < ǫ for α large enough.
By (3.1), there exists a K > 0 such that
kα i=1 A l i ( √ x)
A l i (2) < ǫ for all α ∈ S with k α ≥ K. But by Proposition 1.10 there is also an L > 0 such that
A l ( √ x) A l (2)
< ǫ for all l ≥ L, since this quotient converges to 0. Now let α = a l1 z j1 . . . a l kα ∈ S, with L(α) ≥ LK. Then either k α ≥ K, or there exists i 0 ∈ {1, . . . , k α } such that l i0 ≥ L. In both case we can get (3.2) since
kα i=1 A li ( √ x) A li (2) ≤ A li 0 ( √ x)
A li 0 (2) , each factor of the product being less that one.
Then we can prove the theorem:
Theorem 3.5. The dual of H s+ N has the Haagerup property for all N ≥ 4. Proof. We follow the proof in [6] for O + N . We prove that the dual of H s+ N has the Haagerup approximation property for all N ≥ 4 using both previous propositions. Consider the net (T φx ) x∈IN with I N = (4, N ) if N ≥ 5, I N = (0, 4) if N = 4 and
T φx = α∈Irr(H s+ N ) φ x (χ α ) d α p α
The φ x are states on C(H s+ N ) 0 so, by Theorem 1.2, the T φx are a unital contractions of L 2 (H s+ N ), and their restrictions to L ∞ (H s+ N ) are NUCP h-preserving maps. Moreover, Proposition 3.4 in the case N = 4 and Proposition 3.3 in the cases N ≥ 5, together with the fact that the p α are finite rank operators, show that for each x ∈ I N , the operator T φx is compact. To conclude one has to show that for all x ∈ I N ,
(3.3)
||T φx a − a|| L2 −→ x→N 0 for all a ∈ L ∞ (H s+ N ) (via a ∈ L ∞ (H s+ N ) ֒→ L 2 (H s+ N )). First let us prove that it is true for any element a ∈ P ol(H s+ N ) i.e. any linear combination of matrix coefficients U α ij of irreducible corepresentations of C(H s+ N ) (by linearity, we can do that only for the elements U α ij ). Notice that if α = a l1 z j1 . . . z j kα −1 a l kα then α = a l kα z −j kα−1 . . . z −j1 a l1 = a l kα z s−j kα −1 . . . z s−j1 a l1 . Thus by Proposition 2.1 φ x (χ α ) = ψ x • π(χ α ) = ψ x • π(χ α ) = φ x (χ α ). Hence,
||T φx U α ij − U α ij || L 2 = ||U α ij || L2 1 − kα i=1 A li ( √ x) A li ( √ N ) ,
so let x → N and the assertion (3.3) holds for all these matrix coefficient. Now by L 2 -density of P ol(H s+ N ) and the fact that all T φx , x ∈ I N , are unital contractions (and thus are uniformly bounded), we obtain that (3.3) is true for any a ∈ L 2 (H s+ N ).
Remark 3.6. In [5], it is proved that there is a * -Hopf algebras isomorphism between C(H s+ 2 ) and C * (Z s * Z s × Z 2 ) (see Example 2.5 and thereafter in that paper). Furthermore, the Haar state on C * (Z s ) * w C(S + 2 ) is given by h = h 1 ⊗ h 2 where h 2 is the Haar state on C(S + 2 ) and h 1 is the free product of the Haar states on C * (Z s )). Then, it is clear that H s+ 2 has the Haagerup property by the stability properties of the Haagerup property on groups (see e.g. [9])
The algebra C(H s+ 3 ) is more complicated and does not reduce to a more comprehensive tensor product as for the case N = 2. We are unable at the moment to prove that H s+ 3 has the Haagerup property.
Introduction A (classical) discrete group Γ has the Haagerup property if (and only if) there is a net (ϕ i ) of normalized positive definite functions in C 0 (Γ) converging pointwise to the constant function 1. There are lots of examples of discrete groups with the Haagerup property: all amenable groups have this property. The free groups F N are examples of discrete groups with Haagerup property (see
1. 1 .
1Haagerup property for compact quantum groups of Kac type. Definition 1.1. The dual G of a compact quantum group G = (C(G), ∆) of Kac type has the Haagerup approximation property if the finite von Neumann algebra (L ∞ (G), h) has the Haagerup approximation property i.e. if there exists a net (φ x ) of trace preserving, normal, unital and completely positive maps on L ∞ (G) such that their unique extensions to L 2 (G) are compact operators and (φ x ) converges to id L ∞ (G) pointwise in L 2 -norm.
Theorem 1. 2 .
2[6, Theorem 3.7] Let G = (C(G), ∆) be a compact quantum group of Kac type. Then for any state ψ ∈ C(G) * 0 , the map
Definition 1 .
111.[1, Definition 11.3] Let N ≥ 2, s ≥ 1 be integers. The quantum reflection group H s+ N is the pair (C(H s+ N ), ∆) composed of the universal C * -algebra generated by N 2 normal elements U ij satisfying the following relations
]
Let N ≥ 2, then we have the following isomorphisms of compact quantum groups:
Let us now give the description of the irreducible corepresentations of C(H s+ N ). Theorem 1.18. [4, Theorem 4.3, Corollary 6.4] C(H s+ N ) has a unique family of N -dimensional corepresentations (called basic corepresentations) {U k : k ∈ Z}, satisfying the following conditions:
( 4 )
4U 1 , . . . , U s−1 are irreducible. (5) U 0 = 1 ⊕ ρ 0 , ρ 0 irreducible. (6) ρ 0 , U 1 , . . . , U s−1 are inequivalent corepresentations.Notation 1.19. We will denote the basic irreducible corepresentations of C(H s+ N ) by ρ t , t ∈ {0, . . . , s − 1}, with ρ t = U t ∀t ∈ {1, . . . , s − 1} and ρ 0
Theorem 1.20. [4, Theorem 8.2] Let M be the monoid M = a, z : z s = 1 with involution a * = a, z * = z −1 , and the fusion rules obtained by recursion from the formulae
Lemma 3 . 1 .
31Let ψ x , x ∈ [0, N ] be the states given by the evaluation in x on the central C * -algebra C(S + N ) 0 . Then for all
Proposition 3 . 3 .
33Let N ≥ 5 and let χ α be an irreducible character of C(H s+ N ) associated to the irreducible corepresentation r α with α = a l1 z j1 a l2 . . . a l kα . Then for all x ∈ [0, N ]
AcknowledgementsI am very grateful to my advisors Uwe Franz and Roland Vergnioux for the time they spent discussing the arguments of this paper. I would also like to thank Pierre Fima for discussions on various topics on quantum groups, Amaury Freslon for discussions on averaging methods and approximation properties on quantum groups and Mikael de la Salle for very useful suggestions and commentaries on some special cases of the theorem proved in this paper.
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UFR Sciences et Techniques. Besançon, France E-mail addressFrançois Lemeux, Laboratoire de mathématiques de Besançon ; Université de Franche-Comté,16 route de GrayFrançois Lemeux, Laboratoire de mathématiques de Besançon, UFR Sciences et Tech- niques, Université de Franche-Comté,16 route de Gray, 25000 Besançon, France E-mail address: [email protected]
| []
|
[
"The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity",
"The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity"
]
| [
"Bertrand Souères [email protected] ",
"Dimitrios Tsimpis [email protected] ",
"\nUniversité Claude Bernard\nLyon\n",
"\nUMR 5822\nInstitut de Physique Nucléaire de Lyon\nCNRS/IN2P3\n4 rue Enrico FermiF-69622Villeurbanne CedexFrance\n"
]
| [
"Université Claude Bernard\nLyon",
"UMR 5822\nInstitut de Physique Nucléaire de Lyon\nCNRS/IN2P3\n4 rue Enrico FermiF-69622Villeurbanne CedexFrance"
]
| []
| We develop computational tools for calculating supersymmetric higher-order derivative corrections to eleven-dimensional supergravity using the action principle approach. We show that, provided the superspace Bianchi identities admit a perturbative solution in the derivative expansion, there are at least two independent superinvariants at the eight-derivative order of eleven-dimensional supergravity. Assuming the twelve-superforms associated to certain anomalous Chern-Simons terms are Weil-trivial, there will be a third independent superinvariant at this order. Under certain conditions, at least two superinvariants will survive to all orders in the derivative expansion. However only one of them will be present in the quantum theory: the supersymmetrization of the Chern-Simons terms of eleven-dimensional supergravity required for the cancellation of the M5-brane gravitational anomaly by inflow. This superinvariant can be shown to be unique at the eight-derivative order, assuming it is quartic in the fields. On the other hand, a necessary condition for the superinvariant to be quartic is the exactness, in τcohomology, of X 0,8 -the purely spinorial component of the eight-superform related by descent to the M5-brane anomaly polynomial. In that case it can also be shown that the solution of the Weil-triviality condition of the corresponding twelve-form, which is a prerequisite for the explicit construction of the superinvariant, is guaranteed to exist. We prove that certain highly non-trivial necessary conditions for the τ -exactness of X 0,8 are satisfied. Moreover any potential superinvariant associated to anomalous Chern-Simons terms at the eight-derivative order must necessarily contain terms cubic or lower in the fields. | 10.1103/physrevd.95.026013 | [
"https://arxiv.org/pdf/1612.02021v2.pdf"
]
| 119,371,605 | 1612.02021 | 6b9df87e741e50cf1883d430810b038924986198 |
The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity
16 Dec 2016
Bertrand Souères [email protected]
Dimitrios Tsimpis [email protected]
Université Claude Bernard
Lyon
UMR 5822
Institut de Physique Nucléaire de Lyon
CNRS/IN2P3
4 rue Enrico FermiF-69622Villeurbanne CedexFrance
The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity
16 Dec 2016Preprint typeset in JHEP style -HYPER VERSION
We develop computational tools for calculating supersymmetric higher-order derivative corrections to eleven-dimensional supergravity using the action principle approach. We show that, provided the superspace Bianchi identities admit a perturbative solution in the derivative expansion, there are at least two independent superinvariants at the eight-derivative order of eleven-dimensional supergravity. Assuming the twelve-superforms associated to certain anomalous Chern-Simons terms are Weil-trivial, there will be a third independent superinvariant at this order. Under certain conditions, at least two superinvariants will survive to all orders in the derivative expansion. However only one of them will be present in the quantum theory: the supersymmetrization of the Chern-Simons terms of eleven-dimensional supergravity required for the cancellation of the M5-brane gravitational anomaly by inflow. This superinvariant can be shown to be unique at the eight-derivative order, assuming it is quartic in the fields. On the other hand, a necessary condition for the superinvariant to be quartic is the exactness, in τcohomology, of X 0,8 -the purely spinorial component of the eight-superform related by descent to the M5-brane anomaly polynomial. In that case it can also be shown that the solution of the Weil-triviality condition of the corresponding twelve-form, which is a prerequisite for the explicit construction of the superinvariant, is guaranteed to exist. We prove that certain highly non-trivial necessary conditions for the τ -exactness of X 0,8 are satisfied. Moreover any potential superinvariant associated to anomalous Chern-Simons terms at the eight-derivative order must necessarily contain terms cubic or lower in the fields.
Introduction
Eleven-dimensional supergravity [1] is believed to be the low-energy limit of M-theory [2], the conjectured nonperturbative completion of string theory. As such it is expected to receive an infinite tower of higher-order corrections in an expansion in the Planck length or, equivalently, in the derivative expansion. At present such higher-order corrections cannot be systematically constructed within M-theory, so one must resort to indirect approaches.
One such approach is to calculate the higher-order corrections within perturbative string theory, in particular type IIA in ten dimensions, which is related to eleven-dimensional supergravity by dimensional reduction. The effective action of string theory can be systematically constructed perturbatively in a loop expansion in the string coupling,
S eff = ∞ g=0 g 2g−2 s d 10 x √ GL g ,(1.1)
where g is the loop order (equivalently, the genus of the Riemann surface), g s is the string coupling constant, G is the spacetime metric and L g is the effective action at order g. Each L g admits a perturbative expansion in an infinite series of higher-order derivative terms. Moreover it is expected that each L g should correspond to an independent superinvariant in ten dimensions, see e.g. [3].
The bosonic part of the tree-level effective action takes schematically the following form,
L 0 = L IIA + α ′3 I 0 (R) − 1 8 I 1 (R) + · · · + O(α ′4 ) ,(1.2)
where L IIA is the (two-derivative) Lagrangian of ten-dimensional IIA supergravity, and the ellipses stand for terms which have not been completely determined yet. Unlike the case of N = 1 superstrings, the first higher-derivative correction starts at order α ′3 (eight derivatives). The I 0 , I 1 in (1.2) are defined as follows, I 0 (R) = t 8 t 8 R 4 + 1 2 ε 10 t 8 BR 4 I 1 (R) = −ε 10 ε 10 R 4 + 4ε 10 t 8 BR 4 .
(1.3)
These were constructed in [4], to which we refer for further details, by directly checking invariance under part of the supersymmetry transformations. The terms in (1.3) linear in B are, up to a numerical coefficient, Hodge-dual to the Chern-Simons term B ∧ X 8 [5,6]. The eight-form X 8 , see (4.2) below, is related by descent to the M5-brane anomaly polynomial and is a linear combination of (trR 2 ) 2 and trR 4 . Note that the Chern-Simons term drops out of (1.2).
The R 4 part of the tree-level effective action was determined in [7,8] via four-graviton scattering amplitudes and in [9,10,11] from the vanishing of the worldsheet beta-function at four loops. The NSNS sector of the four-field part of the effective action (common to all superstring theories in ten dimensions) was determined in [12]: it is captured by the simple replacement R →R, whereR is a modified Riemann tensor with torsion which includes the NSNS three-form and the dilaton. 1 The ε 10 ε 10 R 4 term does not contribute to tree-level four-point scattering amplitudes, but gives a nonvanishing contribution to the five-graviton scattering amplitude. The complete tree-level four-point effective action for type II superstrings was first determined in [13] and, in addition to the NSNS sector, consists of terms of the form (∂F ) 2R2 and ∂ 4 F 4 , where F stands for all RR flux.
The superinvariant I 0 can be further decomposed into two separate N = 1 superinvariants in ten dimensions [4], I 0 = −6I 0a + 24I 0b , where, I 0a = (t 8 + 1 2 ε 10 B)(trR 2 ) 2 + · · · I 0b = (t 8 + 1 2 ε 10 B)trR 4 + · · · , (1.4) correspond to the supersymmetrization of the B ∧ (trR 2 ) 2 and B ∧ trR 4 Chern-Simons terms respectively. As we show in the following, if the uplift of I 0a , I 0b gives rise to two separate superinvariants in eleven dimensions, they will necessarily have to be cubic or lower in the fields.
The one-loop effective action takes the following form [7,14],
L 1 = α ′3 I 0 (R) + 1 8 I 1 (R) + · · · + O(α ′4 ) . (1.5)
In particular we see that in this case the Chern-Simons term does not drop out, cf. (1.3). The ellipses above indicate terms which are not completely known, although partial results exist thanks to five-and six-point amplitude computations [15,16,17,18]. Contrary to the tree-level superinvariant L 0 which is suppressed at strong coupling, the uplift of the one-loop superinvariant L 1 is expected to survive in eleven dimensions, and thus to be promoted to an eleven-dimensional superinvariant. We will refer to the latter as the supersymmetrization of the Chern-Simons term C ∧X 8 , the uplift of the ten-dimensional Chern-Simons term, where C is the three-form potential of eleven-dimensional supergravity.
An argument of [19], which we review in the following, guarantees that if the supersymmetrization of the Chern-Simons term is quartic or higher in the fields, then it is unique at the eight-derivative order 2 . The uniqueness of this superinvariant is also supported by the results of [20,21,22] which uses the Noether procedure to implement part of the supersymmetry transformations of elevendimensional supergravity. The results of these references constrain the supersymmetrization of the Chern-Simons term to be of the form, ∆L = l 6 t 8 t 8 R 4 − 1 4! ε 11 ε 11 R 4 − 1 6 ε 11 t 8 CR 4 + R 3 G 2 + · · · + O(l 7 ) , (1.6) 1 Note that [12] contains an error that has unfortunately caused some confusion in the literature: the expansion of the t8t8R 4 terms of eq. (2.11) in that reference indeed has the form of the term in the square brackets on the right-hand side of eq. (2.13) therein. However, if one replaces R by the modified Riemann tensorR, given in eq. (2.12) therein, eq. (2.13) no longer gives the correct expansion of t8t8R 4 . 2 The existence of independent superinvariants starting at order higher than eight in the derivative expansion will of course spoil the uniqueness of the superinvariant at higher orders.
where l is the Planck length. The ellipses indicate terms which were not determined by the analysis of [20,21,22], while the R 3 G 2 terms were only partially determined. The reduction of the above to ten dimensions is consistent, as expected, with the one-loop IIA superinvariant (1.5). In addition the quartic interactions R 2 (∂G) 2 and (∂G) 4 were determined in [23] by elevendimensional superparticle one-loop computations in the light cone, and in [24,25,26] by a different method which uses tree amplitudes instead. 3 The t 8 t 8 R 4 terms have also been obtained by four-graviton one-loop amplitudes in eleven dimensions [27,28], while it can be shown [29] that higher loops do not contribute to the superinvariant (1.6).
In the present paper we reexamine the problem of calculating supersymmetric higher-order derivative corrections to eleven-dimensional supergravity from the point of view of the action principle approach. This method relies on the superspace formulation of the theory and is particularly well suited to the supersymmetrization of Chern-Simons terms. Given an elevendimensional Chern-Simons term there is an associated gauge-invariant twelve-superform obtained by exterior differentiation. The action principle approach can be carried out provided the twelveform is Weil-trivial, i.e. exact on the space of on-shell superfields. Computing the superinvariant then boils down to explicitly solving the Weil-triviality condition for the twelve-form.
We show that, provided the superspace Bianchi identities admit a perturbative solution in the derivative expansion, there will be at least two independent superinvariants at the eightderivative order. If we also assume that the twelve-superforms associated to the anomalous (in the presence of an M5-brane) Chern-Simons terms, C ∧ (TrR 2 ) 2 and C ∧ TrR 4 , are separately Weil-trivial, there will be a third independent superinvariant at this order. Moreover we argue that, under certain conditions, at least two of the superinvariants should be expected to survive to all orders in the derivative expansion. However only one of those would correspond to the supersymmetrization of C ∧ X 8 , cf. (1.6).
As already noted this superinvariant can be shown to be unique, assuming it is quartic in the fields. On the other hand, a necessary condition for the superinvariant to be quartic is the exactness, in the so-called τ -cohomology, of X 0,8 -the purely spinorial component of X 8 . In that case we also show that the solution of the Weil-triviality condition of the corresponding twelveform is guaranteed to exist. Proving the τ -exactness of X 0,8 is the first and arguably most difficult step in obtaining the explicit solution to the Weil-triviality condition of the twelveform, and therefore constructing the superinvariant using the action principle.
To tackle this computationally intensive problem we have built on the computer program [30], to supplement it, among other things, with functionalities related to Young tableaux [31]. By a combination of calculational techniques involving the implementation of Fierz identities and Young tableaux projections we prove that certain highly non-trivial necessary conditions for the τ -exactness of X 0,8 are satisfied. As a corollary of our work, it follows that any potential superinvariant associated to the anomalous Chern-Simons terms, C ∧ (TrR 2 ) 2 and C ∧ TrR 4 , must necessarily contain terms cubic or lower in the fields.
The plan of the rest of the paper is a follows. In section 2 we review the different superspace cohomologies that will be useful in the following. In section 3 we introduce the action principle approach and in section 3.1 we show how to obtain the eleven-dimensional supergravity of [1] in this framework. In section 3.2 we apply the action principle to derive the five-derivative correction. Section 4 considers the eight-derivative correction. In section 4.1 we examine the number of independent superinvariants at the eight-derivative order. Section 4.2 addresses the problem of the τ -exactness of X 0,8 . In section 4.3 we discuss the conditions for the existence of the superinvariants to all orders in the perturbative expansion. We conclude in section 5. Further technical details are included in the appendices.
Cohomology in superspace
In this section we review the various superspace cohomology groups that will be useful in the following. This is not new material, but we are including it here to make the paper self-contained and for the benefit of the readers who may not be familiar with the relevant literature.
Let us start by explaining our conventions: Eleven-dimensional superspace [32,33] consists of eleven even (bosonic) and thirty-two odd (fermionic) dimensions, with structure group the eleven-dimensional spin group. Let A = (a, α) be flat tangent superindices, where a = 0, . . . 10 is a Lorentz vector index and α = 1, . . . 32 is a Majorana spinor index. Curved superindices will be denoted by M = (m, µ), with the corresponding supercoordinates denoted by Z M = (x m , θ µ ). The supercoframe is denoted by E A = (E a , E α ) while the dual superframe is denoted by E A = (E a , E α ). We can pass from the coframe to the coordinate basis using the supervielbein,
E A = dz M E M A .
We shall assume the existence of a connection one-form Ω A B with values in the Lie algebra of the Lorentz group. In particular this implies that,
Ω (a c η b)c = 0 , Ω α β = 1 4 (γ a b ) α β Ω a b , Ω a β = 0 = Ω α b . (2.1)
The associated supertorsion and supercurvature tensors are then given by:
T A = DE A := dE A + E B ∧Ω B A = 1 2 E C ∧E B T BC A R A B = dΩ A B + Ω A C ∧Ω C B = 1 2 E D ∧E C R CDA B ,(2.2)
where the exterior derivative is given by d = dz M ∂ M . The assumption of a Lorentzian structure group implies that the components of the curvature two-form obey a set of equations analogous to (2.1). The super-Bianchi identities (BI) for the torsion and the curvature,
DT A = E B ∧R B A , DR A B = 0 ,(2.3)
follow from the definitions (2.2). Moreover, a theorem due to Dragon [34] ensures that for a Lorentz structure group the second BI above follows from the first and need not be considered separately. Once constraints are imposed the BI cease to be automatically satisfied. As was shown in [33], by imposing the conventional constraint
T c αβ = iγ c αβ ,(2.4)
and solving the torsion BI, one recovers ordinary eleven-dimensional supergravity. In particular one determines in this way all components of the torsion. In addition one can construct a closed super four-form G 4 and a super seven-form G 7 obeying [35,36], 4
dG 4 = 0 , dG 7 + 1 2 G 4 ∧G 4 = 0 ,(2.5)
whose bosonic components correspond to the eleven-dimensional supergravity four-form and its Hodge-dual, respectively:
G m 1 ...m 7 = (⋆G) m 1 ...m 7 . (2.6)
The solution of the eleven-dimensional superspace BI is reviewed in appendix E.
De Rham cohomology and Weil triviality
Let Ω n be the space of n-superforms. Thanks to the nilpotency of the exterior superderivative, one can define de Rham cohomology groups in superspace in the same way as in the case of bosonic space,
H n = {ω ∈ Ω n |dω = 0}/{ω ∼ ω + dλ, λ ∈ Ω n−1 } . (2.7)
The fact that the topology of the odd directions is trivial means that the de Rham cohomology of a supermanifold coincides with the de Rham cohomology of its underlying bosonic manifold, also known as the body of the supermanifold. In the remainder of the paper we shall assume that the body has trivial topology. This is the simplest type of supermanifold, sometimes called a graded manifold. It implies in particular that every d-closed superform is d-exact.
There is an important caveat to the previous statement: it is only valid when the cohomology is computed on the space of unconstrained superfields. Once constraints are imposed it ceases to be automatically satisfied. Adopting the terminology of [37], we shall call Weil-trivial those d-closed superforms which are also d-exact on the space of constrained (also referred to as "onshell", or "physical") superfields. The cohomology groups computed on the space of constrained superfields will be denoted by H n (phys), as in [19]. As already emphasized, there is no a priori reason why H n (phys) should coincide with the cohomology of the body of the supermanifold.
τ -cohomology
The space of superforms can be further graded according to the even, odd degrees of the forms. We denote the space of forms with p even and q odd components by Ω p,q so that,
Ω n = ⊕ p+q=n Ω p,q . (2.8)
A (p, q)-superform ω ∈ Ω p,q can be expanded as follows,
ω = 1 p!q! E βq . . . E β 1 E ap . . . E a 1 ω a 1 ...apβ 1 ...βq . (2.9)
In the following we will use the notation Φ (p,q) ∈ Ω p,q for the projection of a superform Φ ∈ Ω n onto its (p, q) component.
The exterior superderivative, d : Ω p,q → Ω p+1,q + Ω p,q+1 + Ω p−1,q+2 + Ω p+2,q−1 , when written out in this basis will give rise to components of the torsion as it acts on the coframe. Following [38] we split d into its various components with respect to the bigrading,
d = d b + d f + τ + t ,(2.10)
where d b , d f are even, odd derivatives respectively, such that d b : Ω p,q → Ω p+1,q , d f : Ω p,q → Ω p,q+1 . The operators τ and t are purely algebraic and can be expressed in terms of the torsion. Explicitly, for any ω ∈ Ω p,q we have,
(d b ω) a 1 ...a p+1 β 1 ...βq = (p + 1) D [a 1 ω a 2 ...a p+1 ]β 1 ...βq + p 2 T [a 1 a 2 | c ω c|a 3 ...a p+1 ]β 1 ...βq + q(−1) p T [a 1 |(β 1 | γ ω |a 2 ...a p+1 ]γ|β 2 ...βq) (d f ω) a 1 ...apβ 1 ...β q+1 = (q + 1) (−1) p D (β 1 | ω a 1 ...ap|β 2 ...β q+1 ) + q 2 T (β 1 β 2 | γ ω a 1 ...apγ|β 3 ...β q+1 ) + p(−1) p T (β 1 |[a 1 | c ω c|a 2 ...ap]|β 2 ...β q+1 ) (τ ω) a 1 ...a p−1 β 1 ...β q+2 = 1 2 (q + 1)(q + 2) T (β 1 β 2 | c ω ca 1 ...a p−1 |β 3 ...β q+2 ) (tω) a 1 .
..a p+2 β 1 ...β q−1 = 1 2 (p + 1)(p + 2) T [a 1 a 2 γ ω a 3 ...a p+2 ]γβ 1 ...β q−1 .
(2.11)
The nilpotency of the exterior derivative, d 2 = 0, implies the following identities:
τ 2 = 0 d f τ + τ d f = 0 d 2 f + d b τ + τ d b = 0 d b d f + d f d b + τ t + tτ = 0 d 2 b + d f t + td f = 0 d b t + td b = 0 t 2 = 0 .
(2.12)
The first and the last of these equations are algebraic identities and are always satisfied. On the other hand, as a consequence of the splitting of the tangent bundle into even and odd directions, the remaining identities are only satisfied provided the torsion tensor obeys its Bianchi identity.
The first of the equations in (2.12), the nilpotency of the τ operator, implies that we can consider the cohomology of τ , as first noted in [38] (see also [35] for some related concepts). Explicitly we set,
H p,q τ = {ω ∈ Ω p,q |τ ω = 0}/{ω ∼ ω + τ λ, λ ∈ Ω p+1,q−2 } . (2.13)
As in the case of de Rham cohomology, one could make a distinction between cohomology groups computed on the space of unconstrained superfields and those computed on the space of physical fields.
Suppose now that the conventional constraint (2.4) is imposed so that τ reduces to a gamma matrix. It was conjectured in [19], consistently with the principle of maximal propagation of [39], that in this case the only potentially nontrivial τ -cohomology appears as a result of the so-called M2-brane identity, (γ a ) (α 1 α 2 (γ ab ) α 3 α 4 ) = 0 .
(2.14)
Explicitly, for p = 0, 1, 2, one may form the following τ -closed (p, q)-superforms,
ω α 1 ...αq = S α 1 ...αq ; ω aα 1 ...αq = (γ ab ) (α 1 α 2 P b α 3 ...αq) ; ω abα 1 ...αq = (γ ab ) (α 1 α 2 U α 3 ...αq) , (2.15)
with S, P , U , arbitrary superfields. It can be seen using (2.14) that the forms ω above correspond to nontrivial elements of H p,q τ with p = 0, 1, 2. The conjecture of [19] means that all nontrivial cohomology is thus generated, and that all H p,q τ groups are trivial for p ≥ 3. This was subsequently proven in [40] and [41,42,43,44].
Spinorial cohomology
Following [19] To check that this is well-defined, one first shows that d f ω is τ -closed,
τ d f ω = −d f τ ω = 0 , (2.17)
where we used the second equation in (2.12). Morever d s [ω] is independent of the choice of representative,
[d f (ω + τ λ)] = [d f ω − τ d f λ] = [d f ω] . (2.18)
Furthermore it is simple to check that d 2
s = 0, d 2 s [ω] = d s [d f ω] = [d 2 f ω] = −[(d b τ + τ d b )ω] = 0 ,(2.19)
where we took into account the third equation in (2.12). We can therefore define the corresponding spinorial cohomology groups H p,q s as follows,
H p,q s = {ω ∈ H p,q τ |d s ω = 0}/{ω ∼ ω + d s λ, λ ∈ H p,q−1 τ } . (2.20)
The notion of spinorial cohomology was originally introduced in [45,39] and applied in a series of papers with the aim of computing higher-order corrections to supersymmetric theories [46,47,48,49,50], and more recently in [51,52,53]. The spinorial cohomology as presented above was introduced in [19] and is independent of the value of the dimension-zero torsion. It reduces to the spinorial cohomology of [45,39] upon imposing the conventional constraint (2.4).
Pure-spinor cohomology
It was first pointed out by P. Howe [54] and subsequently elaborated in [19], that in the case where the dimension-zero torsion is flat, cf. (2.4), the cohomology groups H 0,q s are isomorphic to Berkovits's pure-spinor cohomology groups [55]. Therefore, in view of what was said in section 2.3, the latter are also isomorphic to the spinorial cohomology groups that had been computed a few months earlier in [39]. In the following we briefly explain the equivalence between the two formulations.
The pure spinor cohomology groups are defined as follows. Consider an eleven-dimensional pure spinor, λ α ,à la Berkovits, i.e. such that it obeys, 5
λ α γ a αβ λ β = 0 . (2.21)
The pure spinor λ α is assigned ghost number one. Furthermore we define a form of ghost number q as a multi-pure spinor,
ω = λ α 1 . . . λ αq ω α 1 ...αq . (2.22)
Note that the above definition implies that ω ∈ [ω] ∈ H 0,q τ : indeed shifting ω α 1 ...αq by a τ -exact term would drop out of the right-hand side above due to the contractions with the pure spinors; moreover ω α 1 ...αq is trivially τ -closed.
The pure-spinor BRST operator is defined as follows,
Q := λ α D α , (2.23)
where D α is the spinor component of the covariant derivative defined in flat superspace. Therefore the action of Q on omega,
Qω = λ α 1 . . . λ αq λ α q+1 D α q+1 ω α 1 ...αq ,(2.24)
corresponds precisely to the action of d s defined in (2.16). Indeed, for flat superspace the torsion terms drop out and d f reduces to D α , cf. the second line of eq. (2.11). Moreover the contraction with the pure spinors on the right-hand side above implies that Qω ∈ [Qω] ∈ H 0,q+1 τ , for the same reasons noted below (2.22). In other words, in the linearized limit the pure-spinor cohomology groups of ghost number q are isomorphic to the spinorial cohomology groups H 0,q s . For an extended review of pure-spinor superfields, see [57].
The action principle
The action principle, also known as ectoplasmic integration, is a way of constructing superinvariants in D spacetime dimensions as integrals of closed D-superforms [58,59]. Indeed if L is a closed D-superform, the following action is invariant under supersymmetry,
S = 1 D! d D x ε m 1 ...m D L m 1 ...m D | , (3.1)
where a vertical bar denotes the evaluation of a superfield at θ µ = 0. This can be seen as follows. Consider an infinitesimal super-diffeomorphism generated by a super-vector field ξ. The corresponding transformation of the action reads,
δL = L ξ L = (di ξ + i ξ d)L = di ξ L ,(3.2)
where we took into account that L is closed. On the other hand, local supersymmetry transformations and spacetime diffeomorphisms are generated by ξ| and, in view of (3.2), the integrand in (3.1) transforms as a total derivative under such transformations. The action is thus invariant assuming boundary terms can be neglected.
This method is particularly well-suited to actions with Chern-Simons (CS) terms and indeed has been used to construct all Green-Schwarz brane actions [60,61], see [62,63] for more recent applications to other theories and [64] for applications to higher-order corrections. The idea is as follows: let Z D be the CS term and W D+1 = dZ D its exterior derivative. Obviously W D+1 is a closed form. On the other hand one might be led to conclude that the de Rham cohomology group of rank D+1 must be trivial on a supermanifold whose body is D-dimensional, hence W D+1 must also be exact. This means that it can be written as W D+1 = dK D where now, contrary to Z D , K D is a globally-defined (gauge-invariant) superform. It follows that L D := Z D − K D is a closed superform, and can therefore be used to construct a supersymmetric action as in (3.1).
Eleven-dimensional supergravity is another example of an action with Chern-Simons terms, and we turn to the application of the action principle to this case in the following sections. Unfortunately there is a caveat to the previous argument that W D+1 is exact. As already noted in section 2.1, this argument can be applied only in the case where the cohomology is computed on the space of unconstrained superfields, but is not a priori true on the space of physical (onshell) superfields. Interestingly it does turn out to be true in all known cases. As we will see in the following this includes the case of ordinary eleven-dimensional supergravity as well as its supersymmetric corrections with five derivatives. In section 4.2 we show that a sufficient condition for the Weil triviality of the eight-derivative correction is the τ -exactness of X 0,8 .
We shall parameterize the derivative expansion in terms of the Planck length l, so that the Cremmer-Julia-Scherk two-derivative action (CJS) corresponds to zeroth order in l. In section 4 we show that, provided the four-and seven-form BI are satisfied at order O(l 6 ), cf. (4.3), there are at least two Weil-trivial twelveforms W 12 and hence at least two independent supersymmetric actions with eight derivatives. Provided the twelve-forms associated to certain anomalous CS terms are Weil-trivial, cf. (4.16) below, there will be a third independent superinvariant at this order. We argue that at least two of those superinvariants will exist to all orders in the derivative expansion.
As we will see in detail in the following, in practice one solves for the flat components of the closed superform L D in a stepwise fashion in increasing engineering dimension. Once all flat components of L D have been determined in this way, the explicit form of the action (3.1) can be extracted using the formula,
L m 1 ...m D = e m D a D · · · e m 1 a 1 L a 1 ...a D + D e m D a D · · · e m 2 a 2 ψ m 1 α 1 L α 1 a 2 ...a D + · · · · · · + ψ m D α D · · · ψ m 1 α 1 L α 1 ...α D ,(3.3)
where ψ α m := E m α | and e m a := E m a | are identified as the gravitino and the vielbein of (bosonic) spacetime respectively. In particular the bosonic terms of the Lagrangian can be read off immediately from L a 1 ...a D .
CJS supergravity in the action principle formulation
The eleven-dimensional supergravity action reads [1],
S = R ⋆ 1 − 1 2 G 4 ∧ ⋆ G 4 − 1 6 C 3 ∧G 4 ∧G 4 ,(3.4)
where dC 3 = G 4 is the threeform potential; it is understood that only the bosonic (11, 0) components of the forms enter the formula above, as in (3.1).
This action can also be understood from the point of view of the action principle as follows. The twelveform corresponding to the CS term reads,
W 12 = − 1 6 G 4 ∧G 4 ∧G 4 = dZ 11 ; Z 11 = − 1 6 C 3 ∧G 4 ∧G 4 . (3.5)
Using the BI (2.5) this can also be written in a manifestly Weil-trivial form,
W 12 = dK 11 ; K 11 = 1 3 G 4 ∧G 7 . (3.6)
Taking L 11 = Z 11 − K 11 we obtain that the following action is invariant under supersymmetry,
S = − 1 3 G 4 ∧G 7 − 1 6 C 3 ∧G 4 ∧G 4 . (3.7)
This can then be put in the form (3.4) by using the on-shell conditions ⋆G 4 = G 7 and G 4 ∧ ⋆ G 4 = 6R ⋆1, cf. appendix A. Therein we also give the details of the solution of the superspace equation W 12 = dK 11 and we show, as a byproduct, that the solution for K 11 given in (3.6) is unique up to exact terms.
The O(l 3 ) correction (five derivatives)
It was shown in [49], by directly computing the relevant spinorial cohomology group, that there is a unique superinvariant at the five derivative level (order l 3 in the Planck length). 6 The modified eleven-dimensional action to order l 3 reads,
S = R ⋆ 1 − 1 2 G 4 ∧ ⋆ G 4 − 1 6 C 3 ∧G 4 ∧G 4 + l 3 C 3 ∧G 4 ∧trR 2 + 2 trR 2 ∧ ⋆ G 4 , (3.8)
where an arbitrary numerical coefficient has been absorbed in the definition of l and trR 2 :
= R a b ∧R b
a ; it is understood that only the bosonic (11, 0) components of the forms enter the formula above. This action can also be easily understood from the point of view of the action principle as follows. Consider the twelve-form corresponding to the CS term at order l 3 ,
W 12 = G 4 ∧G 4 ∧trR 2 = dZ 11 ; Z 11 = C 3 ∧G 4 ∧trR 2 . (3.9)
Using the BI (2.3), (2.5) this can also be written in a manifestly Weil-trivial form,
W 12 = dK 11 ; K 11 = −2 G 7 ∧trR 2 .
(3.10)
Taking L 11 = Z 11 − K 11 we obtain the following superinvariant at order l 3 ,
∆S = C 3 ∧G 4 ∧trR 2 + 2 G 7 ∧trR 2 . (3.11)
This can be seen, using the Hodge duality relation G 7 = ⋆G 4 , to precisely correspond to the order-l 3 terms in (3.8). 7
In appendix B we work out in detail the superspace equation W 12 = dK 11 and confirm that the solution (3.10) for K 11 is unique up to exact terms, in accordance with the spinorial cohomology result of [49]. 6 As explained in [49], on a topologically trivial spacetime manifold this superinvariant can be removed by an appropriate field redefinition of the threeform superpotential. However on a spacetime with nonvanishing first Pontryagin class the superinvariant cannot be redefined away without changing the quantization condition of the fourform field strength. 7 The Hodge duality relation between G7 and G4 is expected to receive higher-order corrections (see below 4.11). These can be neglected here since ∆S is already a higher-order correction.
The O(l 6 ) correction (eight derivatives)
As was shown in [6,65], the requirement that the M5-brane gravitational anomaly is cancelled by inflow from eleven dimensions implies the existence of certain CS terms Z 11 at the eightderivative order in the eleven-dimensional theory. The corresponding twelve-form reads,
W 12 = l 6 G 4 ∧X 8 = dZ 11 ; Z 11 = l 6 C 3 ∧X 8 , (4.1)
where X 8 is related to the M5-brane anomaly polynomial by descent,
X 8 = trR 4 − 1 4 (trR 2 ) 2 ,(4.2)
and we have set (trR 2 ) 2 := trR 2
∧trR 2 , trR 4 := R a b ∧R b c ∧R c d ∧R d a .
At eight derivatives the modified four-and seven-form BI read,
dG 4 = 0 ; dG 7 + 1 2 G 4 ∧G 4 = l 6 X 8 , (4.3)
where a numerical coefficient has been absorbed in the definition of l. We expand the forms perturbatively in l,
G 4 = G (0) 4 + l 6 G (1) 4 + · · · ; G 7 = G (0) 7 + l 6 G (1) 7 + · · · ,(4.4)
and similarly for the supercurvature R A B . Note that in the expansion above the bosonic components of the lowest-order fields, G Solving perturbatively the BI at each order in l, taking into account that the exterior superderiva-
tive d = dz M ∂ M is zeroth-order in l, implies, dG (0) 4 = 0 ; dG (1) 4 = 0 ; dG (0) 7 + 1 2 G (0) 4 ∧G (0) 4 = 0 ; dG (1) 7 + G (1) 4 ∧G (0) 4 = X (1) 8 , (4.5) where we have set l 6 X 8 = l 6 X (1) 8 + · · · . Note that X(1)
8 only involves the lowest-order curvature R (0) . Let us expand the twelve-form W 12 perturbatively in l, W 12 = l 6 W (1) 12 + · · · , so that,
W (1) 12 = X (1) 8 ∧G (0) 4 = dZ 11 ; Z 11 = X (1) 8 ∧C (0) 3 . (4.6)
It then follows from (4.5) that this can also be written in a manifestly Weil-trivial form as follows,
W (1) 12 = dK 11 ; K 11 = G (1) 7 ∧G (0) 4 − 2G (0) 7 ∧G (1) 4 . (4.7)
In particular we see that it suffices to solve the four-and seven-form BI in order to determine the order-l 6 superinvariant corresponding to L 11 = Z 11 − K 11 ,
∆S = l 6 X (1) 8 ∧C (0) 3 − G (1) 7 ∧G (0) 4 + 2G (0) 7 ∧G (1) 4 , (4.8)
where it is understood that only the bosonic (11, 0) components of the forms enter. This is the superinvariant corresponding to the supersymmetrization of the CS term (4.1). The action would then read to this order,
S = R (0) ⋆ 1 − 1 2 G (0) 4 ∧ ⋆ G (0) 4 − 1 6 C (0) 3 ∧G (0) 4 ∧G (0) 4 + l 6 X (1) 8 ∧C (0) 3 − G (1) 7 ∧G (0) 4 + 2G (0) 7 ∧G (1) 4
, (4.9) where R (0) , G (0) are identified with the fieldstrengths of the physical fields in the supergravity multiplet, while the first-order fields R (1) , G (1) should be thought of as gauge-invariant functions of the physical fields. We see that the action above is in agreement with the expectation that the bosonic part of the derivative-corrected supergravity action should be of the form,
S = R ⋆ 1 − 1 2 G 4 ∧ ⋆ G 4 − 1 6 C 3 ∧G 4 ∧G 4 + l 6 X 8 ∧C 3 + ∆L ⋆ 1 , (4.10)
with ∆L a function of R, G and their derivatives. Since ∆L is gauge invariant, we see in particular that the CS terms do not receive higher-order corrections beyond eight derivatives.
Varying (4.10) with respect to C 3 implies,
d ⋆ G 4 + 1 2 G 4 ∧G 4 = X 8 + δ δC 3 (∆L ⋆ 1) . (4.11)
It is straightforward to see that the second term on the right-hand side above is exact by virtue of the fact that ∆L is gauge invariant and thus only depends on C 3 through G 4 . Indeed the variation of the C 3 -dependent terms in the ∆L part of the action (4.10) can be written (possibly up to integration by parts) in the form Φ 7 ∧dδC 3 , for some seven-form Φ 7 . Therefore by appropriately correcting the lowest-order duality relation by higher-derivative terms, G 7 = ⋆G 4 + O(l 6 ), one arrives at the modified BI (4.3).
How many superinvariants?
We have seen that provided the modified BI (4.3) are satisfied, there will be at least one superinvariant at eight derivatives, cf. (4.8). A second independent superinvariant can also be similarly constructed as follows. Consider the twelve-form,
W ′ 12 = 1 6 G 4 ∧G 4 ∧G 4 . (4.12)
Expanding perturbatively to order l 6 we obtain,
W ′(1) 12 = 1 2 G (0) 4 ∧G (0) 4 ∧G(1)4 = dZ 11 ; Z 11 = 1 2 G (1) 4 ∧G (0) 4 ∧C (0) 3 ,(4.13)
The above can also be written in a manifestly Weil-trivial form using (4.5),
W ′(1) 12 = dK 11 ; K 11 = −G (0) 7 ∧G(1)
4 .
(4.14)
The order-l 6 superinvariant corresponding to Z 11 − K 11 then reads,
∆S ′ = l 6 G (1) 4 ∧ 1 2 G (0) 4 ∧C (0) 3 + G (0) 7 , (4.15)
where it is understood that only the bosonic (11, 0) components of the forms enter. The above superinvariant does not contain the correct CS terms required by anomaly cancelation, cf. (4.10), and should therefore be excluded by the requirement of quantum consistency of the theory. However if one is only interested in counting superinvariants at order l 6 in the classical theory, the above superinvariant is perfectly acceptable and its existence is guaranteed provided the BI are obeyed to order l 6 .
Dropping the requirement of quantum consistency, relying on classical supersymmetry alone, one may also consider the following two twelveforms,
U 12 = l 6 G 4 ∧trR 4 ; V 12 = l 6 G 4 ∧(trR 2 ) 2 ,(4.16)
so that U − 1 4 V is the Weil-trivial twelveform corresponding to the CS terms of eleven-dimensional supergravity required for anomaly cancellation, cf. (4.1). It follows that either U , V are both Weil-trivial, or neither U nor V is Weil-trivial. If the former is true, there would exist gaugeinvariant elevenforms K U , K V so that at order l 6 we have U (1) = dK U , V (1) = dK V . One can then construct two corresponding superinvariants using the action principle,
∆S U = l 6 trR 4 ∧C (0) 3 − K U ; ∆S V = l 6 (trR 2 ) 2 ∧C (0) 3 − K V . (4.17)
By the argument at the end of the last section, ∆S U , ∆S V should correspond to a modified BI obtained by replacing the right-hand side of the second equation in (4.3) by trR 4 , (trR 2 ) 2 respectively. Then K U , K V would still be given by (4.7) but with G
4 , G(1)
7 solutions of the new modified BI.
Together with the superinvariant ∆S ′ of (4.15), we would then have a total of at least three independent superinvariants at the eight-derivative order, with only one linear combination thereof, ∆S of (4.8), corresponding to the quantum-mechanically consistent eight-derivative correction. As we will see in section (4.2), if ∆S U , ∆S V exist they must necessarily be cubic or lower in the fields.
τ -exactness of X 8
Based on what is known about superinvariants in D < 11 dimensions [66], it is plausible to assume that the superinvariant (4.8) corresponding to the supersymmetrization of the CS term (4.1) should be quartic or higher in the fields. As pointed out in [19], a necessary condition for the superinvariant to be quartic is that the order-l 6 sevenform should be quartic or higher in the fields. Since G (1) 0,7 cannot be quartic or higher in the fields, as can be seen by dimensional analysis, the order-l 6 seven-form BI (4.5) must be solved for G (1) 0,7 = 0. It then follows that the purely spinorial component of the M5-brane anomaly eightform X 0,8 is τ -exact. Explicitly, the first nontrivial component (at dimension four) of the seven-form BI then reads,
γ f (α 1 α 2 | G (1) f |α 3 ...α 8 ) = X (1) α 1 ...α 8 . (4.18)
As explained in detail in appendix E, taking the form of G (0) into account, cf. (E.1), it follows that the Weil-triviality condition,
W 12 = dK 11 , (4.19)
is solved up to dimension 7/2 for K 0,11 = K 1,10 = K 2,9 = 0. At dimension four, condition (4.19) then takes the form,
γ f (α 1 α 2 | K f ab|α 3 ...α 10 ) = W (1) abα 1 ...α 10 . (4.20)
From this it follows that (4.19) is solved, up to τ -exact terms, for K 3,8 given in terms of G
1,6 , cf. (4.18),
K abcα 1 ...α 8 = 3 γ ab α 1 α 2 G (1) cα 3 ...α 8 , (4.21)
where it is understood that all bosonic (spinor) indices are antisymmetrized (symmetrized). Note that the solution for K 3,8 above relies on the M2-brane identity (2.14).
Moreover, it can be shown that all higher components of K 11 solving (4.19) are automatically guaranteed to exist. To see this, let us define the twelve-form,
I 12 := W 12 − dK 11 ,(4.22)
which is closed by construction,
0 = (dI) p,13−p = τ I p+1,11−p + d f I p,12−p + d b I p−1,13−p + tI p−2,14−p . (4.23)
On the other hand, as we saw above, provided (4.18) holds, condition (4.19) is solved up to dimension four, i.e. I p,12−p = 0 for p = 0, 1, 2. Setting p = 2 in (4.23) then gives τ I 3,9 = 0, which implies I 3,9 = 0 up to a τ -exact piece that can be absorbed in K 4,7 , since all τ -cohomology groups H p,12−p τ are trivial for p ≥ 3, cf. section 2.2. By induction we easily see that I p,12−p = 0, for all p ≥ 3. In other words, provided (4.18) holds, the Weil-triviality condition (4.19) is guaranteed to admit a solution.
In the present paper we provide highly nontrivial evidence corroborating (4.18). We will give the outline of the argument here, relegating the technical details to appendix C. The component X 0,8 of the anomaly polynomial in (4.18) contains a large number of terms of the form G 4 , which can be organized in terms of irreducible representations of B 5 . Using certain Fierz identities, cf. appendix D, we have been able to show that almost all of these terms are indeed τ -exact. There are only three irreducible representations of B 5 corresponding to terms which can potentially be present in X 0,8 and are not τ -exact. These are: (04000), (03002) and (02004), where we use the Dynkin notation for B 5 , see e.g. appendix C of [19].
On the other hand we show that, after Fierzing, X 0,8 can be put in the form,
X 0,8 = (γ a 1 a 2 )(γ a 3 a 4 )(γ a 5 a 6 )(γ a 7 a 8 ) G 4
a 1 a 2 ;a 3 a 4 ;a 5 a 6 ;a 7 a 8 + (γ a 1 a 2 )(γ a 3 a 4 )(γ a 5 a 6 )(γ a 7 ...a 12 ) G 4 a 1 a 2 ;a 3 a 4 ;a 5 a 6 ;a 7 ...a 12 In the above ≈ denotes equality up to τ -exact terms. These two constraints are highly nontrivial, involving seemingly miraculous cancellations between hundreds of terms. We have shown that, remarkably, (4.25) are indeed identically satisfied.
+ (γ a 1 a 2 )(γ a 3 a 4 )(γ a 5 ...a 9 )(γ a 10 ...a 14 ) G 4 a 1 a 2 ;
Furthermore we show that the required cancellations for (4.25) to hold, crucially rely on the relative coefficient between trR 4 and (trR 2 ) 2 in X 8 . In other words we show that the purely spinorial components of trR 4 , (trR 2 ) 2 are not separately τ -exact. Consequently, if the twelveforms U , V are Weil-trivial, the corresponding modified order-l 6 BI will be solved for some G
0,7 would vanish and the purely spinorial components of trR 4 , (trR 2 ) 2 would be τ -exact.) It then follows from (4.7) that also K U , K V will be cubic or lower, and similarly for ∆S U , ∆S V , cf. (4.17).
Integrability
The perturbative expansion of the curved components following from (4.4) reads,
G M 1 ...M 4 = G (0) M 1 ...M 4 + l 6 G (1) M 1 ...M 4 + · · · ,(4.26)
and similarly for G 7 and R A B . Note that in terms of flat components there is a mixing between zeroth order and order l 6 due to,
Φ = E A Φ A = E (0)A Φ (0) A + l 6 (E (0)A Φ (1) A + E (1)A Φ (0) A ) + · · · ,(4.27)
where we have expanded the coframe, E A = E (0)A + l 6 E (1)A + · · · , and we have considered an arbitrary one-form Φ for simplicity. However, if one restricts to the top bosonic component of a superform at θ = 0 as in (3.1), then there is no mixing:
Φ (0) m | = e m a Φ (0) a | + ψ α m Φ (0) α | ; Φ (1) m | = e m a Φ (1) a | + ψ α m Φ (1) α | ,(4.28)
where e m a , ψ α m were defined below (3.3). Indeed the O(l 6 ) corrections to the coframe E A only start at higher orders in the θ-expansion and could be systematically determined as in e.g. [67] once the O(l 6 ) corrections to the torsion components have been determined.
In practice the BI are solved for the flat components of the superforms involved, G
A 1 ...A 4 , G (1) A 1 ...A 4(0)
etc, at each order in l. Consequently the corresponding BI, dG 4 = 0 etc, are only shown to be satisfied up to terms of the next order in l. In principle there may be an integrability obstruction to the solution of the BI at next-to-leading order in the derivative corrections, although that would most probably be prohibitively difficult to check in practice. In the following we shall see that the integrability of a certain superinvariant is guaranteed provided the BI admit solutions to all orders in l. Note however that all-order integrability need not be a consequence of the BI.
The phenomenon of inducing a higher-order correction at next-to-leading order is also well understood at the level of the component action, S = S (0) +l 6 S (1) +· · · . The condition of invariance of the action under supersymmetry transformations δ = δ (0) + l 6 δ (1) + · · · reads, 29) and similarly at higher orders. The term δ (1) S (0) in the second equation above is proportional to the lowest-order equations of motion. Therefore in constructing S (1) we only need to check its invariance with repsect to the lowest-order supersymmetry transformations δ (0) and only up to terms which vanish by virtue of the lowest-order equations of motion. This corresponds, in the superspace approach, to the fact that in solving the first-order BI one uses the zerothorder equations for the various superfields. Once S (1) is thus constructed, the correction δ (1) to the supersymmetry transformations can be read off. Since δ (1) S (1) = 0 in general, this induces a correction S (2) to the action and a corresponding correction δ (2) to the supersymmetry transformations, and so on.
δ (0) S (0) = 0 ; δ (0) S (1) + δ (1) S (0) = 0 ,(4.
The existence of an intergrability obstruction can also be understood in the context of the Noether procedure. Indeed at next-to-leading order we have,
δ (2) S (0) + δ (1) S (1) + δ (0) S (2) = 0 . (4.30)
Therefore there must exist an action S (2) such that its variation with respect to lowest-order supersymmetry transformations is equal to −δ (1) S (1) , up to terms that vanish by virtue of the lowest-order equations of motion. This condition will not be automatically satisfied for every S (1) .
In particular one would like to know how many of the independent superinvariants at order l 6 presented in section 4.1 survive to all orders in the derivative expansion. Assuming M-theory is a non-pertubatively consistent theory, we expect the superinvariant (4.8), corresponding to the supersymmetrization of the CS term required for anomaly cancellation, to be integrable to all orders. Moreover, assuming this superinvariant is at least quartic in the fields, a similar argument as the one detailed below (4.23) shows that it must be unique at order l 6 [19].
In addition, if one assumes that the BI admit a solution to all orders in a perturbative expansion in l, then there is one linear combination of the superinvariants presented in section 4.1 that is guaranteed to exist to all orders in l. Indeed in that case the twelve-form,
W 12 = l 6 X 8 − 1 2 G 4 ∧G 4 ∧G 4 = d G 7 ∧G 4 ,(4.31)
is Weil-trivial by virtue of (4.3), which should now be considered valid to all orders in l. However this is not the superinvariant which corresponds to the supersymmetrization of the anomaly term, cf. (4.8). Indeed by the usual action principle procedure the twelve-form above would give rise to the superinvariant,
∆S = l 6 X 8 ∧C 3 − 1 2 G 4 ∧G 4 ∧C 3 − G 7 ∧G 4 . (4.32)
Expanding to order l 6 and assuming G 4 receives a nonvanishing correction at this order, we see that (4.32) does not coincide with (4.8) and the corresponding l 6 -corrected action is different from (4.9).
In conclusion, under the aforementioned assumptions, we would then expect (at least) two independent superinvariants to exist to all orders in a perturbative expansion in l. Only one of these, the one corresponding to the supersymmetrization of the CS anomaly term, will be quantum-mechanically consistent.
Discussion
We have shown that the highly nontrivial constraints (4.25) are satisfied, corroborating the expectation that the purely spinorial component of X 8 is τ -exact. Furthermore we have seen that the τ -exactness of X 0,8 suffices for the existence of the superinvariant at order l 6 . Solving the τ -exactness of X 0,8 is the first step, and arguably the most difficult, towards the explicit construction, via the action principle approach, of the supersymmetrization of the Chern-Simons term C ∧ X 8 of eleven-dimensional supergravity required for the quantum consistency of the theory.
Conclusively proving the τ -exactness of X 0,8 would in addition require checking that the representation (03002) is absent from X 0,8 . This representation is potentially present in two different Young diagrams. As a consequence, showing the cancellation would involve, after projecting onto the appropriate Young diagram, Fierzing hundreds of four-γ terms. This is equivalent to eight-spinor Fierzing, as opposed to the four-spinor Fierzing which is sufficient in order to show the absence of the (04000) and (02004) representations. At present, this seems prohibitively difficult even with the help of a computer.
As a corollary of this work, we have shown that if the anomalous Chern-Simons terms C ∧ (TrR 2 ) 2 and C∧TrR 4 can be supersymmetrized independently, the corresponding superinvariants must necessarily contain terms cubic or lower in the fields. The existence of eleven-dimensional cubic superinvariants at the eight-derivative order has not been examined in the past. Their existence would presumably imply, by dimensional reduction, the presence of cubic terms in the ten-dimensional superinvariants I 0a , I 0b mentioned in the introduction. This would not be inconsistent with the results of [4] who have excluded from the outset such terms in their analysis. This is an interesting open question to which we hope to return in the future.
dK 11 = − 1 6 G 4 ∧G 4 ∧G 4 , (A.1)
for K 11 gauge-invariant, i.e. function of the fieldstrengths of the physical fields. The explicit construction of K 11 in flat components proceeds by solving the BI at each engineering dimension in a stepwise fashion, from dimension −3 to 2 (i.e. from K α 1 ...α 11 to K a 1 ...a 11 ). In components the BI (A.1) reads,
D [A 1 K A 2 ...A 12 ) + 11 2 T F [A 1 A 2 | K F |A 3 ...A 12 ) = − 11! 6(4!) 3 G [A 1 ...A 4 G A 5 ...A 8 G A 9 ...A 12 ) , (A.2)
where the torsion term arises from the the action of the exterior derivative on the supervielbein. The [ABC) notation stands for symmetrization or antisymmetrization, depending on the bosonic or fermionic nature of the indices. In the following, antisymmetrisation of the indices a i and symmetrisation of the indices α i is always implied.
The engineering (mass) dimensions of the physical fields which will be involved in the construction of K 11 are, At dimension 0, eq. (A.2) reads:
1 2 D α 1 0 K α 2 ...α 6 a 1 ...a 6 + 1 2 D a 1 0 K a 2 .
..a 6 α 1 ...α 6 + 11 2
5 22 T α 1 α 2 f K f α 3 ...α 6 a 1 ...a 6 + 5 22 T a 1 a 2 γ K γa 3 ...a 6 α 1 ...α 6 0 + 12 22 T a 1 α 1 γ K γα 2 ...α 6 a 2 ...a 6 0 = − 11! 6(4!) 3 18 77 G a 1 a 2 α 1 α 2 G a 3 a 4 α 3 α 4 G a 5 a 6 α 5 α 6 .
Most terms vanish and the equation simplifies as follows,
(γ f ) α 1 α 2 K f a 1 .
..a 6 α 3 ...α6 = 90 (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 (γ a 5 a 6 ) α 5 α 6 .
Using the M2-brane identity as well as the so-called M5-brane identity,
(γ e ) α 1 α 2 (γ ea 1 ...a 4 ) α 3 α 4 = 3 (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 , (A.3)
it is easy to check that the solution is given by,
K a 1 ...a 7 α 1 ...α4 = 42 (γ a 1 ...a 5 ) α 1 α 2 (γ a 6 a 7 ) α 3 α 4 . (A.4) Dimension 1/2 -(A 1 . . . A 5 → α 1 . . . α 5 , A 6 . . . A 12 → a 1 . . . a 7 )
At dimension 1/2, eq. (A.2) reads,
12
Dα(γ (5) γ (2) )=0 In the following we will use superscripts to indicate the dimension. This should not be confused with the notation in the main text, e.g. (4.4) where the superscript denotes the order in the derivative expansion. which simplifies to, (γ f ) α 1 α 2 K f a 1 ...a 7 α 3 ...α 5 = 0 .
Since [K f a 1 ...a 7 α 3 ...α 5 ] = 1/2 and there is no gauge-invariant field with that dimension, we conclude,
K a 1 ...a 8 α 1 ...α 3 = 0 . (A.5) Dimension 1 -(A 1 . . . A 4 → α 1 . . . α 4 , A 5 . . . A 12 → a 1 . . . a 8 )
At dimension 1, eq. (A.2) reads,
4 12 D α 1 0 K α 2 ...α 4 a 1 ...a 8 + 8 12 da 1 (γ (5) γ (2) )=0 D a 1 K a 2 ...a 8 α 1 ...α 4 + 11 2 1 11 T α 1 α 2 f K f α 3 α 4 a 1 ...a 8 + 14 33 T a 1 a 2 γ K γa 3 ...a 8 α 1 ...α 4 0 + 16 33 T a 1 α 1 γ K γα 2 ...α 4 a 2 ...a 8 = − 11! 6(4!) 3 12 55 G a 1 a 2 a 3 a 4 G a 5 a 6 α 1 α 2 G a 7 a 8 α 3 α 4 ,
which becomes, using (E.1),
(γ f ) α 1 α 2 K f a 1 ...a 8 α 3 α 4 = − 56 3 i G a 1 a 2 a 3 f (γ f ) α 1 α 2 (γ a 4 ...a 8 ) α 3 α 4 + 7 18 i G f ghi (γ a 1 ...a 6 f ghi ) α 1 α 2 (γ a 7 a 8 ) α 3 α 4 + 70 i (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 G a 5 ...a 8 . (A.6)
The last term above can be expanded as,
70 i (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 G a 5 ...a 8 = 70 3 i (γ f ) α 1 α 2 (γ f a 1 ...a 4 ) α 3 α 4 G a 5 ...a 8 = 42 i (γ f ) α 1 α 2 (γ [f a 1 ...a 4 | ) α 3 α 4 G |a 5 ...a 8 ] − 56 3 i (γ f ) α 1 α 2 (γ [a 1 ...a 5 | ) α 3 α 4 G |a 6 a 7 a 8 ]f .
Similarly, the second term on the right-hand side of (A.6) can be written in a manifestly τ -exact form,
7 18 (γ a 1 ...a 6 f ghi ) α 1 α 2 (γ a 7 a 8 ) α 3 α 4 = − 7 18 ǫ ja 1 ...a 6 f ghi (γ j ) α 1 α 2 (γ a 7 a 8 ) α 3 α 4 = − 1 2 ǫ [ja 1 ...a 6 | f ghi (γ j ) α 1 α 2 (γ |a 7 a 8 ] ) α 3 α 4 + 1 9 ǫ a 1 ...a 7 f ghi (γ j ) α 1 α 2 (γ a 8 j ) α 3 α 4 0 .
Then eq. (A.6) takes the following form,
(γ j ) α 1 α 2 K ja 1 ...a 8 α 3 α 4 = (γ j ) α 1 α 2 42 i (γ [ja 1 ...a 4 ) α 3 α 4 G a 5 ...a 8 ] − 1 2 i ǫ [ja 1 ...a 6 | i 1 ...i 4 (γ |a 7 a 8 ] ) α 3 α 4 G i 1 ...i 4 .
Since the cohomology group H 9,2 τ is trivial, the solution to the above equation reads, K a 1 ...a 9 α 1 α 2 = 42 i (γ a 1 ...a 5 ) α 1 α 2 G a 6 ...a 9 − 1 2 i ǫ a 1 ...a 7 i 1 ...i 4 (γ a 8 a 9 ) α 1 α 2 G i 1 ...i 4 , up to τ -exact terms.
Dimension 3/2 -(A 1 . . . A 3 → α 1 . . . α 3 , A 4 . . . A 12 → a 1 . . . a 9 )
At dimension 3/2, eq. (A.2) reads, 3 12
D α 1 K α 2 α 3 a 1 ...a 9 − 9 12 D a 1 0 K a 2 ...a 9 α 1 ...α 3 − 11 2 1 22 T α 1 α 2 f K f a 1 .
..a 9 α 3 + 6 11 T a 1 a 2 γ K γa 3 ...a 9 α 1 ...α 3 + 9 22
T a 1 α 1 γ K γα 2 α 3 a 2 ...a 9 0 = 0 , which becomes, using (E.1),
(γ f ) α 1 α 2 K f a 1 ...a 9 α 3 = + 252 (γ a 1 ...a 5 ) α 2 α 3 (γ a 6 a 7 ) α 1 γ T a 8 a 9 γ − 3 ǫ a 1 ...a 7 i 1 ...i 4 (γ a 8 a 9 ) α 1 α 2 (γ i 1 i 2 ) α 3 γ T i 3 i 4 γ + 504 (γ a 1 a 2 ) (α 1 α 2 | (γ a 3 ...a 7 ) |α 3 γ) T a 8 a 9 γ . (A.7)
The decomposition of K f a 1 ...a 9 α 3 in irreducible components is given by
(10000) ⊗ (00001) = (10001) ⊕ (00001) ,
whereas T ab α is in the representation (01001). It follows that,
K a 1 ...a 10 α 1 = 0 , (A.8)
and moreover the right-hand side of (A.7) must vanish identically. This can be verified by e.g. taking the Hodge dual of (γ i 1 i 2 ) α 3 γ in the second term of (A.7), and using the γ-tracelessness of T ab γ , cf. (E.3).
Dimension 2 -(A 1 A 2 → α 1 α 2 , A 3 . . . A 12 → a 1 . . . a 10 )
At dimension 2, eq. (A.2) reads, 2 12 D α 1 K α 2 a 1 ...a 10 + 10 12 D a 1 K a 2 ...a 10 α 1 α 2 + 11 2
1 66 T α 1 α 2 f K f a 1 ...a 10 + 10 33 T a 1 a 2 γ K γa 3 ...a 10 α 1 α 2 + 15 22 T a 1 α 1 γ K γα 2 a 2 .
..a 10 = − 11! 6(4!) 3
66
G a 1 a 2 a 3 a 4 G a 5 a 6 a 7 a 8 G a 9 a 10 α 1 α 2 , which becomes, using (E.1), Muliplying by γ (1) and taking the trace leads to,
(γ f ) α 1 α 2 K fK a 1 ...a 11 = 1 72 ǫ a 1 ...a 11 G d 1 ...d 4 G d 1 ...d 4 . (A.10)
On the other hand contracting (A.9) with γ (2) or γ (5) imposes that the contraction of the righthand side must be identically zero. This can indeed be straightforwardly verified using (E.3).
Dimension 5/2 -(A 1 → α 1 , A 2 . . . A 12 → a 1 . . . a 11 )
The equation at dimension 5/2 does not contain any additional information, but serves as a consistency check for the expressions we found for K a 1 ...a 11 . It reads,
1 12 D α 1 K a 1 ...a 11 − 11 12 D a 1 K a 2 ...a 11 α 1 − 11 2 2 12 T a 1 a 2 γ K γa 3 ...a 11 α 1 − 10 12 T a 1 α 1 γ K γa 2 ...a 11 = 0 ,
which becomes, using (E.1) and (A.10),
1 72 ǫ a 1 ...a 11 D α 1 G abcd G abcd − 330 i G a 1 a 2 gh (γ gh a 3 .
..a 9 T a 10 a 11 ) α 1 + 2310 i G a 1 ...a 4 (γ a 5 ...a 9 T a 10 a 11 ) α 1 .
Using (E.2), (D.1) we then obtain the constraint, We have thus constructed the explicit expression of all components of K 11 and have seen that it is unique up to exact terms. Its purely bosonic component in particular takes the following form, 13) where in this subsection we have reverted to bosonic conventions for bosonic forms. Using the action principle then leads to the CJS action of section 3.1.
0 = ǫ a 1 ...a 11 T d 1 d 2 δ (γ d 3 d 4 ) δα 1 G d 1 ...d 4 + 77 4 ǫ a 1 ...a 5 b 1 ...b 6 (γ b 1 ...b 6 T a 6 a 7 ) α 1 G a 8 ...a 11 − 990 ǫ a 1 ...a 7 b 1 b 2 gh (γ b 1 b 2 T a 8 a 9 ) α 1 G aK (2) = 1 72 · 11! ǫ a 1 ...a 11 G d 1 ...d 4 G d 1 ...d 4 dx a 1 ∧ . . . ∧dx a 11 = 1 3 G∧ ⋆ G (A.12) = − R ⋆ 1 + 1 2 G∧ ⋆ G , (A.
The last two equalities in (A.12) above can be seen as follows. The volume element is defined as,
dV = ⋆1 = 1 11! ǫ a 1 ...a 11 dx a 1 ∧ . . . ∧dx a 11 ,
from which it follows that,
G∧ ⋆ G = 1 7! (4!) 2 G a 1 ...a 4 ǫ a 5 ...a 11 b 1 ...b 4 G b 1 ...b 4 −ǫ a 1 ...a 11 dV dx a 1 ∧ . . . ∧dx a 11 = 1 4! G d 1 ...d 4 G d 1 ...d 4 dV .
Moreover, taking the trace of the third relation of (E.3) gives,
R ⋆ 1 = 1 144 G d 1 ...d 4 G d 1 ...d 4 dV = 1 6 G∧ ⋆ G .
B. Weil triviality at l 3
In this section we are looking for the solution to the equation,
dK 11 = G 4 ∧G 4 ∧R ab ∧R ba . (B.1)
We will construct all components of K 11 explicitly and confirm that the solution of section 3.2 is unique up to exact terms. In components the equation above takes the following form,
D [A 1 K A 2 ...A 12 ) + 11 2 T F [A 1 A 2 | K F |A 3 ...A 12 ) = 11! 4(4!) 2 R [A 1 A 2 |c 1 c 2 R |A 3 A 4 | c 2 c 1 G |A 5 ...A 8 G A 9 ...A 12 ) . (B.2)
The dimensions of the physical fields are the same as before, with the addition of [R abcd ] = 2. The dimensions of the various components of K range from −1/2 (K α 1 ...α 11 ) to 5 (K a 1 ...a 11 ).
Dimension 0 to 3/2
Since the dimension of K α 1 ...α 11 is −1/2, it must be set to zero as it cannot be expressed in terms of the physical fields. The equation of dimension 0 then takes the form,
D α 1 K α 2 ...α 12 0 + 11 2 T α 1 α 2 f K f α 3 ...α 12 = 11! 4(4!) 2 R α 1 α 2 c 1 c 2 R α 3 α 4 c 2 c 1 G α 5 ...α 8 G α 9 ...α 12 0
, which simplifies to,
(γ f ) α 1 α 2 K f α 3 ...α 12 = 0 . (B.3)
Since [K f α 3 ...α 12 ] = 0 and H 1,10 τ is nontrivial, a τ -nonexact solution involving only γ-matrices could exist. In that case K f α 3 ...α 12 would necessary transform as a scalar, since the only available gauge-invariant superfield of zero dimension is a constant. On the other hand, (10000) ⊗ (00001) ⊗ S 10 = 1 × (00000) + · · · , i.e. the decomposition of K f α 3 ...α 12 contains a unique scalar combination. It follows that,
K f α 3 ...α 12 ∝ (γ f ) α 3 α 4 (γ a ) α 5 α 6 (γ a ) α 7 α 8 (γ b ) α 9 α 10 (γ b ) α 11 α 12 .Dimension 2 -(A 1 . . . A 8 → α 1 . . . α 8 , A 9 . . . A 12 → a 1 . . . a 4 )
This is the first equation with a non-zero right-hand side,
8 12 D α 1 0 K α 2 ...α 8 a 1 ...a 4 + 4 12 D a 1 0 K a 2 ...a 4 α 1 ...α 8 + 11 2 14 33 T α 1 α 2 f K f a 1 ...a 4 α 3 ...α 8 + 1 11 T a 1 a 2 γ K γα 1 ...α 8 a 3 a 4 0 + 16 33 T a 1 α 1 γ K γα 3 ...α 8 a 2 ...a 4 0 =3 4 55 11! 4(4!) 2 R α 1 α 2 c 1 c 2 R α 3 α 4 c 2 c 1 G a 1 a 2 α 5 α 6 G a 3 a 4 α 7 α 8 ,
which becomes, using (E.1),
(γ f ) α 1 α 2 K f a 1 ...a 4 α 3 ...α 8 = −180 i (γ f ) α 1 α 2 (γ f a 1 ...a 4 ) α 3 α 4 R α 5 α 6 c 1 c 2 R α 7 α 8 c 2 c 1 .
Since H 5,6 τ is trivial, the solution reads,
K (2) a 1 ...a 5 α 1 ...α 6 = −180 i (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 R α 5 α 6 c 2 c 1 ,
up to τ -exact terms.
Dimension 5/2 -(A 1 . . . A 7 → α 1 . . . α 7 , A 8 . . . A 12 → a 1 . . . a 5 )
At dimension 5/2, eq. (B.2) reads,
7 12 D α 1 K α 2 ...α 7 a 1 ...a 5 − 5 12 D a 1 0 K a 2 ...a 5 α 1 ...α 7 − 11 2 7 22 T α 1 α 2 f K f a 1 ...a 5 α 3 ...α 7 + 5 33 T a 1 a 2 γ K γα 1 ...α 7 a 3 ...a 5 0 + 35 66 T a 1 α 1 γ K γα 3 ...α 7 a 2 ...a 5 0 = 11! 4(4!) 2 2 1 11 R α 1 α 2 c 1 c 2 R α 3 a 1 c 2 c 1 G a 2 a 3 α 4 α 5 G a 4 a 5 α 6 α 7 ,
which becomes, using (E.1),
(γ f ) α 1 α 2 K f a 1 ...a 5 α 3 ...α 7 =
− 120 i (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 (γ e 1 e 2 ) α 5 α 6 (γ [c 1 c 2 T e 1 e 2 ] ) α 7 + 1 24 (γ c 1 c 2 e 1 ...e 4 ) α 5 α 6 (γ e 1 e 2 T e 3 e 4 ) α 7
+ 1800 i (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 R α 5 α 6 c 1 c 2 R α 7 a 5 c 2 c 1 . (B.5)
The second term in (B.5) can be written,
1800 i (γ [a 1 a 2 | ) α 1 α 2 (γ |a 3 a 4 | ) α 3 α 4 R α 5 α 6 c 1 c 2 R α 7 |a 5 ]c 2 c 1 (B.6) = 600 i (γ g ) α 1 α 2 (γ g[a 1 a 2 a 3 a 4 | ) α 3 α 4 R α 5 α 6 c 1 c 2 R α 7 |a 5 ]c 2 c 1 = 600 i (γ g ) α 1 α 2 6 5 (γ [ga 1 a 2 a 3 a 4 | ) α 3 α 4 R α 5 α 6 c 1 c 2 R α 7 |a 5 ]c 2 c 1 + 1 5 (γ a 1 a 2 a 3 a 4 a 5 ) α 3 α 4 R α 5 α 6 c 1 c 2 R α 7 gc 2 c 1 .
One can then verify that the second term on the right-hand side of (B.6) cancels with the first term on the right-hand side of (B.5). Since the first term on the right-hand side of (B.6) is in a τ -exact form and H 6,5 τ is trivial, the solution reads,
K (5/2) a 1 ...a 6 α 1 ...α 5 = 720 i (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 R α 5 a 6 c 2 c 1 ,
up to τ -exact terms.
Dimension 3 -(A 1 . . . A 6 → α 1 . . . α 6 , A 7 . . . A 12 → a 1 . . . a 6 )
At dimension 3, eq. (B.2) reads, R α 1 a 1 c 1 c 2 R α 2 a 2 c 2 c 1 G a 3 a 4 α 3 α 4 G a 5 a 6 α 5 α 6
1 2 D α 1 K α 2 ...+ 2 1 154 R α 1 α 2 c 1 c 2 R α 3 α 4 c 2 c 1 G a 1 a 2 α 5 α 6 G a 3 ...a 6 + 2 3 77 R a 1 a 2 c 1 c 2 R α 1 α 2 c 2 c 1 G a 3 a 4 α 3 α 4 G a 5 a 6 α 5 α 6 ,
which becomes, using (E.1),
− 5 4 i (γ f ) α 1 α 2 K f a 1 ...a 6 α 3 ...α 6 = − 1 2 D α 1 K α 2 ...α 6 a 1 ...a 6 − 1 2 d a 1 K a 2 ...a 6 α 1 ...α 6 − 3 T ǫ a 1 α 1 K ǫα 2 ...α 6 a 2 .
..a 6 − 2700 R α 1 a 1 c 1 c 2 R α 2 a 2 c 2 c 1 G a 3 a 4 α 3 α 4 G a 5 a 6 α 5 α 6 + 225 R α 1 α 2 c 1 c 2 R α 3 α 4 c 2 c 1 G a 1 a 2 α 5 α 6 G a 3 ...a 6 + 1350 R a 1 a 2 c 1 c 2 R α 1 α 2 c 2 c 1 G a 3 a 4 α 3 α 4 G a 5 a 6 α 5 α 6 .
Let us now examine separately each group of terms in the equation above with the same type of field content. There are four G 3 terms which read,
− 225 i G a 1 ...a 4 (γ a 5 a 6 ) α 1 α 2 R α 3 α 4 c 1 c 2 R α 5 α 6 c 1 c 2 − 360 (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 T c 2 α 5 ǫ T c 1 ǫ β (γ a 6 ) βα 6 + 720 (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 T c 1 α 5 ǫ T a 6 ǫ β (γ c 2 ) βα 6 − 540 i (γ a 1 ...a 5 ) (α 1 ǫ| T a 6 α 2 ǫ R |α 3 α 4 | c 1 c 2 R |α 5 α 6 )c 2 c 1 . (B.7)
The last term in (B.7) can be split in two parts,
−540 i 2 6 (γ a 1 ...a 5 ) α 1 ǫ T a 6 α 2 ǫ R α 3 α 4 c 1 c 2 R α 5 α 6 c 2 c 1 + 4 6 (γ a 1 ...a 5 ) α 1 α 2 T a 6 α 3 ǫ R ǫα 4 c 1 c 2 R α 5 α 6 c 2 c 1 .
The first one leads to,
(γ g ) α 1 α 2 5 8 i ǫ ga 1 ...a 6 b 1 ...b 4 G b 1 ...b 4 R α 3 α 4 c 1 c 2 R α 5 α 6 c 2 c 1 + 225 i G a 1 ...a 4 (γ a 5 a 6 ) α 1 α 2 R α 3 α 4 c 1 c 2 R α 5 α 6 c 2 c 1 ,
where the first term is τ -exact, and the second term cancels with the first one in (B.7). It can then be verified that the three remaining G 3 terms cancel out.
Moreover there are three terms of the schematic form G(DG),
− 360 i (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 d c 2 T c 1 α 5 β (γ a 6 ) βα 6 − 720 i (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 d c 1 T a 6 α 5 β (γ c 2 ) βα 6 − 180 i (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 d a 6 R α 5 α 6 c 2 c 1 , (B.8)
which cancel out.
There are two RG terms which read, − 1350 (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 R a 5 a 6 c 1 c 2 R α 5 α 6 c 2 c 1
+ 45 (γ a 1 ...a 5 ) α 1 α 2 R α 3 α 4 c 1 c 2 (γ e 1 e 2 γ a 6 ) α 5 α 6 R c 2 c 1 e 1 e 2 − 2 (γ e 1 e 2 γ c 2 ) α 5 α 6 R c 1 a 6 e 1 e 2 . (B.9)
The first term of (B.9)can be put in a τ -exact form,
− 1350 (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 R a 5 a 6 c 1 c 2 R α 5 α 6 c 2 c 1 = (γ g ) α 1 α 2 − 630 i (γ [ga 1 ...a 4 | ) α 3 α 4 R |a 5 a 6 ] c 1 c 2 R α 5 α 6 c 2 c 1 + 180 (γ a 1 ...a 5 ) α 1 α 2 R a 6 g c 1 c 2 R α 5 α 6 c 2 c 1 ,
while the remaining RG terms cancel out.
There are two T 2 terms which read, + 2700 (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 R α 5 a 5 c 1 c 2 R α 6 a 6 c 2 c 1 (B.10) + 1080 i (γ a 1 ...a 5 ) α 1 α 2 (γ e 1 e 2 ) α 3 α 4 (γ [c 1 c 2 T e 1 e 2 ] ) α 5 + 1 24 (γ c 1 c 2 e 1 ...e 4 ) α 3 α 4 (γ e 1 e 2 T e 3 e 4 ) α 5 R α 6 a 6 c 2 c 1 .
The first term can be put in a τ -exact form,
2700 (γ a 1 a 2 ) α 1 α 2 (γ a 3 a 4 ) α 3 α 4 R α 5 a 5 c 1 c 2 R α 6 a 6 c 2 c 1 = 1 3 2700 (γ g ) α 1 α 2 7 5 (γ ga 1 ...a 4 ) α 3 α 4 R α 5 a 5 c 1 c 2 R α 6 a 6 c 2 c 1 + 2 5 (γ a 1 ...a 5 ) α 3 α 4 R α 5 a 6 c 1 c 2 R α 6 gc 2 c 1 ,
while the remaining T T terms cancel out. Taking the triviality of H 7,4 τ into account, the nonvanishing terms extracted from the RG, T 2 , and G 3 terms lead to the solution,
K (3) a 1 ...a 7 α 1 ...α 4 = 504 i (γ a 1 ...a 5 ) α 1 α 2 − R a 6 a 7 c 1 c 2 R α 5 α 6 c 2 c 1 + 2 R α 5 a 6 c 1 c 2 R α 6 a 7 c 2 c 1 − 1 2 i ǫ a 1 ...a 7 b 1 ...b 4 G b 1 ...b 4 R α 1 α 2 c 1 c 2 R α 3 α 4 c 2 c 1 , (B.11)
up to τ -exact terms.
Dimensions 7/2 -(A 1 . . . A 5 → α 1 . . . α 5 , A 6 . . . A 12 → a 1 . . . a 7 )
At dimension 7/2, eq. (B.2) reads,
5 12 D α 1 K α 2 ...α 5 a 1 ...a 7 − 7 12 D a 1 K a 2 ...a 7 α 1 ...α 5 − 11 2 5 33 T α 1 α 2 f K f a 1 ...a 7 α 3 ...α 5 − 7 22
T a 1 a 2 γ K γα 1 ...α 5 a 3 ...a 7 + 35 66 T a 1 α 1 γ K γα 3 ...α 5 a 2 ...a 7 = 11! 4(4!) 2 2 1 11
R α 1 a 1 c 1 c 2 R a 2 a 3 c 2 c 1 G a 4 a 5 α 3 α 4 G a 6 a 7 α 5 α 6
+ 4 1 66 R α 1 a 1 c 1 c 2 R α 2 α 3 c 2 c 1 G a 2 a 3 α 4 α 5 G a 4 ...a 7 .
The right-hand side of the equation above contains terms of the form G(DT ), T (DG), T R, and T G 2 . The first two groups of terms simply vanish (without the use of any equations of motion or BI). Two τ -exact terms can be extracted from RT and T G 2 , and the remaining terms cancel out. This leads to the solution,
K (7/2) a 1 ...a 8 α 1 ...α 3 = 2016 i (γ a 1 ...a 5 ) α 1 α 2 R a 6 a 7 c 1 c 2 R α 3 a 8 c 2 c 1 + 4 ǫ a 1 ...a 7 b 1 ...b 4 G b 1 ...b 4 R α 1 α 2 c 1 c 2 R α 3 a 8 c 2 c 1 ,
up to τ -exact terms.
Dimensions 4 -(A 1 . . . A 4 → α 1 . . . α 4 , A 5 . . . A 12 → a 1 . . . a 8 )
At dimension 4, eq. (B.2) reads,
4 12 D α 1 K α 2 ...α 4 a 1 ...a 8 + 8 12 D a 1 K a 2 ...a 8 α 1 ...α 4 + 11 2 1 11 T α 1 α 2 f K f a 1 ...a 8 α 3 α 4 + 14 33 T a 1 a 2 γ K γα 1 ...α 4 a 3 ...a 8 + 16 33 T a 1 α 1 γ K γα 2 ...α 4 a 2 ...a 8 = 11! 4(4!) 2 1260 R a 1 a 2 c 1 c 2 R a 3 a 4 c 2 c 1 G a 5 a 6 α 3 α 4 G a 7 a 8 α 5 α 6 + 35 R α 1 α 2 c 1 c 2 R α 3 α 4 c 2 c 1 G a 1 a 2 a 3 a 4 G a 5 ...a 8 + 4 · 210 R α 1 α 2 c 1 c 2 R a 1 a 2 c 2 c 1 G a 3 a 4 a 5 a 6 G a 7 a 8 α 3 α 4 − 2 · 840 R α 1 a 1 c 1 c 2 R α 2 a 2 c 2 c 1 G a 3 a 4 α 3 α 4 G a 5 a 6 a 7 a 8 .
The terms in the equation above can be cast in eight groups: R 2 , RG 2 , R(DG), G 4 , G 2 (DG), GT 2 , T (DT ), and G(DR). Parts of the terms of the form R 2 , G 2 R, and GT 2 can be put in a τ -exact form, while the remaining terms cancel out. Taking into account the BI,
D a 1 R a 2 a 3 c 1 c 2 = −T a 1 a 2 γ R γa 3 c 1 c 2 , (B.12)
we see that the term G(DR) cancel against a term from GT 2 . Taking into account the equation of motion of G we see that a term from G 2 (DG) cancels against a term in G 4 ,
ǫ a 1 ...a 7 b 1 ...b 4 D a 8 G b 1 ...b 4 = 1 2 ǫ a 1 ...a 8 b 1 ...b 3 D c G cb 1 ...b 3 = 105 G a 1 ...a 4 G a 5 ...a 8 . (B.13)
We are thus led to the solution,
K (4) a 1 ...a 9 α 1 ...α 2 = −1512 i (γ a 1 ...a 5 ) α 1 α 2 R a 6 a 7 c 1 c 2 R a 8 a 9 c 2 c 1 − 6 ǫ a 1 ...a 7 b 1 ...b 4 G b 1 ...b 4 R α 1 α 2 c 1 c 2 R a 8 a 9 c 2 c 1 + 12 ǫ a 1 ...a 7 b 1 ...b 4 G b 1 ...b 4 R α 1 a 8 c 1 c 2 R α 2 a 9 c 2 c 1 ,
up to τ -exact terms.
Dimensions 9/2 -(A 1 . . . A 3 → α 1 . . . α 3 , A 4 . . . A 12 → a 1 . . . a 9 )
At dimension 9/2, eq. (B.2) reads,
3 12 D α 1 K α 2 α 3 a 1 ...a 9 − 9 12 D a 1 K a 2 ...a 9 α 1 ...α 3 − 11 2 1 22 T α 1 α 2 f K f a 1 .
..a 9 α 3 + 6 11 T a 1 a 2 γ K γα 1 ...α 3 a 3 ...a 9 + 9 22 T a 1 α 1 γ K γα 2 α 3 a 2 ...a 9 = 11! 4(4!) 2 2
1 110 R α 1 α 2 c 1 c 2 R α 3 a 1 c 2 c 1 G a 2 .
..a 5 G a 6 ...a 9 + 4 3 55 R a 1 a 2 c 1 c 2 R α 1 a 3 c 2 c 1 G α 2 α 3 a 4 a 5 G a 6 ...a 9 .
The terms in the equation above can be cast in seven groups: R(DT ), RT G, G 2 (DT ), G 3 T , T 3 , T G(DG) and T (DR). One term of the form RT G is τ -exact, while all the remaining terms can be seen to cancel out, using (B.12) and (B.13) to convert a term of the form T (DR) to the form T 3 , and a term of the form T G(DG) to the form G 3 T . Up to τ -exact terms, the component of dimension 9/2 then reads,
K (9/2) a 1 ...a 9 α 1 α 2 = 60 i ǫ a 1 ...a 7 b 1 ...b 4 G b 1 ...b 4 R a 8 a 9 c 1 c 2 R α 1 a 10 c 2 c 1 . Dimensions 5 -(A 1 A 2 → α 1 α 2 , A 3 . . . A 12 → a 1 . . . a 10 )
At dimension 5, eq. (B.2) reads,
2 12 D α 1 K α 2 a 1 ...a 9 + 10 12 D a 1 K a 2 .
..a 10 α 1 α 2
+ 11 2 1 66 T α 1 α 2 f K f a 1 ...a 10 + 15 22 T a 1 a 2 γ K γα 1 α 2 a 3 ...a 10 + 10 33 T a 1 α 1 γ K γα 2 a 2 ...a 10 = 11! 4(4!) 2 2 1 66 R α 1 α 2 c 1 c 2 R a 1 a 2 c 2 c 1 G a 3 ...a 6 G a 7 ...a 10 − 1 2 33 R α 1 a 1 c 1 c 2 R α 2 a 2 c 2 c 1 G a 3 a 4 a 5 a 6 G a 7 ...a 10 + 2 1 11 R a 1 a 2 c 1 c 2 R a 3 a 4 c 2 c 1 G a 5 a 6 α 1 α 2 G a 7 a 8 α 3 α 4 .
The terms in the equation above can be cast in nine groups: RT 2 , GT (DT ), G 2 T 2 , GR 2 , GR(DG), RG 3 , R(DR), G 2 (DR), and T 2 (DG). One term in GR 2 is τ -exact, while all the remaining terms cancel out, as can be seen using eq. (B.12) and (B.13) to convert a term of the form R(DR) to the form RT 2 , a term of the form G 2 (DR) to the form G 2 T 2 , and a term of the form T 2 (DG) to the form G 2 T 2 . Up to τ -exact terms, the component of dimension 5 then reads,
K (5) a 1 ...a 11 = −165 ǫ a 1 ...a 7 b 1 ...b 4 G b 1 ...b 4 R a 8 a 9
c 1 c 2 R a 10 a 11 c 2 c 1 .
(B.14)
Dimensions 11/2 -(A 1 → α 1 , A 2 . . . A 12 → a 1 . . . a 11 )
Since there is no new component of K appearing, this equation should be satisfied automatically,
1 12 D α 1 K a 1 ...a 11 − 11 12 D a 1 K a 2 ...a 11 α 1 − 11 2 1 6 T a 1 α 2 f K f a 2 ...a 11 − 5 6 T a 1 a 2 γ K γα 1 a 3 ...a 11 = 11! 4(4!) 2 2 6 R α 1 a 1 c 1 c 2 R a 2 a 3 c 2 c 1 G a 4 ...a 7 G a 8 ...a 11 .
The equation contains six types of terms: T R 2 , GR(DT ), G 2 T R, GT (DR), RT (DG), and T 3 G.
As expected all the terms cancel out, as can be seen using (B.12) and (B.13) to convert a term of the form GT (DR) to the form T 3 G, and a term of the form RT (DG) to the form G 2 T R.
Action at O(l 3 )
We have constructed the explicit expression of each component of K 11 and showed that it is unique up to exact terms. In particular the top component, given in eq. (B.14), precisely agrees with (3.10), leading to the superinvariant of section 3.2.
C. Weil triviality at l 6
The same method will be used to generate the corrections at l 6 -order, cf. section 4. We will look for the solution to the equation
dK 11 = G 4 ∧X(1)
8 . In components this reads,
D [A 1 K A 2 ...A 12 ) + 11 2 T F [A 1 A 2 | K F |A 3 ...A 12 ) = 11! (4!)4 2 G [A 1 ...A 4 R |A 5 A 6 |c 1 c 2 R |A 7 A 8 | c 2 c 3 R |A 9 A 10 |c 3 c 4 R |A 11 A 12 ) c 4 c 1 − 1 4 G [A 1 ...A 4 R |A 5 A 6 |c 1 c 2 R |A 7 A 8 | c 1 c 2 R |A 9 A 10 |d 1 d 2 R |A 11 A 12 ) d 1 d 2 . (C.1)
The dimensions of the various components of K 11 now range from [K α 1 ...α 12 ] = 5 2 to [K a 1 ...a 11 ] = 8.
Dimension 3 and 7/2
If we assume that the superivariant at O(l 6 ) is quartic or higher in fields, the first potentially nonvanishing component of K 11 appears at dimension 4 (it is of the form G 4 ). We thus obtain,
K (5/2) α 1 ...α 11 = K (3) a 1 α 1 ...α 10 = K (7/2) a 1 a 2 α 1 ...α 9 = 0 .
This is consistent with (C.1), whose right-hand side vanishes for dimensions lower than 4.
Dimension 4 -(A 1 . . . A 10 → α 1 . . . α 10 , A 11 A 12 → a 1 a 2 )
Eq. (C.1) takes the following form,
= 11! 4!4 2 G a 1 a 2 α 1 α 2 R α 3 α 4 c 1 c 2 R α 5 α 6 c 2 c 3 R α 7 α 8 c 3 c 4 R α 9 α 10 c 4 c 1 − 1 4 R α 3 α 4 c 1 c 2 R α 5 α 6 c 1 c 2 R α 7 α 8 d 1 d 2 R α 9 α 10 d 1 d 2 ,
which simplifies to,
(γ f ) α 1 α 2 K (4) f a 1 a 2 α 3 ...α 10 = 2520 (γ a 1 a 2 ) α 1 α 2 R α 3 α 4 c 1 c 2 R α 5 α 6 c 2 c 3 R α 7 α 8 c 3 c 4 R α 9 α 10 c 4 c 1 − 1 4 R α 3 α 4 c 1 c 2 R α 5 α 6 c 1 c 2 R α 7 α 8 d 1 d 2 R α 9 α 10 d 1 d 2 = (γ a 1 a 2 ) α 1 α 2 X (8) α 3 ...α 10 . (C.2)
Explicitly, the term (trR 2 ) 2 reads (omitting the factor −1/4),
1 6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ u 4 u 5 )(γ u 6 u 7 ) G u 0 u 1 y 0 y 1 G u 2 u 3 y 0 y 1 G u 4 u 5 z 0 z 1 G u 6 u 7 z 0 z 1 4 24·6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ u 4 u 5 )(γ v 0 ...v 5 ) G u 0 u 1 y 0 y 1 G u 2 u 3 y 0 y 1 G u 4 u 5 v 0 v 1 G v 2 ...v 5 2 24 2 6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ v 0 ...v 3 x 0 x 1 )(γ w 0 ...w 3 x 0 x 1 ) G u 0 u 1 y 1 y 2 G u 2 u 3 y 1 y 2 G v 0 ...v 3 G w 0 ...w 3 4 24 2 6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ v 0 ...v 5 )(γ w 0 ...w 5 ) G u 0 u 1 v 0 v 1 G u 2 u 3 w 0 w 1 G v 2 ...v 5 G w 2 ...w 5 4 24 3 6 4 (γ u 0 u 1 )(γ v 0 ...v 5 )(γ w 0 ...w 3 y 0 y 1 )(γ x 0 ...x 3 y 0 y 1 ) G u 0 u 1 v 0 v 1 G v 2 ...v 5 G w 0 ...w 3 G x 0 ...x 3 1 24 4 6 4 (γ u 0 ...u 3 y 0 y 1 )(γ v 0 ...v 3 y 0 y 1 )(γ w 0 ...w 3 z 0 z 1 )(γ x 0 ...x 3 z 0 z 1 ) G u 0 ...u 3 G v 0 ...v 3 G w 0 ...w 3 G x 0 ...x 3 ,
while the term trR 4 reads,
1 6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ u 4 u 5 )(γ u 6 u 7 ) G u 0 u 1 y 0 y 1 G u 2 u 3 y 0 z 0 G u 4 u 5 y 1 z 1 G u 6 u 7 z 0 z 1 4 24·6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ u 4 u 5 )(γ v 0 ...v 5 ) G u 0 u 1 y 0 y 1 G u 2 u 3 v 0 y 0 G u 4 u 5 v 1 y 1 G v 2 ...v 5 2 24 2 6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ v 0 ...v 5 )(γ w 0 ...w 5 ) G u 0 u 1 v 0 w 0 G u 2 u 3 v 1 w 1 G v 2 ...v 5 G w 2 ...w 5 4 24 2 6 4 (γ u 0 u 1 )(γ u 2 u 3 )(γ v 0 ...v 4 x 0 )(γ w 0 ...w 4 x 0 ) G u 0 u 1 v 0 y 0 G u 2 u 3 w 0 y 0 G v 1 ...v 4 G w 1 ...w 4 4 24 3 6 4 (γ u 0 u 1 )(γ v 0 ...v 4 y 0 )(γ w 0 ...w 4 y 1 )(γ x 0 ...x 3 y 0 y 1 ) G u 0 u 1 v 0 w 0 G x 0 ...x 3 G v 1 ...v 4 G w 1 ...w 4 1 24 4 6 4 (γ u 0 ...u 3 y 0 y 1 )(γ v 0 ...v 3 y 0 z 0 )(γ w 0 ...w 3 y 1 z 1 )(γ x 0 ...x 3 z 0 z 1 ) G u 0 ...u 3 G v 0 ...v 3 G w 0 ...w 3 G x 0 ...x 3 .
Suppose now that the purely femionic component of X 8 can be cast in the τ -exact form of eq. (4.18). The right-hand side of eq. (C.2) would then take the form,
(γ a 1 a 2 ) α 1 α 2 X (8) α 3 ...α 10 = (γ a 1 a 2 ) α 1 α 2 (γ f ) α 3 α 4 G f α 5 ...α 10 = (γ f ) α 1 α 2 3 (γ [a 1 a 2 | ) α 3 α 4 G |f ]α 5 ...α 10 − 2 (γ f a 1 ) α 3 α 4 G a 2 α 5 ...α 10 = (γ f ) α 1 α 2 3 (γ [a 1 a 2 | ) α 3 α 4 G |f ]α 5 ...α 10 ,
which yields,
K f a 1 a 2 α 3 ...α 10 = 3 (γ [a 1 a 2 | ) α 3 α 4 G |f ]α 5 ...α 10 .
In the following we will examine whether X 0,8 can be τ -exact. Since (C.2) contains many different types of terms, it is useful to reduce this expression by simplifying every pair of γ-matrices whose bosonic indices contain contractions, using the decompositions in appendix D. When applied to (trR 2 ) 2 , this method will give three terms of the form γ (2) γ (2) γ (2) γ (2) , γ (2) γ (2) γ (2) γ (6) and γ (2) γ (2) γ (6) γ (6) , together with several manifestly τ -exact terms. Applied to trR 4 , this method will give several of terms of the form previously encountered, plus some new terms of the form γ (2) γ (2) γ (5) γ (5) , which are equivalent to γ (2) γ (2) γ (6) γ (6) by Hodge duality. In order to compare (trR 2 ) 2 with trR 4 , all the γ (6) γ (6) terms must be converted into the form γ (5) γ (5) . This creates new γ-matrices with contracted bosonic indices, which are simplified as before using appendix D. At the end of this process all the terms have the form γ (2) γ (2) γ (2) γ (2) , γ (2) γ (2) γ (2) γ (6) or γ (2) γ (2) γ (5) γ (5) contracted with G 4 (without any contractions among γ-matrices), h G a 4 f gh + · · · G 4 a 1 a 2 ;a 3 a 4 ;a 5 a 6 ;a 7 ...a 12 = 25 2 9 3 4 G a 1 a 7 f g G a 2 a 11 a 12 f G a 3 a 4 a 9 a 10 G a 5 a 6 a 8 g + · · · G 4 a 1 a 2 ;a 3 a 4 ;a 5 ...a 9 ;a 10 ...a 14 = 1 2 11 3 6 G a 1 a 2 a 10 f G a 3 a 5 a 11 f G a 4 a 12 a 13 a 14 G a 6 a 7 a 8 a 9 + · · · ,
(γ a 1 a 2 )(γ a 3 a 4 )(γ a 5 a 6 )(γ a 7 a 8 ) G 4 a 1 a 2 ;(C.6)
where the ellipses stand for more than a hundred terms of this form. No obvious cancellations appear between these three types of terms at this point.
Let us further analyse how X 0,8 is decomposed into irreducible components. First, the product of four γ-matrices contains a symmetric product of eight spinor indices which can be decomposed as follows in irreps of B 5 , (00001) ⊗ S 8 = 1(00000) ⊕ · · · ⊕ 1(40000) 45 .
Each irrep on the right-hand side above corresponds to a γ-structure which can be thought of as a Clebsch-Gordan coefficient: the γ-structure corresponding to (00000) can be thought of as a Clebsch-Gordan coefficient from the scalar to (00001) ⊗ S 8 , etc.
Next, the product of four four-forms G can be decomposed as follows in irreps of B 5 , (00010) ⊗ S 4 = 4(00000) ⊕ · · · ⊕ 6(00004) ⊕ · · · ⊕ 3(40000)
terms, various multiplicites
, and all 95 terms except (00006), (00008), (01006), and (10006) can be found in (00001) ⊗ S 8 . This analysis implies that the contraction of four γ-matrices with four fourforms G can be decomposed into 51 γ-structures, each contracted with (multiple) G 4 terms corresponding to the same irrep of B 5 .
For example, the term (00000) in the decomposition of (00001) ⊗ S 8 gives rise to a single γstructure contracted with the four possible G 4 terms giving rise to a scalar. Explicitly we have,
(γ e 1 )(γ e 1 )(γ e 2 )(γ e 2 ) α 1 G a 1 ...a 4 G a 1 ...a 4 G b 1 ...b 4 G b 1 ...b 4 + α 2 G a 1 a 2 b 1 b 2 G b 1 b 2 c 1 c 2 G c 1 c 2 d 1 d 2 G d 1 d 2 a 1 a 2 + α 3 G a 1 b 1 c 1 c 2 G c 1 c 2 d 1 f 1 G a 1 d 1 g 1 g 2 G g 1 g 2 b 1 f 1 + α 4 G a 1 b 1 ...b 3 G b 1 ...b 3 c 1 G a 1 d 1 ...d 3 G c 1 d 1 ...d 3 ,
for some constants α 1 , . . . , α 4 . Similarly, the (00004) gives rise to the following term,
β 1 (γ e 1 )(γ e 1 )(γ a 1 ...a 6 )(γ b 1 ...b 6 ) + β 2 (γ [a 1 )(γ a 2 ...a 6 ] )(γ [b 1 )(γ b 2 ...b 6 ] ) × α 1 G a 1 a 2 b 1 e 1 G a 3 b 2 b 3 e 1 G a 4 ...a 6 e 2 G b 4 ...b 6 e 2 + α 2 G a 1 a 2 b 1 e 1 G a 3 a 4 b 2 e 2 G a 5 b 3 b 4 e 1 G a 6 b 5 b 6 e 2 + α 3 G a 1 ...a 4 G a 5 b 3 b 4 e 1 G a 6 b 5 b 6 e 2 G b 1 b 2 e 1 e 2 + α 4 G a 1 b 1 b 2 e 1 G a 2 ...a 5 G a 6 b 6 e 1 e 2 G b 3 ...b 5 e 1 + α 5 G a 1 ...a 4 G a 5 b 1 e 1 e 2 G a 6 b 2 e 1 e 2 G b 3 ...b 6 + α 6 G a 1 ...a 3 b 3 G a 4 ...a 6 b 2 G b 1 b 2 e 1 e 2 G b 5 b 6 e 1 e 2 ,
for some constants β 1 , β 2 , α 1 , . . . , α 6 . The 51 γ-structures involved in the decomposition of X (8) can all be found explicitly, and only three of them are not τ -exact: (04000), (03002), and (02004). In other words, except for the structures corresponding to these three irreps all other γ-structures appearing in X 0,8 involve at least one contraction with a γ (1) .
Going back to (C.3): the G 4 a 1 a 2 ;...;a 7 a 8 term, by virtue of its contraction with the four γ-matrices, transforms in the symmetrized product of four Young diagrams , cf. appendix F. Decomposing in irreducible representations of S 8 ,
⊗ S 4 = ⊕ ⊕ ⊕ ⊕ Y T 1 (5 terms) . (C.7)
At the same time G 4 a 1 a 2 ;...;a 7 a 8 admits a decomposition into modules of B 5 × S 8 , R V R × R, where V R is the plethysm of the module V = (10000) of B 5 with respect to the Young diagram R of S 8 . Moreover only the plethysms corresponding to the right-hand side of (C.7) will appear in the decomposition of G 4 a 1 a 2 ;...;a 7 a 8 under B 5 × S 8 . On the other hand we can compute the module V R corresponding to each R on the right-hand side of (C.7), using [68], with the result that only the plethysm corresponding to Y T 1 contains (04000), while neither (02004) nor (03002) is contained in any of the plethysms corresponding to the Young diagrams on the right-hand side of (C.7).
The G 4 a 1 a 2 ;...;a 7 ...a 12 term of (C.4) admits the following decomposition in irreps of S 12 ,
⊗ S 3 ⊗ = Y T 2 ⊕ . . . (16 terms) . (C.8)
Only the plethysms corresponding to the Young diagrams on the right-hand side of (C.8) will appear in the decomposition of G 4 a 1 a 2 ;...;a 7 ...a 12 under B 5 × S 12 . On the other hand it can be shown that only the plethysm corresponding to Y T 2 contains (03002), while neither (04000) nor (02004) is contained in any of the plethysms corresponding to the Young diagrams on the right-hand side of (C.8).
Finally, the G 4 a 1 a 2 ;...;a 10 ...a 14 term of (C.5) admits the following decomposition in irreps of S 14 ,
⊗ S 2 ⊗ ⊗ S 2 = Y T 3 ⊕ Y T 4 ⊕ . . . (23 terms) . (C.9)
Moreover only the plethysms corresponding to the Young diagrams on the right-hand side of (C.9) will appear in the decomposition of G 4 a 1 a 2 ;...;a 10 ...a 14 under B 5 × S 14 . On the other hand it can be shown that only the plethysm corresponding to Y T 3 contains (02004); only the plethysm corresponding to Y T 4 contains (03002), while (04000) is not contained in any of the plethysms corresponding to the Young diagrams on the right-hand side of (C.9).
Using the method of appendix F, the γ-matrices in (C.3) and (C.5) can be projected respectively onto Y T 1 and Y T 3. The terms (C.3), (C.5) can thus be shown to vanish. Moreover, it can be seen that the cancellations are sensitive to the relative coefficient between (trR 2 ) 2 and trR 4 inside X 8 . In other words, it can be shown that (trR 2 ) 2 and trR 4 are not separately τ -exact.
D. Eleven-dimensional γ-matrices
In this section we give our conventions for the eleven-dimensional γ-matrices, and list a number of Fierz identities used in the analysis presented in the main text.
Hodge duality for γ-matrices is defined as follows, where γ (0) is identified with the charge conjugation matrix.
⋆γ (n) = −(−1)
The following Fierz identities were used in the analysis. Antisymmetrisation over the a i and b j indices is always understood, as well as symmetrization over all fermionic indices of the γmatrices (which are suppressed here to avoid cluttering the notation),
(γ a 1 ...a 5 e 1 )(γ b 1 ...b 5 e 1 ) = + 120 δ a 1 ...a 5 b 1 ...b 5 (γ e 1 )(γ e 1 ) + 1 (γ a 1 ...a 5 )(γ b 1 ...b 5 ) − 600 δ a 1 ...a 3 b 1 ...b 3 (γ e 1 )(γ e 1 a 4 a 5 b 4 b 5 ) + 25 δ a 1 b 1 (γ e 1 )(γ e 1 a 1 ...a 4 b 1 ...b 4 ) − 150 1 2 δ a 1 b 1 (γ a 2 a 3 )(γ a 4 a 5 b 2 ...b 5 ) + (a ↔ b) + 600 δ a 1 ...a 3 b 1 ...b 3 (γ a 4 a 5 )(γ b 4 b 5 ) (γ a 1 ...a 4 e 1 e 2 )(γ b 1 ...b 4 e 1 e 2 ) = − 12 1 2 (γ a 1 a 2 )(γ a 3 a 4 b 1 ...b 4 ) + (a ↔ b) + 288 δ a 1 a 2 b 1 b 2 (γ a 3 a 4 )(γ b 3 b 4 ) − 96 1 2 δ a 1 b 1 (γ a 2 )(γ a 3 a 4 b 2 ...b 4 ) + (a ↔ b) + 192 δ a 1 ...a 3 b 1 ...b 3 (γ a 4 )(γ b 4 ) + 2 (γ e 1 )(γ e 1 a 1 ...a 4 b 1 ...b 4 ) − 144 δ a 1 a 2 b 1 b 2 (γ e 1 )(γ e 1 a 3 a 4 b 3 b 4 ) + 48 δ a 1 ...a 4 b 1 ...b 4 (γ e 1 )(γ e 1 ) (γ a 1 ...a 3 e 1 ...e 3 )(γ b 1 ...b 3 e 1 ...e 3 ) = + 36 δ a 1 ...a 3 b 1 ...b 3 (γ e 1 )(γ e 1 ) − 108 δ a 1 b 1 (γ e 1 )(γ e 1 a 2 a 3 b 2 b 3 ) + 216 δ a 1 b 1 (γ a 2 a 3 )(γ b 2 b 3 ) − 36 1 2 (γ a 1 )(γ a 2 a 3 b 1 ...b 3 ) + (a ↔ b) + 324 δ a 1 a 2 b 1 b 2 (γ a 3 )(γ b 3 )
(γ a 1 a 2 e 1 ...e 4 )(γ b 1 b 2 e 1 ...e 4 ) =
+ 48 δ a 1 a 2 b 1 b 2 (γ e 1 )(γ e 1 ) − 96 (γ e 1 )(γ e 1 a 1 a 2 b 1 b 2 ) + 168 (γ a 1 a 2 )(γ b 1 b 2 ) + 672 δ a 1 b 1 (γ a 1 )(γ b 1 ) (γ a 1 e 1 ...e 5 )(γ b 1 e 1 ...e 5 ) = + 240 δ a 1 b 1 (γ e 1 )(γ e 1 ) + 1680 (γ a 1 )(γ b 1 ) (γ e 1 ..
E. Eleven-dimensional superspace
In this section we review the properties of on-shell eleven-dimensional superspace at lowest order in the Planck length [33]. The theory thus obtained is equivalent to CJS supergravity [1].
The non-zero superfield components are as follows,
G abαβ = −i (γ ab ) αβ T αβ f = −i (γ f ) αβ T aα β = − 1 36 (γ bcd ) α β G abcd + 1 8 (γ a bcde ) α β G bcde R αβab = i 6 (γ gh ) αβ G ghab + 1 24 (γ ab ghij ) αβ G ghij R αabc = i 2 (γ a T bc ) α − 2(γ [b T c]a ) α . (E.1)
The action of the spinorial derivative on the superfields reads,
D α G abcd = 6 i (γ [ab| ) αǫ T |cd] ǫ D α R abcd = d [a| R α|b]cd − T ab ǫ R ǫαcd + 2 T [a|α ǫ R ǫ|b]cd D α T ab β = 1 4 R abcd (γ cd ) α β − 2 D [a T b]α β − 2 T [a|α ǫ T [b]ǫ β .
(E.
2)
The equations of motion for the field-strengths G,R and T are given by,
D f G f a 1 a 2 a 3 = − 1 1152 ǫ a 1 a 2 a 3 b 1 ...b 4 c 1 ...c 4 G b 1 ...b 4 G c 1 ...c 4 (γ a ) αǫ T ab ǫ = 0 R ab − 1 2 η ab R = 1 12 G af gh G b f gh − 1 8 η ab G f ghi G f ghi . (E.3)
F. Tensor representation of a Young diagram
A Young diagram with n boxes, see [69] for a review, represents an irreducible representation of the symmetric group S n . It is possible to give explicit expressions for Young diagrams in the form of tensors. The method is more easily understood using a specific example. Consider a tensor T a 1 a 2 a 3 a 4 without any a priori symmetry properties, and let us construct its projection onto . Several symmetry operations will have to be applied on the tensor, but the Young diagram does not state which indices correspond to its different boxes. First one must determine all the standard tableaux, i.e. all the Young diagrams with numbered boxes, with increasing numbers in all rows and columns. Different Young tableaux corresponding to the same Young diagram give equivalent but distinct representations of the symmetric group. The diagram has three standard tableaux, 1 2 3 4 , 1 3 4 2 , and 1 2 4 3 , to which correspond three tensors, T (1) , T (2) and T (3) respectively.
To obtain the tensor corresponding to a given standard tableau, one must first symmetrize over the indices indicated in each row, and then antisymmetrize over the indices indicated in each column. For example, (T (1) ) a 1 a 2 a 3 a 4 will be obtained by first symmetrizing over the indices a 1 , a 2 and a 3 , T a 1 a 2 a 3 a 4 + T a 1 a 3 a 2 a 4 + T a 2 a 1 a 3 a 4 + T a 2 a 3 a 1 a 4 + T a 3 a 1 a 2 a 4 + T a 3 a 2 a 1 a 4 , and then antisymmetrizing over a 1 and a 4 , (Π (1) T ) a 1 a 2 a 3 a 4 = (T (1) ) a 1 a 2 a 3 a 4 = 1 8 T a 1 a 2 a 3 a 4 + T a 1 a 3 a 2 a 4 + T a 2 a 1 a 3 a 4 + T a 2 a 3 a 1 a 4 + T a 2 a 3 a 4 a 1 + T a 2 a 4 a 3 a 1 + T a 3 a 1 a 2 a 4 + T a 3 a 2 a 1 a 4 + T a 3 a 2 a 4 a 1 + T a 3 a 4 a 2 a 1 + T a 4 a 2 a 3 a 1 + T a 4 a 3 a 2 a 1 .
The overall normalization above can be straightforwardly determined by imposing Π (T (1) ) a(bc)d = (T (1) ) abcd (T (2) ) ab(cd) = (T (2) ) abcd .
More generally, each T (i) has exactly three independent orderings of indices, which can be taken to be T A tensor T projected onto a non-standard tableau can be expressed as a linear combination of the three standard ones. For example it is straightforward (but tedious) to check that the projection onto the non-standard tableau 3 Every other tableau (corresponding to the same Young diagram ) and any symmetry operation on the indices can be expressed as a linear combination of those nine elements. The automatization of general decompositions onto Young tablaux, such as the one above, has been implemented in the computer program [31].
More generally a tensor T a 1 a 2 a 3 a 4 without any a priori symmetry properties can be decomposed into ten Young tableaux,
⊗4 T = T S ⊕ 3 T (1,2,3) ⊕ 2 T ′(1,2) ⊕ 3 T ′′(1,2,3) ⊕ T A , (F.2)
where T (1) , T (2) and T (3) are the Young tableaux appearing on the right-hand side of (F.1) above, and correspond to the term 3 . The remaining Young tableaux in the decomposition can be explicitly constructed using the same method.
Consider now a tensor T with a symmetry structure given by, e.g., ⊗ . The previous decomposition of ⊗4 can also be used to decompose T into its irreducible components. Indeed, a tensor with structure ⊗ can be viewed as a particular set of symmetry operations performed on the indices of a tensor without any symmetry (i.e. with structure ⊗4 ). Therefore T can be expressed as a linear combination of the tensors already used in the decomposition (F.2).
The following example shows the decomposition of the symmetric product of two threeforms H, T a 1 ...a 6 := H a 1 a 2 a 3 H a 4 a 5 a 6 −→
⊗ S 2 =
⊕ .
There are five standard tableaux corresponding to each of the Young diagrams , . The tensors corresponding to these Young tableaux can be denoted by T (1) , . . . T (5) and T ′(1) , . . . T ′(5) , respectively. In the particular example above, it can be shown that, H a 1 a 2 a 3 H a 4 a 5 a 6 = T (1) a 1 a 2 a 3 a 4 a 5 a 6 + T ′(1) a 1 a 2 a 3 a 4 a 5 a 6 + T ′(1) a 1 a 2 a 3 a 4 a 6 a 5 − T ′(1) a 1 a 2 a 3 a 5 a 6 a 4 ,
i.e. only the tensors T (1) and T ′(1) , corresponding to
...m 4 etc, are identified with the fieldstrengths of the supergravity multiplet, while the higher-order fields G (1) 4 etc, are composite higher-derivative fields which are polynomial in the fieldstrengths of the supergravity fields.
a 3 a 4
4;a 5 ...a 9 ;a 10 ...a 14 , a 2 ;...;a 7 a 8 , G 4 a 1 a 2 ;...;a 7 ...a 12 , G 4 a 1 a 2 ;...;a 10 ...a 14 denote certain sums of G 4 terms with 8,4,2 indices contracted respectively, cf. (C.6), and we have supressed spinorial indices for simplicity of notation. Furthermore we show that (04000) can only be potentially present in the projection of G 4 a 1 a 2 ;...;a 7 a 8 onto the Young diagram associated to the partition [4, 4], while (02004) can only be potentially present in the projection of G 4 a 1 a 2 ;...;a 10 ...a 14 onto the Young diagram [4, 4, 2, 2, 2]. Therefore a necessary condition for X 0,8 to be τ -exact is that the two aforementioned projections should vanish identically up to τ -exact terms, Π G 4 a 1 a 2 ;...;a 7 a 8 ≈ 0 ; Π G 4 a 1 a 2 ;...;a 10 ...a 14 ≈ 0 . (4.25)
cubic or lower in the fields. (Indeed if G (1) 7 were quartic or higher, G
[D a 1
1From dimension −3 (12 odd indices) to −1/2 (7 odd and 5 even indices), the right hand side of (A.1) always vanishes. Given the dimensions of the fieldstrengths of the physical fields, the first non-vanishing component of K 11 is K α 1 ...α 4 a 1 ...a 7 , appearing for the first time in the α 1 K α 2 ...α 7 a 1 ...a 5 − 5 12D a 1 K a 2 ...a 5 α 1 ...K f α 3 ...α 7 a 1 ...a 5 − 7 22 T a 1 a 2 γ K γa 3 ...a 5 α 1 ...α 7 − 35 66 T a 1 α 1 γ K γα 2 ...α 7 a 2 ...a 5 = 0 , ...α 7 a 1 ...a 4 and K (−2) α 1 ...α 8 a 1 ...a 3 , which cannot be expressed in terms of the physical fields: the equation is thus trivially satisfied. Dimension 0 -(A 1 . . . A 6 → α 1 . . . α 6 , A 7 . . . A 12 → a 1 . . . a 6 )
D α 1
1K α 2 ...α 5 a 1 ..α 1 α 2 f K f α 3 ...α 5 a 1 ...a 7 − 35 66 T a 1 a 2 γ K γa 3 ...a 7 α 1 ...α 5 0 − 7 22T a 1 α 1 γ K γα 2 ...α 6 a 2 ..
a 1 ...a 10 = − 10 i 42 i (γ a 2 ...a 6 ) α 1 α 2 d a 1 G a 7 ...a 10 − 1 2 i ǫ a 2 ...a 8 i 1 ...i 4 (γ a 9 a 10 ) α 1 α 2 d a 1 G i 1 ...i 4 − 20 i T a 1 α 1 ǫ 42 i (γ a 2 ...a 6 ) ǫα 2 G a 7 ...a 10 − 1 2 i ǫ a 2 ...a 8 i 1 ...i 4 (γ a 9 a 10 ) ǫα 2 G i 1 ...i 4 − 1575 G a 1 ...a 4 G a 5 ...a 8 (γ a 9 a 10 ) α 1 α 2 . (A.9)
10 a 11 gh , (A.11) which can be seen to be automatically satisfied by contracting (A.11) with ǫ a 1 ...a 11 . The next equation (of dimension 3) is trivially satisfied, since the purely bosonic component of a twelveform vanishes automatically in eleven dimensions. Action at O(l 0 )
can be verified that this expression does not satisfy eq. (B.3), unless K a 1 α 3 ...α 12 a 2 α 1 ...α 9 will be set to zero because there is no gauge-invariant field of dimension 1/2. The components K a 2 a 3 a 4 α 1 ...α 7 will be set to zero, up to exact terms, as a consequence of the triviality of H 3,8 τ , H 4,7 τ .
α 6 a 1 .
1.α 1 α 2 f K f a 1 ...a 6 α 3 ...α 6 + 5 22 T a 1 a 2 γ K γα 1 ...α 6 a 3 ..a 1 α 1 γ K γα 3 ...α 6 a 2 ..
a 3 a 4 ;a 5 a 6 ;a 7
467a 1 a 2 )(γ a 3 a 4 )(γ a 5 a 6 )(γ a 7 ...a 12 ) G 4 a 1 a 2 ;a 3 a 4 ;a 5 a 6 ;a 7 ...a 12(C.4) (γ a 1 a 2 )(γ a 3 a 4 )(γ a 5 ...a 9 )(γ a 10 ...a 14 ) G 4 a 1 a 2 ;a 3 a 4 ;a 5 ...a 9 ;a 10 ...a 14 , (C.5)up to manifestly τ -exact terms which we do not need to write out explicitly. In the above, G 4 a 1 a 2 ;...;a 7 a 8 , G 4 a 1 a 2 ;...;a 7 ...a 12 , G 4 a 1 a 2 ;...;a 10 ...a 14 , denote certain sums of G 4 terms with 8,4,2 indices contracted respectively. More explicitly, G 4 a 1 a 2 ;a 3 a 4 ;a 5 a 6 ;a 7 a 8 = 7 2 7 3 3 G a 1 ef g G a 2 a 7 a 8 e G a 3 a 5 a 6
)
definition of the Hodge operator reads,(⋆S) a 1 ...a k = 1 (11 − k)! ǫ a 1 ...a k b 1 ...b 11−k S b 1 ...b 11−k .The symmetry properties of the γ-matrices are given by, (γ a 1 ...an ) αβ = ((γ a 1 ...an ) βα ,
b 1
1.e 6 )(γ e 1 ...e 6 ) = + 4320 (γ e 1 )(γ e 1 ) (γ a 1 a 2 e 1 ...e 3 )(γ b 1 b 2 e 1 ...e 3 ) = − 36 δ a 1 a 2 b 1 b 2 (γ e 1 )(γ e 1 ) + 24 (γ e 1 )(γ e 1 a 1 a 2b 1 b 2 ) − 42 (γ a 1 a 2 )(γ b 1 b 2 ) + 168 δ a 1 b 1 (γ a 2 )(γ b 2 ) (γ a 1 e 1 ...e 4 )(γ b 1 e 1 ...e 4 ) = − 96 δ a 1 b 1 (γ e 1 )(γ e 1 ) + 336 (γ a 1 )(γ b 1 ) (γ e 1 ...e 5 )(γ e 1 ...e 5 ) = − 720 (γ e 1 )(γ e 1 ) (γ a 1 ...a 4 e 1 )(γ b 1 ...b 4 e (γ a 2 )(γ a 3 a 4 b 2 ...b 4 ) + (a ↔ b) + 96 δ a 1 ...a 3 b 1 ...b 3 (γ a 4 )(γ b 4 ) − 1 (γ e 1 )(γ e 1 a 1 ...a 4 b 1 ...b 4 ) + 72 δ a 1 a 2 b 1 b 2 (γ e 1 )(γ e 1 a 3 a 4 b 3 b 4 ) − 24 δ a 1 ...a 4 b 1 ...b 4 (γ e 1 )(γ e 1 ) (γ a 1 ...a 3 e 1 e 2 )(γ b 1 ...b 3 e 1 e 2 ) = − 24 δ a 1 ...a 3 b 1 ...b 3 (γ e 1 )(γ e 1 ) + 36 δ a 1 b 1 (γ e 1 )(γ e 1 a 2 a 3 b 2 b 3 ) − 54 δ a 1 b 1 (γ a 2 a 3 )(γ b 2 b 3 ) a 1 )(γ a 2 a 3 b 1 ...b 3 ) + (a ↔ b) + 108 δ a 1 a 2 b 1 b 2 (γ a 3 )(γ b 3 ) (γ a 1 e 1 )(γ b 1 ...b 4 e 1 ) = + 1 (γ e 1 )(γ e 1 a 1 b 1 ...b 4 ) + 12 δ a 1 b 1 (γ b 2 )(γ b 3 b 4 ) (γ a 1 e 1 )(γ b 1 e 1 ) = + 1 (γ a 1 )(γ b 1 ) − 1 δ a 1 b 1 (γ e 1 )(γ e 1 ) (γ a 1 ...a 4 e 1 )(γ b 1 ...b 5 e 1 ) = − 60 η a 1 b 1 (γ a 2 a 3 )(γ a 4 b 2 ...b 5 ) − 60 η a 1 b 1 (γ a 2 )(γ a 3 a 4 b 2 ...b 5 ) − 720 δ a 1 ...a 3 c 1 ...c 3 η c 1 b 1 η c 2 b 2 η c 3 b 3 (γ a 4 )(γ b 4 b 5 ) + 240 δ a 1 ...a 3 c 1 ...c 3 η c 1 b 1 η c 2 b 2 η c 3 b 3 (γ a 4 )(γ b 4 b 5 ) + 140 η a 1 b 1 (γ [a 2 )(γ a 3 a 4 b 2 ...b 5 ] ) − 120 δ a 1 a 2 c 1 c2 η c 1 b 1 η c 2 b 2 (γ e 1 )(γ e 1a 3 a 4 b 3 ...b 5 ) (γ a 1 e 1 )(γ b 1 ...b 5 e 1 ) = − 6 (γ [a 1 )(γ b 1 ...b 5 ] ) − 5 η a 1 b 1 (γ e 1 )(γ e 1 b 2 ...b 5 ) + 1 (γ a 1 )(γ b 1 ...b 5 )
( 1 )( 1 )
11Π (1) T = Π (1) T , where Π (1) T = T(1) is the projection of the tensor T onto the Young tableau 1 ) [a|bc|d] = (T (1) ) abcd (T (2) ) [ab]cd = (T(2) ) abcd
a 3 a 1 a 4 . Any symmetry operation on the indices of T (i) can be expressed as a linear combination of these three orderings, e.2 a 1 a 4 a 3 + 0 T (1) a 2 a 3 a 1 a 4 .
( 4 )
4T ) a 1 a 2 a 3 a 4 = T (1) a 1 a 4 a 3 a 2 + T (1) a 2 a 1 a 4 a 3 + 0 T (1) a 2 a 3 a 1 a 4 + 0 T (2) a 1 a 4 a 3 a 2 − T (2) a 2 a 1 a 4 a 3 + T (2) a 2 a 3 a 1 a 4 + T (3) a 2 a 1 a 3 a 4 + 0 T (3) a 2 a 1 a 4 a 3 − T (3) a 2 a 3 a 1 a 4 .(F.1)
, let us now define a spinorial derivative d s which acts on elements of τ -cohomology,d s : H p,q
τ → H p,q+1
τ
. For any ω ∈ [ω] ∈ H p,q
τ
we set,
d s [ω] := [d f ω] .
(2.16)
terms with multiplicity 1⊕ 2(00004) ⊕ 2(10002) ⊕ 2(01002)
3 terms with multiplicity 2
There is disagreement between[23] and[25] concerning part of the (∂G) 4 terms.
The G7 BI receives a correction at the eight-derivative order, cf. (4.3) below.
This definition is different from an eleven-dimensional pure spinorà la Cartan, used in[56], which obeys λ α γ ab αβ λ β = 0 in addition to(2.21).
AcknowledgementsWe would like to thank Paul Howe for valuable discussions. D.T. would like to thank the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work.A. Weil triviality at l 0In this section we give the details of the solution of the superspace equation W 12 = dK 11 at lowest order in the Planck length. As a byproduct we will see that the solution for K 11 given in (3.6) is unique up to exact terms. We will look for the solution to,
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| []
|
[
"Generalized Sparse Precision Matrix Selection for Fitting Multivariate 1 Gaussian Random Fields to Large Data Sets",
"Generalized Sparse Precision Matrix Selection for Fitting Multivariate 1 Gaussian Random Fields to Large Data Sets"
]
| [
"S Davanloo Tajbakhsh \nThe Ohio State University\n\n",
"N S Aybat \nThe Pennsylvania State University\n\n",
"E Del Castillo \nThe Pennsylvania State University\n\n"
]
| [
"The Ohio State University\n",
"The Pennsylvania State University\n",
"The Pennsylvania State University\n"
]
| []
| We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed byDavanloo et al. (2015)for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance matrix and for the estimated parameters of the underlying spatial correlation function are established. Numerical tests using simulated and real datasets validate our theoretical findings. Data segmentation is used to handle large data sets. | 10.5705/ss.202017.0091 | [
"https://arxiv.org/pdf/1605.03267v2.pdf"
]
| 188,757 | 1605.03267 | cd0f5d9353ca177a1b5cc66888e2ea42ac44ce07 |
Generalized Sparse Precision Matrix Selection for Fitting Multivariate 1 Gaussian Random Fields to Large Data Sets
6 Jul 2017
S Davanloo Tajbakhsh
The Ohio State University
N S Aybat
The Pennsylvania State University
E Del Castillo
The Pennsylvania State University
Generalized Sparse Precision Matrix Selection for Fitting Multivariate 1 Gaussian Random Fields to Large Data Sets
6 Jul 2017and phrases: Multivariate Gaussian ProcessesGaussian Markov Random FieldsSpatial StatisticsCovariance SelectionConvex Optimization
We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed byDavanloo et al. (2015)for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance matrix and for the estimated parameters of the underlying spatial correlation function are established. Numerical tests using simulated and real datasets validate our theoretical findings. Data segmentation is used to handle large data sets.
Introduction. Gaussian Random Field (GRF) models are very popular in Machine
Learning, e.g., (Rasmussen et al., 2006), and are widely used in Geostatistics, e.g. (Cressie et al., 2011). They also have applications in meteorology to model satellite data for forecasting or to solve inverse problems to tune weather models (Cressie et al., 2011), or to model outputs of expensive-to-evaluate deterministic Finite Element Method (FEM) computer codes, e.g., (Santner et al., 2003). More recently, there have been applications of GRF to model stochastic simulations, e.g., queuing or inventory control models (Ankenman et al., 2010;Kleijnen, 2010), or to model free-form surfaces of manufactured products from noisy measurements for inspection or quality control purposes . In a GRF model, a key role is played by the covariance or kernel function which determines how the covariance between the process values at two locations changes as the locations change across the process domain. There are many valid parametric covariance functions, e.g., Exponential, Squared Exponential, or Matern; and Maximum Likelihood (ML) is the dominant method to estimate their parameters from data (Santner et al. (2003)). However, the ML fitting procedure suffers from two main challenges: i) the negative loglikelihood is a nonconvex function of the covariance matrix; therefore, the covariance parameters may be poorly estimated, ii) the problem is computationally hard when the number of spatial locations n is big. This is known as the "big-n" problem in the literature. Along with some other approximation methods, there is an important class that approximates the Gaussian likelihood using different forms of conditional independence assumptions which reduces the computational complexity significantly, e.g., (Snelson and Ghahramani, 2006;Pourhabib et al., 2014) and references therein.
In we proposed the Sparse Precision Selection (SPS) algorithm for univariate processes to deal with the first challenge by providing theoretical guarantees on the SPS parameter estimates, and presented a segmentation scheme on the training data to be able to solve big-n problems. Given the nature of SPS, the segmentation does not result in discontinuities in the predicted process. In contrast, localized regression methods also rely on segmentation to reduce the computational cost; but, these methods may suffer from discontinuities on the predicted surface at the boundaries of the segments. In this paper, we present a Generalized SPS (GSPS) method for fitting a multivariate GRF process that deals with the two aforementioned challenges when there are possibly cross-correlated 2 multiple responses that occur at each spatial location.
Compared to SPS (and also to GSPS), the likelihood approximation type GRF methods, e.g., (Snelson and Ghahramani, 2006), have the advantage of computational efficiency; but, there are no guarantees on the quality of the parameter estimates as only an approximation to the likelihood function is optimized (compared to MLE, this is a small dimensional problem; but, still non-convex). On the other hand, SPS has theoretical error bound guarantees on hyper-parameter estimates (this is also the case for GSPS, see Theorem 4 below) -note that these bounds also imply error guarantees on prediction quality through the mean of the predictive distribution.
There is a wide variety of applications that require the approximation of a vector of correlated responses obtained at each spatial or spatial-temporal location. Climate models are
classic Geostatistical examples where environmental variables such as atmospheric CO 2 concentration, ocean heat uptake and global surface temperature are jointly modeled (a simple such model is studied in Urban et al. (2010)). Another classical application is environmental monitoring, for instance, Lin (2008) uses a Multivariate GRF model to map spatial variations of five different heavy metals in soil. This is an application sharing a similar aim with Kriging in mining engineering where the spatial occurrence of two metals may be cross-correlated, e.g., silver and lead. Multivariate GRFs are also popular in multi-task learning (Bonilla et a., 2008), an area of machine learning where multiple related tasks need to be learned so that simultaneously learning them can be better than learning them in isolation without any transfer of information between the tasks. The joint modeling of spatial responses is also useful in metrology when conducting multi-fidelity analysis (Forrester et al., 2008), where an expensive, high fidelity spatial response needs to be predicted from predominantly low fi-3 delity responses, which are inexpensive -see also (Boyle et al., 2004). Likewise, multivariate GRFs have been used to reconstruct 3-dimensional free-form surfaces of manufactured products through modeling each of the 3 coordinates of a measured point as a parametric surface response . Other applications of multivariate GRF include: (Wang and Chen , 2015) to model the response surface of a catalytic oxidation process with two highly correlated response variables; (Castellanos et al. , 2015) to estimate low dimensional spatio-temporal patterns of finger motion in repeated reach-to-grasp movements; (Bhat et al. , 2010) to study a multi-output GRF for computer model calibration with multivariate spatial data to infer parameters in a climate model. Note that in many of such applications multiple realizations of the GRF are sensed/measured over time (N > 1) over a fixed set of locations. GRF applications with N > 1 commonly arise in practice, including those i) in "metamodeling" of stochastic simulations for modeling an expensive-to-evaluate queuing or inventory control model, ii) in modeling product surfaces for inspection or quality control purposes, and iii) in models for which we observe a spatial process over time at the same locations for a system known to be static with respect to time.
Rather than considering each response independently, using the between-response covariance can significantly enhance the prediction performance. As mentioned by Cressie (2015), the principle of exploiting co-variation to improve mean-squared prediction error goes back to Kolmogorov and Wiener in the first half of the XX century. It is well-known that the minimum-mean-square-error predictor of a single response component of a multivariate GRF involves the between-response covariances of all responses (Santner et al., 2003), a result that lies at the basis of the so-called Co-Kriging technique in Geostatistics (Cressie, 2015).
In this paper, we adopted a separable cross-covariance structure -see (3.2) -which has 4 been already adopted in the literature: Mardia and Goodall (1993) proposed separability to model multivariate spatio-temporal data, and Bhat et al. (2010) used separable crosscovariance for computer model calibration. This structure is also well known in the literature, see (Gelfand et al., 2004;Banerjee et al., 2014;Gelfand and Banerjee , 2010) and (Genton and Kleiber , 2015); moreover, Li et al. (2008) even proposed a technique to test the separability assumption for a multivariate random process. Furthermore, Gelfand and Banerjee (2010) mention one additional use of a separable covariance structure: "A bivariate spatial process model using separability becomes appropriate for regression with a single covariate X(s) and a univariate response Y (s). In fact, we treat this as a bivariate process to allow for missing X(s) for some observed Y (s) and for inverse problems, inferring about X(s 0 ) for a given Y (s 0 )". As an example of this type of application, Banarjee and Gelfand have employed such separable models in (Banerjee and Gelfand , 2002;Banerjee et al., 2014) to analyze the relationship between shrub density and dew duration for a dataset consisting of 1129 locations in a west-facing watershed in the Negev desert in Israel.
However, fitting multivariate GRFs not only suffers from the two challenges mentioned above; in particular, the parametrization of the matrix-valued covariance functions requires a higher-dimensional parameter vector which aggravates the difficulty of the GRF estimation problem further (Banerjee et al., 2014;Cressie et al., 2011). The goal of this paper is to extend the theory of the univariate SPS method to include the hyper-parameter estimation of multivariate GRF models for which the error bounds on the approximation quality can be established. The paper is organized as follows: Section 1.1 introduces the notation, and Section 2 provides some preliminary concepts related to the SPS method. In Section 3, GSPS, the multivariate generalization of the SPS method is described 5 and compared with other methods for fitting multivariate GRF, and theoretical guarantees of the GSPS estimates are discussed. Section 4 includes numerical results. Finally, we summarize the main results in the paper and provide some future research directions in Section 5.
1.1. Notation. Throughout the paper, given x ∈ R n , x , x 1 , x ∞ denote the Euclidean, ℓ 1 , and ℓ ∞ norms, respectively. For x ∈ R n , diag(x) ∈ S n denotes a diagonal matrix with its diagonal equal to x. Given X ∈ R m×n , we denote the vectorization of X using vec(X) ∈ R np , obtained by stacking the columns of the matrix X on top of one another.
Moreover, let r = rank(X), and σ = [σ i ] r i=1 ⊂ R r ++ (positive orthant) denote the singular values of X; then, X F := σ , X 2 := σ ∞ , and X * := σ 1 denote the Frobenius, spectral, and nuclear norms of X, respectively. Given X, Y ∈ R m×n , X, Y := Tr(X ⊤ Y ) denotes the standard inner product. Let V be a normed vector space with norm . a . For
x ∈ V and r > 0, B . a (x, r) := {x ∈ V :
x −x a < r} denotes the open ball centered atx with radius r > 0, andB . a (x, r) denotes its closure.
2. Preliminaries: the SPS method for a scalar GRF. Let X ⊆ R d and y : X → R be a GRF, where y(x) denotes the value of the process at location x ∈ X . Let m(x) = E(y(x)) for x ∈ X , and c(x, x ′ ) be the spatial covariance function denoting the covariance between y(x) and y(x ′ ), i.e., c(x, x ′ ) = cov (y(x), y(x ′ )) for all x, x ′ ∈ X . Without loss of generality, we assume that the GRF has a constant mean equal to zero, i.e., m(x) = 0. Suppose the training data D = {(x i , y For simplicity in estimation, the covariance function, c(x, x ′ ), is typically assumed to be-
|P * ij | vec(P * ) ∞ C * ij vec(C * ) ∞ |P * ij | vec(P * ) ∞ C * ij vec(C * ) ∞ |P * ij | vec(P * ) ∞ C * ij vec(C * ) ∞long to some parametric family {c(x, x ′ ; θ, ν) : θ ∈ Θ, ν ≥ 0} and c(x, x ′ , θ) := νρ(x, x ′ , θ),
where ρ(x, x ′ , θ) is a parametric correlation function where θ and ν denote the spatial correlation and variance parameters, respectively, and Θ ⊂ R q is a set that contains the true spatial correlation parameters -see e.g. Cressie (2015). Let θ * and ν * denote the unknown true parameters of the process. Given a set of locations D x = {x i } n i=1 , let C(θ, ν) ∈ S n ++ be such that its (i, j) th element is c(x i , x j ; θ, ν) -throughout, S n ++ and S n + denote the set of n-by-n symmetric, positive definite and positive semidefinite matrices, respectively.
Let C * = C(θ * , ν * ) denote the true covariance matrix corresponding to locations in
D x = {x i } n i=1
, and P * = (C * ) −1 denote the true precision matrix. In Davanloo et al. (2015),
we proposed a two-stage method, SPS, to estimate the unknown process parameters θ * and ν * . The method is motivated by the results in numerical linear algebra which demonstrate that if the elements of a matrix show a decay property, then the elements of its inverse also 7 show a similar behavior -see Benzi (2016);Jaffard (1990). In particular, consider the two decay classes defined in Jaffard (1990):
Definition 2.1. Given {x i } n i=1 ⊂ X and a metric d : X × X → R + , a matrix A ∈ R n×n belongs to the class E γ for some γ > 0 if for all γ ′ < γ there exists a constant K γ ′ such that
|A ij | ≤ K γ ′ exp − γ ′ d(x i , x j ) for all 1 ≤ i, j ≤ n. Moreover, A belongs to the class Q γ for some γ > 1 if there exists a constant K such that |A ij | ≤ K 1 + d(x i , x j ) −γ for all 1 ≤ i, j ≤ n. Theorem 2.2. Given {x i } n i=1 ⊂ X and a metric d : X × X → R + , let A ∈ R n×n be an invertible matrix. If A ∈ E γ for some γ > 0, then A −1 ∈ E γ ′ for some γ ′ > 0. Moreover, if A ∈ Q γ for some γ > 0, then A −1 ∈ Q γ .
Proof. See Proposition 2 and Proposition 3 in Jaffard (1990). This fast decay structure in the precision (inverse covariance) matrix of a GRF makes it a compressible signal (Candes, 2006); hence, one can argue that it can be well-approximated by a sparse matrix -compare it with the covariance matrix depicted in Figure 1. For all stationary GRFs tested, we observed that for a finite set of locations, the magnitudes of the off-diagonal elements of the precision matrix decay to 0 much faster than the elements of the covariance matrix.
Let a * and b * be given constants
such that 0 ≤ a * ≤ σ min (P * ) ≤ σ max (P * ) ≤ b * ≤ ∞.
In the first stage of the SPS algorithm, we proposed to solve the following convex loglikelihood problem penalized with a weighted ℓ 1 -norm to estimate the true precision matrix corresponding to the given data locations D x :
P := argmin{ S, P − log det(P ) + α G, |P | : a * I P b * I}, (2.1) where S = 1 N N r=1 y (r) y (r) ⊤ ∈ S n+
is the sample covariance matrix. The weight matrix G ∈ S n is chosen as the matrix of pairwise distances:
G ij = x i − x j , if i = j, G ii = min{ x i − x j : j ∈ I \ {i}}, (2.2)
for all (i, j) ∈ I × I, where I = {1, 2, ..., n} and |.| is the elementwise absolute value operator. The sparsity structure of the estimated precision matrixP encodes the conditional independence structure of a Gaussian Markov Random Field (GMRF) approximation to the GRF. Using ADMM, the Alternating Direction Method of Multipliers, see (Boyd et al., 2011), (2.1) can be solved efficiently. Indeed, since − log det(.) is strongly convex and has a Lipschitz continuous gradient for 0 < a * ≤ b * < ∞, ADMM iterate sequence converges linearly to the optimal solution with a linear rate (Deng and Yin, 2015).
In the second stage of the SPS method, we proposed to solve a least-square problem (2.3)
to estimate the unknown parameters θ * and ν * :
(θ,ν) = argmin θ∈Θ, ν≥0 C(θ, ν) −P −1 2 F . (2.3)
In Davanloo et al. (2015), we showed how to solve each optimization problem, and also established theoretical convergence rate of the SPS estimator.
SPS is therefore based on a Gaussian Markov Random Field (GMRF) approximation to the GRF. While a GMRF on a lattice can represent exactly a GRF under the conditional independence assumption, this representation of a GRF can only be an approximation in a general continuous location space. The index set is countable for the lattice data, but the index set X for a GRF is uncountable; hence, in general GMRF models cannot represent GRFs exactly. Lindgren et al. (2011) recently established that the Matern GRFs are Markovian; in particular, they are Markovian when the smoothing parameter ν is such that ν − d/2 ∈ Z + , where d is the dimension of the input space -see Lindgren et al. 9 (2011) and Fulgstad et al. (2015) for using this idea in the approximation of anisotropic and non-stationary GRFs. Rather than using a triangulation of the input space as proposed by Lindgren et al. (2011), or assuming a lattice process, the first stage of SPS lets the data determine the near-conditional independence pattern between variables through the precision matrix estimated via a weighted ℓ 1 -regularization. Furthermore, this first stage helps to "zoom into" the area where the true covariance parameters are located; hence, it helps not to get trapped in local optimum solutions in the second stage of the method.
3. Multivariate GRF Models. From now on, let y(x) ∈ R p be the response vector at
x ∈ X ⊆ R d of a multivariate Gaussian Random Field (GRF) y : X → R p with zero mean and a cross-covariance function c(x, x ′ ) = cov (y(x), y(x ′ )) ∈ S p ++ . The cross-covariance
function is a crucial object in multivariate GRF models which should converge to a symmetric and positive-definite matrix as x − x ′ → 0. Similar to the univariate case, the process is second-order stationarity if c(., .) depends on x and x ′ only through x − x ′ , and it is isotropic
if c(., .) depends on x and x ′ only through x − x ′ .
The parametric structure of the cross-covariance matrix should be such that the resulting cross-covariance matrix is a positive-definite matrix. Gelfand et al. (2004) and Banerjee et al. (2014) review some methods to construct a valid cross-covariance function. In these methods, parameter estimation involves solving nonconvex optimization problems.
In this study, we assume a separable cross-covariance function belonging to a parametric family, and propose a two-stage procedure for estimating the unknown parameters. The separable model assumes that the cross-covariance function is a multiplication of a spatial correlation function and a positive-definite between-response covariance matrix (see Gelfand and Banerjee (2010); Gelfand et al. (2004) and the references therein):
c(x, x ′ ) = ρ(x, x ′ ) Γ * ∈ S p + , (3.1)
where ρ : X × X → [0, 1] is the spatial correlation function, and Γ * ∈ S p ++ is the betweenresponse covariance matrix. Furthermore, let y = [y(x 1 ) ⊤ , ..., y(x n ) ⊤ ] ⊤ ∈ R np denote the process values in long vector form corresponding to locations in D
x := {x i } n i=1 ⊂ X .
Given the cross-covariance function (3.1), and the set of locations D x , y follows a multivariate Gaussian distribution with zero mean and covariance matrix equal to
C * = R * ⊗ Γ * , (3.2) where R * ∈ S n ++ is the spatial correlation matrix such that R * ij = ρ(x i , x j ) for i, j ∈ I := {1, .
. . , n}, and ⊗ denotes the Kronecker product. Hence,
y ∼ N (0, C * ). (3.3) Let D = {(x i , y(r)
i ) : i ∈ I, r = 1, ..., N} be the training data set that contains N realizations of the process over n distinct locations D x ⊂ X , i.e., for each r ∈ {1, . . . , N},
y (r) = [y (r) i ] i∈I ∈ R np is an independent realization of y = [y(x i )] i∈I . Hence, {y (r) } N r=1 are i.i.d. according to (3.3).
As in the univariate case, suppose the correlation function belongs to a parametric family
{ρ(x, x ′ ; θ) : θ ∈ Θ},
where Θ is a closed convex set containing the true parameter vector,
θ * , of the correlation function ρ. Given D x = {x i } i∈I , define R * := R(θ * ), where R(θ) ∈ S n ++ is such that R(θ) = [r ij (θ)] i,j∈I , r ij (θ) = ρ(x i , x j ; θ) ∀i, j ∈ I. (3.4)
Consider a GRF model with all its parameters known, the best linear unbiased prediction 11 at a new location x 0 is given by the mean of the conditional distribution
p(y(x 0 )|{y (r) } N r=1 , D x ) which iŝ y(x 0 ) = (r(x 0 ; θ * ) ⊤ ⊗ Γ * )(R(θ * ) ⊗ Γ * ) −1 N r=1 y (r) /N, (3.5)
where r(x 0 ; θ * ) ∈ R n contains the spatial correlation between the new point x 0 and n observed data points -see (Santner et al., 2003). It is important to note that the prediction equation is a continuous function of the parameters θ * and Γ * ; hence, biased estimation of the parameters will translate to poor prediction performance. Finally, the prediction formula (3.5) shows the importance of considering the between-response covariance matrix Γ * rather than using p independent univariate GRFs for prediction. Indeed, predicting each response independently of the others will result in suboptimal predictions.
The sample covariance matrix S ∈ S np + is calculated as S = 1 N N r=1 y (r) y (r) ⊤ . Furthermore, let G ∈ S n be such that G ij > 0 for all i, j ∈ I; in particular, we fix G as in (2.2) based on inter-distances. Let P * = (C * ) −1 be the true precision matrix corresponding to locations in D x , and let a * and b * be some given constants such that 0 ≤ a * ≤ σ min (P * ) ≤ σ max (P * ) ≤ b * ≤ ∞. To estimate P * , we propose to solve the following convex program:
P = argmin a * I P b * I S, P − log det(P ) + α G ⊗ (1 p 1 ⊤ p ), |P | , (3.6)
where |.| is the element-wise absolute value operator, and 1 p ∈ R p denotes the vector of all ones. This objective penalizes the elements of the precision matrix with weights proportional to the distance between their locations. Problem (3.6) can be solved efficiently using the ADMM implementation proposed in Davanloo et al. (2015). Indeed, for 0 < a * ≤ b * < ∞, the function − log det(.) is strongly convex and has a Lipschitz continuous gradient; therefore, the ADMM sequence converges linearly to the optimal solution -see Deng and Yin (2015).
12
LetĈ :=P −1 , and for all (i, j) ∈ I × I define block matrices S ij ∈ S p ,Ĉ ij ∈ S p and
Σ ij ∈ S p such that S = [S ij ],Ĉ = [Ĉ ij ] and C * = [Σ ij ], i.e., S ij ∈ S p ,Ĉ ij ∈ S p and Σ ij ∈ S p
are the sample, estimated and true covariance matrices between the locations x i and x j . The following establishes a probability bound for the estimation errorP − P * .
Theorem 3.1. Let {y (r) } N r=1 ⊂ R nq be independent realizations of a GRF with zeromean and stationary covariance function c(x, x ′ ; θ * ) observed over n distinct locations {x i } i∈I with I := {1, ..., n}; furthermore, let C * = R(θ * ) ⊗ Γ * be the true covariance matrix, and P * := C * −1 be the corresponding true precision matrix, where R(θ) is defined in (3.4).
Finally, letP be the GSPS estimator computed as in (3.6) for some G ∈ S n such that G ij ≥ 0 for all (i, j) ∈ I × I. Then for any given M > 0, N ≥ N 0 := 2 (M + 2) ln(np) + ln 4 , and b * ≥ σ max (P * ),
Pr P − P * F ≤ 2b * 2 p(n + G F )α ≥ 1 − (np) −M ,(3.
7)
for all α such that 40 max i=1,...,p
(Γ * ii ) N 0 N ≤ α ≤ 40 max i=1,...,p (Γ * ii ). Proof. See the appendix.
Given that C * = R * ⊗ Γ * , and the diagonal elements of the spatial correlation matrix R * are equal to one, we have Σ ii = Γ * . Therefore, we propose to estimate the between-response covariance matrix Γ * by taking the average of the p × p matrices along the diagonal ofĈ, i.e.,Γ := 1 n n i=1Ĉ ii ∈ S n ++ .
(3.8)
Note that (3.6) implies thatP ∈ S np ++ ; hence,Ĉ ∈ S np ++ as well. Therefore, all its blockdiagonal elements are positive definite, i.e.,Σ ii ∈ S n ++ for i = 1, ..., n. SinceΓ is a convex combination ofΣ ii ∈ S n + , i = 1, ..., n and the cone of positive definite matrices is a convex set, we also haveΓ ∈ S n ++ . A probability bound in the estimation error of the covariance matrices is shown in the following theorem.
Theorem 3.2. Given M > 0, N ≥ N 0 := 2 (M + 2) ln(np) + ln 4 , and a * , b * such that 0 < a * ≤ σ min (P * ) ≤ σ max (P * ) ≤ b * < ∞, letP be the SPS estimator as in (3.6). Then
Γ, defined in (3.8), andĈ =P −1 satisfy Pr max{ Ĉ − C * 2 , Γ − Γ * 2 } ≤ 2 b * a * 2 p(n + G F )α ≥ 1 − (np) −M , for all α such that 40 max i=1,...,p (Γ * ii ) N 0 N ≤ α ≤ 40 max i=1,...,p (Γ * ii ). Proof. From (3.7), we have Ĉ − C * 2 ≤ 1 a * 2 P − P * 2 ≤ 1 a * 2 P − P * F ≤ 2 b * a * 2 p(n + G F )α,
where the first inequality follows from the Lipschitz continuity of P → P −1 on the domain P a * I with respect to the spectral norm . 2 . Hence, given that Γ * = Σ ii for all i ∈ I, we have Ĉ ii − Γ * 2 ≤ 2 b * a * 2 p(n + G F )α for all i ∈ I. Therefore, from convexity of
X → X − Γ * 2 , it follows that Γ − Γ * 2 ≤ i∈I 1 n Ĉ ii − Γ * 2 ≤ 2 b * a * 2 p(n + G F )α.
Remark. For Theorems 2 and 3 to hold, α should belong to the interval 40 max i=1,...,p
(Γ * ii ) N 0 N ≤ α ≤ 40 max i=1,...,p (Γ * ii ); for N ≥ N 0 this interval is non-empty.
The trade-off here is such that smaller α makes the estimation error bounds inside the probabilities tighter -hence, desirable; however, at the same time, smaller α makes the estimated precision matrix less sparse which would require more memory to store a denser estimated precision matrix.
Although the upper-bound on α is fixed, one can play with the lower bound; in particular, one can make it smaller by requiring more realizations N.
Given D x = {x i } i∈I ⊂ X , define R : R q → S n over Θ ⊂ R q as in (3.4), i.e., R(θ) = [r ij (θ)] i,j∈I ∈ S n and r ij (θ) = ρ(x i , x j ; θ) for all (i, j) ∈ I×I. To estimate the true parameter vector of the spatial correlation function, θ * , we propose to solvê
θ ∈ argmin θ∈Θ 1 2 i,j∈I r ij (θ)Γ −Ĉ ij 2 F . (3.9)
The objective function of (3.9) can be written in a more compact form as the parametric function below, with parameters Γ ∈ S p and C ∈ S np :
f (θ; Γ, C) := 1 2 R(θ) ⊗ Γ − C 2 F . (3.10) Let θ = [θ 1 , . . . , θ q ] ⊤ , and R ′ k : R q → S n such that R ′ k (θ) = [ ∂ ∂θ k r ij (θ)] i,j∈I for k = 1, . . . , q. Similarly, R ′′ kℓ : R q → S n such that R ′′ kℓ (θ) = [ ∂ 2 ∂θ k ∂θ ℓ r ij (θ)] i,j∈I for 1 ≤ k, ℓ ≤ q. Let Z(θ; Γ, C) := R(θ) ⊗ Γ − C; hence, f (θ; Γ, C) = Z(θ; Γ, C) 2 F /2; and define Z ′ k (θ; Γ) := R ′ k (θ) ⊗ Γ for k = 1, . . . , q.
Lemma 3.3. Suppose ρ(x, x ′ ; θ) is twice continuously differentiable in θ over Θ for all x, x ′ ∈ X , then there exists γ * > 0 such that ∇ 2 θ f (θ * ; Γ * , C * ) γ * I if and only if {vec(R ′ k (θ * ))} q k=1 ⊂ R n 2 are linearly independent.
Proof. Clearly, ∇ θ f (θ; Γ, C) = Z ′ 1 (θ; Γ), Z(θ; Γ, C) , . . . , Z ′ q (θ; Γ), Z(θ; Γ, C) ⊤ .
Hence, it can be shown that for
1 ≤ k ≤ q ∂ ∂θ k f (θ; Γ, C) = Γ 2 F R ′ k (θ), R(θ) − C, R ′ k (θ) ⊗ Γ ,(3.11)
and from the product rule for derivatives, it follows that for 1
≤ k, ℓ ≤ q ∂ 2 ∂θ k ∂θ ℓ f (θ; Γ, C) = Γ 2 F R ′ k (θ), R ′ ℓ (θ) + R ′′ kℓ (θ) ⊗ Γ, R(θ) ⊗ Γ − C .
(3.12)
Thus, since C * = r(θ * ) ⊗ Γ * , we have
∂ 2 ∂θ k ∂θ ℓ f (θ; Γ * , C * ) = Γ * 2 F R ′ k (θ * ), R ′ ℓ (θ * ) .
Therefore, ∇ 2 θ f (θ * ; Γ * , C * ) = Γ * 2 F J(θ * ) ⊤ J(θ * ), where J(θ) ∈ R n 2 ×q such that J(θ) := [vec(R ′ 1 (θ)) . . . vec(R ′ q (θ))]. Hence, there exists γ * > 0 such that ∇ 2 θ f (θ * ; Γ * , C * ) γ * I when {vec(R ′ k (θ * ))} q k=1 ⊂ R n 2 are linearly independent.
Remark. We comment on the linear independence condition stated in Lemma 3.3. For illustration purposes, consider the anisotropic exponential correlation function ρ(
x, x ′ , θ) = exp − (x − x ′ ) ⊤ diag(θ)(x − x ′ ) , where q = d, and Θ = R d + . Let X = [−β, β] d for some
β > 0, and suppose {x i } i∈I is a set of independent identically distributed uniform random samples inside X . Then it can be easily shown that for the anisotropic exponential correlation function, the condition in Lemma 3.3 holds with probability 1, i.e., {vec(R ′ k (θ * ))} d k=1 are linearly independent w.p. 1.
The next result builds on Lemma 3.3, and it shows the convergence of the GSPS estimator as the number of samples per location, N, increases.
Theorem 3.4. Suppose θ * ∈ int Θ, and ρ(x, x ′ ; θ) is twice continuously differentiable in θ over Θ for all x, x ′ ∈ X . Suppose {vec(R ′ k (θ * ))} q k=1 ⊂ R n 2 are linearly independent.
For any given M > 0 and N ≥ N 0 := ⌈2(M + 2) ln(np) + ln 16⌉, letθ (N ) be the GSPS estimator of θ * , i.e.,θ = argmin θ∈Θ f (θ;Γ,Ĉ), andΓ be computed as in (3.8). Then for any sufficiently small ǫ > 0, there exists N ≥ N 0 satisfying N = O(N 0 /ǫ 2 ) such that
setting α = 40 max i=1,...,p (Γ * ii ) N 0 N in (3.6) implies θ (N ) − θ * ≤ ǫ and Γ − Γ * = O(ǫ)
with probability at least 1 − (np) −M ; moreover, the STAGE-II function f (·;Γ,Ĉ) is strongly convex around the estimatorθ.
Proof. See the appendix.
Remark. In Theorem 4, α is explicitly set equal to the lower bound, i.e., A summary of the proposed algorithm for fitting multivariate GRFs models is provided in Algorithm 1.
Algorithm 1 GSPS algorithm to fit multivariate GRFs
input: D = {(x i , y (r) i )} n i=1 ⊂ X × R p , i ∈ I, r = 1, ..., N } /* Compute the sample covariance and distance matrices*/ y (r) ← [y(x 1 ) T , ..., y(x n ) T ] T ∈ R np , r = 1, ..., N S ← 1 N N r=1 y (r) y (r) ⊤ G ij ← x i − x j 2 , if i = j, G ii ← min{ x i − x j 2 : j ∈ I \ {i}
/* Compute the precision matrix and its inverse */ P ← argmin{ S, P − log det(P ) + α G ⊗ (1 q 1 T q ), |P | : a * I P b * I} C ←P −1 /* Compute the between response covariance matrix */ Γ ← 1 n i∈IĈ ii /* Compute the spatial correlation parameter vector*/ θ ← argmin θ∈Θ 1 2 i,j∈I ρ ij (θ)Γ −Ĉ ij 2 F return:Γ andθ 3.1. Connection to SPS. The main difference between the SPS method and GSPS is hoŵ Γ, the estimator for Γ * , is computed (when p = 1, Γ * ∈ R ++ corresponds to the variance parameter ν * > 0 in SPS), and this difference in the way Γ * is estimated has significant implications on: a) the numerical stability of solving STAGE-II problem, and b) the proof technique to show consistency of the hyperparameter estimate as the number of process realization, N, increases.
In the derivation of SPS, we considered the estimateν(θ) as an optimal response to the spatial correlation parameter θ, and show thatν(θ) can be written in a closed form. In the second stage problem of SPS, given in (2.3), we solve a least squares problem over θ, i.e.,
θ = argmin θ∈R d + 1 2 i,j (ν(θ)ρ(x i , x j , θ) −Ĉ ij ) 2 .
Onceθ is computed, we estimate ν * using the best response function:ν =ν(θ). The problem we observed with this approach in Davanloo et al. (2015) when applied to hyper-parameter estimation of a multivariate GRF is that the second stage problem becomes challenging due to its strong nonconvexity, which is significantly aggravated relative to the univariate case due to the multiplicative structure ofΓ(θ)ρ(x i , x j , θ) (when there is a single response, p = 1, this was not a problem for SPS). However, when p > 1, this same structure causes numerical problems in the STAGE-II problem as one would need to solve
min θ∈R d + 1 2 R(θ) ⊗Γ(θ) −Ĉ 2 F . (3.13)
Compared to the above problem, the STAGE-II problem we proposed in (3.9) for GSPS,
i.e., min θ∈R d + 1 2 R(θ) ⊗Γ −Ĉ 2 F , behaves much better (although it is also non-convex in general), whereΓ = 1 n n i=1Ĉ ii -note that Theorem 3.4 shows that the STAGE-II objective of GSPS is strongly convex around a neighborhood of the estimator. In all our numerical tests, standard nonlinear optimization techniques were able to compute a point close to the global minimizer very efficiently; however, this was not the case for the problem in (3.13) when p > 1 -the same nonlinear optimization solvers we used for GSPS get stuck at a local minimizer far away from the global minimum. This is why we propose GSPS using (3.9) in this paper. Moreover, this new step of estimatingΓ = 1 n n i=1Ĉ ii also helps us to give a much simpler proof for Theorem 4.
We now comment on using GSPS to fit a multivariate GRF as opposed to using SPS to fit p independent univariate GRFs to p responses. As mentioned earlier, the latter can only be suboptimal in the presence of cross-covariances between the responses. Furthermore, fitting a multivariate anisotropic GRF requires estimating p(p + 1)/2 parameters for the between-response covariance matrix Γ * ∈ S p ++ and d parameters for the anisotropic spatial correlation function θ ∈ Θ ⊆ R d ++ . On the other hand, fitting p independent univariate anisotropic GRF requires estimating p(d + 1) parameters, i.e., for each univariate GRF one needs to estimate d spatial correlation parameters and 1 variance parameter. Therefore, if d > p 2 , then fitting p univariate GRF requires estimating more hyperparameters. Indeed, for some machine learning problems we have d ≫ p, e.g., the classification problem for text categorization (Joachims, 1998) with p > 1 related classes, and for these type of problems d could be ≈ 10000 and estimating pd hyper-parameters will lead to overfitting; hence, its prediction performance on test data will be worse compared to the prediction performance for multivariate GRF using (3.5) with θ * and Γ * replaced byθ andΓ which are computed as in (3.9) and (3.8), respectively -see Theorem 3.4 for bounds on hyper-parameter approximation quality. In Sections 4.2 and 4.3, the numerical tests conducted on simulated and real-data also show that the proposed GSPS method performs significantly better than modeling each response independently.
Computational Complexity. The computational bottleneck of GSPS method is the singular value decompositions (SVD) that arises when solving the STAGE-I problem using
the ADMM algorithm. The per-iteration complexity is O((np) 3 ). However, we should note that the STAGE-I problem is strongly convex; and ADMM has a linear rate (Deng and Yin, 2015). Therefore, an ǫ-optimal solution can be computed within O(log(1/ǫ)) iterations of ADMM. Thus, the overall complexity of solving STAGE-I is O((np) 3 log(1/ǫ)). Note that likelihood approximation methods do not have such iteration complexity results due to the non-convexity of the approximate likelihood problem being solved, even though they have 19 cheaper per-iteration-complexity. In case of an isotropic process, the STAGE-II problem in (3.9) is one dimensional and it can simply be solved by using bisection. If the process is anisotropic, then (3.9) is non-convex in general. That said, this problem is low dimensional due to d ≪ n; hence, standard nonlinear optimization techniques can compute a local minimizer very efficiently -note that we also show that STAGE-II objective is strongly convex around a neighborhood of the estimator. In all our numerical tests, STAGE-II problem is solved in much shorter time compared to STAGE-I problem; hence, it does not affect the overall complexity significantly. In our code, we use golden-section search for isotropic processes, and Knitro's nonconvex solver to solve (3.9) for general anisotropic processes.
To eliminate O((np) 3 ) complexity due to an SVD computation per ADMM iteration and due to computingĈ, we used a segmentation scheme. We partition the data to K segments, each one composed of ≈ n/K points chosen uniformly at random among n locations, and assuming conditional independence between blocks. In , we discussed two blocking/segmentation schemes: Spatial Segmentation (SS) and Random Selection (RS).
Solving the STAGE-I problem with blocking schemes assumes a conditional independence assumption between blocks. In SS scheme such conditional independence assumption is potentially violated for points along the common boundary between two blocks. The RS scheme, however, works numerically better for "big-n" scenarios. We believe that with RS scheme the infill asymptotics make the blocks conditionally independent to a reasonable degree. Using such blocking schemes, the bottleneck complexity reduces to O((np/K) 3 ) by solving STAGE-I problem for each block; hence, solving STAGE-I and computingĈ, which we assume to be block diagonal, requires a total complexity of O(log(1/ǫ) (np) 3 /K 2 ) and this bottleneck complexity can be controlled by properly choosing K.
Numerical results
In this section, comprehensive simulation analyses are reported for the study of the performance of the proposed method. N realizations of a zero-mean p-variate GRF with anisotropic spatial correlation function are simulated in a square domain X = [0, 10] d over n distinct points. The separable covariance function is the product of an anisotropic exponential spa-
tial correlation function ρ(x, x ′ , θ * ) = exp − (x − x ′ ) ⊤ diag(θ * )(x − x ′ )
and a p-variate
between-response covariance matrix Γ * ∈ S p ++ . The correlation function parameter vector θ * ℓ is sampled uniformly from the surface of a hyper-sphere in R d in the positive orthant for each replication ℓ ∈ {1, ..., L}. The between-response covariance matrix is Γ * ℓ = A ⊤ A for A ∈ R w×p such that w > p, where the elements of A are sampled independently from N (0, 1) per replication. To solve the STAGE-I problem, the sparsity parameter α in (2.1) is set equal to c log(np)/N for some constant c. After some preliminary cross-validation studies, we set c equal to 10 −2 . In our code, we use golden-section search for isotropic processes which requires a univariate optimization in STAGE-II, and use Knitro's nonconvex solver to solve (3.9) for general anisotropic processes.
Parameter estimate consistency
We first compare the quality of GSPS parameter estimate with the Maximum Likelihood Estimate (MLE). For 10 different replicates, we simulated N independent realizations of GRF described above under different scenarios, and the mean of { θ ℓ −θ * } 10 ℓ=1 and { Γ ℓ −Γ * F } 10 ℓ=1 are reported.
To deal with the nonconcavity of the likelihood, the MLEs are calculated from 10 random initial solutions and the best final solutions are reported. To solve problem in (3.6) for the scenarios with np > 2000, we used the Random Selection (RS) blocking scheme as described 21 in Davanloo et al. (2015). Tables 1 and 2 show the results for p-variate GRF models with p = 2 and p = 5, respectively. For fixed n, the parameter estimation error increases with the dimension of the input space d, which is reasonable due to higher number of parameters in the anisotropic correlation 22 function. Furthermore, the errors increase with p, the number of responses. As expected, increasing the point density n helps in improving the estimation of the parameters, i.e., reducing the errors, a result in accordance to the expected effect of infill asymptotics.
N=1 N=10 N=40 d n Method θ l − θ * 2 Γ ℓ − Γ * F θ l − θ * 2 Γ ℓ − Γ * F θ l − θ * 2 Γ ℓ − Γ * F Time(θ l − θ * 2 Γ ℓ − Γ * F θ l − θ * 2 Γ ℓ − Γ * F θ l − θ * 2 Γ ℓ − Γ * F
Overall, the GSPS method results in better parameter estimates compared to MLE with relative performance improvements becoming more obvious as p and d increase. Furthermore, as the number of realizations N increases GSPS performs consistently better than MLE. Note that the robust performance of the proposed method is theoretically guaranteed for N ≥ N 0 from Theorem 3.4.
Prediction consistency
To evaluate prediction performance, we compared the GSPS method against using multiple univariate SPS (mSPS) fits and against the Convolved Multiple output Gaussian Process (CMGP) method by Alvarez and Lawrence (2011). Given the size of the training data n, none of the approximations in (Alvarez and Lawrence, 2011) with induced points were used, this corresponds to what Alvarez and Lawrence refer as the CMGP method.
For 10 different replicates, we simulated N independent realizations of the same GRF, which is defined at the beginning of Section 4, under different scenarios to learn the model parameters. We also simulated the p-variate response over a fixed set of n 0 = 1000 test locations per replicate. The mean of the conditional distribution p(y(x 0 )|{y (r) } N r=1 , D x ) is used to predict at these test locations and, then, the mean of Mean Squared Prediction Error (MSPE) over 10 replicates, p outputs, and n 0 test points are reported for p = 2 and p = 5
in Tables 3 and 4, respectively.
One important observation is that the prediction performance of GSPS is almost ubiquitously better than mSPS method. This means that learning the cross-covariance between different responses provides additional useful information that helps improve the prediction performance of the joint model, GSPS, over mSPS. Comparing GSPS vs. CMGP, we observe relatively better performance of CMGP over GSPS when N = 1 in a lower dimensional input space, e.g., (N, d) = (1, 2). However, as n, the number of locations, increases, the GSPS predictions become better than CMGP even if N = 1, e.g., for (N, d) = (1, 5), GSPS does better than CMPG for n = 400. The prediction performance of GSPS improves significantly with increasing N, the number of realizations of the process. In d = 10 dimensional space, GSPS is performing consistently better, even when N = 1 for both p = 2 and p = 5. However, we should note that CMGP with 50 inducing points is significantly faster than GSPS in the learning phase.
Real data set
We now use a real data set to compare the prediction performance of GSPS with the naive method of using multiple univariate SPS (mSPS) fits, and with the two approximation methods proposed in Alvarez and Lawrence (2011). The data set consists of n=9635 (x, y, z) measurements obtained by a laser scanner from a free-form surface of a manufactured product. Del proposed modeling each coordinate, separately, as a function of the corresponding (u, v) surface coordinates (obtained using the ISOMAP algorithm by Tenenbaum et al. (2000)). These (u, v) coordinates are selected such that their pairwise Euclidean distance is equal to the pairwise geodesic distances between their corresponding (x, y, z) points along the surface. We first model (x(u, v), y(u, v), z(u, v)) as a multivariate GRF using GSPS and compare against fitting p = 3 independent univariate GRF using the SPS method (mSPS).
Given the large size of the data set, n=9635, we use the Random Selection blocking scheme as described in Davanloo et al. (2015) for varying number of blocks; hence, there are different number of observations per block. According to the results reported in Theoretical convergence rates for the estimated between-response covariance matrix and the estimated correlation function parameter are established with respect to the number of process realizations. Numerical studies confirm the theoretical results. From a statistical perspective, the first stage provides a Gaussian Markov Random Field (GMRF) approximation to the underlying GRF without discretizing the input space or assuming a sparsity structure for the precision matrix. From an optimization perspective, the first stage helps to "zoom into" the region where the global optimal covariance parameters exist, facilitating the second stage least-squares optimization.
In this research, we considered separable covariance functions. Future research may consider non-separable covariance functions, e.g., convolutions of covariance functions, or kernel convolutions. As another potential future work, we also propose estimating the crosscovariance matrixΓ at the outset by solvingΓ = argmin Γ { Γ − 1 n n i=1 S ii F : Γ ǫI}.
Then we propose solving the following problem as the new STAGE-I:
Pρ = argmin Pρ S, Pρ ⊗Γ −1 − log det(Pρ ⊗Γ −1 ) + α G ⊗ (1p1 ⊤ p ), |Pρ ⊗Γ −1 | s.t.
a * λmax(Γ)I Pρ b * λmin(Γ)I.
Note that log det(P ρ ⊗Γ −1 ) = p log det(P ρ )−n log det(Γ). Hence, there exists some S ρ , G ρ ∈ S n , which can be computed very efficiently, such that Pρ = argmin Pρ Sρ, Pρ − p log det(Pρ) + α Gρ, |Pρ| : a * λmax(Γ)I Pρ b * λmin(Γ)I .
Such an approach would be much easier to solve in terms of computational complexitythe overall complexity is O(log(1/ǫ)n 3 ) for this STAGE-I problem. Further work could be devoted to proving consistency of the resulting estimator and its rate could be compared with the log(1/ǫ 2 ) of GSPS.
Appendix.
Proof of Theorem 3.1
The proof given below is a slight modification of the proof of Theorem 3.1 in to obtain tighter bounds. For the sake of completeness, we provide the proof.
Through the change of variables ∆ := P − P * , we can write (2.1) in terms of ∆ aŝ ∆ = argmin{F (∆) := S, ∆ + P * − log det(∆ + P * ) + α G ⊗ (1p1 ⊤ p ), |∆ + P * | : ∆ ∈ F},
where F := {∆ ∈ R np×np : ∆ = ∆ ⊤ , a * I ∆ + P * b * I}. Note that∆ =P − P * .
Define g(∆) := − log det(∆ + P * ) on F . g(.) is strongly convex over F with modulus 1/b * 2 ; hence, for any ∆ ∈ F , it follows that g(∆) − g(0) ≥ − P * −1 , ∆ + 1 2b * 2 ∆ 2 F . Let H(∆) := F (∆) − F (0) and S ∆ := {∆ ∈ F : ∆ F > 2b * 2 p(n + G F )α}. Under probability event Ω = { vec(S ij − Σ ij ) ∞ ≤ α, ∀(i, j) ∈ I × I}, for any ∆ ∈ S ∆ ⊂ F ,
H(∆) ≥ S, ∆ − P * −1 , ∆ + 1 2b * 2 ∆ 2 F + α G ⊗ (1p1 ⊤ p ), |∆ + P * | − α G, |P * | ≥ 1 2b * 2 ∆ 2 F + ∆, S − C * − α G ⊗ (1p1 ⊤ p ), |∆| ≥ 1 2b * 2 ∆ 2 F − αp(n + G F ) ∆ F > 0,
where the second inequality follows from the triangle inequality, the third one holds under the probability event Ω and follows from the Cauchy-Schwarz inequality, and the final strict one follows from the definition of S ∆ . Since F (0) is a constant,∆ = argmin{H(∆) : ∆ ∈ F }.
Hence, H(∆) ≤ H(0) = 0. Therefore,∆ ∈ S ∆ under the probability event Ω. It is important to note that∆ satisfies the first two conditions given in the definition of S ∆ . This implies ∆ F ≤ 2b * 2 p(n + G F )α whenever the probability event Ω is true. Hence,
Pr P − P * F ≤ 2b * 2 p(n + G F )α ≥ Pr vec(S ij − Σ ij ) ∞ ≤ α, ∀(i, j) ∈ I × I = 1 − Pr max i,j∈I vec(S ij − Σ ij ) ∞ > α ≥ 1 − i,j∈I
Pr vec(S ij − Σ ij ) ∞ > α .
Recall that S = 1 N N r=1 y (r) y (r) ⊤ and y (r) = [y (r) i ] i∈I for r = 1, . . . , N. Note Σ ii = Γ * 30 for i ∈ I; hence, y (r) i ∼ N (0, Γ * ), i.e., multivariate Gaussian with mean 0 and covariance matrix Γ * , for all i and r. Therefore, Lemma 1 in Ravikumar et al. (2011) implies Pr ( vec(S ij − Σ ij ) ∞ > α) ≤ B α for α ∈ (0, 40 max i Γ * ii ), where
B α := 4p 2 exp −N 2 α 40 max i Γ * ii 2 .
Hence, given any M > 0, by requiring N ≥ 40 max i Γ * ii α 2 N 0 , we get B α ≤ 1 n 2 (np) −M . Thus, for any N ≥ N 0 , we have
i,j∈I Pr vec(S ij − Σ ij ) ∞ > α ≤ (np) −M for all 40 max i Γ * ii N 0 N ≤ α ≤ 40 max i Γ * ii .
Proof of Theorem 3.4
For the sake of simplicity of the notation let Φ = (Γ, C) ∈ S n × S np , and define (Γ, C) a := max{ Γ 2 , C 2 } over the product vector space S n × S np ; also let Ψ = (θ, Γ, C) ∈ R q × S n × S np , and define (θ, Γ, C) b := θ + (Γ, C) a over the product vector space R q × S n × S np .
Throughout the proofΦ := (Γ,Ĉ), Φ * := (Γ * , C * ), andΨ := (θ,Φ), Ψ * := (θ * , Φ * ).
As θ * ∈ int(Θ), there exists δ 1 > 0 such that B . 2 (θ * , δ 1 ) ⊂ Θ. Moreover, since ρ(x, x ′ ; θ)
is twice continuously differentiable in θ over Θ for all x, x ′ ∈ X , R : Θ → S n is also twice continuously differentiable. Hence, from (3.12), it follows that ∇ 2 f (θ; Γ, C) is continuous in Ψ = (θ, Γ, C); and since eigenvalues of a matrix are continuous functions of matrix entries, λ min (∇ 2 f (θ; Γ, C)) is continuous in Ψ on B . b (Ψ * , δ 1 ) as well. Therefore, it follows from Lemma 3.3 that there exists 0 < δ 2 ≤ δ 1 such that ∇ 2 θ f (θ; Γ, C) γ * 2 I for all Ψ = (θ, Γ, C) ∈ B . b (Ψ * , δ 2 ).
i
) : i = 1, ..., n, r = 1, ..., N} contains N realizations of the GRF at each of n distinct locations in D x := {x i } n i=1 ⊂ X . Let y (r) = [y (r) i ] n i=1 ∈ R n denote the vector of r-th realization values for locations in D x .
Fig. 1 :
1Decaying behavior of elements of the Precision and Covariance matrices for GRFs. The largest 1000 off-diagonal elements of the precision and covariance matrices (scaled by their maximums) plotted in descending order. The underlying GRF was evaluated over 100 randomly selected points in X = {x ∈ R 2 : −50 ≤ x ≤ 50} for three covariance functions with range and variance parameters equal to 10, and 1, respectively.
α
= 40 max i=1,...,p (Γ * ii ) N 0 N = 40 max i=1,...,p (Γ * ii ) 2 (M +2) ln(np)+ln 4 N . Note that M controls the 16 probability bound; hence, the only unknown is max i=1,...,p (Γ * ii ) -we implicitly assume that this quantity can be estimated empirically or we have a prior knowledge about it. Moreover, Theorem 4 also guides us how to select α. Indeed, both θ (N ) − θ * ≤ ǫ and Γ − Γ * = O(ǫ)whenever N = O(N 0 /ǫ 2 ); therefore, this implies we should set α = O(ǫ). In the simulations provided in Section 4, α is set equal to c log(np)/N where c is chosen 10 −2 after some preliminary cross-validation studies.
Table 1 :
1Comparison of GSPS vs. MLE for p=2 response variablesN=1
N=10
N=40
d
n
Method
Table 2 :
2Comparison of GSPS vs. MLE for p=5 response variables
Table 3 :
3MSPE comparison for p = 2 response vari-
ables
d
n
Method
N = 1
N = 10
N = 40
2
100
mSPS
7.02
2.68
2.08
GSPS
6.71
2.12
1.44
CMGP
6.40
2.39
1.61
400
mSPS
6.76
2.22
1.87
GSPS
5.53
1.89
0.91
CMGP
5.16
2.04
1.33
5
100
mSPS
7.12
3.09
2.39
GSPS
6.98
2.45
1.52
CMGP
6.74
2.95
1.99
400
mSPS
7.34
3.04
2.24
GSPS
5.88
2.45
1.05
CMGP
6.32
2.89
1.73
10
100
mSPS
7.83
4.15
3.23
GSPS
7.11
3.34
2.02
CMGP
6.97
3.67
2.39
400
mSPS
7.65
3.53
2.65
GSPS
6.13
2.96
1.22
CMGP
6.63
3.32
2.28
Table 4 :
4MSPE comparison for p = 5 response vari-
ables
d
n
Method
N = 1
N = 10
N = 40
2
100
mSPS
7.83
4.42
3.08
GSPS
7.05
3.89
2.11
CMGP
6.74
3.71
2.49
400
mSPS
7.51
3.78
2.18
GSPS
6.81
2.96
1.32
CMGP
6.23
3.36
2.03
5
100
mSPS
8.54
5.30
3.32
GSPS
7.19
4.43
2.01
CMGP
7.10
4.97
2.86
400
mSPS
8.22
4.15
2.63
GSPS
7.00
3.10
1.45
CMGP
7.45
4.04
2.65
10
100
mSPS
9.23
5.67
3.43
GSPS
7.23
4.68
2.19
CMGP
8.53
5.25
3.24
400
mSPS
8.54
4.24
2.94
GSPS
7.08
3.23
1.63
CMGP
7.82
4.20
2.87
Table 5
5reports the MSPE and the corresponding standard errors (std. error) obtained from 10-fold cross validation.Method n/block MSPE std. error
mSPS
100
0.0932
0.0047
mSPS
500
0.0621
0.0021
mSPS
1000
0.0842
0.0013
GSPS
100
0.0525
0.0023
GSPS
500
0.0167
0.0012
GSPS
1000
0.0285
0.0019
Table 5 :
510-fold cross validation to evaluate prediction performance of multiple SPS (mSPS) and GSPS for the metrology data set with n=9635 data points.
Table 5 ,
5the best predictions are obtained when the number of observations per block is 500. We compare the GSPS method with 500 data points per block against the two approximation methods developed inAlvarez and Lawrence (2011), namely the Full Independent Training Conditional (FITC) method and the Partially Independent Training Conditional (PITC) method. For different number of inducing points K ∈ {100, 500, 1000}, we ran both methods on the data set. The locations of the inducing points along with the hyper-parameters of their model are found by maximizing the likelihood through a scaled conjugate gradient method as proposed byAlvarez and Lawrence (2011).Initially, the inducing points are located completely at random.Method
MSPE std. error
mSPS (n/block=500) 0.0621
0.0021
GSPS (n/block=500) 0.0167
0.0012
FITC (K=100)
0.0551
0.0042
FITC (K=500)
0.0463
0.0011
FITC (K=1000)
0.0174
0.0010
PITC (K=100)
0.0698
0.0062
PITC (K=500)
0.0421
0.0021
PITC (K=1000)
0.0197
0.0007
Table 6 :
610-fold cross validation to compare prediction performance of mSPS, GSPS vs. FITC and PITC methods byAlvarez and Lawrence (2011) for the metrology data set with n=9635 data points Intuitively, the best prediction performance for both FITC and PITC approximations are obtained for the larger K values as this represents a better approximation of the underlying GRF. The GSPS method is performing better than FITC and PITC for all K parameter choice. Finally, as expected, fitting p univariate GRF models (mSPS) is performing worse than the multivariate methods.4. Conclusions and future research. A new two-stage estimation method is proposed to fit multivariate Gaussian Random Field (GRF) models with separable covariance functions.
This research is based on the dissertation of the first author, conducted under the guidance of the second and the third authors, Drs. Aybat and del Castillo.
Let Q :=B . a (Φ * , 1 2 δ 2 ) and Θ ′ := Θ ∩B . 2 (θ * , 1 2 δ 2 ), i.e., Q = {(Γ, C) : max{ Γ − Γ * 2 , C − C * 2 } ≤ 1 2 δ 2 }, (5.1)Clearly f is strongly convex in θ over Θ ′ with convexity modulus γ * 2 for all (Γ, C) ∈ Q.Define the unique minimizer over Θ ′ :Since Θ ′ is a convex compact set and f (θ; Γ, C) is jointly continuous in Ψ = (θ, Γ, C) on Θ ′ × Q, from Berge's Maximum Theorem -seeOk (2007), θ(Γ, C) is continuous at (Γ * , C * ) and θ(Γ * , C * ) = θ * . Therefore, for any 0 < ǫ ≤ 1 2 δ 2 , there exists δ(ǫ) > 0 such thatFix some arbitrary ǫ ∈ (0, 1 2 δ 2 ]. LetP (ǫ) be computed as in (3.6) withwhere sample size N(ǫ) denotes the number of process realizations (chosen depending on ǫ > 0). Hence, Theorem 3.2 implies that by choosing N(ǫ) sufficiently large, we can guarantee thatĈ(ǫ) =P (ǫ) −1 , andΓ(ǫ) defined as in (3.8) satisfyi.e., Φ − Φ * a < δ(ǫ), with high probability. In the rest of the proof, for the sake of notational simplicity, we do not explicitly show the dependence on the fixed tolerance ǫ;instead we simply writeP ,Ĉ, andΓ.Note that due to the parametric continuity discussed above, (5.4) implies that θ(Γ,Ĉ)− θ * < ǫ ≤ 1 2 δ 2 . Hence, the norm-ball constraint in the definition of Θ ′ will not be tight whenThis implies thatθ ∈ int Θ; thus, ∇ θ f (θ;Γ,Ĉ) = 0.Although one can establish a direct relation between δ(ǫ) and ǫ by showing that θ(Γ, C)is Lipschitz continuous around θ * , we will show a more specific result by upper boundingwhere the equality follows from the fact that ∇ θ f (θ * ; Γ * , C * ) = ∇ θ f (θ;Γ,Ĉ) = 0. Next, from (3.11) it follows thatwhere the second inequality uses the following basic inequalities and identities: GivenNote that since R(θ * ) ∈ S n ++ , 33 R(θ * ) * = Tr(R(θ * )) = n. Moreover, (5.4) implies that Γ * ≤ Γ * * + p 2 δ 2 , and Ĉ * ≤ C * * + np 2 δ 2 . Hence,Therefore, for κ := 3n Γ * * + C * * + (np+1)Applying Cauchy Schwarz inequality to (5.6), we have(5.7)Thus, choosing N(ǫ) ≥ N 0 := 2 (M + 2) ln(np) + ln 4 such thati.e., N(ǫ) = O( 1 ǫ 2 ), implies that θ − θ * ≤ ǫ, and Γ − Γ * 2 ≤ γ * 2κ ǫ with probability at least 1 − (np) −M .34
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Integrated Systems Engineering. Dept, Columbus OH 43210 USA [email protected]. of Integrated Systems Engineering, Columbus OH 43210 USA E-mail: [email protected]
| []
|
[
"Burst Oscillation Periods from 4U 1636-53: A Constraint on the Binary Doppler Modulation",
"Burst Oscillation Periods from 4U 1636-53: A Constraint on the Binary Doppler Modulation"
]
| [
"A B Giles \nSchool of Mathematics and Physics\nUniversity of Tasmania GPO\nBox 252-217001Hobart TasmaniaAustralia\n\nSpurion Technology Pty. Ltd\n200 Mt. Rumney Road, Mt. Rumney Tasmania7170Australia\n",
"K M Hill \nSchool of Mathematics and Physics\nUniversity of Tasmania GPO\nBox 252-217001Hobart TasmaniaAustralia\n",
"T E Strohmayer \nLaboratory for High Energy Astrophysics\nMail Code 662\n\nNASA Goddard Space Flight Center Greenbelt\n20771MDUSA\n",
"N Cummings \nLaboratory for High Energy Astrophysics\nMail Code 662\n\nNASA Goddard Space Flight Center Greenbelt\n20771MDUSA\n"
]
| [
"School of Mathematics and Physics\nUniversity of Tasmania GPO\nBox 252-217001Hobart TasmaniaAustralia",
"Spurion Technology Pty. Ltd\n200 Mt. Rumney Road, Mt. Rumney Tasmania7170Australia",
"School of Mathematics and Physics\nUniversity of Tasmania GPO\nBox 252-217001Hobart TasmaniaAustralia",
"Laboratory for High Energy Astrophysics\nMail Code 662",
"NASA Goddard Space Flight Center Greenbelt\n20771MDUSA",
"Laboratory for High Energy Astrophysics\nMail Code 662",
"NASA Goddard Space Flight Center Greenbelt\n20771MDUSA"
]
| []
| The burst oscillations seen during Type I X-ray bursts from low mass X-ray binaries (LMXB) typically evolve in period towards an asymptotic limit that likely reflects the spin of the underlying neutron star. If the underlying period is stable enough, measurement of it at different orbital phases may allow a detection of the Doppler modulation caused by the motion of the neutron star with respect to the center of mass of the binary system. Testing this hypothesis requires enough X-ray bursts and an accurate optical ephemeris to determine the binary phases at which they occurred. We present here a study of the distribution of asymptotic burst oscillation periods for a sample of 26 bursts from 4U 1636-53 observed with the Rossi X-ray Timing Explorer (RXTE). The burst sample -2includes both archival and proprietary data and spans more than 4.5 years. We also present new optical light curves of V801 Arae, the optical counterpart of 4U 1636-53, obtained during 1998-2001. We use these optical data to refine the binary period measured byAugusteijn et al. (1998)to 3.7931206(152) hours. We show that a subset of ∼ 70% of the bursts form a tightly clustered distribution of asymptotic periods consistent with a period stability of ∼ 1 × 10 −4 . The tightness of this distribution, made up of bursts spanning more than 4 years in time, suggests that the underlying period is highly stable, with a time to change the period of ∼ 3 × 10 4 yr. This is comparable to similar numbers derived for X-ray pulsars. We investigate the period and orbital phase data for our burst sample and show that it is consistent with binary motion of the neutron star with v ns sin i < 55 and 75 km s −1 at 90 and 99% confidence, respectively. We use this limit as well as previous radial velocity data to constrain the binary geometry and component masses in 4U 1636-53. Our results suggest that unless the neutron star is significantly more massive than 1.4 M ⊙ the secondary is unlikely to have a mass as large as 0.36 M ⊙ , the mass estimated assuming it is a main sequence star which fills its Roche lobe. We show that a factor of 2-3 increase in the number of bursts with asymptotic period measurements should allow a detection of the neutron star velocity. | 10.1086/338890 | [
"https://arxiv.org/pdf/astro-ph/0109294v1.pdf"
]
| 119,406,047 | astro-ph/0109294 | 154e411b97b4e699a606420ed8f69cb967fe22c9 |
Burst Oscillation Periods from 4U 1636-53: A Constraint on the Binary Doppler Modulation
arXiv:astro-ph/0109294v1 18 Sep 2001
A B Giles
School of Mathematics and Physics
University of Tasmania GPO
Box 252-217001Hobart TasmaniaAustralia
Spurion Technology Pty. Ltd
200 Mt. Rumney Road, Mt. Rumney Tasmania7170Australia
K M Hill
School of Mathematics and Physics
University of Tasmania GPO
Box 252-217001Hobart TasmaniaAustralia
T E Strohmayer
Laboratory for High Energy Astrophysics
Mail Code 662
NASA Goddard Space Flight Center Greenbelt
20771MDUSA
N Cummings
Laboratory for High Energy Astrophysics
Mail Code 662
NASA Goddard Space Flight Center Greenbelt
20771MDUSA
Burst Oscillation Periods from 4U 1636-53: A Constraint on the Binary Doppler Modulation
arXiv:astro-ph/0109294v1 18 Sep 2001Subject headings: Binariesgeneral -Stars: individual (4U 1636-53) -Starsneutron -X-raysstars
The burst oscillations seen during Type I X-ray bursts from low mass X-ray binaries (LMXB) typically evolve in period towards an asymptotic limit that likely reflects the spin of the underlying neutron star. If the underlying period is stable enough, measurement of it at different orbital phases may allow a detection of the Doppler modulation caused by the motion of the neutron star with respect to the center of mass of the binary system. Testing this hypothesis requires enough X-ray bursts and an accurate optical ephemeris to determine the binary phases at which they occurred. We present here a study of the distribution of asymptotic burst oscillation periods for a sample of 26 bursts from 4U 1636-53 observed with the Rossi X-ray Timing Explorer (RXTE). The burst sample -2includes both archival and proprietary data and spans more than 4.5 years. We also present new optical light curves of V801 Arae, the optical counterpart of 4U 1636-53, obtained during 1998-2001. We use these optical data to refine the binary period measured byAugusteijn et al. (1998)to 3.7931206(152) hours. We show that a subset of ∼ 70% of the bursts form a tightly clustered distribution of asymptotic periods consistent with a period stability of ∼ 1 × 10 −4 . The tightness of this distribution, made up of bursts spanning more than 4 years in time, suggests that the underlying period is highly stable, with a time to change the period of ∼ 3 × 10 4 yr. This is comparable to similar numbers derived for X-ray pulsars. We investigate the period and orbital phase data for our burst sample and show that it is consistent with binary motion of the neutron star with v ns sin i < 55 and 75 km s −1 at 90 and 99% confidence, respectively. We use this limit as well as previous radial velocity data to constrain the binary geometry and component masses in 4U 1636-53. Our results suggest that unless the neutron star is significantly more massive than 1.4 M ⊙ the secondary is unlikely to have a mass as large as 0.36 M ⊙ , the mass estimated assuming it is a main sequence star which fills its Roche lobe. We show that a factor of 2-3 increase in the number of bursts with asymptotic period measurements should allow a detection of the neutron star velocity.
(see Strohmayer 2001 for a review). All of these results are based on observations with the Proportional Counter Array (PCA) on the Rossi X-ray Timing Explorer (RXTE) except for the evidence for burst oscillations from the accreting millisecond pulsar SAX J1808-369 which is based on SAX Wide Field Camera (WFC) data (see in't Zand et al. 2001). A large body of evidence supports the hypothesis that these oscillations are produced by rotational modulation of a hot spot (or possibly a pair of hot spots) induced on the neutron star surface by inhomogeneous nuclear burning. In particular, the large modulation amplitudes, high coherence and long term stability of the frequency are fully consistent with the rotational modulation scenario (see Strohmayer, Zhang & Swank 1997;Strohmayer et al. 1998a;Strohmayer & Markwardt 1999;Muno et al. 2000 andStrohmayer et al. 1998b).
The oscillation frequency during a burst is usually not constant. Often the frequency is observed to increase by ≈ 1 −3 Hz in the cooling tail, reaching a plateau or asymptotic limit. have suggested that the time evolution of the burst oscillation frequency results from angular momentum conservation of the thermonuclear shell. The burst expands the shell, increasing its rotational moment of inertia and slowing its spin rate.
Near burst onset the shell is thickest and thus the observed frequency lowest. The shell spins back up as it cools and recouples to the underlying neutron star. Cumming & Bildsten (2000) studied this mechanism in some detail and concluded that it appeared to be viable.
However, more recent work by Cumming et al (2001) which corrects an error in the previous work and includes general relativistic effects suggests that it may not be able to account for all of the observed frequency evolution. Spitkovsky, Levin & Ushomirsky (2001), however, suggest that geostrophic effects due to the coriolis force may account for the additional frequency evolution. Nevertheless, this scenario suggests that the limiting frequency is the neutron star spin frequency. We note, however, that not all bursts exhibit this behavior. For example, Strohmayer (1999) and Miller (2000) identified a burst from 4U 1636-53 (burst 4 in Table 2) with a spin down of the oscillations in the decaying tail. This burst also had an unusually long decaying tail which may have been related to the spin down episode. Muno et al. (2000) also reported an episode of spin down in a burst from KS 1731-260.
The long term (over year timescales) stability of burst oscillations from 4U 1728-34 and 4U 1636-53 has been studied by Strohmayer et al. (1998b). For three bursts from 4U 1728-34 separated in time by ≈ 1.6 years they found the 363 Hz burst frequency to be highly stable, with an estimated time scale to change the oscillation period of about 23,000 years. Based on a study of three bursts from 4U 1636-53 (bursts number 1, 2 & 3 in Table 2) spanning a much shorter time interval (about 1 day) they suggested that the observed changes in the limiting frequency of the 581 Hz oscillation might be due to orbital motion of the neutron star, which could provide a way of deriving or constraining the X-ray mass function of the system. However, with only three bursts available at the time it was not possible to test this hypothesis definitively nor draw any strong conclusions on the mass function. 4U 1636-53 is perhaps the best system in which to search for such an effect since the orbital period is known and a large sample of bursts have now been obtained with RXTE. For plausible system parameters and the orbital period of ∼3.8 hours the expected Doppler shifts are of order a part in 10 −4 .
The optical counterpart of 4U 1636-53, V801 Arae, has been observed many times since its identification in 1977 (McClintock et al. 1977) and a collection of photometric data, Augusteijn et al. (1998) who identified a cycle miss count by reanalysing all the old data and incorporating newer observations made between June 1992 and August 1993. Augusteijn et al. (1998) also reported some spectroscopic measurements of emission and absorbtion line features.
In this paper we report new photometric light curves of 4U 1636-53 obtained over the period 1998 March to 2001 May and use them to revise the ephemeris of Augusteijn et al. (1998). We then use this new emphemeris to derive the binary phases of RXTE Xray bursts and examine the possibility that the distribution of observed asymptotic burst oscillation periods is consistent with Doppler modulation caused by the orbital velocity of the neutron star. The paper is organized as follows. In §2 we begin with a discussion of our new optical observations. We then explore in §3 the implications of our new observations for the ephemeris of maximum light from V801 Arae. We show that our data suggest a small correction to the orbital period of Augusteijn et al. (1998). In §4 we describe the sample of X-ray bursts from 4U 1636-53 and we study in detail the observed distribution of asymptotic burst oscillation periods. We show that a subset of ≈ 70% of bursts with asymptotic period measurements form a tightly clustered distribution consistent with having been generated by a highly stable underlying period. We then fit this distribution to models of the period -phase distribution expected from binary motion of the neutron star and show that it is consistent with circular orbital motion of the neutron star with v sin i < 55 km s −1 (90% confidence).
In §5 we summarize our findings and discuss their implications for the component masses and binary geometry of 4U 1636-53. We conclude with a discussion of future improvements to our constraints expected from a larger sample of X-ray bursts.
Optical Observations
All the optical observations described in this paper were made using the Mt. Canopus 1-m telescope at the University of Tasmania observatory. The observations used standard V & I filters and the CCD reduction procedure was identical to that described in Giles, Hill & Greenhill (1999). All times presented in this paper have been corrected to Heliocentric Julian Dates (HJD) and a complete journal of the observations is given in Figure 1 which plots the differential magnitudes with respect to a brighter star that can be located on the finder chart in McClintock et al. (1977). This secondary standard is at the western end of the 20 ′′ scale bar (see Figure 2 on their 2S1636-536 chart). 4U 1636-53 is star number 3 on this same chart and is ∼ 1.8 V magnitudes dimmer than our secondary standard. For the 1999 and later observations the telescope was equipped with an SITe CCD camera having 512 x 512 pixels with an image scale of 0.42 ′′ pixel −1 . The reduction procedures for these observations were similar to the 1998 data and the same local secondary standard was used.
In Figure 2 we show the light curves for the nights of 1999 June 9 and 2001 May 7 & 8. We do not show plots for the remaining nights listed in Table 1 since the individual time spans are rather limited.
Optical Ephemeris
The ephemeris for maximum optical light given by Augusteijn et al. (1998) There is a small phase shift evident between our data and the prediction after extrapolating have previously commented on multi-humped profiles which they had eliminated from their analysis procedure. Our small change to the binary period would be expected to have a fairly minimal effect on the earlier light curves analysed by Augusteijn et. al. (1998) particularly for the older data. We have not attempted to revise the epoch of phase zero or its error as quoted by Augusteijn et al. (1998) since we do not have all the old raw data and phase zero is hard to define for this system where the light curve is quite variable and has no sharp features.
In any case there is still an unknown relationship between the optical & true orbit phase zero and this will likely remain so at least until more extensive radial velocity observations are available. Throughout this paper phase zero is defined as the optical maximum when superior conjunction of the companion star is thought to occur (neutron star closest to the Earth).
Asymptotic Oscillation Periods of RXTE X-ray Bursts
A total of 30 X-ray bursts from 4U 1636-53 are available to us as public or PI data from the PCA experiment on RXTE and information about them relevant to this study are listed in Table 2. A comprehensive description of the properties of these bursts will be given elsewhere (Cummings & Strohmayer 2001). Here we will be primarily interested in the asymptotic burst oscillation periods and inferred binary orbital phases of the bursts. The 1.72 ms (581 Hz) oscillation in most of these bursts exhibits a characteristic evolution towards a limiting (shortest) period in the tail of the burst. It was our aim to try and measure this limiting period for each burst in the sample. For most of these bursts we had event mode data with a time resolution of 1/8192 seconds across the entire 2 -90 keV PCA bandpass. In a few cases we had binned data with the same time resolution. We began by correcting the event arrival times to the solar system barycenter using the JPL DE200 ephemeris and the standard RXTE analysis tools (either fxbary, or faxbary for the most recent data). We then calculated dynamic variability spectra using the Z 2 1 statistic (see Strohmayer & Markwardt 1999 for a discussion and example). Such spectra are essentially similar to standard FFT dynamic power spectra except that we oversample in frequency. We used 2 s intervals and start a new interval every 0.125 s. We oversample in frequency by a factor of 16. For each burst we calculated two dynamic spectra, one using data across the entire bandpass, and a second using only a hard band from 7 − 20 keV. We did this because burst oscillation amplitudes are often stronger at higher energies (see for example ).
To determine the asymptotic period we searched the pair of dynamic power spectra of each burst and determined the shortest period detectable during each burst. By detectable we mean that the signal peak had to be larger than Z 2 1 > 16, which corresponds to a single trial significance of 3.4 × 10 −4 . As an example Figure 3 shows a typical dynamic spectrum from one of our bursts and the power spectrum from which the asymptotic period was deduced (burst 20 in Table 2, in this case the spectrum from the hard band). In most cases a clear frequency track of the oscillation could be seen in the dynamic power spectrum, and the procedure was straightforward. In several cases, either the oscillations were very weak or the frequency evolution was "anomalous" (meaning the frequency was observed to decrease with time), and in these cases we judged that an asymptotic period could not be reliably measured. An example of this is the burst which occurred on 1996 December 31 (burst 4 in Table 2) and has been discussed in detail by Strohmayer (1999). We note that this was the case for only 4 bursts in our sample, so that in the majority of cases the asymptotic period was reasonably well defined. Although these bursts could not be used for the present investigation, for completeness, we also include them in Table 2. We selected the shortest asymptotic period measured in either power spectra as the asymptotic value for that burst.
These periods are also listed in Table 2.
The column in Table 2 showing the burst binary phases has been derived using the new optical ephemeris described in the previous section. The phase error for each burst is dominated by the ability to determine the optical phase zero for any particular epoch but is typically < ±0.05. Relative phase errors are much smaller given the >48,000 cycle time span of the optical observations and the fact that the X-ray bursts used here all occur within a time interval of ∼ 4.4 years (only 10,000 cycles) ending in 2001 May.
Period Measurement Uncertainty
An important quantity to understand is the characteristic error, σ P , in our period measurements. To estimate this we have carried out a series of simulations which mimic the conditions of our asymptotic period measurements. To do this we first generate a count rate model comprised of a constant plus a sinusoidal modulation of fixed period and amplitude.
We then generate random realizations of this model using the same temporal resolution as our burst data. We model a 2 s interval of data since this was the interval length we used for all our dynamic spectra. We use a count rate and modulation amplitude typical of the intervals in the tails of bursts where we actually measure the asymptotic periods. We then compute the Z 2 1 spectra for each of the simulated data sets and determine the centroid period of the signal. Since typically we follow the signal in a real burst down to or near a limiting threshold (in this case Z 2 1 = 16), we only keep simulated period measurements for which the peak signal power was close to our limiting threshold. In practice we found that 16 < Z 2 1 < 24 was characteristic of our actual asymptotic period measurements. We then determine how these simulated periods are distributed around the true period. Specifically we fit a gaussian to the distribution of simulated periods and identify the width of this gaussian with the characteristic uncertainty, σ P , in any one of our period measurements. Figure 4 shows the period distribution and best fitting gaussian derived from one of these simulations. We find that the typical measurement error associated with one of our periods is ∼ 2.2 × 10 −4 ms. Note that this is purely a statistical uncertainty. Another source of possible systematic error is associated with the assumption that the last period detected in a dynamic spectra represents a limiting value. We will have more to say on this in a later section.
The Observed Distribution of Asymptotic Periods
We used the period measurements from Table 2 to construct a distribution of asymptotic periods. Figure 5 shows a histogram representation of the distribution. Although the range of all observed periods is rather large, a subset of ∼ 70% of the bursts form a tightly peaked distribution. Also shown in Figure 5 is the gaussian model which best fits this cluster of periods. The gaussian is centered at 1.71929 ± 1.0 × 10 −4 ms, has a width of 2.3 × 10 −4 ± 1.2 × 10 −4 and gives an excellent fit to the data. This subset is comprised of bursts from all epochs of our sample, and suggests that a highly stable underlying period is responsible for this component of the asymptotic period distribution. Note also that the width of this distribution is comparable to our estimate above of the typical width which would be produced by statistical uncertainties alone. This suggests that any systematic error associated with our measurements not reflecting a true limiting value are small, at least within this subset of the entire sample.
A Constraint on the Orbital Doppler Modulation
Assuming that the burst oscillations do reflect the spin of the neutron star the binary motion should imprint doppler modulations on the measured periods. We use the values from Table 2 to construct in Figure 6 a plot of asymptotic period against photometric (orbital) phase. Visual inspection of this plot reveals no strong indication of a sinusoidal modulation that might be produced by a sufficiently strong Doppler modulation. Such a modulation would likely have a peak on Figure 6 at a phase of ∼ 0.25, when the neutron star has a maximum recession velocity assuming that photometric maximum occurs at superior conjunction of the secondary. We tested this conclusion quantitatively by fitting a periodphase model to the data. We used the model
P i = P 0 (1 + (v sin i/c) sin(2π(φ i − 1.0))) ,(1)
where P 0 , v sin i, and φ i are the period measured at inferior conjunction of the neutron star (neutron star nearest to observer), the projected orbital velocity of the neutron star with respect to the center of mass of the binary, and the orbital phase at which the burst occurred, respectively. Figure 6 also shows the results of such fits. The model prefers a
small v sin i/c = 6 × 10 −5 , with χ 2 = 19.6 for 16 degrees of freedom. This model is the solid curve in Figure 6. However, the difference between this fit and one with v sin i/c = 0 is not statistically significant (∆χ 2 = 0.53), hence the data are consistent with no doppler modulation. The 90% and 99% confidence upper limits (∆χ 2 = 2.71 and 6.63) on v sin i are 55 and 75 km s −1 , respectively. The model with v ns sin i = 55 km s −1 is the dashed curve in Figure 6. Note that these fits assumed that the relative phase of the modulation is known based on the photometric ephemeris. If we relax this assumption and allow the phase of the peak modulation to be a parameter we find a better fit with v sin i = 59.3 km s −1 , with 90% confidence range of 15.8 < v sin i < 102.7 km s −1 (dotted curve in Figure 6). However, the phase offset required would be 0.2 away from that implied under the assumption that phase zero (photometric maximum) is at superior conjunction of the secondary. Although this seems large it might be possible if X-ray heating of the disk bulge and accretion stream interaction region contribute to the observed optical modulations. We discuss this further below.
Although we do not detect any doppler modulation we were able to place an upper limit on v sin i from the period -phase data. Since there was no strong evidence for a modulation with orbital phase we also investigated the upper limit using only the expected distribution of periods for a given v ns sin i and σ P . To do this we generated an expected period distribution by sampling a large number of random periods from the model. Samples were drawn uniformly in orbital phase and the random period was selected from a gaussian distribution with width σ P centered on the model period for that phase. We then binned the sample periods in the same manner as the data and computed a χ 2 goodness of fit
statistic χ 2 = j (O j − M j ) 2 /M j .
Since our data have small numbers of events in each bin we computed the upper limit for v ns sin i using monte carlo simulations. Our resulting upper limit using this method is in good agreement with our result from the period versus phase fits.
Summary and Discussion
We have investigated the asymptotic period distribution of burst oscillations in a large sample of bursts from 4U 1636-53. We find that ∼ 70% of these bursts form a tight distribution consistent with being produced by a highly stable mechanism such as rotation of the neutron star. The fact that the distribution is made up of bursts spanning a time scale of 4.4 years and has a characteristic width of ∆P/P = 1.3 × 10 −4 indicates that the time scale to change the underlying period is τ > ∆T P/∆P = 3.4 × 10 4 yr. This is comparable to the overall period stability estimated for the 363 Hz oscillations in 4U 1728-34 (see Strohmayer et al. 1998b), and is a number characteristic of other rotating neutron stars such as X-ray pulsars. This provides further evidence that rotation of the neutron star sets the burst oscillation period.
Why do some of the bursts fall well outside this distribution? It seems likely that several effects may be at work here. One problem is that the oscillation in some bursts does not remain strong enough to detect for a long enough time interval within the burst, so that the asymptotic limit is not reached. This results because burst oscillation properties are not identical from burst to burst. Another possible effect was discussed by Cumming & Bildsten (2000). They argued that as long as the burning shell was not recoupled to the neutron star the frequency observed in the burst tail would deviate slightly (by about 1 part in 10 −4 ) from the neutron star spin frequency. This comes about because the thickness of the cooling atmosphere in the tail is different to the initial thickness by about 1 m, though the exact amount depends on the mean molecular weight of the burned material which in turn depends on how complete the burning was and would be expected to vary from burst to burst. Although this could conceivably be a source of additional scatter in the asymptotic periods the fact that our observed distribution has a width comparable to that expected based on statistical uncertainty alone suggests that if operating at all it must be small. If the asymmetry on the star is created by a nonradial oscillation mode (see for example Bildsten & Cutler 1995;Heyl 2001), then the observed oscillation frequency would always be close to the spin frequency or perhaps a multiple mΩ of it, but it could change by ∼ 1Hz due to long term changes in the surface layers of the neutron star. This could produce outliers in the period distribution, but would also tend to produce a tight component as long as surface conditions were similar for enough bursts.
Recently, Spitkovsky, Levin & Ushomirsky (2001) have also studied mechanisms which can cause frequency drift. They suggest that Coriolis forces can have an important effect and might introduce shifts in the observed frequency comparable to those expected from radial uplift.
Constraints on the Binary Geometry
In general the optical flux from LMXBs is thought to be dominated by the accretion optical maximum therefore occurs when the companion is on the far side of the neutron star (superior conjunction) but there may be some asymmetry or variation about the mean profile due to gas flows causing various X-ray shielding effects (Pedersen et al. 1982a).
In order to explore the implications for the binary geometry of our radial velocity limit for the neutron star we have created in Figure 7 a plot of the Roche geometry for 4U 1636-53. For the neutron star we assumed a mass of 1.6M ⊙ . For the secondary we use a mass of 0.36M ⊙ (see Smale & Mukai 1988;Patterson 1984). With these masses and the known 3.8 hr orbital period the binary separation is ∼ 1.58R ⊙ . The velocities of the neutron star and secondary with respect to the center of mass would be 91 and 390 km s −1 , respectively. Figure 7 shows a view looking down on the orbital plane of the system. The numbers circling the system denote orbital phase positions assuming phase zero occurs at superior conjunction of the secondary (photometric maximum). The dashed circle shows the extent of a disk which fills 90 % of the Roche lobe, a radius at which tidal effects will likely truncate it (see for example Frank, King & Lasota 1987). Constraints from analysis of optical reprocessing of X-ray bursts also indicate a large accretion disk in 4U 1636-53 (Pedersen et al. 1982a).
We also show on the plot inferred locations of the radial velocity components measured by Augusteijn et al. (1998) and given in their Table 6. Since the inferred velocity amplitudes from their three sets of fits were all rather similar we just used the average velocity as well as the average uncertainty. We plotted with triangles the ±1σ average velocity amplitude at the phases of superior conjunction given for each of their three fits. The phase locations were deduced by assuming that the entire binary system rotates rigidly about the center of mass. We also shaded the region enclosed by the triangles to further highlight its location. Augusteijn et al. (1998) suggested the radial velocity components could be identified with the bulge region associated with the interaction of the accretion stream with the disk. Our plot certainly supports this suggestion, since the shaded region is consistent with where the accretion stream would likely impact the disk. The location of the shaded region also suggests that the bulge might be a significant component with regard to optical modulations.
In particular, if photometric maximum occurs closer to a phase of 0.8 in Figure 7, when the X-ray illuminated portion of the bulge is facing the observer, then the implied phase shift is in the same sense as that suggested by the fits to the period -phase data of the X-ray bursts with the phase shift left as a free parameter. More detailed modelling would be required to determine if the bulge can indeed effect the optical modulations at this level, but the period -phase fits are suggestive.
We also note that although the three simultaneous X-ray & optical bursts discussed by Pedersen et al. (1982a) (see page 336) have relatively large error bars on the optical time delays we have re-examined them in the light of our new ephemeris and the system model shown in Figure 7. The optical delays in these bursts appear more consistent, both in delay and phase, with the reprocessed X-ray burst optical flux coming from the outer parts of our shaded region in Figure 7 than from the facing hemisphere of the companion star. Although there is no evidence of a second optical pulse from the companion in the many optical bursts studied by Pedersen et al. (1982b) a weaker following pulse might easily be lost. Such a pulse might only be evident at optimum binary phases, around phase 0.85, with reprocessing delays always tending to broaden and confuse the light curve features.
Although the radial velocities of the neutron star and secondary are not well measured in 4U 1636-53, as Figure 7 suggests the system is rather well constrained.
i (M 1 + M 2 ) 2/3 km s −1 ,(2)
with v ns set to either our 90 or 99% limit (see §4.3 above). To derive mass constraints from the radial velocity data we required that the inferred location of the radial velocity components (determined from the velocity amplitude and phase of superior conjunction data of Augusteijn et al. 1998, see discussion above) must fit within 90% of the Roche lobe radius of the neutron star (a likely size for the accretion disk). This further assumes that the entire binary rotates rigidly around the center of mass. Our constraints are summarized in Figure 8. We show allowed regions in the component mass plane for a pair of different inclinations (50 and 60 • ) for our 90 and 99% neutron star velocity limits. Indeed for v ns 55 km s −1 the mass of the secondary must be significantly less than the 0.36M ⊙ estimate based on the mass -radius relation for main sequence stars. Further, if the secondary is ∼ 0.3M ⊙ then the neutron star must be quite massive M ns > 1.8M ⊙ . The radial velocity constraints essentially exclude i 40 • for any reasonable masses of the components. This is because the disk cannot be big enough to allow high radial velocities if the inclination is too low.
Although this conclusion is dependent on our assumptions for deriving the radial velocity constraints, observations of large amplitude oscillations on the rising edge of bursts from this source also indicate that the inclination cannot be too low (see Nath, Strohmayer & Swank 2001). These arguments suggest a likely range for the inclination of 50 • < i < 60 • . With this inclination a likely range of masses for the neutron star and secondary are, in solar units, 1.4 < M ns < 1.6 and 0.2 < M sec < 0.25. More precise limits on the radial velocity of either component will allow more precise mass limits to be inferred.
Clearly additional optical photometry and spectroscopy are required for 4U 1636-53, and at some time in the future it would prove worthwhile to collect together all the optical observations of 4U 1636-53 to derive a fully consistent ephemeris. As more burst data become available it should become possible to measure the neutron star velocity. For example, with a factor of 2-3 increase in the number of bursts with reliable asymptotic periods and with a burst oscillation period measurement uncertainty of 2.2 × 10 −4 ms, our simulations suggest that a velocity of 55 km s −1 (equal to our current 90% upper limit) can be detected at better than 3σ confidence. Figure 9 shows the results of such a simulation for 36 burst asymptotic period measurements. The bursts listed in Table 2 were found in observations totaling ∼ 1.2 Msec of exposure. Based on this X-ray burst rate the presently approved RXTE observing time on 4U 1636-53 (1.15 Msec in AO6) can be expected to provide another ∼ 28 X-ray bursts, which should roughly double the sample. Since RXTE provides much higher quality X-ray burst profiles than did Hakucho, further attempts to get simultaneous X-rayoptical burst observations are clearly worthwhile but this requires and is dependant on the availability of a large optical telescope.
We thank Holger Pedersen for re-examining and confirming the dates and times of observations of 4U 1636-53 made in 1980. Archive data was obtained from the High Energy Astrophysics Science Archive Research Center Online Service provided by the NASA / God-dard Space Flight Center. We also thank the referees for their informative comments. Table 2 (top). The horizontal dashed line marks the asymptotic period inferred for this burst. The burst lightcurve is overlaid (right axis). The gaps in the lightcurve are due to telemetry limitations for this data mode. Also shown is the Z 2 1 spectrum in the tail of the burst from which the asymptotic period was measured (bottom). In this case the vertical dashed line marks the asymptotic period. See the text ( §4.1) for a discussion of the simulations. The width of the gaussian is 2.2×10 −4 ms and represents the characteristic uncertainty in our asymptotic period measurements. We also show the best fitting orbital doppler model. The neutron star velocity is detected at better than 3σ confidence in this simulation. This suggests that a doubling of the number of observed asymptotic periods should enable a detection of the neutron star velocity. Table 2 (top). The horizontal dashed line marks the asymptotic period inferred for this burst. The burst lightcurve is overlaid (right axis). The gaps in the lightcurve are due to telemetry limitations for this data mode. Also shown is the Z 2 1 spectrum in the tail of the burst from which the asymptotic period was measured (bottom). In this case the vertical dashed line marks the asymptotic period. See the text ( §4.1) for a discussion of the simulations. The width of the gaussian is 2.2×10 −4 ms and represents the characteristic uncertainty in our asymptotic period measurements. -32 - Figure 9: Period versus orbital phase simulation using 36 simulated asymptotic periods sampled with the same statistical uncertainty as we estimated for our real measurements.
Figure Captions
We also show the best fitting orbital doppler model. The simulation used v sin i = 55 km s −1 , our 90% confidence upper limit. The neutron star velocity is detected at better than 3σ confidence in this simulation. This roughly suggests that a doubling of the number of observed asymptotic periods should enable a detection of the neutron star velocity. We note, however, that due to the fact that only ∼ 70% of bursts fall within the asymptotic distribution, this would correspond to more than a doubling of the current burst sample.
is HJD = 2446667.3183(26) ± [ N × 0.15804738(42) ] where the errors are indicated in the round brackets. This ephemeris was based on observations made between 1980 July 11 and 1993 July 12 and covers a total of 30048 binary periods. The predictions for this ephemeris are shown on Figures 1 & 2 as the dotted trace in the lower section of each light curve panel.
forward in time for the additional ∼18080 binary periods to 2001 May. We have fitted a sine curve to the 2001 May 7 & 8 data and determined that the phase difference at this epoch is ∼ 0.15. This can be eliminated by reducing the period of Augusteijn et. al. (1998) by a very small amount and, assuming that there is no cycle mis-count which is very unlikely, this period change corresponds to 1.65 times their quoted error. We therefore adopt the following new ephemeris for the time of maximum light HJD = 2446667.3183(26) ± [ N × 0.15804669(24) ]. After adjusting the modulation amplitude and mean level this ephemeris is plotted as the solid line in Figures 1 & 2 and although derived from fitting only the data from 2001 May 7 & 8 it appears reasonably consistent with the other four light curves. The night of 1998 April 3 in Figure 1 does have an odd profile but van Paradijs et al. (1990)
disk (see van Paradijs & McClintock 1995 for a review). There are three regions of a LMXB system which might contribute to its optical variability due to X-ray heating. These are the accretion disk itself, a bright spot or bulge on the outer edge of the accretion disk formed by interaction of the accretion stream with the disk, and the hemisphere of the companion facing the neutron star which is not shadowed by the accretion disk. In LMXBs with relatively low inclinations (i 60 • ) it is this last region which is thought to dominate the optical modulations from the rest of the system (vanParadijs 1983, van Paradijs & McClintock 1995. These systems generally produce sinusoidal optical modulations. The
Fig. 1 .
1-The V band light curves for 4U 1636-53 on 1998 March 25 & 27 and 1998 April 3. The solid sine curve marks our new ephemeris with a period of 3.70312064 hours. The dotted curve shows the ephemeris prediction of Augusteijn et al. (1998) for the same three nights with an arbitrary offset and amplitude. The number in each panel refers to the HJD starting at zero hours within each light curve.
Fig. 2 .
2-The V band light curves for 4U 1636-53 on 1999 June 9 and 2001 May 7 & 8. The solid sine curve marks our new ephemeris and the first optical maximum on 7 May occurs at HJD = 2452036.954706 The dotted curve shows the ephemeris prediction of Augusteijn et al. (1998) for the same three nights with an arbitrary offset and amplitude. The number in each panel refers to the HJD starting at zero hours within each light curve.
Fig. 3 .
3-Dynamic Z 2 1 spectrum in the hard X-ray band (7 -20 keV) for burst number 20 in
Fig. 4 .
4-Histogram of simulated period measurements and best fitting gaussian distribution.
Fig. 5 .
5-Histogram of measured asymptotic burst oscillation periods for 4U 1636-53. The periods are corrected to the solar system barycenter. Note the cluster of 18 periods centered near 1.7192 ms. A gaussian distribution centered at 1.71929 ms, of width σ = 2.3 × 10 −4 ms fits these data well and is shown by the thick solid curve. Note the presence of outliersFig. 6.-Plot of asymptotic period versus orbital phase for the subset of 18 bursts which have a tightly clustered period distribution. The solid curve is the best fitting doppler model with v ns sin i = 18 km s −1 . This fit, however, is not statistically significant compared to a v ns sin i = 0 model (see §4.3). The dashed curve shows the model with v ns sin i = 55 km s −1 , which is equal to our 90% confidence upper limit. The dotted curve shows the best fitting model when the phase of maximum is added as an additional parameter. This fit has v ns sin i = 59.3 km s −1 .
Fig. 7 .
7-Diagram of the Roche geometry for 4U 1636-53. The figure was drawn assuming neutron star and secondary masses of 1.6M ⊙ and 0.36M ⊙ , respectively. The numbers circling the components correspond to orbital phases under the assumption that phase zero corresponds to superior conjunction of the secondary (V801 Arae). The center of mass (CM) is denoted by a square symbol. The dashed circle around the neutron star marks the likely extent of an accretion disk under the assumption that it fills 90% of the neutron star Roche lobe. The triangles and shaded region mark the inferred locations of the radial velocity components measured byAugusteijn et al. (1998).
Fig. 8 .
8-Constraints on the component masses in 4U 1636-53 derived from our upper limit on v ns sin i and the radial velocity data ofAugusteijn et al. (1998). The regions with horizontal hatching are excluded by the neutron star velocity limit, while the vertical hatched regions are excluded by the radial velocity data. We show constraints for i = 50 • and v ns sin i < 55 km s −1 (a), i = 50 • and v ns sin i < 75 km s −1 (b), i = 60 • and v ns sin i < 55 km s −1 (c), i = 60 • and v ns sin i < 75 km s −1 (d). The thick lines denote M ns = 1.4M ⊙ and M sec = 0.36M ⊙ , respectively. See §5.1 for a discussion on how the constraints were derived.
Fig. 9 .
9-Period versus orbital phase simulation using 36 simulated asymptotic periods -23sampled with the same statistical uncertainty as we estimated for our real measurements.
Figure 1 :Figure 2 :
12The V band light curves for 4U 1636-53 on 1998 March 25 & 27 and 1998 April 3. The solid sine curve marks our new ephemeris with a period of 3.70312064 hours. The dotted curve shows the ephemeris prediction of Augusteijn et al. (1998) for the same three nights with an arbitrary offset and amplitude. The number in each panel refers to the HJD starting at zero hours within each light curve. The V band light curves for 4U 1636-53 on 1999 June 9 and 2001 May 7 & 8. The solid sine curve marks our new ephemeris and the first optical maximum on 7 May occurs at HJD = 2452036.954706 The dotted curve shows the ephemeris prediction of Augusteijn et al. (1998) for the same three nights with an arbitrary offset and amplitude. The number in each panel refers to the HJD starting at zero hours within each light curve.
Figure 3 :
3Dynamic Z 2 1 spectrum in the hard X-ray band (7 -20 keV) for burst number 20 in
Figure 4 :
4Histogram of simulated period measurements and best fitting gaussian distribution.
Figure 5 :Figure 6 :
56Histogram of measured asymptotic burst oscillation periods for 4U 1636-53. The periods are corrected to the solar system barycenter. Note the cluster of 18 periods centered near 1.7192 ms. A gaussian distribution centered at 1.71929 ms, of width σ = 2.3 × 10 −4 ms fits these data well and is shown by the thick solid curve. Note the presence of outliers towards longer period, but none shortward of the gaussian. Plot of asymptotic period versus orbital phase for the subset of 18 bursts which have a tightly clustered period distribution. The solid curve is the best fitting doppler model with v ns sin i = 18 km s −1 . This fit, however, is not statistically significant compared to a v ns sin i = 0 model (see §4.3). The dashed curve shows the model with v ns sin i = 55 km s −1 , which is equal to our 90% confidence upper limit. The dotted curve shows the best fitting model when the phase of maximum is added as an additional parameter. This fit has v ns sin i = 59.3 km s −1 .
Figure 7 :
7Diagram of the Roche geometry for 4U 1636-53. The figure was drawn assuming neutron star and secondary masses of 1.6M ⊙ and 0.36M ⊙ , respectively. The numbers circling the components correspond to orbital phases under the assumption that phase zero corresponds to superior conjunction of the secondary (V801 Arae). The center of mass (CM) is denoted by a square symbol. The dashed circle around the neutron star marks the likely extent of an accretion disk under the assumption that it fills 90% of the neutron star Roche lobe. The triangles and shaded region mark the inferred locations of the radial velocity components measured by Augusteijn et al. (1998).
Figure 8 :
8Constraints on the component masses in 4U 1636-53 derived from our upper limit on v ns sin i and the radial velocity data ofAugusteijn et al. (1998). The regions with horizontal hatching are excluded by the neutron star velocity limit, while the verticalhatched regions are excluded by the radial velocity data. We show constraints for i = 50 • and v ns sin i < 55 km s −1 (a), i = 50 • and v ns sin i < 75 km s −1 (b), i = 60 • and v ns sin i < 55 km s −1 (c), i = 60 • and v ns sin i < 75 km s −1 (d). The thick lines denote M ns = 1.4M ⊙ and M sec = 0.36M ⊙ , respectively. See §5.1 for a discussion on how the constraints were derived.
from July 1980 to May 1988, was compiled by van Paradijs et al. (1990) (see references therein). The van Paradijs et al. (1990) ephemeris was later revised by
Table 1 .
1For the 1998 observations the telescope was equipped with an SBIG CCD camera having 375 x 242 pixels with an image scale of 0.42 ′′ × 0.49 ′′ pixel −1 . On the nights of 1998 March 25 & 27 continuous pairs of V & I integrations were obtained but the I band data are not discussed further in this paper. Three V band light curves from 1998 are shown in
Table 1 :
1Optical observations of 4U 1636-53Date
HJD Start HJD End
Filter Int. No.
-2450000
-2450000
(s) Exp.
3/25/98 0898.01619 0898.30426 V & I 180
65
3/27/98 0899.99518 0900.23595 V & I 180
56
4/ 3/98 0907.09065 0907.32071
V
300
65
3/26/99 1264.02547 1264.30913
V
300
32
3/28/99 1266.15552 1266.26166
V
300
10
3/31/99 1269.15566 1269.29098
V
300
16
4/ 2/99 1271.08660 1271.32108
V
300
33
4/ 4/99 1273.25977 1273.31819
V
300
8
6/ 9/99 1338.88939 1339.14694
V
180
81
6/10/99 1340.18839 1340.27722
V
180
12
5/ 7/01 2036.89853 2037.34667
V
300
72
5/ 8/01 2037.97337 2038.28175
V
300
74
Table 2 :
2X-ray bursts detected from 4U 1636-53 by RXTEBurst
RXTE
Date
HJD
Period
Binary
No.
Obs. ID.
-2450000
(ms)
phase
± 0.00040 ± 0.05
1
10088-01-07-02 12/28/96 0446.439466
1.71940
0.42
2
10088-01-07-02 12/28/96 0446.491308
1.71910
0.75
3
10088-01-08-01 12/29/96 0447.472404
1.71925
0.96
4
10088-01-08-030 12/31/96 0449.229474
-
0.07
5
10088-01-09-01 2/23/97 0502.913912
1.72028
0.75
6
30053-02-02-02 8/19/98 1044.991053
1.72083
0.60
7
30053-02-01-02 8/20/98 1045.654542
1.72161
0.80
8
30053-02-02-00 8/20/98 1045.719849
1.71954
0.22
9
40028-01-02-00 2/27/99 1236.865609
1.71954
0.64
10
40028-01-04-00 4/29/99 1297.575867
1.71925
0.77
11
40028-01-06-00 6/10/99 1339.751875
1.72240
0.63
12
40028-01-08-00 6/18/99 1348.493173
1.71943
0.94
13
40030-03-04-00 6/19/99 1349.234723
1.72260
0.63
14
40031-01-01-06 6/21/99 1351.300601
1.71930
0.70
15
40028-01-10-00 9/25/99 1447.360320
-
0.49
16
40028-01-13-00 1/22/00 1565.570419
-
0.44
17
40028-01-13-00 1/22/00 1565.703136
1.72043
0.28
18
40028-01-14-01 1/30/00 1573.506255
-
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| []
|
[
"Double-Directional Information Azimuth Spectrum and Relay Network Tomography for a Decentralized Wireless Relay Network",
"Double-Directional Information Azimuth Spectrum and Relay Network Tomography for a Decentralized Wireless Relay Network"
]
| [
"Yifan Chen [email protected] \nSchool of Engineering\nInstitute for Infocomm Research\nUniversity of Greenwich Chatham Maritime\n1 Fusionopolis Way, 21-01ME4 4TB, 138632Kent, ConnexisSingapore\n",
"Chau Yuen [email protected] \nSchool of Engineering\nInstitute for Infocomm Research\nUniversity of Greenwich Chatham Maritime\n1 Fusionopolis Way, 21-01ME4 4TB, 138632Kent, ConnexisSingapore\n"
]
| [
"School of Engineering\nInstitute for Infocomm Research\nUniversity of Greenwich Chatham Maritime\n1 Fusionopolis Way, 21-01ME4 4TB, 138632Kent, ConnexisSingapore",
"School of Engineering\nInstitute for Infocomm Research\nUniversity of Greenwich Chatham Maritime\n1 Fusionopolis Way, 21-01ME4 4TB, 138632Kent, ConnexisSingapore"
]
| []
| A novel channel representation for a two-hop decentralized wireless relay network (DWRN) is proposed, where the relays operate in a completely distributive fashion. The modeling paradigm applies an analogous approach to the description method for a double-directional multipath propagation channel, and takes into account the finite system spatial resolution and the extended relay listening/transmitting time. Specifically, the double-directional information azimuth spectrum (IAS) is formulated to provide a compact representation of information flows in a DWRN. The proposed channel representation is then analyzed from a geometrically-based statistical modeling perspective. Finally, we look into the problem of relay network tomography (RNT), which solves an inverse problem to infer the internal structure of a DWRN by using the instantaneous doubledirectional IAS recorded at multiple measuring nodes exterior to the relay region.Index Terms-Wireless relay networks, decentralized relays, information flows, cooperative communications, information azimuth spectrum, relay network tomography | 10.1109/isita.2010.5649651 | [
"https://arxiv.org/pdf/1004.1045v1.pdf"
]
| 9,149,304 | 1004.1045 | a72b1f9ef20633cd96d0c991f68d243b7cc7640b |
Double-Directional Information Azimuth Spectrum and Relay Network Tomography for a Decentralized Wireless Relay Network
7 Apr 2010
Yifan Chen [email protected]
School of Engineering
Institute for Infocomm Research
University of Greenwich Chatham Maritime
1 Fusionopolis Way, 21-01ME4 4TB, 138632Kent, ConnexisSingapore
Chau Yuen [email protected]
School of Engineering
Institute for Infocomm Research
University of Greenwich Chatham Maritime
1 Fusionopolis Way, 21-01ME4 4TB, 138632Kent, ConnexisSingapore
Double-Directional Information Azimuth Spectrum and Relay Network Tomography for a Decentralized Wireless Relay Network
7 Apr 2010
A novel channel representation for a two-hop decentralized wireless relay network (DWRN) is proposed, where the relays operate in a completely distributive fashion. The modeling paradigm applies an analogous approach to the description method for a double-directional multipath propagation channel, and takes into account the finite system spatial resolution and the extended relay listening/transmitting time. Specifically, the double-directional information azimuth spectrum (IAS) is formulated to provide a compact representation of information flows in a DWRN. The proposed channel representation is then analyzed from a geometrically-based statistical modeling perspective. Finally, we look into the problem of relay network tomography (RNT), which solves an inverse problem to infer the internal structure of a DWRN by using the instantaneous doubledirectional IAS recorded at multiple measuring nodes exterior to the relay region.Index Terms-Wireless relay networks, decentralized relays, information flows, cooperative communications, information azimuth spectrum, relay network tomography
I. INTRODUCTION
Recently a great deal of research has been devoted to cooperative wireless communications [1]- [3], which reap spatial diversity benefits from a virtual antenna array formed with multiple relay nodes. In a cooperative scheme, one user terminal (UT) partners with another UT to send its signal to the access point (AP) or some other final destinations. The partner UT serves as a relay, forwarding the message from the source to the destination. For most of the existing works, implementation of the schemes assumes central coordination among relays. Nevertheless, the overhead to set up the cooperation may drastically reduce the useful throughput [4]. Furthermore, a centralized operation may not be feasible for a distributed system with a dynamic infrastructure (e.g., a wireless ad hoc network) or with a energy-constrained operational mode (e.g., a wireless sensor network). In this paper, we consider a decentralized wireless relay network (DWRN) without link combining and joint scheduling among relays [5].
In our previous work [6], the performance of a DWRN was analyzed by realizing the noteworthy analogy between a virtual DWRN and a physical propagation channel. The information delivered from the source to the destination flows through a DWRN just as the signal power flows through a physical channel. The concept of information azimuth-delay spectrum (IADS) was defined, which is parallel to the power azimuth-delay spectrum (PADS) used in describing a singledirectional physical channel [7], [8]. However, the methodology in [6] is only applicable to a perfect receiver, which has an infinite angle-resolving capability and incurs zero data processing delay. Furthermore, the relay is supposed to listen and forward the data within very short intervals, comparable to the propagation delays of wireless signals in a DWRN. The current work revisits the analysis in [6] by looking into more general operation scenarios with finite system sensitivity and extended relay listening and transmitting periods. This can be achieved by using a double-directional description of the relay network, parallel to the double-directional characterization of the propagation channel [8]. Moreover, the DWRN is sampled at uniformly-spaced angles-of-departure (AODs) and anglesof-arrival (AOAs), leading to a discrete information azimuth spectrum (IAS). The problem of interest is thus to determine the parameters of the discrete IAS to achieve both accuracy with respect to the continuous benchmark model and coherence with respect to the resolution limit of the systems. Subsequently, the novel channel representation will be analyzed from a geometrically-based statistical modeling perspective [6], [9], which assumes certain geometric distributions of relay nodes and then derives the IAS by applying the fundamental information theory.
Provided with the network description preliminaries, the concept of relay network tomography (RNT) is introduced, which is applied to identify relay locations in an unknown DWRN. Probing signals are transmitted from several exterior sources to illuminate the service region, and relayed signals at several exterior destinations are measured. As a result, the spatial distribution of active relays is obtained from the received instantaneous IAS by solving an inverse problem, which provides critical insight into the performance bottlenecks in a network. Apparently, the basic principle of RNT bears a strong resemblance to other inverse problems, in which key aspects of a system are not directly observable. The term tomography is coined to link the problem of interest here, in concept, to other processes that infer the internal characteristics of an object from external observation, as done in medical tomography [10].
The rest of the paper is organized as follows. In Section II, we introduce the system model of a DWRN with realistic operation scenarios and formulate several principal quantities for description of such networks. Subsequently, the proposed channel representation under the framework of geometric network modeling is investigated in Section III, for which the relevant channel quantities are derived. In Section IV, we provide a general formulation of the RNT problem. Section V demonstrates the properties of the model parameters and the efficacy of the RNT methodology through several numerical examples. Finally, some concluding remarks are drawn in Section VI.
II. DOUBLE-DIRECTIONAL DESCRIPTION OF DWRN
A. System Model
We will consider a DWRN with L + 2 nodes: A source S, a destination D, and L non-collocated relays R 1 , R 2 , · · · , R L as illustrated in Fig. 1. The relays are randomly scattered in a given region R, which are due to the irregular nature of wireless sensor or ad hoc networks. In a sensor network, a large number of relays with sensing capabilities may be spread over the site under investigation in a distributive and randomized manner. On the other hand, each UT in an ad hoc network can serve as an assisting relay and the nonstationary network architecture may cause uncertainty to the relay positions [11]. Subsequently, the following assumptions on the transmission strategy are imposed.
1) Relay Distribution: The locations of R l (l = 1, 2, · · · , L) are described by a statistical density function f R (R l ) (R l ∈ R), which represents the ensemble of a large number of randomly distributed relays. 2) Relay Selection and Cooperation: The network geometry is unknown to either S or D. Thus, S broadcasts the same message to all R l and coherent relaying is not employed (i.e., without link combining and joint scheduling among relays). 3) Decode-and-forward Processing: Each R l receives information from S in the first hop (backward channel) and then forwards the decoded signal to D in the second hop (forward channel). Two orthogonal frequency tones are used at these two transmission phases, respectively. 4) Medium Access: Due to the decentralized and unregulated nature of relay positions, a simplified time-division protocol is applied, which requires no knowledge of the network geometry at either S or D. At the initial time t = t 0 , S transmits its signal to all R l (l = 1, 2, · · · , L).
For the lth relay, it listens to the transmission from S in the time interval t 0 , t 0 + ∆t
(B) l
. After processing the received signals from S, R l forwards the message to D within t 0 + ∆t
t 0 + k × ∆t (B) l + ∆t (F) l , t 0 + (k + 1) × ∆t (B) l + ∆t (F) l (k = 1, 2, · · · )
. For simplicity, each relay spends equal times in its listening and forwarding phases. Apparently, R l introduces an AOD Ω l and an AOA Ψ l during each transmission cycle as shown in Fig. 1. It is worth emphasizing that the time-of-arrival (TOA)
τ l = dS,R l +dR l ,D c
, which is one of the key quantities used in the multipath channels [8], has lost its physical meaning in the current context. Here d S,R l and d R l ,D are the distances of the links S → R l and R l → D, respectively, and c is the speed of electromagnetic waves. 5) Frequency-Tone Assignment: In general, mutual interference among the forward channels may incur due to the simultaneously transmitting relays. One way to eliminate the interference is to assign L frequency tones drawn from the pool of available bandwidth [ω min , ω max ] to L relay-to-destination pairs, all of which are also orthogonal to the carrier allocated to the backward channels.
B. Double-Directional Representation of DWRN
For a two-hop DWRN, multiple information pipelines are laid via the L links S → R 1 → D, · · · , S → R L → D. Furthermore, the intersection of each AOD and each AOA uniquely determines the location of a relay. Therefore, as an ideal reference model, the information flows in the network are continuously distributed in the AOD-AOA domain as
I D (Ω, Ψ) = L l=1 I l δ (Ω − Ω l ) δ (Ψ − Ψ l )(1)
where I D (Ω, Ψ) is defined as the double-directional IAS and I l is the outage capacity given an outage probability of P out for the link S → R l → D. The terminology double-directional is adopted according to [8].
Eq. (1) assumes perfect receivers that are able to resolve the departing and arriving information flows with infinite sensitivity in the angular domain. Therefore, the formulation represents an ideal benchmark model. In practice, both the source and destination have limited system sensitivity. Consider two data streams via two relays R l1 and R l2 (1≤l 1 =l 2 ≤L). Due to the finite length of the transmit (receive) antenna aperture, the AODs (AOAs) of these two information flows cannot be successfully distinguished if the separation of their departing (impinging) angles, |Ω l1 −Ω l2 | (|Ψ l1 −Ψ l2 |), is within the resolution limit of the source (destination) antenna, ∆Ω (∆Ψ). Therefore, the DWRN channel should be sampled at uniformly-spaced AODs and AOAs:
Ω i = i∆Ω, Ω min ∆Ω ≤i≤ Ω max ∆Ω (2) Ψ j = j∆Ψ, Ψ min ∆Ψ ≤j≤ Ψ max ∆Ψ(3)
where ⌊·⌋ and ⌈·⌉ are the floor and ceiling functions, respectively. Subsequently, the discrete double-directional IAS can be expressed as
I (d) D (Ω i , Ψ j ) = Ξi,j I D (Ω, Ψ)f (i,j) Ω,Ψ (Ω, Ψ)dΩdΨ, where Ξ i,j (Ω, Ψ) : Ω i − ∆Ω 2 ≤ Ω ≤ Ω i + ∆Ω 2 & Ψ j − ∆Ψ 2 ≤ Ψ ≤ Ψ j + ∆Ψ 2(4)
In (4), The superscript (d) denotes a discrete model. Fig. 1.
The local joint AOD-AOA pdf f (i,j) Ω,Ψ (Ω, Ψ) satisfies Ξi,j f (i,j) Ω,Ψ (Ω, Ψ)dΩdΨ = 1, where the integration domain Ξ i,j is illustrated in
III. GEOMETRICALLY-BASED STATISTICAL MODEL FOR A RANDOM DWRN
We will discuss the proposed analytical framework in Section II from a geometrically-based statistical modeling perspective [6], where a large number of relays are randomly located in the two-dimensional space according to a specified relay density function. This approach is useful when the spatial structure observed in a large DWRN is far from being regular [11]. The medium access arrangement follows the protocol in Section II-A. Moreover, all channels experience independent frequency-flat block fading with the complex channel gain between the link M → N, h M,N (M ∈ {S, R 1 , · · · , R L } , N ∈ {D, R 1 , · · · , R L }) following the Nakagami distribution with shape parameter m Subsequently, the average received signal-to-noise ratio (SNR) for each link is SNR = P BN0 , where P is the average transmit power for all transmitting terminals. Finally, the channel state information is assumed to be known at the receivers but unavailable to the transmitters [1].
Let the x-y coordinate system be defined such that the destination is at the origin and the source lies on the x axis, as shown in Fig. 2. It is assumed that there are many relay nodes, the locations of which are described by the statistical relay density function f x,y (x, y). By applying the law of sines to the triangle SRD in Fig. 2, the relay-to-destination distance d R,D can be expressed in terms of the AOD Ω and the AOA Ψ as
d R,D = d S,D sin Ω sin(Ω + Ψ)(5)
The source-to-relay distance d S,R is similarly derived as
d S,R = d S,D sin Ψ sin(Ω + Ψ)(6)
Proposition 1 (Derivation of f Ω,Ψ (Ω, Ψ)):
The joint AOD-AOA pdf of information flows within a DWRN is given by
where d S,D is the distance between the source and the destination.
Proof: The proof is omitted here for simplicity.
= 1 − 1 − γ m, m(4 I −1) SNR×d ν S,R Γ(m) 1 − γ m, m(4 I −1) SNR×d ν R,D Γ(m) (8)
when P (I) = P out . In (8), γ(α, β) = β 0 t α−1 e −t dt is the lower incomplete gamma function and Γ(·) is the gamma function.
Proof: The proof is omitted here for simplicity. Finally, the local joint AOD-AOA pdf in (4), f Ξi,j f Ω,Ψ (Ω, Ψ)dΩdΨ (9) Substituting (7)-(9) into (4) gives the discrete doubledirectional IAS.
IV. RNT FOR A DWRN
A. Formulation of RNT
In the general RNT problem depicted in Fig. 3, L relays (R 1 , · · · , R L ) are randomly scattered in a finite area R and Q measuring nodes (N 1 , · · · , N Q ) are placed exterior to R. Nevertheless, the following analysis is also applicable to any other measurement network orientation with respect to R. Each measuring node can serve as the probing source as well as the information destination, which has a transmit/receive azimuthal resolution of ∆Θ. As shown in where the superscript (1) denotes N 1 . All other nodes N 2 ∼ N Q record an information flow. Each receive node N q (q = 2, · · · , Q) scans its entire angular range
j∆Θ j = Ψ (q) min ∆Θ , Ψ (q) min ∆Θ + 1, · · · , Ψ (q) max ∆Θ
, where the superscript (q) denotes N q . At each j∆Θ, N q searches the whole frequency band within which the relays are operating. When N q locks on the lth frequency component ω l , which yields a detectable information flow associated with the relay R l , it then monitors the instantaneous channel capacity over an observation period and records the time-variant capacityÎ (t).
From the set ofÎ (t), the channel outage capacityÎ (1,q) l can be estimated empirically, where the superscript (1,q) denotes the information pathway from N 1 to N q . The similar process continues until all the active relays are identified. Then the second node N 2 sends the probing message and all the other nodes N 1 , N 3 , · · · , N Q receive the data, and so on.
A procedure to calculate the double-directional IAS for a DWRN with given sources, relays and destinations is called the direct problem. A procedure to obtain relay distribution knowing sources/destinations and measuring the information flows at multiple receivers is called the inverse problem. The Fig. 3. Formulation of RNT direct problem has already been discussed in Section III. The corresponding inverse problem can be formulated as follows.
RNT Problem: Given the set of measuring nodes are the empirical AOA and channel outage capacity associated with the information pipeline N q1 → R l → N q2 , respectively. Note that due to the channel reciprocity in a two-hop DWRN, it is expected that Ψ
N 1 , · · · , N Q , the estimated (Q − 1) 2 ×L single-directional AOA matrix Ψ = Ψ 1Ψ2 · · ·Ψ L = Ψ (1,2) 1Ψ (1,2) 2 · · ·Ψ(q1,q2) l =Ω (q2,q1) l andÎ (q1,q2) l =Î (q2,q1) l
. Therefore, (11) is equivalent to a double-directional AOD-AOA matrix, where half of the AOAs correspond to the AODs of the reversed information pathways. Identify the locations of R 1 , · · · , R L that satisfy the following two constraint functions
e 1 Ψ l − Ψ l ≤ ǫ 1 , l = 1, 2, · · · , L(12)
and e 2 Î l − I l ≤ ǫ 2 , l = 1, 2, · · · , L
Ψ l and I l are the AOA and channel outage capacity vectors solved in the direct problem for all the information paths via R l assuming a specific location of R l .
B. Algorithms to Solve The RNT Problem
In the current work, we will consider the ℓ 2norm of the two error vectors, i.e., e 1 Ψ l − Ψ l = q1 q2 =q1
Ψ (q1,q2) l − Ψ (q1,q2) l 2 and e 2 Î l − I l = q1 q2 =q1 Î (q1,q2) l − I (q1,q2) l 2 .
The objectives are to minimize the values of e 2 and meanwhile ensure that e 1 = 0 for all the detectable information flows. The optimization process can be realized through the following steps:
1) Discretize the solution domain of the relay locations R into W sufficiently small square cells, i.e., R = R (d) {x 1 , · · · , x W }, where x w = (x w , y w ) (w = 1, 2, · · · , W ) represents the Cartesian coordinates of the center of the wth cell;
2) For each identified information pipeline
N q1 → R l → N q2 , determine the subset of R, R (q1,q2) l ⊆ R, such that ∀x ∈ R (q1,q2) l ,Ψ (q1,q2) l = Ψ (q1,q2) l (x);
3) Determine the joint of all the sets obtained in Step 2 as
R l = R (1,2) l R (1,3) l · · · R (Q,Q−1) l ; 4) Estimate the location of R l as x (R l ) = arg min x∈R l q1 q2 =q1 Î (q1,q2) l − I (q1,q2) l (x) 2(14)
It is not always possible to obtain an accurate estimate of the outage capacityÎ (q1,q2) l due to the limited observation data at each destination. In such a case, the inverse problem could be approached from a hypothesis-testing perspective. An alternative optimization procedure to Step 4 can be formulated as follows.
4 ⋆ ) Let there be K relay-position hypotheses denoted H 1 : x(R l ) =x 1 ,H 2 : x(R l ) =x 2 , · · · ,H K : x(R l ) =x K , where K = |R l | and R l = {x 1 ,x 2 , · · · ,x K }. Furthermore, let ε k1,k2 be the desired probability of incorrectly selecting H k2 given that H k1 is true. The sequential probability ratio test procedures will be applied for statistical decision making, where the decision at each stage is based on the likelihood ratio. In particular, the multi-hypothesis sequential test is considered and summarized below [12]. The likelihood ratio including prior probabilities for a pair of hypotheses k 1 and k 2 after the oth data observation is
η o k1,k2 = f H k 1 (I 1 )f H k 1 (I 2 )· · ·f H k 1 (I o )Pr (H k1 ) f H k 2 (I 1 )f H k 2 (I 2 )· · ·f H k 2 (I o )Pr (H k2 )(15)
where I o = I H k1 , which is given by
f H k 1 (I o ) (a) = q1 q2 =q1 f I I (q1,q2) l,o ; x(R l ) =x k1 (b) = ln 4 × q1 q2 =q1 ρ 1 I (q1,q2) l,o ,x k1 m × exp −ρ 1 I (q1,q2) l,o ,x k1 × 1 − γ m, ρ 2 I (q1,q2) l,o ,x k1 Γ(m) + ρ 2 I (q1,q2) l,o ,x k1 m × exp −ρ 2 I (q1,q2) l,o ,x k1 × 1 − γ m, ρ 1 I (q1,q2) l,o ,x k1 Γ(m) (16)
where (a) follows the fact that the instantaneous outage capacities are sequentially observed for all the information paths via R l during the measurement process. Therefore, they can be assumed to be independently distributed. The equality in (b) is obtained by differentiating the cdf in (8) with respect to the variable I. In (16),
ρ 1 I (q1,q2) l,o ,x k1 = m 4 I (q 1 ,q 2 ) l,o − 1 SNR× [d S,R (x k1 )] ν and ρ 2 I (q1,q2) l,o ,x k1 = m 4 I (q 1 ,q 2 ) l,o − 1 SNR× [d R,D (x k1 )] ν
If the number of observations is not fixed, a decision on the relay-position state (i.e., Hk is selected) is made when the condition
η ô k,k2 > 1 − εk ,k2 εk ,k2 ∀k =k 2(17)
is met. The threshold in (17) to which the likelihood ratio is compared is designed to ensure that the average rate of making a wrong decision in favor of H k2 when H k1 is true is no more than ε k1,k2 . Nevertheless, the inequality in (17) may not always be satisfied due to the insufficient observation data.
In such a case, a hypotheses Hk is selected if its associated likelihood has the maximum value after a fixed number of O observations, i.e.,
k = arg max k f H k (I 1 )f H k (I 2 )· · ·f H k (I O )Pr (H k ) (18)
Eqs. (17) and (18) provide alternative objective functions to the one given in (14). We consider the high SNR regime (SNR = 30 dB) for a fixed outage probability P out = 0.01. The shape parameter of the Nakagami distribution is m = 1 (Rayleigh fading). The distance between the source and the destination d S,D is assumed to be 100 √ 3 m (see also Fig. 2) and the path loss exponent is ν = −3. The relay region is assumed to be a circle centered at O(50 √ 3 m, 50 m) with a radius r = 40 m. Finally, the spatial resolutions of both the source and destination antennas are 10 • . Fig. 4 illustrates the double-directional IAS for both the continuous reference model in (1) and its discrete counterpart in (4). Apparently, both I D (Ω, Ψ) and I (d) D (Ω, Ψ) decrease as the AOD Ω and the AOA Ψ increase. This certainly holds an intuitive appeal because larger values of Ω and Ψ lead to more severe path loss, thereby reducing the outage capacity of a relay path. As a practical concern, Fig. 4 provides an intuitive and quantitative answer to the following two important questions: how will the information flows be scattered in the AOD and AOA domains and what will be the beam-scanning ranges of the source and destination antennas in order to achieve sufficiently high outage capacities?
Next, we look into the RNT problem discussed in Section IV. A three-node measurement network is deployed as illustrated in Fig. 5. A number of relays are uniformly distributed within a circular region. The procedures presented in Section IV-B are applied to estimate the relay locations, where the objective function of (18) is used. The number of observations of the instantaneous channel capacity for each information pipeline is O = 10. Furthermore, no a priori knowledge of the probability of each location under hypothesis test is available. As can be observed from Fig. 5, most of the estimated relay locations agree well with the actual ones, which demonstrates the efficacy of the proposed RNT algorithm by using a relatively small number of measuring nodes and observations. VI. CONCLUSIONS We have presented a double-directional description method for analyzing two-hop DWRN channels, which is based on an analogous approach to the double-directional physical wireless channels. The proposed model has been investigated using a geometrically-based statistical modeling approach with uniformly distributed relay nodes. We have also investigated the RNT problem by applying the proposed network representation. Finally, numerical examples have been used to demonstrate the practical significance of the proposed analytical framework.
Fig. 1 .
1Double-directional DWRN channel in the backward and forward phases, respectively. The same time-division process repeats in the subsequent time slots
[8]. The associated instantaneous power |h M,N | 2 is gammadistributed with the same shape parameter, i.e., |h M,N | 2 ∼ G m,λM,N m , where G(κ, θ) represents the gamma distribution with scale θ and shape κ. The mean value λ M,N = d ν M,N with ν being the path loss exponent. The ambient noise at the relays and the destination, Z N ∼ N (0, N 0 ), where N (µ, σ 2 )
Fig. 2 .
2Geometrically-based statistical model for a DWRN represents a normal distribution with mean µ and variance σ 2 . The communication bandwidth is B for any channel.
Proposition 2 (
2Derivation of I D (Ω, Ψ)): The double-directional IAS I D (Ω, Ψ) is the argument of the function P (I)
Fig. 3 ,
3the first node N 1 transmits over its full
estimated (Q − 1) 2 ×L outage capacity matrix I = Î 1Î2 · · ·Î L
o , · · · , I (Q,Q−1) l,o T is the instantaneous outage capacity vector at the oth observation for all the information routes via R l . f H k 1 (I o ) is the pdf of I o under
Fig. 4 .Fig. 5 .
45Double-directional IAS: (a) continuous reference model and (b) discrete model for systems with an angular resolution of 10 • Estimation of relay locations using the proposed RNT algorithm V. NUMERICAL EXAMPLES
Cooperative diversity in wireless networks: Efficient protocols and outage behavior. J N Laneman, D N C Tse, G W Wornell, IEEE Trans. Inform. Theory. 5012J. N. Laneman, D. N. C. Tse, and G. W. Wornell, "Cooperative diversity in wireless networks: Efficient protocols and outage behavior," IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3062-3080, 2004.
Achievable rates and scaling laws of power-constrained wireless sensory relay networks. B Wang, J Zhang, L Zheng, IEEE Trans. Inform. Theory. 529B. Wang, J. Zhang, and L. Zheng, "Achievable rates and scaling laws of power-constrained wireless sensory relay networks," IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 4084-4104, 2006.
On the power efficiency of cooperative broadcast in dense wireless networks. B Sirkeci-Mergen, A Scaglione, IEEE J. Select. Areas Commun. 252B. Sirkeci-Mergen and A. Scaglione, "On the power efficiency of cooperative broadcast in dense wireless networks," IEEE J. Select. Areas Commun., vol. 25, no. 2, pp. 497-507, 2007.
Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks. A Özgür, O Lévêque, D N C Tse, IEEE Trans. Inform. Theory. 53A.Özgür, O. Lévêque, and D. N. C. Tse, "Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks," IEEE Trans. Inform. Theory, vol. 53, pp. 3549-3572, Oct. 2007.
Decentralized two-hop opportunistic relaying with limited channel state information. S Cui, A M Haimovich, O Somekh, H V Poor, Proc. IEEE International Symposium on Information Theory. IEEE International Symposium on Information TheoryTorontoS. Cui, A. M. Haimovich, O. Somekh, and H. V. Poor, "Decentralized two-hop opportunistic relaying with limited channel state information," in Proc. IEEE International Symposium on Information Theory, Toronto, July 2008.
Decentralized wireless relay network channel modeling: An analogous approach to mobile radio channel characterization. Y Chen, P Rapajic, IEEE Trans. Commun. 58Y. Chen and P. Rapajic, "Decentralized wireless relay network channel modeling: An analogous approach to mobile radio channel characteri- zation," IEEE Trans. Commun., vol. 58, pp. 467-473, Feb. 2010.
A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments. K I Pedersen, P E Mogensen, B H Fleury, IEEE Trans. Veh. Technol. 492K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, "A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments," IEEE Trans. Veh. Technol., vol. 49, no. 2, pp. 437-447, 2000.
A F Molisch, Wireless Communications. Chichester, West SussexJohn Wiley and SonsA. F. Molisch, Wireless Communications, John Wiley and Sons, Chichester, West Sussex, 2005.
Angle and time of arrival statistics for circular and elliptical scattering models. R B Ertel, J H Reed, IEEE J. Sel. Areas Commun. 1711R. B. Ertel and J. H. Reed, "Angle and time of arrival statistics for circular and elliptical scattering models," IEEE J. Sel. Areas Commun., vol. 17, no. 11, pp. 1829-1840, 1999.
A C Kak, M Slaney, Principles of Computerized Tomographic Imaging. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Society of Industrial and Applied Mathematics, 2001.
Rethinking information theory for mobile ad hoc networks. J G Andrews, N Jindal, M Haenggi, R Berry, S Jafar, D Guo, S Shakkottai, R Heath, M Neely, S Weber, A Yener, IEEE Commun. Mag. 46J. G. Andrews, N. Jindal, M. Haenggi, R. Berry, S. Jafar, D. Guo, S. Shakkottai, R. Heath, M. Neely, S. Weber, and A. Yener, "Rethinking information theory for mobile ad hoc networks," IEEE Commun. Mag., vol. 46, pp. 94-101, Dec. 2008.
Adaptive waveform design and sequential hypothesis testing for target recognition with active sensors. N A Goodman, P R Venkata, M A Neifeld, IEEE Journal of Selected Topics in Signal Processing. 11N. A. Goodman, P. R. Venkata, and M. A. Neifeld, "Adaptive waveform design and sequential hypothesis testing for target recognition with active sensors," IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 1, pp. 105-113, 2007.
| []
|
[
"Genetic Algorithm for Chromaticity Correction in Diffraction Limited Storage Rings",
"Genetic Algorithm for Chromaticity Correction in Diffraction Limited Storage Rings"
]
| [
"M P Ehrlichman \nPaul Scherrer Institut\nVilligenSwitzerland\n"
]
| [
"Paul Scherrer Institut\nVilligenSwitzerland"
]
| []
| A multi-objective genetic algorithm is developed for optimizing nonlinearities in diffraction limited storage rings. This algorithm determines sextupole and octupole strengths for chromaticity correction that deliver optimized dynamic aperture and beam lifetime. The algorithm makes use of dominance constraints to breed desirable properties into the early generations. The momentum aperture is optimized indirectly by constraining the chromatic tune footprint and optimizing the off-energy dynamic aperture. The result is an effective and computationally efficient technique for correcting chromaticity in a storage ring while maintaining optimal dynamic aperture and beam lifetime. This framework was developed for the Swiss Light Source (SLS) upgrade project[1]. | 10.1103/physrevaccelbeams.19.044001 | [
"https://arxiv.org/pdf/1603.02459v2.pdf"
]
| 118,472,148 | 1603.02459 | b99b132796fcbf922610b8b9ce630ee55aaa691f |
Genetic Algorithm for Chromaticity Correction in Diffraction Limited Storage Rings
15 Apr 2016
M P Ehrlichman
Paul Scherrer Institut
VilligenSwitzerland
Genetic Algorithm for Chromaticity Correction in Diffraction Limited Storage Rings
15 Apr 2016(Dated: April 18, 2016)arXiv:1603.02459v2 [physics.acc-ph]
A multi-objective genetic algorithm is developed for optimizing nonlinearities in diffraction limited storage rings. This algorithm determines sextupole and octupole strengths for chromaticity correction that deliver optimized dynamic aperture and beam lifetime. The algorithm makes use of dominance constraints to breed desirable properties into the early generations. The momentum aperture is optimized indirectly by constraining the chromatic tune footprint and optimizing the off-energy dynamic aperture. The result is an effective and computationally efficient technique for correcting chromaticity in a storage ring while maintaining optimal dynamic aperture and beam lifetime. This framework was developed for the Swiss Light Source (SLS) upgrade project[1].
I. INTRODUCTION
Multi-objective genetic algorithms have found much success providing non-intuitive solutions to problems that are not adequately solved by analytic methods. Such algorithms have been successfully applied to various aspects of accelerator design [2], [3], [4], [5], and [6]. In this paper, a genetic algorithm is developed for optimizing dynamic aperture and beam lifetime using sextupole and octupole strengths in next-generation diffraction limited storage rings (DLSR).
The 1-turn map for a DLSR contains strong nonlinearities. Such machines use stronger focusing and lower dispersion to achieve an emittance that is below the diffraction limit for some wavelengths of light. The strong focusing of these machines leads to a large chromaticity, which must be corrected by placing strong sextupoles in dispersive regions. Furthermore, the low dispersion necessitates that the sextupole strengths be even higher. Such strong sextupole moments add strong nonlinearities to the lattice.
These nonlinearities must be carefully designed to maintain adequate injection efficiency and beam lifetime. Nonlinearities induce tune shifts and increase the sensitivity of particles to machine imperfections. This reduces the dynamic aperture, which is the volume in 6D phasespace containing stable particle trajectories. Touschek scattering and residual gas scattering excite the phase space coordinates of particles stored in a ring. A particle is lost if it exits the dynamic aperture. This imposes a beam lifetime. A reduced dynamic aperture also complicates injection by reducing the capture efficiency of the machine. The problem of nonlinear optics in a storage ring is to correct the chromaticity without inducing nonlinearities that degrade the other lattice properties too much.
The established technique for optimizing nonlinearities in a storage ring is resonant driving term minimization and was developed for the original SLS [7]. It is a Lie algebra expansion of the transfer map in driving terms * [email protected] that are functions of the sextupole strength. These terms drive higher order chromatic tune shifts, amplitude dependent tune shifts, and resonances. A gradient optimizer is used to minimize a weighted vector of the driving terms. The weights are determined by judging from the tune diagram and frequency maps which resonances and tune shifts are most important. This technique is straightforward to wield up to second order in sextupole strength. Beyond second order, the number of driving terms which need to be minimized makes the method less robust. The complexity of the optimization space necessitates much trial and error to locate good local minima, and the procedure becomes a bit of a dark art.
A storage ring is a complicated nonlinear oscillator. Its transfer map is the concatenation of many hundreds of linear and nonlinear maps. Explicit algorithms yield only an incomplete control of particle motion in a storage ring. This lack of analytic clarity makes accelerators good candidates for genetic algorithms, which typically do not depend on knowledge of the system. Genetic algorithms optimize a system by selectively breeding trial solutions according to their fitness.
One example where a genetic algorithm performs better than perturbation techniques is in confining the chromatic and amplitude-dependent tune shifts. Second order perturbation theory is unable to bend the chromatic and amplitude dependent tune shifts beyond second order. The genetic algorithm developed here fits tune footprints into tighter areas using higher orders of correction.
A well-designed genetic algorithm will encourage behaviors in the evolving population that, further down the road, lead to solutions with optimal objective functions. For example, the algorithm presented here includes a dominance constraint (See Sec. III C) on the amplitude of the nonlinear dispersion. A small-amplitude nonlinear dispersion is not one of the objective functions, but it is a property that a lattice with good objective functions will have. By applying a dominance constraint to the nonlinear dispersion, we are, in a sense, breeding characteristics into the early generations, that in later generations will yield improved objective functions.
Genetic algorithms for the optimization of storage ring nonlinearities have been developed and evaluated elsewhere [4], [8], [9], [10], and [11]. The present applica-tion stands out in its use of dominance constraints to more efficiently evolve the population. The algorithm requires only modest computing resources. On a Linux cluster consisting of 64 E5-2670 Xeon cores, it delivers solutions to 10 variable problems in one or two days, and 20 variable problems in two or three days. The scheme consistently does as well or better than Lie algebra approaches, and repeated attempts with different random seeds on different lattice variants suggest that the solutions it finds are globally optimal. So the optimization scheme presented here allows for a lattice development cycle on the order of a couple days, and does not consume expensive computing resources.
Section II of this paper gives an overview of the system, listing the components out of which this optimization scheme is built. Section III introduces multi-objective genetic algorithms, and Sec. IV describes the optimization problem, including the physics behind the calculations of the objectives and constraints. In Sec. V the optimization scheme is applied to upgrade prototypes of SLS and the proposed Armenian light source CANDLE [12]. Misalignment studies are conducted on the SLS upgrade lattice in Sec. VI.
II. SYSTEM ARCHITECTURE
The optimizer is built within the PISA framework [13], which specifies that the sorting algorithm be separated from the rest of the optimizer. The sorting algorithm is implemented as a stand-alone binary that communicates with the rest of the optimizer using a text-file based API. This separation simplifies the coding and makes it trivial to switch between different sorting algorithms, such as SPEA2 [14] or NSGA2 [15].
We use the aPISA variant [6] of PISA. aPISA modifies the original PISA framework by supporting dominance constraints. aPISA was originally developed for the design of the Cornell ERL injector.
Accelerator physics calculations are handled by calls to the Bmad library [16]. The top-level coding, which includes population management and breeding, parallelization, and additional physics calculations, was developed at PSI and is coded in Fortran90. The parallelization paradigm is master-slave and is implemented using Coarrays, which in Intel's Fortran compiler is implemented as a high-level language on top of MPI.
The cluster management software is Sun Grid Engine (SGE). The cluster is composed of several 16-core E5-2670 compute nodes running 64-bit Scientific Linux. Typically 4 of these nodes are used in an optimization run.
III. MULTI-OBJECTIVE GENETIC ALGORITHMS
The multi-objective optimization problem is formulated as [17]:
Minimize f m (x) , m = 1, 2, ..., M ; subject to g j (x) ≥ 0, j = 1, 2, ..., J; h k (x) = 0, k = 1, 2, ..., K; x (L) i ≤ x i ≤ x (U) i i = 1, 2, ..., n.
(1) f m are the objectives, which generally are competing. g j are inequality constraints and h k are equality constraints.
x (L) i and x (U) i are upper and lower bounds on variables. A vector of variable strengths x = (x 1 , x 2 , ..., x n ) T is called an individual.
A genetic algorithm manages a population of individuals. Every individual i in the population is represented by a vector of variables x (i) . For our purposes x (i) is realvalued, but in general it could contain integer, logical, and complex variables. Each individual i has an associated vector of objective values f (i) and constraint values g (i) and h (i) .
The output of a multi-objective optimizer is of a different nature than that of a single-objective optimizer. A single objective optimizer, or equivalently, an optimizer which reduces f (i) to a single value by weighting the individual objectives, gives the user one particular x that is the best solution to the optimization problem it could find. A multi-objective optimizer, on the other hand, returns a population of x's. This returned population is an optimal surface in the objective space called a Pareto front. For any individual on the Pareto front, no improvement to any one of its objectives can be achieved without worsening the others. The user of a multi-objective optimizer typically applies additional criteria when selecting a particular solution from the Pareto front.
A. Ranking the Population
The ranking of individuals in a single objective optimization problem is straightforward: the individual with the better objective value is preferred. Ranking in multi-objective optimization problem is more complicated. The core concept is the dominance relationship, which is a way of comparing any two individuals, say x (i) and x (j) . They are compared by asking the question: "Does x (i) dominate x (j) ?" The dominance relationship is defined as [17]:
Definition 1 An individual x (i) is said to dominate an- other x (j)
, if both of the following conditions are true:
1. x (i) is no worse than x (j) in all objectives.
2. x (i) is strictly better than x (j) in at least one objective.
A sorting algorithm applies the dominance relationship to sort the population from best to worst. There exist many different sorting algorithms. Two algorithms that we have used are NSGA2 and SPEA2. We obtain similar results with both algorithms, but find that the populations resulting from SPEA2 span the objective space more evenly.
For details on SPEA2 see Ref. [14]. In short summary, dominance is determined for every ordered pair of individuals in the population. Each individual in the population is assigned a strength which is the number of individuals it dominates. Then, each individual is assigned a fitness which is the sum of the strengths of all the individuals that dominate it. A lower fitness is better. A 'clumping' penalty is added to this fitness based on the shortest distance (in objective space) of the individual to another individual. This encourages the population to span a wider region of the objective space. Incidentally, the clumping penalty makes it unlikely for any two individuals to have the exact same fitness, even if they are both not dominated by any other individual.
Individuals are ordered according to their fitness value. The lowest ranked individuals are deleted from the population. This is typically the worst half or three-quarters of the population each generation. The population is replenished by mating the surviving individuals.
Mating pairs are determined by drawing integers in a procedure called a tournament. Two or more random integers are drawn between 1 and N , where N is the number of surviving individuals. 1 corresponds to the most fit, and N to the least fit. The individual corresponding to the smallest of the drawn integers is chosen for mating. Thus, individuals with better fitness are more likely to reproduce. Its mate is chosen through the same process. This is repeated until enough pairs have been selected to replenish the population. Each pair generates two offspring.
A two-step process generates two new children from each mating pair.
The first step is simulated binary cross-over [18].
Say the two parents are
x (p1) = x (p1) 1 , x (p1) 2 , ..., x (p1) n and x (p2) = x (p2) 1 , x (p2) 2 , ..., x (p2) n .
They will produce two offspring, x (c1) and x (c2) .
Start with variable x (·) 1 and draw a random real t between 0 and 1. Compare t to P c , which is a parameter between 0 and 1 that determines how likely it is that a variable is simply copied from parent to child, as opposed to applying a stochastic function. If t > P c , then variable x (p1) 1 is simply copied to the child x (c1)
1 , and x (p2) 1 copied to x (c2)
1 . If t ≤ P c , then draw another random real q between 0 and 1. Then,
β q = (2q) κ if q ≤ 0.5 0.5 1 − q κ otherwise,(2)= 1 2 (1 + β q ) x (p1) 1 + (1 − β q ) x (p2) 1 (3) x (c2) 1 = 1 2 (1 − β q ) x (p1) 1 + (1 + β q ) x (p2) 1 ,(4)
where κ is a parameter that controls the width of the distribution. As depicted in Fig. 1, smaller values of κ cause the offspring to explore a broader parameter space. A typical value for P c is 0.8. This process is repeated for all n variables. Next a mutator is applied separately to x (c1) and x (c2) . For each variable i in x (c1) , draw a random real t between 0 and 1. Compare t to P m , a parameter that determines how likely it is for a variable to undergo mutation. If t > P m , the variable is not mutated. If t ≤ P m , then draw a second random real m. Variable x (c1) i is adjusted according to
β m = (2m) κ − 1 if m ≤ 0.5 1 − (2 (1 − m)) κ otherwise,(5)
and
x (c1) i = x (c1) i + σ mut × β m .(6)
A typical value for P m is n −1 , so that on average one variable is mutated per child. σ mut is typically set to about 10% of the reasonable variable strength. β m is depicted in Fig. 2.
After x (c1) , the mutation process is repeated on each variable in x (c2) and both are added to the population. New individuals are generated until the population is fully replenished.
B. Progression of Generations
Once the population is fully replenished, the worker processes calculate the objectives and constraints of the newly generated individuals. The population, consisting of both the parents and children, is resorted and the least fit are deleted. Some of the children will be more fit and displace the older individuals in the surviving population. So is the overall fitness of the population improved. This process of evaluation, sorting, deletion, and replenishment is looped continually. Each loop is referred to as a generation.
As implemented, there is not actually a clear line between one generation and the next. The algorithm is modified to improve computational efficiency. By far, the most time-consuming step is when the objectives and constraints of an individual are calculated. It could often be the case that many nodes in the cluster sit idle while the last few individuals of a given generation are being evaluated. To avoid this situation, the optimizer initially generates extra individuals equal to the number of cores in the cluster. For example, if the population is set to 300 and there are 64 nodes in the cluster, the optimizer will initially generate 364 individuals. Whenever a core finishes evaluating an individual, the individual is added to the population and the core is always immediately given a new unevaluated individual to process. Whenever the population reaches 300, it is resorted, culled, and new individuals are bred. These new individuals replenish the pool of individuals awaiting evaluation. This modification to the algorithm improves cluster efficiency. The loss of a clear demarcation between the generations does not seem to negatively impact the evolution of the population.
C. Dominance Constraints
Dominance constraints are a powerful type of constraint that is unique to multi-objective optimization. It is implemented by modifying the dominance relationship. We apply the dominance relationship as implemented in aPISA [6].
In addition to calculating objective values f x (i) for each individual i, we also calculate a vector of constraint values c x (i) . For example, say c 1 is a dominance constraint for the off-momentum horizontal closed orbit x inj at the injection point, and we wish to constrain |x inj | to be less than x max . Then, c 1 = x max − |x inj |. If c 1 is negative, it indicates that the constraint is violated. The magnitude of c 1 represents the degree to which it is violated. Dominance constraints are implemented by replacing the ordinary definition for dominance in Def. 1 with Def. 2. An individual is called infeasible if it violates any of its dominance constraints, else it is called feasible.
Definition 2 An individual x (i) is said to constraint- dominate an individual x (j)
if any of the following conditions are true:
1. x (i) is feasible and x (j) is not. 2. x (i) and x (j) are both feasible, and x (i) dominates x (j) as in Def. 1.
3. x (i) and x (j) are both infeasible, and both
(a) x (i) is no worse than x (j) in all constraints. (b) x (i) is strictly better than x (j) in at least one constraint.
The behavior of a genetic algorithm implementing dominance constraints flows through three phases:
1. Random population, possibly containing no feasible individuals, spans variable space, sorted according to severity of constraint violations.
2. Entire population is feasible and spans feasible region of variable space, sorted according to objective values.
3. Population spans variable space that approaches the Pareto optimal objective space.
Notice that if an individual violates any of its dominance constraints, then its objective values are not taken into account during ranking. Therefore, to save computing time, objective values are calculated only for individuals which do not violate any dominance constraints. Dominance constraints are based on variable bounds, closed orbit amplitudes, and chromatic tune shifts. These quantities are orders of magnitude quicker to evaluate than objective values, which are based on particle tracking.
IV. EVALUATION OF INDIVIDUALS
The design problem is to correct the chromaticity while maintaining acceptable dynamic aperture and beam lifetime. The beam lifetime would be maximized by maximizing the momentum aperture, which is the largest momentum kick that an initially on-axis particle can receive The aperture is found using a binary search for particle loss.
In this plot N angle = 7.
without being lost downstream. The momentum aperture can vary throughout the lattice and is typically calculated element-by-element or in fixed steps. It is computationally expensive to calculate, so instead we optimize the off-energy dynamic aperture and apply a dominance constraint to the chromatic tune footprint. The results in Sec. V show that this is an effective proxy for the momentum aperture.
A. Objectives
The dynamic apertures are calculated relative to the linear aperture. The linear aperture is the smallest aperture found by projecting the beam chamber from each point in the lattice to the injection point using the linearization of the map about the particle momentum. The linear aperture, in general, depends on the particle momentum. The on-energy linear aperture does not depend on the sextupole and multipole strengths, but the offenergy linear aperture does. The objective function is formulated relative to the linear aperture such that 1.0 is perfectly bad and 0.0 is perfectly good,
f (x) = 1 N angle N angle L l (x) − L da (x) L l (x) 2 , if L da < L l 0, otherwise,(7)
where N angle is the number of rays along which the aperture is calculated. L l is the length of the linear aperture ray. L da is the length of the dynamic aperture ray. L l and L da are depicted in Fig. 3. The conditional in Eq. 7 reflects our design philosophy that a machine with optimized nonlinearities should behave as if it were linear. The objective function is not rewarded when the dynamic aperture exceeds the linear aperture.
Three objectives are used in our multi-objective optimization problem:
f 0 (x) = on energy dynamic aperture (8) f + (x) = dynamic aperture at ∆E +DA (9) f − (x) = dynamic aperture at ∆E -DA ,(10)
where ∆E +DA and ∆E -DA specify the energy offset where dynamic aperture is evaluated. Typical energy offsets are 2% or 3%.
B. Constraints
Three constraint techniques are used: dominance constraints, modified objective functions, and variable space projection.
Five constraints are implemented as dominance constraints. They are 1. c mag : Boundary on sextupole and multipole strength.
2. c +co : Global bound on nonlinear dispersion at ∆E +DA .
3. c −co : Global bound on nonlinear dispersion at ∆E -DA .
4. c +χ : Confine chromatic tune footprint between 0 and ∆E +χ .
5. c −χ : Confine chromatic tune footprint between 0 and ∆E −χ .
The constraint on sextupole and multipole strength is calculated as,
c mag = magnets i x (U) i − K i x (U) + x (L) , K i > x (U) i K i − x (L) i x (U) + x (L) , K i < x (L) i 0, otherwise,(11)
where K i is the strength of magnet i, and x
c ±co = 1 x (co) x (co) − max i∈all elements abs (x co,i − η i × ∆E ±DA ) ,(12)
where x (co) is the maximum allowed closed orbit, usually set to millimeter or so. x co,i and η i are the closed orbit and ordinary dispersion at element i.
The positive chromatic footprint between 0 and ∆E +χ is constrained to cross neither the half-integer nor the integer resonances. It is calculated by dividing the interval from ∆E = 0 to ∆E = ∆E +χ into N χ equal segments. The horizontal and vertical tunes are calculated by linearizing the optics about each ∆E j ∈ {∆E 1 , ∆E 2 , ..., ∆E +χ }. The smallest of these energy offsets that results in an unstable transfer matrix, or a horizontal or vertical tune that crosses an integer or halfinteger resonance, is used to calculate the value of the dominance constraint. The value is calculated as,
c +χ = −1 + j − 1 N χ .(13)
If all ∆E +j are stable and do not cross the half-integer or integer, c +χ is set to 0. A similar procedure is used to constrain the negative chromatic tune footprint c −χ . Two constraints are implemented by modifying the objective functions. The first is a constraint on the minimum size of the off-energy linear aperture and it modifies the off-energy objective functions f + and f − . The constraint prevents a failure condition where the optimizer improves the off-energy objectives by making the linear aperture tiny, rather than by growing the dynamic aperture. Setting this constraint to 2 or 3 mm is usually sufficient to avoid the condition. If the shortest linear aperture ray at ∆E +DA or ∆E -DA is shorter than this constraint, then f + or f − is set to a perfectly bad value of 1.
The second constraint implemented by modifying the objective functions modifies the on-energy objective function f 0 . It confines the on-energy amplitude dependent tune shift (ADTS). Along the two DA rays closest to the horizontal axis, the horizontal and vertical tunes are calculated. Along the vertical ray, the vertical tune is calculated. This is because the horizontal motion for large vertical and small horizontal offsets is dominated by coupling, rather than by the horizontal optics. The tunes are calculated by summing the element-by-element phase advance in normal mode coordinates. If the tunes of the particle cross the half-integer or integer resonances, then the tracking code considers the particle lost. The apertures along the two horizontal rays and one vertical ray define a 'clipping' box. When f 0 is calculated, all DA rays are clipped at the box.
The chromaticity correction is applied by projecting the N cs -dimensional space of chromatic sextupole strengths onto the (N cs − 2)-dimensional surface on which the horizontal and vertical chromaticities have the desired values. This is possible because chromaticity depends linearly on the sextupole strength. N cs is the number of chromatic sextupole families in the lattice. A chromaticity response matrix A is determined numerically,
A = dχx dK1 dχx dK2 · · · dχx dKNcs dχy dK1 dχy dK2 · · · dχy dKNcs .(14)
The Moore-Penrose pseudoinverse A p of A is calculated via singular value decomposition (SVD) [19]. Then the thin-QR decomposition [19] Q 1 of I − A p A is taken, where I is the N cs × N cs identity matrix. Note that Q 1 ∈ R Ncs×Ncs−2 . With A p and Q 1 in hand, take any vector ω ∈ R N cs−2 . The chromatic sextupole strengths K given by
K = A p χ x0 χ y0 + Q 1 ω(15)
result in chromaticities of χ x0 and χ y0 . The algorithm does not operate directly on the chromatic sextupole strengths. Instead it operates on ω, thus constraining the chromaticities to the desired values and reducing the dimension of the variable space by 2.
V. APPLICATION
Here the genetic algorithm is applied to a prototype lattice for the SLS upgrade, and also the proposed Armenian light source CANDLE.
The SLS upgrade is a 2.4 GeV storage ring built of 12 arcs which consist of 5 longitudinal gradient bends (LGB) plus 2 half-bend longitudinal gradient dispersion suppressors. There are 3 types of straight (short, medium, and long) which reduce the periodicity to 3. The lattice uses anti-bends to focus the dispersion into the LGBs to minimize the radiation integral I 5 [20]. The lattice parameters are summarized in Tab. I. For the SLS upgrade lattice, the objective functions are the on-energy dynamic aperture and the dynamic aperture at −3% and +3%. The chromatic footprint between −5% and +5% is constrained such that 37.0 < Q x < 37.5 and 10.0 < Q x < 10.5. The amplitude-dependent tune shift as described in Sec. IV is constrained to this same region.
For CANDLE, the objective functions are the onenergy dynamic aperture and the dynamic aperture at −2% and +2%. The chromatic footprint between −3% and +3% is constrained such that 24.5 < Q x < 25.0 and 14.0 < Q x < 14.5. The amplitude-dependent tune shift as described in Sec. IV is constrained to these same regions.
The optimization parameters for both lattices are summarized in Tab A. SLS Upgrade Lattice
Layout and Linear Lattice Considerations
The following were taken into consideration during the design of the layout and linear optics of the SLS upgrade lattice in order to improve the nonlinearities.
The ADTS is suppressed if the horizontal tune is close to
Q opt = 2q + 1 2 N,(16)
where N is the periodicity of the machine and q is some integer [22,Chapter 14.3.1]. The periodicity of SLS is 3. Thus ADTS is reduced by selecting a horizontal tune near one of the following: {..., 34.5, 37.5, 40.5, ...}.
2. Chromatic sextupoles are placed where dispersion is large, and where either β x << β y or β x >> β y . Harmonic sextupoles are placed in dispersion-free regions where β x >> β y , β x << β y , or β x ≈ β y .
3. The arcs are constructed from five identical unit cells. The horizontal and vertical phase advances per unit cell are 0.4 and 0.1 radians, respectively. Over the five unit cells, the lowest order chromatic and geometric resonant driving terms are canceled out [7].
Optimization
The population size for the SLS upgrade optimization is 300 and begins with a pool of 364 unevaluated, randomly generated, individuals. Each individual is described by 21 variables representing the 23 magnet strengths. The strengths are bounded by c mag,sext. and c mag,oct. , as given in Tab. II. Each seed is farmed via MPI to a CPU which evaluates its constraint and objective values. As soon as 300 individuals have been evaluated, the population is sorted and the 150 least fit are deleted. A four-way tournament is used to select parent pairs from the surviving population. From each pair, simulated binary crossover plus mutation is used to generate two new children. The new children are added to the pool of unevaluated individuals, and the process repeats.
The initial random population contains no individuals which satisfy the dominance constraints (i.e. all individuals are infeasible). Therefore the population members initially compete for who has the least-bad constraint violations. For the particular optimizer run shown here, the first feasible individual appears at generation 21, and at generation 41 the population consists entirely of individuals which satisfy all of the dominance constraints. From generation 1 to 20 requires 2 minutes, and from generation 20 to 41 requires 14 minutes. This first stage of the optimization proceeds quickly because infeasible individuals are evaluated only for off-momentum closed orbit amplitudes and tune footprints. During the remainder of the optimization run, individuals are competing based on dynamic aperture. The entire optimization completes after 45 hours to converge at generation 616. Computing resources are 64 E5-2670 Xeon cores. These compute time requirements are generally indicative of the resources required by the algorithm on the SLS upgrade.
Following the optimization run, the seeds with the most promising objective function values are selected by hand for further evaluation. Further evaluation includes on-and off-energy rastered survival plots, higher resolution chromatic and ADTS tune footprints, momentum aperture, and Touschek lifetime evaluation. Using these additional evaluations, the lattice designer selects a best individual.
The dynamic aperture of this individual at 0%, −3%, and +3% is shown in Fig. 4. Figure 5 compares the momentum aperture and linear momentum aperture. The linear momentum aperture is calculated by linearizing the one-turn map at each element. The reference Touschek lifetime is calculated from the linear RF bucket height, and is taken as the benchmark against which to judge the effectiveness of the algorithm.
The assumptions used for the Touschek lifetime calculation are shown in Tab. III. The lifetime exceeds the reference lifetime because the nonlinearity of the longitudinal phase space causes it to exceed the dimensions of the linear RF bucket, and the momentum aperture is not otherwise limited by the transverse nonlinearities.
Recall that the genetic algorithm does not directly optimize the Touschek lifetime nor momentum aperture. Rather, it constraints the chromatic and amplitudedependent tune footprints and maximizes the dynamic aperture area at 0% and ±3%. The element-by-element variation in the momentum aperture is small. This indicates that the Touschek lifetime is limited by the longitudinal dynamics, and not by nonlinearities in the transverse optics. Judging by this result, an off-momentum dynamic aperture optimization plus tune footprint constraint is a valid proxy for optimizing the Touschek lifetime and momentum aperture. Fig. 6 are x-p x and y-p y phase space portraits for the optimized SLS upgrade lattice. The portraits are calculated using 4D tracking for 100 turns. The lack of large resonance islands and lack of thick chaotic layers Momentum aperture of SLS upgrade. Linear aperture is calculated using an element-by-element linearization of the optics. The 6D aperture is calculated with full 6D tracking including radiation damping and synchrotron oscillation for 2.5 periods of the linear synchrotron tune. The asymmetry of the 6D momentum aperture is the influence of nonlinear momentum compaction. The RF voltage is set such that the linear calculation gives a ±5% bucket. inside the stable region is a positive result that should contribute to the robustness of the solution when misalignments are added. Figure 7 shows the chromatic tune footprint and ADTS along ±x. The chromatic tunes are calculated by linearizing the off-energy optics. The ADTS is calculated by summing element-by-element phase advances in normal mode coordinates. The ADTSs along +x and −x mostly overlap.
Shown in
B. CANDLE
CANDLE is a proposed 3 GeV, 216 m Armenian light source project [21] providing 8.54 nm horizontal beam emittance. According to the recent developments in storage ring lattice design and magnet technologies a new upgrade prototype has been designed [12], which is constructed of sixteen 4BA cells and provides 1.1 nm horizontal beam emittance. The study shown here is on this new 1.1 nm prototype. Each cell is composed of combined function bends with both quadrupole and sextupole moments. Some properties are shown in Tab. I. Two features which contribute to CANDLE's nonlinearities are: 1) The sextupole moments are spread out across a broad phase advance. 2) All sextupole moments are in dispersive regions. The dynamic aperture on-energy and at ±2% are shown in Fig. 8. The horizontal ADTS and chromatic footprint out to ±3% are shown in Fig. 9.
Touschek lifetime results and assumptions are shown in Tab. III. The momentum aperture calculated from 6D tracking is shown in Fig. 10. The aperture is determined entirely by the RF bucket and not limited by the transverse optics. The phase space portraits are shown in Fig. 11. Optimizing the ±2% dynamic aperture and con- straining the chromatic tune footprint to ±3% has successfully optimized the global momentum aperture to at least ±3%.
VI. TOLERANCE TO MACHINE MISALIGNMENTS
The tolerance of the optimized SLS upgrade lattice to machine misalignments is tested. The genetic algorithm is applied to the ideal lattice and a single solution is selected by the lattice designer. The lattice is then misaligned according to Tab. IV and corrected as described below. The beam lifetime and on-energy dynamic aperture of the resulting misaligned and corrected lattice is calculated. This procedure is repeated for many misalignment seeds. The correction procedure begins by flattening the horizontal and vertical orbits using an orbit response matrix and SVD. Then a simultaneous horizontal phase, vertical phase, and horizontal dispersion correction is applied using a combined phase and dispersion response matrix. The residual coupling after these corrections ranges from 0.4 to 3.2%. A dedicated coupling correction is not included in this study.
So 30 misalignment seeds are generated and corrected. One of these seeds fails to have a closed orbit and is discarded. The on-energy dynamic aperture and momentum aperture for the remaining 29 are calculated and the results are shown in Fig. 12 and Fig. 13. 50% of the misaligned and corrected lattices have a lifetime longer than 3.8 hr, and 95% have a lifetime longer than 3.6 hr. The reference lifetime is 4.4 hr.
The misalignment and correction procedure applied here is pessimistic. The fully developed SLS upgrade misalignment model will take into account that the LGBs will be aligned relative to the girders, and the girders relative to one another. Furthermore, coupling correction and vertical dispersion correction will be applied in the actual machine. Despite the pessimistic scenario, the calculated dynamic aperture and lifetimes of the misaligned and corrected lattices are acceptable. From this we conclude that the sextupole and octupole scheme generated by the genetic algorithm is sufficiently robust against imperfections. The sensitivity of the chromaticity correction scheme to beta beating is tested by applying gradient errors to the quadrupole moments in quadrupoles and gradient bends. Errors with RMS values of 0.05%, 0.10%, 0.15%, and 0.20%, subject to a 2-σ cutoff, are tested. No corrections are applied. 1000 seeds are generated for each of the four cases. For each seed, the on-energy dynamic aperture and mean percent beta beat is calculated. Plotted in Fig. 14 is the mean for each case and the convex hulls that contain 50% and 90% of the seeds closest to the mean. In the original SLS, beta-beating is measured to be 2% [23]. We therefore anticipate a reduction in the on-energy dynamic aperture area in the SLS upgrade due to beta-beating of less than 20%.
VII. CONCLUSION
The genetic algorithm presented here offers a robust and computationally affordable technique for generating globally optimal chromaticity correction schemes for FIG. 14. Area of the +y dynamic aperture versus percent beta beating induced by quadrupole gradient errors. The solid enclosure is the convex hull containing 50% of the seeds nearest the mean, the dashed enclosure contains 90% of the seeds.
diffraction limited light sources. The resulting correction schemes have good on-energy dynamic aperture which should help injection efficiency and give a wide momentum aperture for long beam lifetime. The schemes are sufficiently robust against misalignments.
One feature of this algorithm is the use of dominance constraints to encourage individuals in the early population to take on properties that will later on contribute to healthy objective values.
A second feature is the use of off-energy dynamic aperture along with tune footprint constraints as a proxy for the computationally expensive direct momentum aperture calculation.
Based on development efforts at SLS and results shared by the CANDLE collaboration [12], this genetic algorithm delivers results that are as good or better than those obtained by applying 2nd order resonant driving term minimization. The genetic algorithm converges in a couple days on commonly available computing resources. This turn-around time is comparable to that required for a lattice designer to develop a scheme using resonant driving term minimization. Thus the genetic algorithm presented here is a practical solution for optimizing sextupole and octupole strengths in a diffraction limited light source project.
ACKNOWLEDGMENTS
I would like to thank David Sagan of Cornell University for supporting the development of this genetic algorithm through his work on the Bmad [16] library. Many aspects of this work benefited greatly from discussions with Masamitsu Aiba and Michael Böge. I would like to thank Andreas Streun for his mentorship in storage ring nonlinearities, and for the well-designed SLS upgrade lattice. I would like to thank Artsrun Sargsyan for the opportunity to contribute to the development of CANDLE.
FIG. 1 ..
1The crossover operation takes the parent values x Larger values of the parameter κ result in 'near-parent' offspring.
FIG. 2 .
2The mutator operation takes a random real m and adjusts an offspring variable by an amount M × βm (m). M is a settable parameter which can be customized for each variable type. Larger values of κ make large mutations less likely.
FIG. 3 .
3Dynamic aperture is calculated using an element-byelement tracking code. Particles are tracked for 200 turns.
upper and lower bounds on the magnet strength. The two constraints on global nonlinear dispersion are calculated as,
FIG. 4 .
4Dynamic aperture for SLS upgrade at 0%, −3%, and 3% energy offset. Dashed lines are dynamic aperture, solid lines are linear aperture. At the injection point βx = 3.3 m and βy = 6.5 m. The maxima throughout the machine are βx = 8.7 m and βy = 11.7 m.
FIG. 5. Momentum aperture of SLS upgrade. Linear aperture is calculated using an element-by-element linearization of the optics. The 6D aperture is calculated with full 6D tracking including radiation damping and synchrotron oscillation for 2.5 periods of the linear synchrotron tune. The asymmetry of the 6D momentum aperture is the influence of nonlinear momentum compaction. The RF voltage is set such that the linear calculation gives a ±5% bucket.
FIG. 6 .
6Horizontal and vertical phase space portraits for SLS upgrade on-energy, and at −3% and +3% energy defect.
FIG. 8 .
8Dynamic aperture (DA) and linear aperture (LA) for CANDLE at 0%, −2%, and 2% energy offset. At the injection point βx = 3.3 m and βy = 1.8 m. The maxima throughout the machine are βx = 9.1 m and βy = 8.4 m.
FIG. 9 .
9CANDLE: Chromatic footprint from −3% to +3% and ADTS along x from the −x DA to the +x DA. ADTS is calculated with a small vertical offset to allow for accurate calculation of vertical tune. Higher order resonance lines excluded by periodicity 16 are not shown.
FIG. 10 .
10Momentum aperture and linear momentum aperture for one period of the CANDLE lattice. Linear momentum aperture is calculated from Twiss optics linearized about the on-energy particle. The RF voltage is set such that the linear calculation of the RF bucket depth is ±3%.
FIG. 12 .
12On-energy dynamic aperture for ideal lattice and 30 misaligned and corrected lattices. At the injection point βx = 3.3 m and βy = 6.5 m. The maxima throughout the machine are βx = 8.7 m and βy = 11.7 m.
FIG. 13 .
13Momentum aperture from 6D tracking for ideal lattice and 30 misaligned and corrected lattices.
. II.SLS upgrade
CANDLE
# nonlin. mag. fam.
23
8
# variables
21
6
cmag,sext. (Ksext) *
500.0
500.0
cmag,oct. (KoctL) *
500.0
500.0
∆E+DA
3%
2%
∆E-DA
−3%
−2%
∆E+χ
5%
3%
∆E−χ
−5%
−3%
Footprint * * Qx,min, Qx,max
37.0, 37.5
24.5, 25.0
Footprint * * Qy,min, Qy,max
10.0, 10.5
14.0, 14.5
TABLE II. Parameters for Genetic Optimizer. * The sex-
tupole and octupole quantities have been normalized by n!.
* * The footprint constraints apply to both chromatic tune shift
and on-energy ADTS.
FIG. 11. Horizontal and vertical phase space portraits for CANDLE on-energy, and at −2% and +2% energy defect.8
-6
-4
-2
0
2
4
6
8
p
y
y (mm)
+3% Energy Offset
Property
Relative to Distribution σ
Quad. & sext. tilt
girder
50 µrad
Quad. & sext. horiz. offset
girder
20 µm
Quad. & sext. vert. offset
girder
20 µm
Bend & anti-bend tilt
girder
50 µrad
Bend & anti-bend horiz. offset
girder
20 µm
Bend & anti-bend vert. offset
girder
20 µm
Girder tilt
lab
50 µrad
Girder horiz. offset
lab
50 µm
Girder vert. offset
lab
50 µm
LGB tilt
lab
50 µrad
LGB horiz. offset
lab
20 µm
LGB vert. offset
lab
20 µm
TABLE IV .
IVMisalignments are drawn from a random Gaussian distribution, subject to a 2-σ cutoff. Misalignments which exceed the cutoff are re-drawn.
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| []
|
[
"Generic instabilities of non-singular cosmologies in Horndeski theory: a no-go theorem",
"Generic instabilities of non-singular cosmologies in Horndeski theory: a no-go theorem"
]
| [
"Tsutomu Kobayashi \nDepartment of Physics\nRikkyo University\n171-8501ToshimaTokyoJapan\n"
]
| [
"Department of Physics\nRikkyo University\n171-8501ToshimaTokyoJapan"
]
| []
| The null energy condition can be violated stably in generalized Galileon theories, which gives rise to the possibilities of healthy non-singular cosmologies. However, it has been reported that in many cases cosmological solutions are plagued with instabilities or have some pathologies somewhere in the whole history of the universe. Recently, this was shown to be generically true in a certain subclass of the Horndeski theory. In this short paper, we extend this no-go argument to the full Horndeski theory and show that non-singular models (with flat spatial sections) in general suffer from either gradient instabilities or some kind of pathology in the tensor sector. This implies that one must go beyond the Horndeski theory to implement healthy non-singular cosmologies. | 10.1103/physrevd.94.043511 | [
"https://arxiv.org/pdf/1606.05831v2.pdf"
]
| 118,469,364 | 1606.05831 | 68bbfcac44ce734d3165f23830cd668bc954d674 |
Generic instabilities of non-singular cosmologies in Horndeski theory: a no-go theorem
9 Aug 2016
Tsutomu Kobayashi
Department of Physics
Rikkyo University
171-8501ToshimaTokyoJapan
Generic instabilities of non-singular cosmologies in Horndeski theory: a no-go theorem
9 Aug 2016PACS numbers: 9880Cq, 0450Kd
The null energy condition can be violated stably in generalized Galileon theories, which gives rise to the possibilities of healthy non-singular cosmologies. However, it has been reported that in many cases cosmological solutions are plagued with instabilities or have some pathologies somewhere in the whole history of the universe. Recently, this was shown to be generically true in a certain subclass of the Horndeski theory. In this short paper, we extend this no-go argument to the full Horndeski theory and show that non-singular models (with flat spatial sections) in general suffer from either gradient instabilities or some kind of pathology in the tensor sector. This implies that one must go beyond the Horndeski theory to implement healthy non-singular cosmologies.
I. INTRODUCTION
Inflation [1][2][3] is now the strongest candidate of the early universe scenario that explains current cosmological observations consistently. Nonetheless, alternative scenarios deserve to be considered as well. First, in order to be convinced that inflation indeed occurred in the early stage of the universe, all other possibilities must be ruled out. Second, even inflation cannot resolve the problem of the initial singularity [4]. It is therefore well motivated to study how good and how bad alternative possibilities are compared to inflation. Non-singular stages in the early universe, such as contracting and bouncing phases [5], cannot only be something that replaces inflation, but also "early-time" completion of inflation just to get rid of the initial singularity. In this paper, we address whether healthy non-singular cosmologies can be implemented in the framework of general scalar-tensor theories.
If gravity is described by general relativity and the energy-momentum tensor T µν of matter satisfies the null energy condition (NEC), that is, T µν k µ k ν ≥ 0 for every null vector k µ , then (assuming flat spatial sections) it follows from the Einstein equations that dH/dt ≤ 0, where H is the Hubble parameter. This implies that an expanding universe yields a singularity in the past, while NEC violation could lead to singularity-free cosmology. However, violating the NEC in a healthy manner turns out to be challenging. The NEC is by construction satisfied for a canonical scalar field, T µν k µ k ν =φ 2 ≥ 0. In a general non-canonical scalar-field theory whose Lagrangian is dependent on φ and its first derivative [6,7], the NEC can be violated, but NEC-violating cosmological solutions are unstable because the curvature perturbation has the wrong sign kinetic term.
Galileon theory [8] and its generalizations [9,10] involve the scalar field whose Lagrangian contains second * Email: tsutomu"at"rikkyo.ac.jp derivatives of φ while maintaining the second-order nature of the equation of motion and thus erasing the Ostrogradsky instability. In contrast to the previous case, it was found that the NEC and the stability of cosmological solutions are uncorrelated in Galileon-type theories [11]. This fact gives rise to healthy NEC-violating models of Galilean genesis [11][12][13][14][15][16][17] and stable non-singular bouncing solutions [18][19][20], as well as novel dark energy and inflation models with interesting phenomenology [21,22]. See also a recent review [23].
Although the Galileon-type theories do admit a stable early stage without an initial singularity, the genesis/bouncing universe must be interpolated to a subsequent (possibly conventional) stage and the stable early stage does not mean that the cosmological solution is stable at all times during the whole history. Several explicit examples [24][25][26][27][28][29][30] show that the sound speed squared of the curvature perturbation becomes negative at around the transition between the genesis/bouncing phase and the subsequent phase, leading to gradient instabilities. In some cases the universe can experience a healthy bounce, but then the solution has some kind of singularity in the past or future [19]. Although the gradient instabilities can be cured by introducing higher spatial derivative terms [29,30] and there are some models in which the strong coupling scale cuts off the instabilities [31], it would be preferable if the potential danger could be removed from the beginning. The next question to ask therefore is whether the appearance of instabilities is generic or a model-dependent nature. For general dilation invariant theories a no-go theorem was given in Ref. [32]. (A counterexample was presented in Ref. [33], but it has an initial singularity.) Recently, it was clearly shown in Ref. [34] that bouncing and genesis models suffer from instabilities or have singularities for the scalartensor theory whose Lagrangian is of the form
L = R 2κ + G 2 (φ, X) − G 3 (φ, X)✷φ, X := − 1 2 g µν ∂ µ φ∂ ν φ,(1)
where R is the Ricci scalar. This Lagrangian is widely used in the attempt to obtain non-singular stable cosmology. The Lagrangian (1) forms a subclass of the most general scalar-tensor theory with second-order field equations, i.e., the Horndeski theory [35]. The goal of this short paper is to generalize the no-go argument of Ref. [34] to the full Horndeski theory.
II. NO-GO THEOREM
We consider the Horndeski theory [35] in its complete form,
S = d 4 x √ −gL H ,(2)
where
L H = G 2 (φ, X) − G 3 (φ, X)✷φ + G 4 (φ, X)R + G 4,X (✷φ) 2 − (∇ µ ∇ ν φ) 2 + G 5 (φ, X)G µν ∇ µ ∇ ν φ − 1 6 G 5,X (✷φ) 3 − 3✷φ(∇ µ ∇ ν φ) 2 + 2(∇ µ ∇ ν φ) 3 .(3)
(The Lagrangian here is written in the form of the generalized Galileon [10], but the two theories are in fact equivalent [36].) In the full Horndeski theory, we have four arbitrary functions of the scalar field φ and X = −g µν ∂ µ φ∂ ν φ/2. The scalar field is coupled to the Ricci scalar R and the Einstein tensor G µν in the particular way shown above. The structure of the Lagrangian (3) guarantees the second-order nature of the field equations. The equations of motion governing the background cosmological evolution can be obtained by substituting ds 2 = −N 2 (t)dt 2 + a 2 (t)δ ij dx i dx j and φ = φ(t) to the Horndeski action and varying it with respect to N , a, and φ [36]. In this paper, we only consider a spatially flat universe.
Linear perturbations around a spatially flat FLRW spacetime in the Horndeski theory were studied in Ref. [36]. Taking the unitary gauge, δφ = 0, the spatial part of the metric can be written as γ ij = a 2 (t)e 2ζ e h ij , where ζ is the curvature perturbation and h ij is the tensor perturbation. The quadratic actions for h ij and ζ are given, respectively, by [36]
S (2) h = 1 8 dtd 3 x a 3 G Tḣ 2 ij − F T a 2 (∂h ij ) 2 ,(4)
and
S (2) ζ = dtd 3 x a 3 G Sζ 2 − F S a 2 (∂ζ) 2 .(5)
Here, the coefficients are written as
F T := 2 G 4 − X φ G 5,X + G 5,φ ,(6)G T := 2 G 4 − 2XG 4,X − X HφG 5,X − G 5,φ ,(7)
where a dot denotes differentiation with respect to cosmic time t, while F S and G S have more complicated dependence on the functions G 2 , G 3 , G 4 , and G 5 , the explicit forms of which are found in Ref. [36]. It is reasonable to assume that F T , G T , F S , and G S are smooth functions of time. To avoid ghost and gradient instabilities, we require that
F T > 0, G T > 0, F S > 0, G S > 0.(8)
If φ is minimally coupled to gravity, we have G 4 = const and G 5 = 0, and hence F T = G T = const. In other words, the time evolution of F T and G T is caused by non-minimal coupling to gravity. The crucial point for the no-go argument is that F S is generically of the form
F S = 1 a dξ dt − F T ,(9)
where
ξ := aG 2 T Θ ,(10)
with
Θ := −φXG 3,X + 2HG 4 − 8HXG 4,X − 8HX 2 G 4,XX +φG 4,φ + 2XφG 4,φX + 2HX (3G 5,φ + 2XG 5,φX ) − H 2φ 5XG 5,X + 2X 2 G 5,XX .(11)
Since Θ is something written in terms of φ and H, it is supposed to be a smooth function of time which is finite everywhere. This then implies that ξ can never vanish except at a singularity, a = 0. The absence of gradient instabilities is equivalent to
dξ dt > aF T > 0.(12)
Integrating Eq. (12) from some t i to t f , we obtain
ξ f − ξ i > t f ti aF T dt.(13)
This is the key equation for the following argument, and it was used to prove the no-go theorem in the subclass of the Horndeski theory with G 4 = const and G 5 = 0 in Ref. [34]. Remarkably, it turns out that essentially the same equation holds in the full Horndeski theory. Now, consider a non-singular universe which satisfies a > const (> 0) for t → −∞ and is expanding for large t. The integral in the right hand side of Eq. (13) may be convergent or not as one takes t f → ∞ and t i → −∞, depending on the asymptotic behavior of F T . To allow the integral to converge, it is necessary that F T approaches zero sufficiently fast in the asymptotic past or future. For the moment let us focus on the case where the integral is not convergent.
Suppose that ξ i < 0. Equation (13) reads
−ξ f < |ξ i | − t f ti aF T dt.(14)
Since the integral is an increasing function of t f , the right hand side becomes negative for sufficiently large t f . We therefore have ξ f > 0, which means that ξ crosses zero. 1 This is never possible in a non-singular universe. It is therefore necessary to have ξ > 0 everywhere. Writing Eq. (13) as
−ξ i > −ξ f + t f ti aF T dt,(15)
we see that the right hand side will be positive for t i → −∞ and hence ξ i < 0. However, this is in contradiction to the assumption that ξ is always positive. Thus, we have generalized the no-go argument of Ref. [34] to the full Horndeski theory. The same no-go theorem holds even in the presence of another field, provided at least that the field is described by
L χ = P (χ, Y ), Y := − 1 2 g µν ∂ µ χ∂ ν χ,(16)
which is not coupled to the Horndeski field φ directly. Now there are two degrees of freedom in the scalar sector of cosmological perturbations. In terms of
y := ζ, Θ G T δχ χ ,(17)
the quadratic action can be written in the form [37][38][39]
S (2) = dtd 3 xa 3 ˙ y G˙ y − 1 a 2 ∂ y F ∂ y + · · · ,(18)
where
G = G S + Z −Z −Z Z , F = F S −c 2 s Z −c 2 s Z c 2 s Z ,(19)
with
c 2 s := P ,Y P ,Y + 2Y P ,Y Y , Z := G T Θ 2 Y P ,Y c 2 s .(20)
Here, G S and F S were defined previously and c s is the sound speed of the χ field. We have the relation 2Y P ,Y = ρ + P , where ρ is the energy density of χ and P corresponds to the pressure of χ.
Ghost instabilities can be evaded if G is a positive definite matrix. The condition amounts to
G S > 0, Y P ,Y c 2 s > 0.(21)
1 We do not allow for discontinuity in ξ because F S is supposed to be smooth. (This means that Θ cannot cross zero.)
The propagation speed v can be determined by solving
det(v 2 G − F) = 0,(22)
yielding the condition for the absence of gradient instabilities,
c 2 s > 0, F S − c 2 s Z G S > 0.(23)
We thus have the inequality
F S > 1 2 G T Θ 2 (ρ + P ) > 0,(24)
and taking the same way we can show the no-go theorem for the Horndeski + k-essence (or a perfect fluid) system. The no-go theorem we have thus established can be circumvented only if F T approaches zero sufficiently fast either in the asymptotic past or the future, given the assumption that the evolution of the scale factor is nonsingular. 2 The normalization of vacuum quantum fluctuations tells us that they would grow and diverge as F T → 0, and hence the tensor sector is pathological in the asymptotic past or future. 3 In the next section, we will demonstrate that, in contrast to the cases in Refs. [29,30], it is indeed possible to construct a model that exhibits a stable transition from the Galilean genesis to inflation by allowing for some kind of pathology in the tensor sector due to vanishing F T .
III. STABLE TRANSITION FROM GENESIS TO DE SITTER WITH PATHOLOGIES IN THE PAST
Let us turn to study a specific setup as an example: Galilean genesis followed by inflation. Such an expansion history was proposed in Refs. [29,30] as early-time completion of the inflationary universe, and there it was pointed out that the sound speed squared (or more specifically F S ) becomes negative at the transition from the genesis phase to inflation. This is consistent with the no-go theorem, because in the genesis phase we have a → const as t → −∞ and F T = const. The resultant gradient instability is cured by the introduction of higher order spatial derivatives arising in the effective field theory approach [29] or in theories beyond Horndeski [30,40,41].
Working within the second-order theory, i.e., the Horndeski theory, we are going to show in this section that the stable transition is indeed possible if F T → 0 as t → −∞ so that the integral in Eq. (13) is convergent. To do so it is more convenient to use the ADM form of the action rather than the original covariant one [30]. The ADM decomposition of the Horndeski Lagrangian leads to [40]
L = A 2 (t, N ) + A 3 (t, N )K + A 4 (t, N ) K 2 − K 2 ij + A 5 (t, N ) K 3 − 3KK 2 ij + 2K 3 ij + B 4 (t, N )R (3) + B 5 (t, N )K ij G (3) ij ,(25)
where φ = const hypersurfaces are taken to be constant time hypersurfaces, and K ij , R
ij , and G
ij are the extrinsic curvature, the Ricci tensor, and the Einstein tensor of the spatial slices. The functions of φ and X in the covariant Lagrangian are now the functions of t and the lapse function N . Two of the six functions in the ADM Lagrangian (25) are subject to the constraints
A 4 = −B 4 − N ∂B 4 ∂N , A 5 = N 6 ∂B 5 ∂N ,(26)
in accordance with the fact that there are four arbitrary functions in the Horndeski theory.
The specific example we are going to study is given by the functions of the form
A 2 = f −2(α+1)−δ a 2 (N ), A 3 = f −2α−1−δ a 3 (N ), A 4 = −B 4 = −f −2α , A 5 = B 5 = 0,(27)
where f = f (t) is dependent only on t, and α and δ are constant parameters satisfying 2α > 1 + δ > 1. This class of models is similar to but different from that in Ref. [30]. The covariant form of the Lagrangian can be recovered by re-introducing the scalar field, e.g., through −t = e −φ and N −1 = e −φ √ 2X and using the Gauss-Codazzi equations. In terms of G 2 (φ, X), G 3 (φ, X), ..., the Lagrangian is written in a slightly more complicated form [42]. Without moving to the covariant description, one can derive the equations of motion for the homogeneous background directly from variation of the ADM action with respect to N and the scale factor a.
The evolution of the Hubble parameter, H := N −1 d ln a/dt, is dependent crucially on the choice of f (t), and to describe the genesis to de Sitter transition we take f (t) such that f ∼ c(−t) ≫ 1 (c > 0) in the past and f ∼ const in the future. In the early time, we have an approximate solution of the form
H ≃ const (−t) 1+δ ,(28)
and hence the universe starts expanding from Minkowski,
a ≃ 1 + const (−t) δ ,(29)
with N ≃ const. In the late time where f ≃ const, we have an inflationary solution H ≃ const, again with N ≃ const. For all the models described by (27), we have and hence the stability conditions for the tensor modes are fulfilled. Since aF T ∼ (−t) −2α with 2α > 1 as t → −∞, F T possesses the desired property to evade the nogo theorem. As a concrete example, we consider
F T = G T = f −2α > 0,(30)a 2 = − 1 N 2 + 1 3N 4 , a 3 = 1 4N 3 ,(31)
with α = 1, δ = 1/2, and
f (t) = c 2 −t + ln(2 cosh(st)) s + f 1 ,(32)
where the parameters are taken to be c = 10 −1 , f 1 = 10, and s = 2 × 10 −3 . The background equations are solved numerically to give the evolution of H and N as shown in Fig. 1. It can be seen that the universe indeed undergoes the genesis phase followed by inflation. For this background solution we evaluate F S and G S numerically to judge its stability. As presented in Fig. 2, we find that F S and G S remain positive in the whole expansion history. This is in contrast to the similar example in Refs. [29,30] which has F S < 0 around the transition. Although the present model can circumvent the gradient instability at the genesis-de Sitter transition, some pathologies arise in the t → −∞ limit. We see that F T , G T ∼ (−t) −2α and F S , G S ∼ (−t) −2α+δ in the genesis phase, leading to the vanishing quadratic action for tensor and scalar fluctuations in the t → −∞ limit. This implies that the validity of the perturbative expansion is questionable early in the genesis phase, which is, in fact, worse than what is required for violating the no-go theorem, i.e., F T → 0 as t → −∞.
IV. SUMMARY
In this paper, we have generalized the no-go argument of Ref. [34] to the full Horndeski theory and shown that non-singular cosmological models with flat spatial sections are in general plagued with gradient instabilities or some pathological behavior of tensor perturbations. We have presented an explicit example which is free from singularities and instabilities, but has a vanishing quadratic action for the tensor perturbations (and for the curvature perturbation as well) in the asymptotic past. To circumvent the no-go theorem, it is therefore necessary to go beyond the Horndeski theory. One direction is to consider a (yet unknown) multi-field extension of the Horndeski theory [39,[43][44][45][46][47] in which scalar fields are coupled nontrivially to each other. Another is extending further the single-field Horndeski theory as has been done recently e.g. in Refs. [40,41,[48][49][50][51][52][53]. It would be interesting to explore to what extent the no-go argument for non-singular cosmologies can be generalized.
FIG. 1 :
1Evolution of the Hubble parameter and the lapse function around the genesis-de Sitter transition. FIG. 2: FS and GS around the genesis-de Sitter transition.
The "modified genesis" model proposed in Ref.[34] evades the no-go theorem by the use of the vanishing scale factor in the asymptotic past. In contrast, we are assuming that the expansion history is non-singular everywhere, i.e., a ≥ const.3 One could resolve this issue by the particular, fine-tuned evolution of G T , which would offer a loophole.
AcknowledgmentsThis work was supported in part by the JSPS Grants-in-Aid for Scientific Research No. 16H01102 and No. 16K17707.
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Derivative-dependent metric transformation and physical degrees of freedom. G Domènech, S Mukohyama, R Namba, A Naruko, R Saitou, Y Watanabe, arXiv:1507.05390Phys. Rev. D. 92884027hep-thG. Domènech, S. Mukohyama, R. Namba, A. Naruko, R. Saitou and Y. Watanabe, "Derivative-dependent met- ric transformation and physical degrees of freedom," Phys. Rev. D 92, no. 8, 084027 (2015) [arXiv:1507.05390 [hep-th]].
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Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations. J Ben Achour, D Langlois, K Noui, arXiv:1602.08398Phys. Rev. D. 9312124005gr-qcJ. Ben Achour, D. Langlois and K. Noui, "Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations," Phys. Rev. D 93, no. 12, 124005 (2016) [arXiv:1602.08398 [gr-qc]].
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| []
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[
"The influence of small scale magnetic field on the heating of J0250+5854 polar cap",
"The influence of small scale magnetic field on the heating of J0250+5854 polar cap"
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| [
"D P Barsukov \nIoffe Institute, Saint Petersburg\nRussia\n\nPeter the Great St. Petersburg Polytechnic University\nSaint PetersburgRussia\n",
"M V Vorontsov \nPeter the Great St. Petersburg Polytechnic University\nSaint PetersburgRussia\n",
"I K Morozov \nIoffe Institute, Saint Petersburg\nRussia\n\nPeter the Great St. Petersburg Polytechnic University\nSaint PetersburgRussia\n"
]
| [
"Ioffe Institute, Saint Petersburg\nRussia",
"Peter the Great St. Petersburg Polytechnic University\nSaint PetersburgRussia",
"Peter the Great St. Petersburg Polytechnic University\nSaint PetersburgRussia",
"Ioffe Institute, Saint Petersburg\nRussia",
"Peter the Great St. Petersburg Polytechnic University\nSaint PetersburgRussia"
]
| []
| The influence of surface small scale magnetic field on the heating of PSR J0250+5854 polar cap is considered. It is assumed that polar cap is heated only by reverse positrons, accelerated in pulsar diode. It is supposed that pulsar diode is in stationary state with lower plate nearby the star surface (polar cap model), occupies all pulsar tube crosssection and operates in regime of steady space charge limited electron flow. The influence of small scale magnetic field on electric field inside pulsar diode is taken into account. To calculate the electron-positron pairs production rate we take into account only the curvature radiation of primary electrons and its absorption in magnetic field. It is assumed that part of electropositron pairs may be created in bound state (positronium). And later such positroniums are photoionized by thermal photons from star surface. | 10.1088/1742-6596/2103/1/012034 | [
"https://arxiv.org/pdf/2004.05135v1.pdf"
]
| 215,737,312 | 2004.05135 | 0e725272ce393903a75aaf16d20c140cf4b80eb8 |
The influence of small scale magnetic field on the heating of J0250+5854 polar cap
10 Apr 2020
D P Barsukov
Ioffe Institute, Saint Petersburg
Russia
Peter the Great St. Petersburg Polytechnic University
Saint PetersburgRussia
M V Vorontsov
Peter the Great St. Petersburg Polytechnic University
Saint PetersburgRussia
I K Morozov
Ioffe Institute, Saint Petersburg
Russia
Peter the Great St. Petersburg Polytechnic University
Saint PetersburgRussia
The influence of small scale magnetic field on the heating of J0250+5854 polar cap
10 Apr 2020arXiv:2004.05135v1 [astro-ph.HE]
The influence of surface small scale magnetic field on the heating of PSR J0250+5854 polar cap is considered. It is assumed that polar cap is heated only by reverse positrons, accelerated in pulsar diode. It is supposed that pulsar diode is in stationary state with lower plate nearby the star surface (polar cap model), occupies all pulsar tube crosssection and operates in regime of steady space charge limited electron flow. The influence of small scale magnetic field on electric field inside pulsar diode is taken into account. To calculate the electron-positron pairs production rate we take into account only the curvature radiation of primary electrons and its absorption in magnetic field. It is assumed that part of electropositron pairs may be created in bound state (positronium). And later such positroniums are photoionized by thermal photons from star surface.
Introduction
Radiopulsar J0250+5854 rotates with period P = 23.54 s [1] and is the slowest pulsar among rotation powered pulsars [2]. It is old pulsar with spin down age τ = 13.7 · 10 6 years, P = 2.71 · 10 −14 , its spin down energy loss rateĖ is equal toĖ = 8.2 · 10 28 erg/s, the strength B dip of dipolar magnetic field at pole estimated by pulsar slowdown is B dip = 5.1 · 10 13 G, distance D DM estimated by dispersion measure is D DM = 1.56 kpc [2]. Such pulsars lie beyond conventional pulsar "death line" , see, for example, [3,4], and usually its radio radiation is explained by the presence of small scale surface magnetic field, see, for example, [5,6,7,8]. It is worth to note that the radio radiation of J0250+5854 also may be explained in case of pure dipolar magnetic field if it is take into account that the B dip value is calculated at assumption aligned pulsar χ = 0 • and braking due to magneto-dipolar losses [9], where χ is inclination angle i.e. angle between vector of magnetic dipole momentum m and vector of angular velocity of star rotation Ω, Ω = 2π/P , see fig. 1 and fig. 2. In case of pulsar braking due to current losses and nearby orthogonal pulsars χ ≈ 90 • dipolar magnetic field is substantially larger than B dip value estimated by slow down [10] that moves the pulsar to "life zone" [9]. In this paper we will not consider a such possibility and assume that the B dip value is the right estimation of dipolar magnetic field strength. In case of large surface magnetic field B surf 4.4 · 10 12 G electron-positron pairs may be produced in bound state (positronium) [11]. The influence of this process on pulsar electrodynamics, pair generation and polar cap X-ray luminosity of radio pulsars has been thoroughly considered in many papers, see, for example, [12,11]. In this paper we consider the influence of small scale magnetic field on polar cap heating by reverse positron current with taking into account positronium generation and its photoionization by thermal photons from star surface. Pulsar is considered in inner gap model with free electron emission from neutron star surface. We assume that pulsar diode is in stationary state and take into account only positron generation due to curvature radiation of primary electrons and its absorption in magnetic field.
Model
Let the neutron star have a radius r ns and dipolar magnetic moment m (its field at magnetic pole is B dip = 2m/r 3 ns ). We assume also that a small-scale magnetic field with strength B sc and characteristic scale ℓ presents nearby the polar cap. For simplicity we model small-scale magnetic field by by additional magnetic moment m sc locating in the polar region of the neutron star at depth ℓ [13,14,15]: where ρ = r − (r ns − ℓ) e z , m = m e z , B sc = m sc /ℓ 3 -small-scale field strength at (dipolar) magnetic pole. For simplicity we suppose that the vector m sc lies parallel to surface (and m sc · m = 0) in the plane containing m and Ω and is directed "along" Ω, see fig. 1. Hence surface small-scale magnetic field is directed "opposite" pulsar rotation velocity Ω, i.e. ψ Ω = 0, see fig. 2. Also we assume that inclination angle χ is equal to χ = 60 • . We consider only the case of inner gap [16] and assume that the inner gap occupies the entire pulsar tube cross section and resides as low as possible. Let us denote the altitudes of inner gap lower plate (cathode) and upper plate (anode) by z lo and is z hi respectively, see fig. 1. In most cases the inner gap resides exactly on neutron star surface (z lo = 0), see [17] for details. We suppose that the inner gap is stationary and operates in the regime of charge limited steady flow [18]. Hence in the reference frame rotating with the star we can write, see [19] for details:
B = 3 r ( r · m) − mr 2 r 5 + 3 ρ ( ρ · m sc ) − m 1 ρ 2 ρ 5(1)∆Φ = −4π(ρ − ρ GJ ), E = − ∇Φ (2) E || z=z lo = 0 and E || z=z hi = 0(3)
Φ| z=z lo = 0 and Φ| side = 0
where z is the altitude above star surface, Φ is electrostatic potential, Φ| side is its value at pulsar tube boundary, ρ GJ = ΩB 2πcρ GJ is Goldreich-Julian density [20], ρ = ΩB 2πcρ is total charge density,ρ =ρ − +ρ + ,ρ,ρ − ,ρ + are total charge density, electron and positron densities in units ΩB 2πc correspondingly. We assume that inside pulsar diode the particles move along field lines v B with relativistic velocity v ≈ c. So continuity equation div (ρ ± v) = 0 may be rewritten as ( B · ∇)ρ ± = 0 and hence densitiesρ ± are constant along field lines. Also it is worth to note that without frame draggingρ GJ ( x) ≈ − cosχ, whereχ is the angle between field vector B( x) at point x and angular velocity vector Ω.
For simplicity we take into account the generation of electron-positron pairs only by curvature radiation of primary electrons in magnetic field. We also take into account the generation of pairs in bound state (positronium). For simplicity we assume that the probability P b that a pair is created in bound state is defined as follows [11]:
P b = 0 if B < B low (no positroniums are created), P b = (B − B low )/(B high − B low ) if B low ≤ B ≤ B high and P b = 1 if B > B high (all pairs are created in bound state),
where B is magnetic strength at point of pair creation, B low = 0.04 B cr , B high = 0.15 B cr , B cr ≈ 4.41 · 10 13 G [11]. In order to simplify the calculation we assume that the pair generation and its properties do not depend on photon polarization. However, we take into account the photoionization of positronium by thermal photons from hot polar cap. The photoionization rate is estimated by formula [11]
dN dt ( x) = W 0 10 2 Γ 3 T ns 10 6 K 2 (1 − cos θ ns ),(5)
where Γ is positronium Lorentz factor, T ns is neutron star surface temperature, θ ns is angular radius of neutron star at point x, W 0 = 6 · 10 5 s −1 [11]. Due to small polar cap size we neglect positronium photoionization by thermal photons from hot polar cap. In this paper we does not take into account photon splitting and positronium decay. In order to crude estimate the effect of these processes we assume that (1− f ) part of positroniums immediately decays after creation and f part of positroniums does not decay at all.
The calculations of reverse positron current are performed in two models based on extreme assumptions about the rate of parallel electric field E || = ( E · B)/B screening: the model of rapid screening [21] according to which the electron-positron plasma screens parallel electric field almost immediately and the model of gradually screening [22,23], which allows the parallel electric field penetrates deep into electron-positron plasma, see details of calculation in [19]. For simplicity we assume that anode altitude z hi is determined by equatioñ
ρ pair | z=z hi = max 1 10 ,ρ r +(6)
whereρ pair is number unbound electron-positron pairs generated at central field line,ρ r + is reverse positron density calculated according rapid screening model. The input of reverse positron heating to polar cap temperature T pc is estimated as
σ B T 4 pc = Φ| z=z hi · ΩB 2π z=0 ·ρ +(7)
where altitude z = 0 corresponds to star surface, σ B is Stefan-Boltzmann constant and all values calculated at the same field line. Polar cap luminosity due to reverse positron heating is estimated as
L pc = Spc σ B T 4 pc dS ≈ Φ| z=z hi · ΩB dip 2π ·ρ + · π (θ 0 r ns ) 2(8)
where we integrate over polar cap surface and θ 0 = Ωr ns /c, see [19] for details.
Results
The dependence of primary electron currentρ − and diode lower plate (cathode) altitude z lo on strength of small scale magnetic field B sc is shown in fig. 3. According to used model altitude z lo does not depend on pair production at all [17]. And because of pulsar tube radius is very small θ 0 r ns ≪ z hi primary electron densityρ − also does not depend on pair production. The dependence of energy of primary electrons and diode upper plate (anode) altitude z hi on strength of small scale magnetic field B sc are shown in fig. 4 and fig. 5 correspondingly. At B sc 0.5B dip altitude z hi decreases with increasing B sc because of increasing total magnetic field strength and, most importantly, due to increasing field line curvature. Later according considered magnetic field model cathode altitude z lo begins to increase. Hence primary electrons are accelerated at larger altitude where field strength and its curvature are less. And consequently pair production becomes less effective and anode altitude z hi grows with B sc at B sc 0.6B dip . And the increasing of the altitude z hi causes an increase in the energy of primary electron. The dependence of reverse positron currentρ + and the polar cap luminosity L pc caused by this current on strength of small scale magnetic field B sc in case of rapid and gradually screening model are shown in figures 6-9. The dependence of total number of produced unbound or photoionized pairsρ pair on strength of small scale magnetic field B sc is shown in fig. 10. It is worth to note that number of pairs ρ pair produced in case of T ns = 3 · 10 5 K and f = 1 is larger than in case of W 0 = +∞.
Discussion
We consider inner gap model with stationary space charge limited flow in J0250+5854 pulsar and show that this pulsar may lye upper than pulsar "death line" in case of two assumption: the presence of surface magnetic field with very small characteristic scale ℓ ≈ 500 m and neutron star surface temperature T ns ∼ (1 − 3) · 10 5 K. Main problem is that the pulsar is very old τ = 13.7 · 10 6 years. Hence it is difficult to explain why field with so small scale has survived and why star is so hot.
It is worth to note that radiopulsar B0950+08 has spin down age τ = 17.5 · 10 6 years and star surface temperature T ns ∼ (1 − 3) · 10 5 K [24]. A such temperature may be related to internal heating mechanisms like rotochemical heating and heating due to vortex friction [25]. We also may speculate that magnetic field decay event with Hall cascade has occur not so long ago in this pulsar [26]. Hence small scale magnetic field may be generated during Hall cascades and accompanying field decay may heat up the star.
In the paper we does not take into account the photon polarization and, consequently, we can not estimate input of photon splitting effect and positronium decay, see, for example, [27,28,29]. Hence, our conclusion that magnetic field with characteristic scale ℓ ≈ 500 m is enough to explain radio radiation of the pulsar may be too optimistic. But we hope that field with ℓ ≈ 300 m would be enough. Also it is worth to note that we take into account only curvature radiation of primary electrons and resonant compton scattering may give a similar quantity of pairs [30].
Our choice of inclination angle χ = 60 • does not motivated by anything. Although we find that in considered field configuration the pulsar lye down "pulsar death" line in case of χ = 0 • and χ = 30 • . But we guess that it is only artifact of used small scale field model.
Figure 1 .Figure 2 .
12A sketch of the vicinity of an inner gap. Neutron star is shown by gray area, boundaries of pulsar tube are shown by black lines, the inner gap is shown by brown area. The definition of angles χ and φ Ω .
Figure 3 .
3The dependence of the primary electron densityρ − (in units ΩB 2πc ) on small scale field strength B sc is shown on the left panel. The dependence of the altitude z lo (in units r ns ) of diode lower plate (cathode) on small scale field strength B sc is shown on the right panel.
Figure 4 .
4The dependence of Lorentz factor Γ = e Φ| z=z hi /mc 2 of primary electrons at central field line on small scale field strength B sc is shown. Dot-dashed blue line corresponds to T ns = 3 · 10 5 K and f = 1, dashed cyan line corresponds to T ns = 3 · 10 5 K and f = 0.3, solid violet line corresponds to T ns = 3 · 10 5 K and f = 0.1, dot-dashed red line corresponds to T ns = 10 5 K and f = 1, dashed green line corresponds to T ns = 10 5 K and f = 0.3. solid orange line corresponds to T ns = 1 · 10 5 K and f = 0.1, The case W 0 = +∞ (all positroniums are photoionized immediately) is shown by solid black line.
Figure 5 .
5The same asfig. 4, but the dependence of the altitude z hi (in units r ns ) of diode upper plate (anode) on small scale field strength B sc is shown. The altitude of diode lower plate (cathode) z lo is shown by black solid line on both graphs. Left and right graphs differ only in scale.
Figure 6 .
6The same as fig. 4, but the dependence of the reverse positron currentρ + (in units ΩB 2πc ) calculated with rapid screening model on small scale field strength B sc is shown. Left and right graphs differ only in scale.
Figure 7 .
7The same asfig. 4, but the dependence of the input of reverse positron heating to polar cap surface temperature T pc at polar cap center on small scale field strength B sc is shown on the left panel. The dependence of corresponding polar cap luminosity L pc on small scale field strength B sc is shown on the right panel. Both panels correspond to rapid screening model.
Figure 8 .
8The same asfig. 6, but the case of gradually screening model is shown. Left and right graphs differ only in scale.
Figure 9 .
9The same asfig. 7, but the case of gradually screening model is shown.
Figure 10 .
10The same as fig. 4, but the dependence of the total number of produced unbound or photoionized pairsρ pair (in units ΩB 2πc ) on small scale field strength B sc is shown. Left and right graphs differ only in scale.
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| []
|
[
"PIXEL-WISE DEEP LEARNING FOR CONTOUR DETEC- TION",
"PIXEL-WISE DEEP LEARNING FOR CONTOUR DETEC- TION"
]
| [
"Jyh-Jing Hwang [email protected] \nInstitute of Information Science\nAcademia Sinica Taipei\nTaiwan\n",
"Tyng-Luh Liu \nInstitute of Information Science\nAcademia Sinica Taipei\nTaiwan\n"
]
| [
"Institute of Information Science\nAcademia Sinica Taipei\nTaiwan",
"Institute of Information Science\nAcademia Sinica Taipei\nTaiwan"
]
| []
| We address the problem of contour detection via per-pixel classifications of edge point. To facilitate the process, the proposed approach leverages with DenseNet, an efficient implementation of multiscale convolutional neural networks (CNNs), to extract an informative feature vector for each pixel and uses an SVM classifier to accomplish contour detection. In the experiment of contour detection, we look into the effectiveness of combining per-pixel features from different CNN layers and verify their performance on BSDS500. | null | [
"https://arxiv.org/pdf/1504.01989v1.pdf"
]
| 14,612,342 | 1504.01989 | 709c509e97c1d6e2361ea30568690f63f8ac2bc6 |
PIXEL-WISE DEEP LEARNING FOR CONTOUR DETEC- TION
8 Apr 2015
Jyh-Jing Hwang [email protected]
Institute of Information Science
Academia Sinica Taipei
Taiwan
Tyng-Luh Liu
Institute of Information Science
Academia Sinica Taipei
Taiwan
PIXEL-WISE DEEP LEARNING FOR CONTOUR DETEC- TION
8 Apr 2015Accepted as a workshop contribution at ICLR 2015
We address the problem of contour detection via per-pixel classifications of edge point. To facilitate the process, the proposed approach leverages with DenseNet, an efficient implementation of multiscale convolutional neural networks (CNNs), to extract an informative feature vector for each pixel and uses an SVM classifier to accomplish contour detection. In the experiment of contour detection, we look into the effectiveness of combining per-pixel features from different CNN layers and verify their performance on BSDS500.
INTRODUCTION
We consider deep nets to construct a per-pixel feature learner for contour detection. As the task is essentially a classification problem, we adopt deep convolutional neural networks (CNNs) to establish a discriminative approach. However, one subtle deviation from typical applications of CNNs should be emphasized. In our method, we intend to use the CNN architecture, e.g., AlexNet (Krizhevsky et al., 2012), to generate features for each image pixel, not just a single feature vector for the whole input image. Such a distinction would call for a different perspective of parameter fine-tuning so that a pre-trained per-image CNN on ImageNet (Deng et al., 2009) can be adapted into a new model for per-pixel edge classifications. To investigate the property of the features from different convolutional layers, we carry out a number of experiments to evaluate their effectiveness in performing contour detection on the benchmark BSDS Segmentation dataset (Martin et al., 2001).
PER-PIXEL CNN FEATURES
Learning features by employing a deep architecture of neural net has been shown to be effective, but most of the existing techniques focus on yielding a feature vector for an input image (or image patch). Such a design may not be appropriate for vision applications that require investigating image characteristics in pixel level. For contour detection, the central task is to decide whether an underlying pixel is an edge point or not. Thus, it would be convenient that the deep network could yield per-pixel features. To this end, we extract per-pixel CNN features in AlexNet (Krizhevsky et al., 2012) using DenseNet (Iandola et al., 2014), and pixel-wise concatenate them to feed into a support vector machine (SVM) classifier.
Our implementation uses DenseNet for CNN feature extraction owing to its efficiency, flexibility, and availability. DenseNet is an open source system that computes dense and multiscale features from the convolutional layers of a Caffe CNN based object classifier. The process of feature extraction proceeds as follows. Given an input image, DenseNet computes its multiscale versions and stitches them to a large plane. After processing the whole plane by CNNs, DenseNet would unstitch the descriptor planes and then obtain multiresolution CNN descriptors.
The dimensions of convolutional features are ratios of the image size, e.g., one-fourth for Conv1, and one-eighth for Conv2. We rescale feature maps of all the convolutional layers to the image size. That is, there is a feature vector in every pixel. The dimension of the resulting feature vector is 1376 × 1, which is concatenated by Conv1 (96 × 1), Conv2 (256 × 1), Conv3 (384 × 1), Conv4 (384 × 1), and Conv5 (256 × 1). For classification, we use the combined per-pixel feature vectors to learn a binary linear SVM. It is worth noting that, in our multiscale setting, we train the SVM based on only the original resolution.
In test time, we classify test images using both the original and the double resolutions. We average the two resulting edge maps for the final output of contour detection.
EXPERIMENTAL RESULTS
We test our method on the Berkeley Segmentation Dataset and Benchmark (BSDS500) (Martin et al., 2001;Arbelaez et al., 2011). To better assess the effects of the features of different layers, we report their respective performance of contour detection. The BSDS500 dataset contains 200 training, 100 validation, and 200 testing images. Boundaries in each image are labeled by several workers and are averaged to form the ground truth. The accuracy of contour detection is evaluated by three measures: the best F-measure on the dataset for a fixed threshold (ODS), the aggregate F-measure on the dataset for the best threshold in each image (OIS), and the average precision (AP) on the full recall range (Arbelaez et al., 2011). Prior to evaluation, we apply a standard non-maximal suppression technique to edge maps to obtain thinned edges (Canny, 1986).
In Table 1, we see that features in Conv2 contribute the most, and then Conv3 and Conv4. These suggest that low-to mid-level features are most useful for contour detection, while the lowestand higher-level features provide additional boost. Although features in Conv1 and Conv5 are less effective when employed alone, we achieve the best results by combining all five streams. It indicates that the local edge information in low-level features and the object contour information in higherlevel features are both necessary for achieving high performance in contour detection tasks.
ACKNOWLEDGMENTS
This work was supported in part by NSC 102-2221-E-001-021-MY3.
Table 1 :
1Contour detection results of using CNN features from different layers.Conv1 Conv2 Conv3 Conv4 Conv5 Conv1-5
ODS .627
.699
.655
.654
.604
.741
OIS .660
.718
.670
.667
.620
.759
AP
.625
.712
.619
.615
.546
.757
Contour detection and hierarchical image segmentation. Pattern Analysis and Machine Intelligence. Pablo Arbelaez, Maire, Michael, Charless Fowlkes, Jitendra Malik, IEEE Transactions on. 335Arbelaez, Pablo, Maire, Michael, Fowlkes, Charless, and Malik, Jitendra. Contour detection and hierarchical image segmentation. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33(5):898-916, 2011.
A computational approach to edge detection. Pattern Analysis and Machine Intelligence. John Canny, IEEE Transactions on. 6Canny, John. A computational approach to edge detection. Pattern Analysis and Machine Intelli- gence, IEEE Transactions on, (6):679-698, 1986.
ImageNet: A large-scale hierarchical image database. Jia Deng, Dong, Wei, Socher, Richard, Li, Li-Jia, Kai Li, Li Fei-Fei, CVPR. Deng, Jia, Dong, Wei, Socher, Richard, Li, Li-Jia, Li, Kai, and Fei-Fei, Li. ImageNet: A large-scale hierarchical image database. In CVPR, pp. 248-255, 2009.
Forrest Iandola, Moskewicz, Matt, Karayev, Sergey, Girshick, Ross, Trevor Darrell, Kurt Keutzer, Densenet, arXiv:1404.1869Implementing efficient convnet descriptor pyramids. arXiv preprintIandola, Forrest, Moskewicz, Matt, Karayev, Sergey, Girshick, Ross, Darrell, Trevor, and Keutzer, Kurt. Densenet: Implementing efficient convnet descriptor pyramids. arXiv preprint arXiv:1404.1869, 2014.
Imagenet classification with deep convolutional neural networks. Alex Krizhevsky, Ilya Sutskever, Geoffrey E Hinton, Advances in neural information processing systems. Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geoffrey E. Imagenet classification with deep convo- lutional neural networks. In Advances in neural information processing systems, pp. 1097-1105, 2012.
A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. David Martin, Fowlkes, Charless, Doron Tal, Jitendra Malik, Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on. IEEE2Martin, David, Fowlkes, Charless, Tal, Doron, and Malik, Jitendra. A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecolog- ical statistics. In Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on, volume 2, pp. 416-423. IEEE, 2001.
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[
"Comparison of results for the electromagnetic form factors of the proton at low Q 2 Evangelos Matsinos",
"Comparison of results for the electromagnetic form factors of the proton at low Q 2 Evangelos Matsinos"
]
| []
| []
| []
| The goal in this work is the comparison of five parameterisations of the Sachs form factors of the proton G p E (electric) and G p M (magnetic) at low 4-momentum transfer Q 2 . It will be shown that a simple model, based on two dipoles which admit as parameters the rms electric charge radius r E;p and the rms magnetic radius r M ;p of the proton, suffices for the purposes of the phase-shift analyses (PSAs) of the low-energy pion-nucleon (πN ) data. The replacement of the electromagnetic form factors, currently used in the ETH model of the πN interaction, by the parameterisation of this work will enable the removal from the PSAs of this research programme of the largest part of the dependence on extraneous sources. | null | [
"https://arxiv.org/pdf/2009.04156v1.pdf"
]
| 221,558,148 | 2009.04156 | 3fd3c960d64e13b05bc3bcf4ae68fdd3c50de9ae |
Comparison of results for the electromagnetic form factors of the proton at low Q 2 Evangelos Matsinos
9 Sep 2020
Comparison of results for the electromagnetic form factors of the proton at low Q 2 Evangelos Matsinos
9 Sep 2020Electromagnetic processes, electromagnetic form factors of the nucleon
The goal in this work is the comparison of five parameterisations of the Sachs form factors of the proton G p E (electric) and G p M (magnetic) at low 4-momentum transfer Q 2 . It will be shown that a simple model, based on two dipoles which admit as parameters the rms electric charge radius r E;p and the rms magnetic radius r M ;p of the proton, suffices for the purposes of the phase-shift analyses (PSAs) of the low-energy pion-nucleon (πN ) data. The replacement of the electromagnetic form factors, currently used in the ETH model of the πN interaction, by the parameterisation of this work will enable the removal from the PSAs of this research programme of the largest part of the dependence on extraneous sources.
Introduction
To determine the scattering amplitude of electromagnetic (EM) processes, pertaining to the elastic scattering of charged particles on proton targets, an expression for the EM transition current of the proton is required. The general form of the proton vertex, fulfilling Lorentz invariance and charge conjugation, can be found in Chapter 8.8.2 of Ref. [1], p. 259:
p ′ , s ′ | J µ EM |p, s = eū(p ′ , s ′ ) F p 1 γ µ + i κ p F p 2 2m p σ µν q ν u(p, s) ,(1)
where • e is the electric charge of the proton; • m p is the proton mass;
• p and s stand for the 4-momentum and spin of the initial-state proton;
• p ′ and s ′ stand for the 4-momentum and spin of the final-state proton;
• u(p, s) is the Dirac spinor associated with the plane-wave of a proton with 4-momentum p and spin s; • the quantities γ µ (µ = 0, 1, 2, 3) are the standard Dirac 4 × 4 matrices, satisfying the relation {γ µ , γ ν } = 2g µν I 4 , g µν being the Minkowski metric, with signature '+ − − −'; • the matrices σ µν are defined by the relation: σ µν = i 2 [γ µ , γ ν ]; and • q = p ′ − p represents the 4-momentum transfer (i.e., the 4-momentum of the EM current). The standard Mandelstam variable t is defined as: t := q µ q µ = q 2 . As t ≤ 0 in the physical region for elastic scattering, widely used in Particle Physics is the 4-momentum transfer in the form Q 2 := −t ≥ 0.
The quantity κ p := µ p − 1 is known as 'anomalous magnetic moment' of the proton 1 , where µ p is the numerical value of the magnetic moment of the proton, when expressed in units of the nuclear magneton µ N := e /(2m p ). Recommended by the Particle Data Group (PDG) [2] is µ p = 2.79284734462(82), taken from Ref. [3], a value which will be used in the numerical results of this work.
In Eq. (1), the Dirac F p 1 and Pauli F p 2 form factors are t-dependent functions. According to an older convention (which is also followed in Ref. [1]), these two form factors were taken to satisfy the normalisation conditions: F p 1 (0) = F p 2 (0) = 1. Another convention has gained popularity at more recent times: the constant κ p is now usually absorbed in F p 2 (t), thus yielding the normalisation condition: F p 2 (0) = κ p . As this redefinition of F p 2 (t) (i.e., κ p F p 2 (t) → F p 2 (t)) somewhat simplifies the expressions for scattering, the recent convention will be adopted in this paper.
In the Born approximation (one-photon exchange), the differential cross section, describing electron-proton (ep) elastic scattering in the laboratory frame of reference, was put into the form
dσ dΩ = dσ dΩ ns (F p 1 ) 2 + τ (F p 2 ) 2 + 2τ (F p 1 + F p 2 ) 2 tan 2 (θ/2)(2)
by Rosenbluth [4] (also see Eq. (8.207) in Ref. [1], p. 259), where (dσ/dΩ) ns represents the so-called 'no-structure' differential cross section, frequently referred to as 'Mott cross section', see Eq. (8.49) in Ref. [1], p. 229. (Not included in Eq. (2) are the proton-recoil effects.) The quantity τ in Eq. (2) is defined as the ratio Q 2 /(4m 2 p ). Finally, θ is the scattering angle of the projectile (electron).
The two form factors of the neutron F n 1 and F n 2 are defined similarly, and (of course) are also t-dependent. The Dirac and Pauli form factors of the nucleons were first expressed in terms of the so-called EM Sachs (electric G N E and magnetic G N M ) form factors in the early 1960s [5]:
F N 1 = G N E + τ G N M 1 + τ(3)
and
F N 2 = G N M − G N E 1 + τ ,(4)
where N = p or n. From Eqs. (3,4), one obtains
G N E = F N 1 − τ F N 2(5)
and
G N M = F N 1 + F N 2 .(6)
Evidently, the two Sachs form factors of the proton satisfy the normalisation conditions: G p E (0) = 1 and G p M (0) = µ p . For the corresponding quantities of the neutron: G n E (0) = 0 (the neutron has no net electric charge) and G n M (0) = µ n (equal to −1.91304273(45) [6], according to the PDG recommendation [2]).
Relevant in the context of a pion-nucleon (πN) interaction model [7,8,9,10] (ETH model, henceforth), which accounts for the strong-interaction (hadronic) part of the s-and p-wave scattering amplitudes on the basis of s-, u-, and tchannel Feynman diagrams (see Fig. 1), are the proton (and pion) form factors at pion laboratory kinetic energy T ≤ 100 MeV. The restriction on T (necessitated by several reasons, e.g., see Section 1 of Ref. [10]) imposes an upper limit on the Q 2 values involved in the phase-shift analyses (PSAs) of the ETH model. The Q 2 max value is attained at T = 100 MeV and backward scattering:
Q 2 max ≈ 98941.3 MeV 2 ≈ 0.1 GeV 2 .
On the other hand, the available experimental data, used as input in determinations of the form factors G N E and G N M , span a Q 2 domain exceeding 10 GeV 2 ; therefore, of interest to the ETH model is very small part of the Q 2 domain within which ep experimental data are available.
f 0 (500) π π N N π π N N N, ∆(1232) π π ρ(770) π π N N N, ∆(1232) N N Fig. 1.
The main Feynman graphs of the ETH model: scalar-isoscalar I G (J P C ) = 0 + (0 ++ ) and vector-isovector I G (J P C ) = 1 + (1 −− ) t-channel graphs (upper part), and N and ∆(1232) s-and u-channel graphs (lower part). Not shown in this figure, but also analytically included in the model, are the small contributions from all other scalar-isoscalar and vector-isovector mesons [11] with rest masses below 2 GeV, as well as from all well-established (four-star) s and p higher baryon resonances (HBRs) [12] in the same mass range.
In this work, the masses and the 4-momenta will be expressed in energy units. As a result, some of the expressions herein, involving the 4-momentum of the EM current, will contain the conversion constant c, rather than the reduced Planck constant .
2 Parameterisation schemes at low Q 2
The t-dependence of the Sachs form factors cannot be derived from first principles [13]. Although this is not particularly problematic as far as the description of the experimental data is concerned, it becomes problematic when one attempts to extract from the fitted parameters (of an assumed parameterisation) physical quantities pertaining to properties of the nucleons (see Section 5.3 of Ref. [14]). For the sake of example, regarding the extraction of the rms electric charge radius of the proton from parameterisations of G p E , criticism has appeared in Refs. [14,15]. As Sick and Trautmann pointed out [15], while such parameterisations are "valid representations of the data in the q-region where they have been measured, they are not suitable for an extrapolation to q = 0 where the proton rms-radii are extracted." Section 3 of Ref. [14] provides a detailed account of the complexity in this issue, whereas Section 5.3 therein starts with the remark: "Due to the complications mentioned in Section 3, most authors analyzing the electron scattering data employ parameterizations in q-space only to get the q = 0 slope, without ever worrying what these parameterizations would imply in r-space."
The conundrum lies in the fact that the functional behaviour of G p E (q) (a momentum-space attribute) and of the charge density ρ(r) (a configurationspace attribute), representing the particle proton, are related via a Fourier transformation. Regarding the former, one can write [14]:
G p E (q) = 4π c q ∞ 0 ρ(r) sin qr c rdr ,(7)
whereas for the latter, the inverse transformation yields:
ρ(r) = 1 2π 2 r( c) 2 ∞ 0 G p E (q) sin qr c qdq .(8)
Although both expressions are valid when the recoil velocity of the proton is small (β ≪ 1), relativistic corrections are available, see Section 3 of Ref. [14].
In several parameterisations of G p E , the large-r tail of ρ, obtained from Eq. (8), clashes with our understanding of the proton [14].
In fact, starting 'from the other end' appears to be promising [15]: one could assume ρ(r) distributions, which make physical sense, and obtain the corresponding parameterisations of G p E via Eq. (7). The recoil effects could then be treated as described in Section 3 of Ref. [14].
My main interest in this work concerns the parameterisation of the Sachs form factors of the proton at low Q 2 for the purposes of PSAs conducted with the ETH model. At present, I have no intention to make contributions to the problem of the determination of the rms radii of the proton. Therefore, I will next list some of the available parameterisations (in chronological order of appearance), without entering the subject of the large-r behaviour of ρ. The parameterisations of Sections 2.1 and 2.5 follow dipole models, known to provide a good description of the nucleon form factors at low Q 2 values, and also (at least in case of G n M ) at moderate ones, e.g., see Ref. [16].
The 'standard dipole' parameterisation
Being the product of systematic experimentation by Hofstadter 2 and collaborators at Stanford University, the 'standard dipole' form [5] f
D (t) = 1 − t Λ 2 −2 ,(9)
with Λ 2 = 0.71 GeV 2 , enabled the routine parameterisation of the Sachs form factors of the nucleon G N E and G N M up to the 1980s.
One notable application of the 'standard dipole' forms, relevant to the development of the ETH model in the early 1990s, was in the programme of the NORDITA team, which led to the extraction of the EM corrections [17,18,19], suitable for the analysis of the π ± p scattering data; these corrections extend up to a momentum (in the centre-of-momentum (CM) frame of reference) equal to three times the charged-pion mass, equivalent to T ≈ 531 MeV. Regarding their formulae (one-photon-exchange contribution to the EM scattering amplitude), one needs to pay some attention as Tromborg, Waldestrøm, and Øverbø abide by the NORDITA definition of the Pauli form factor, which is different to everyone else's:
(F N 2 ) NORDITA = F N 2 /(2m p ), where F N 2
is the Pauli form factor of this work 3 . The NORDITA parameterisation of the Sachs form factors of the nucleon followed the scheme [20]:
G p E (t) = G p M (t) µ p = G n M (t) µ n = f D (t) , G n E (t) = 0 .(10)
For the pion form factor F π , NORDITA used [20]:
F π (t) = F p 1 (t) − F n 1 (t).
A parameterisation from the A1 Collaboration
In 2010, the A1 Collaboration published the results of an analysis of differential cross-section and polarisation ep measurements acquired at the Mainz Microtron (MAMI) [21]. Their more detailed 2014 paper [22] included important information on their parameterisations of the Sachs form factors of the proton G p E and G p M . The results of their fits had appeared already in 2010 as material supplementing their first paper. Chosen herein are the fitted values of G p E (t) and G p M (t), obtained on the basis of spline fits to the World data;
according to Table IV of Ref. [22], their spline fits yield the best description of the input database. Some criticism about the results of Ref. [21] is expressed in Section 4 of Ref. [14].
The VAMZ parameterisation
In 2007, Arrington, Melnitchouk, and Tjon [23] extracted G p E (t) and G p M (t) from constrained fits to the available cross-section and polarisation ep measurements, including corrections accounting for two-photon-exchange effects [24]. The functions G p E and G p M were parameterised using a Padé approximant of order [3/5], namely
G p E (t), G p M (t)/µ p = 1 + 3 i=1 a i τ i 1 + 5 i=1 b i τ i .(11)
The fitted values of the parameters a i and b i for the two form factors were presented in tabulated form (see Table I of Ref. [23]).
A few years later, Venkat, Arrington, Miller, and Zhan [25] used again the Padé parameterisation of Eq. (11) -with renamed parameters (q i for G p E and p i for G p M /µ p ) -as well as an improved theoretical background. The PSAs of the low-energy πN data with the ETH model since 2015 have been based on the Sachs form factors of the proton obtained in Ref. [25]. For the sake of brevity, this parameterisation will be named 'VAMZ' henceforth.
The YAHL parameterisation
Arrington's team employed another parameterisation of the Sachs form factors of the nucleon G N E and G N M in 2018 [26], this time in terms of the so-called zexpansion; for the sake of brevity, this parameterisation will be named 'YAHL' henceforth. Their model rests upon high-degree polynomials (twelve-degree polynomials are used for the proton form factors, ten-degree polynomials for the neutron form factors) in the variable z, representing a conformal mapping of t onto the unit circle:
z = √ t cut − t − √ t cut − t 0 √ t cut − t + √ t cut − t 0 ,(12)
where t cut = 4m 2 c (two-pion cut), whereas the free parameter t 0 (the root of z(t) = 0) was globally fixed in Ref. [26] to −0.7 GeV 2 . The authors tabulated their results in the document 'Explanation of Supplementary Material', also taking the trouble to detail the input to their optimisation scheme and make it available to others as supplementary material to their paper. Albeit straightforward, the authors also provided the code relating to the implementation of their results in two standard computer languages.
The parameterisation of this work
To model the t-dependence of the Sachs form factors of the proton G p E and G p M , two dipoles will be introduced in this work, featuring two parameters, namely the rms electric charge radius r E;p of the proton and its rms magnetic radius r M ;p . The two radii are defined on the basis of the Sachs form factors of the proton according to the relations:
r 2 E,M ;p ≡ r 2 E,M p := 6( c) 2 G p E,M (0) dG p E,M (t) dt t=0 .(13)
As
G p E,M (t) = G p E,M (0) 1 − t Λ 2 E,M ;p −2 ,(14)
one obtains
Λ E,M ;p = 2 √ 3 c r E,M ;p .(15)
For the sake of completeness, regarding the pion form factor F π (which is also required in the EM part of the πN interaction), a monopole is being used in the ETH model since 2015. (In fact, it does not matter much whether the pion form factor is parameterised at low Q 2 according to the dipole or to the monopole model.)
F π (t) = 1 − t Λ 2 E;π −1 ,(16)
where
Λ E;π = √ 6 c r E;π ,(17)
which is the equivalent of Eqs. (15) in case of a monopole model.
Results
The The maximal symmetric mean absolute differences 4 in G p E among these results range (in the Q 2 domain of the plot) between 0.23 % (A1-YAHL) and 2.76 % ('standard dipole'-VAMZ). Regarding G p M , the differences range between 0.51 % ('standard dipole'-A1) and 2.05 % (A1-VAMZ). In comparison with the other three solutions, the 'standard dipole' systematically overestimates G p E in the Q 2 domain of Fig. 2. A similar effect is observed in Fig. 3 for the two solutions originating from Arrington's team. Regarding the low-Q 2 behaviour of the A1 solution for G p M , one notices that it exceeds all other so- lutions, including the one corresponding to the 'standard dipole' up to about Q 2 max . This behaviour of the A1 solution for G p M is reflected in a sizeable difference (to all other solutions) in r M ;p ; as inspection of Table 1 (which will be introduced shortly) reveals, the r M ;p value in the A1 solution is about 10 % lower than the values associated with the parameterisations of Sections 2.1, 2.3, and 2.4.
Q 2 (MeV 2 ) G p E (Q 2 )/f D (Q 2 ) A1 VAMZ YAHLQ 2 (MeV 2 ) G p M (Q 2 )/(µ p f D (Q 2 )) A1 VAMZ YAHL
To be able to include in the comparison the proton form factors from the parameterisation of this work, one would first need to fix the model parameters r E;p and r M ;p . Before that, however, one test might be helpful: one could first assess the differences between (on the one hand) the parameterisations of Sections 2.1-2.4 and (on the other hand) the parameterisation of this work after fixing the model parameters r E;p and r M ;p to the appropriate values corresponding to each of the former four parameterisation schemes. The r E;p and r M ;p values which have been used in this comparison, along with the results for the maximal symmetric mean absolute difference d max in the Q 2 domain of this work, are given in Table 1. Regarding the A1, VAMZ, and YAHL parameterisations, the r E;p and r M ;p results can be estimated from the parameter values found in Refs. [21,25,26]. For the sake of example, r E;p = Table 1 Comparison of the results obtained with the parameterisation of the Sachs form factors of the proton G p E and G p M of this work (see Section 2.5) with the form factors detailed in Sections 2.1-2.4. The maximal symmetric mean absolute differences d max correspond to the Q 2 domain of this work, namely up to about 0.1 GeV 2 . In this comparison, the model parameters r E;p and r M ;p have been fixed (separately for the purposes of each comparison) to the quoted values per case, which correspond to the results of the parameterisations of Sections 2.1-2.4. Before taking the r E,M ;p values of this table too seriously, the reader should bear in mind the criticism expressed in Ref. [14], as outlined at the beginning of Section 2 of this work.
Parameterisation c 6(q 2 − q 6 )/(2m p ) in the VAMZ parameterisation, and a similar expression holds for r M ;p after the replacement of the parameters: q 2,6 → p 2,6 .
r E;p (fm) r M ;p (fm) d max for G p E (%) d max for G p M (%) '
Of course, given that a single-dipole model is used in the 'standard dipole' parameterisations of G p E and G p M , the perfect agreement between the results of this work and those obtained with the first parameterisation in Table 1 is expected. As it does not assume that r E;p = r M ;p , the parameterisation of this work is more general. The interest in Table 1 lies in the comparisons of the results in the remaining three cases.
It appears that the agreement between the parameterisation of this work and the VAMZ parameterisation is close to the 1 % level. Regarding the comparison with the YAHL parameterisation, the difference is larger in case of G p E , whereas it remains close to 1 % in case of G p M . The maximal differences in case of the comparison with the A1 solution are between 2.0 and 2.6 %, larger in case of G p M . To summarise, the differences between the solutions of Refs. [21,25,26] and the results obtained with the parameterisation of this work (after the appropriate fixation of the model parameters r E;p and r M ;p ) are comparable to the differences among the solutions of Refs. [21,25,26] themselves in the Q 2 domain of this work.
In representative PSAs of the low-energy πN data with the ETH model, the median relative uncertainty in the fitted values of the model parameters is about 5 %. In addition, the maximal relative uncertainties in the model predictions for the π + p differential cross sections between 20 and 45 MeV range between 5.4 and 6.1 %; the corresponding uncertainties in the model predic-tions for the π − p elastic-scattering differential cross sections are considerably larger, reaching 30 % in backward scattering. In comparison, the aforementioned form-factor effects are small. The results of Table 1 indicate that the PSAs of the low-energy πN data with the ETH model can be made selfsufficient by replacing the imported parameterisations of the proton form factors by the simpler scheme of Section 2.5. Provided that reliable r E;p and r M ;p values are supplied, the parameterisation of this work suffices for the purposes of the PSAs of the low-energy πN data with the ETH model. At present however, the fixation of r E;p and r M ;p is not as straightforward as one might expect one century after the proton was given a name. I will next elaborate on this issue.
Before 2010, the estimates for the rms electric charge radius of the proton were predominantly based on the results of analyses of experimental data -(mostly) differential cross sections and (occasionally also) polarisation measurements -from ep elastic scattering; those estimates hovered around 0.875 fm. To the best of my knowledge, the first indications that something might be amiss about the values, recommended both by the PDG as well as by NIST (the former were mostly fixed from the CODATA compilations of the latter), appeared in a 2007 paper by Belushkin, Hammer, and Meißner [27]. After using two theoretical approaches in a dispersion-relation analysis of the ep experimental data, also employing the theoretical constraints of analyticity and unitarity, the authors reported two results (0.830 +0.005 −0.008 and 0.844 +0.008 −0.004 fm), neither of which tallied well with the recommended (at that time) r E;p values, namely 0.8750(68) fm (PDG2006) and 0.8768(69) fm (CODATA2006).
A few years later, the pioneering experiment by Pohl and collaborators enabled the extraction of a precise r E;p value from muonic hydrogen [28]. Being heavier than electrons, muons come closer to the hydrogen nucleus, the consequence being that several small effects (e.g., effects pertaining to vacuum polarisation, to the fine/hypefine splitting, to the proton size, etc.) yield a larger (in comparison with the electronic hydrogen) difference in the binding energies of the 2S and 2P states (Lamb shift). The result of that experiment was: r E;p = 0.84184(36)(56) fm. A second, even more precise value from a follow-up experiment (also on muonic hydrogen) became available in 2013 [29], confirming the earlier result: r E;p = 0.84087(26)(29) fm. Several turbulent years followed, during which attempts were made towards a resolution of what became known as 'the proton-radius puzzle' on the basis of established or (more frequently) exotic Physics.
Between 2013 and 2019, the PDG retained the neutral (and somewhat awkward) stand of reporting both results (i.e., the CODATA2012/2016 results, as well as the results of Ref. [29]) in their compilations, encouraging the experimentalists to settle the obvious discrepancy. As the 2016 results from muonic deuterium [30], from electronic hydrogen 5 [32,33] in 2017 and 2019, as well as from re-analyses of the ep elastic-scattering data [34,35,36,37,38] all pointed in the direction of a 'low' r E;p value, the CODATA2016 result was dropped in the recent PDG compilation; recommended now by the PDG is r E;p = 0.8409(4) fm [2], an average obtained from the results of Refs. [29,33,39]; these three results originate from measurements of the Lamb shift in muonic and electronic hydrogen, as well as from a recent ep elastic-scattering experiment ('PRad' -Proton Radius Experiment at the Thomas Jefferson National Accelerator Facility) at low Q 2 , respectively. The NIST also adapted their recommendation to r E;p = 0.8414 (19) fm (CODATA2018 [40]). All would have been perfect, had it not been for one hitch. In a 2019 paper, Hagelstein and Pascalutsa [41] demonstrated that a lower bound for r E;p can be obtained from the ep elastic-scattering data; with 95 % confidence, the lowest acceptable r E;p value appears to be equal to 0.847 fm, i.e., a value exceeding both results from muonic hydrogen [28,29] by several standard deviations.
There is no doubt that the incompatibility between the results of Refs. [28,29] and the lower bound obtained in Ref. [41] calls for further investigation. In addition, it is imperative to understand the source of the discrepancy between the former results from the ep elastic-scattering data and the currentlyrecommended values. These two comments have appeared in several other works.
Equally confusing is the available information on the rms magnetic radius of the proton r M ;p . The recommended value by the PDG between 2011 and 2015 was the 2010 result by the A1 Collaboration [21], whereas between 2016 and 2018 the PDG favoured a similar result obtained in Ref. [42] also from an analysis of the data from Mainz. In fact, two results had been obtained in Ref. [42]: 0.776(34)(17) fm from the data acquired in Mainz and 0.914(35) fm from the data acquired elsewhere 6 . Although these two results are incompatible (the p-value, corresponding to their reproduction by one constant, is equal to about 7.57 · 10 −3 ), Ref. [42] reported "a simple average" of 0.851 (26) fm, which (surprisingly) the PDG adopted in their 2019 compilation. However, when fitting a constant to incompatible measurements, it is imperative to correct the fitted uncertainties for the (poor) quality of the fit via the application of the Birge factor (which, in this case, comes out equal to about 2.67); if not, the resulting uncertainty is not representative of the variation of the input values. The correct weighted average of the two incompatible r M ;p results of Ref. [42] is not 0.851 (26), but 0.851(69) fm! For the sake of completeness, I will next give some results for the pion form factor F π , which also enters the EM part of the πN scattering amplitude of the ETH model. As Fig. 4 demonstrates, the agreement between the monopole model of this work (using r E;π = 0.659 fm [2]) and the NORDITA parameterisation of F π is satisfactory. The data shown in the figure have been taken from Refs. [43,44], which reported |F π | 2 at forty (in total) Q 2 values below 0.1 GeV 2 .
The treatment of the normalisation effects in the experiment of Ref. [43] according to the Arndt-Roper method [45] (see Appendix A) yields r E;π = 0.663(23) fm, which is the value reported in Ref. [43]. In an obvious attempt to demonstrate the near model-independence of their r E;π result, several estimates were extracted (and reported) in Ref. [44], all accompanied by smaller uncertainties than those obtained in this work from the same set of data: from a monopole fit with constrained normalisation, the authors obtained 0.657 (8) fm; from a monopole fit with free normalisation, they obtained 0.653(8) fm; from a dipole fit, they obtained 0.637(8) fm; finally, using a Padé-type parameterisation (which they evidently favoured), they obtained 0.663(6) fm (this result was unquestionably imported into the PDG database [2]). Although plurality can be desirable on several occasions, I find it confusing in this case. The use of the Arndt-Roper formula in the optimisation of the data of Ref. [44], following the monopole approximation, yields the result: r E;π = 0.664(11) fm, end of story. I subsequently pursued a common analysis of the two datasets [43,44] and obtained the result: r E;π = 0.664(10) fm, as well as fitted scale factors which were close to 1, namely equal to 0.9987 and 0.9938 for the datasets of Refs. [43,44], respectively. Owing to the fact that the normalisation uncertainties in the two experiments were reported as 1.0 % [43] and 0.9 % [44], the differences of two fitted scale factors to 1 are well within the reported normalisation uncertainties.
Conclusions
The first goal in this work was the comparison of the results obtained from four parameterisations of the Sachs form factors of the proton G p E (electric) and G p M (magnetic) in the region of interest to a pion-nucleon (πN) interaction model (ETH model) [7,8,9,10], namely for 4-momentum transfer Q 2 0.1 GeV 2 . Compared were the results obtained
• from the 'standard dipole' (Section 2.1), which had provided the routine parameterisation of the nucleon form factors up to the 1980s; • from a spline fit to electron-proton cross-section and polarisation measure- Fig. 4. The pion form factor F π . The NORDITA team defined [20]: F π (t) = F p 1 (t) − F n 1 (t). The parameterisation of this work rests upon Eqs. (16,17) with r E;π = 0.659 fm [2]. The data shown come from two experiments: the filled points have been taken from Ref. [43], the open ones from Ref. [44]. In both cases, only the statistical uncertainties are shown. ments (Section 2.2); and • from two solutions from Arrington's teams (Sections 2.3 and 2.4).
The relative differences between these solutions remain smaller than about 2.8 % in the aforementioned Q 2 region of interest.
The second goal herein was to introduce -and, to an extent, test -a parameterisation of the Sachs form factors of the proton G p E and G p M based on two dipoles, one pertaining to G p E (admitting as parameter the rms electric charge radius of the proton r E;p ), the other to G p M (admitting as parameter the rms magnetic radius of the proton r M ;p ). Comparisons between (on the one hand) the four aforementioned parameterisation schemes of G p E and G p M at low Q 2 and (on the other hand) the parameterisation of this work were enabled after fixing the model parameters r E;p and r M ;p to the appropriate values corresponding to each of the four parameterisations of Sections 2.1-2.4.
The resulting differences in the aforementioned Q 2 region of interest remained below about 2.6 % (see Table 1), i.e., slightly below the maximal differences found when comparing the four parameterisations of Sections 2.1-2.4 among themselves.
The conclusion from these comparisons is that the parameterisation of this work suffices for the purposes of the ETH model. The replacement of the form factors, which the ETH model used after 2015, by the simple parameterisation of this work will remove from the phase-shift analyses (PSAs) of the low-energy πN data with the ETH model the largest part of the dependence on extraneous sources.
The decision regarding the fixation of the two model parameters, r E;p and r M ;p , may be postponed to the time when the next PSA of the ETH model will be conducted.
A Formal treatment of datasets with known normalisation uncertainty
The formal procedure for treating datasets, which are subject to normalisation uncertainty, rests upon the use of the Arndt-Roper formula [45], see Refs. [8,10] (and several other references therein). According to this method, one parameter is introduced per dataset, to account for the fact that the absolute normalisation of each dataset is known with a finite (non-zero) uncertainty. This parameter, named normalisation parameter in Ref. [45] and scale factor in Refs. [8,10], is applied to each input dataset as a whole: all datapoints of a dataset are affected by the same (relative) amount.
The determination of the absolute normalisation of each dataset involves a 'calibration' procedure, resting upon a comparison of experimental results of the reaction at issue with those of a reaction whose absolute normalisation is more accurately known. This comparison introduces one additional uncertainty, the normalisation uncertainty, which encompasses all known uncertainties associated with the calibration procedure. In practice, the fixation of the absolute normalisation of each dataset may be thought of as resting upon one measurement, which is accompanied by an uncertainty, as all other datapoints of the dataset. Consequently, the treatment of the normalisation uncertainty in a manner which is different to that of the uncertainties of any of the datapoints of the dataset is hardly justifiable.
As I have found several statements in the literature, expressing discomfort at the 'free' floating of the datasets and the introduction of 'the additional parameters' it entails, I rather doubt that it is generally understood that the 'floating' of the datasets is not 'free', but 'controlled', in that it is accompanied by an appropriate compensation to the minimisation function. The χ 2 contributions of each dataset arise from the differences between the rescaled fitted values and the input values, as well as from a term taking account of the departure of the scale factor of the dataset from 1. As each scale factor appears only in the χ 2 contribution of one dataset, the minimisation of the overall χ 2 (with respect to each scale factor) is equivalent to the minimisation of the χ 2 contribution of each dataset (with respect to its own scale factor). This requirement fixes each scale factor from the fitted and the input values at each step of the optimisation.
To conclude, it is true that one additional parameter per dataset is introduced in the optimisation when the floating of the datasets is allowed. However, this parameter is fixed at each step of the optimisation. Consequently, one ends up with exactly the same number of degrees of freedom in the problem as when the normalisation effects are altogether ignored (i.e., when no floating of the input datasets is allowed). The interested reader is referred to Refs. [8,10].
parameterisations of the proton form factors of Sections 2.1-2.4 are 'fixed in time', in that they have been obtained on the basis of certain educated guesses for their parameters ('standard dipole') or from fits to available data (A1, VAMZ, YAHL), where several other physical constants had been imported from extraneous sources. The results for the Sachs form factors of the
Fig. 2 .
2The dependence of the Sachs form factor of the proton G p E on the square of the 4-momentum transfer Q 2 := −t, where t is the standard Mandelstam variable. The Q 2 range corresponds to the domain of interest in the context of the ETH model of the πN interaction, corresponding to pion laboratory kinetic energy T ≤ 100 MeV. To reduce the range of variation of G p E , ratios are shown of the parameterisations of Sections 2.2-2.4 to the values obtained with the 'standard dipole' of Section 2.1. proton, obtained from these parameterisations, are compared in Figs. 2 and 3.
Fig. 3 .
3The equivalent of Fig. 2 for the Sachs form factor of the proton G p M .
The word 'anomalous' indicates that κ p is the magnetic moment in excess of 1; for a structureless proton, κ p vanishes.
For "his pioneering studies of electron scattering in atomic nuclei and for his thereby achieved discoveries concerning the structure of the nucleons," Hofstadter was awarded the Nobel Prize in Physics 1961.3 For the 'nucleon mass' m N , the NORDITA team assumed in their works: m N = m p ; others favour: m N = (m p + m n )/2.
The symmetric mean absolute difference between two values v 1 and v 1 is defined equal to 2|v 1 − v 2 |/(|v 1 | + |v 2 |).
A 2018 result[31] is in favour of a high r E;p value, thus contradicting the results obtained from the other two experiments on electronic hydrogen.6 Incidentally, 'World data' as a description of 'data acquired anywhere but Mainz' is misleading.
AcknowledgementsI am indebted to M. Horbatsch and to J.C. Bernauer for their prompt response to my questions.
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[
"The 1-Jettiness DIS Spectrum: Factorization, Resummation, and Jet Algorthim Dependence",
"The 1-Jettiness DIS Spectrum: Factorization, Resummation, and Jet Algorthim Dependence"
]
| [
"Zhong-Bo Kang \nTheoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n",
"Xiaohui Liu \nMaryland Center for Fundamental Physics\nUniversity of Maryland\n20742College Park, MarylandUSA\n\nCenter for High Energy Physics\nPeking University\n100871BeijingChina\n",
"Sonny Mantry \nDepartment of Physics\nUniversity of North Georgia\n30597DahlonegaGAUSA\n",
"Jianwei Qiu \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n\nC.N. Yang Institute for Theoretical Physics\nStony Brook University\n11794Stony BrookNYUSA\n"
]
| [
"Theoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNMUSA",
"Maryland Center for Fundamental Physics\nUniversity of Maryland\n20742College Park, MarylandUSA",
"Center for High Energy Physics\nPeking University\n100871BeijingChina",
"Department of Physics\nUniversity of North Georgia\n30597DahlonegaGAUSA",
"Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA",
"C.N. Yang Institute for Theoretical Physics\nStony Brook University\n11794Stony BrookNYUSA"
]
| []
| The 1-Jettiness (τ1) event shape for Deep Inelastic Scattering (DIS), allows for a quantitative and global description of the pattern of QCD radiation for single jet (J) production in electron-nucleus (NA) collisions e − + NA → e − + J + X. It allows for precision studies of QCD and is a sensitive probe of nuclear structure and dynamics. The large transverse momentum (PJ T ) of the final state jet J, characterizes the hard scale in the problem. The region of phase space where τ1 PJ T , corresponds to configurations where energetic radiation (E ∼ PJ T ) is only along either the single jet direction or the beam direction with only soft radiation (E ∼ τ1 PJ T ) in between. Thus, the restriction τ1 PJ T corresponds to a veto on additional jets and leads to large Sudakov logarithms of τ1/PJ T that must be resummed. Based on a factorization framework, derived using the Soft Collinear Effective Theory (SCET), we provide resummation results at the NNLL level of accuracy and match them onto the NLO result in fixed order perturbation theory, appropriate in the τ1 ∼ PJ T region where additional jets and hard radiation are allowed. The τ1-distribution depends on the jet algorithm used to find the leading jet in the region τ1 ∼ PJ T , unlike the resummation region where this dependence is power suppressed in τ1/PJ T 1. We give results for the entire τ1 spectrum, with a smooth matching between the resummation region and fixed-order region, where we make use of the anti-kt jet algorithm. The 1-Jettiness event shape can be a powerful probe of nuclear and QCD dynamics at future electron-ion colliders and by analyzing existing HERA data.Event shapes probe QCD dynamics, providing a quantitative global measure to characterize QCD radiation in the final state and allowing good analytic control over the corresponding calculations. Various event shapes for Deep Inelastic Scattering (DIS) processes were first studied in Refs.[1,2,3,4]. Thrust [1] and Broadening [3] distributions were studied at the next-toleading logarithmic (NLL) level of accuracy and matched at O(α s ) to fixed-order results. A numerical comparison was also done against O(α 2 s ) results[5,6]. Thrust distributions were measured at HERA by the H1 [7, 8] and ZEUS [9, 10] collaborations.In Refs. [11,12,13,14], single jet (J) production in the DIS process1 | null | [
"https://arxiv.org/pdf/1503.04210v1.pdf"
]
| 118,501,858 | 1503.04210 | ed5ee8c4e88cb9293fb515e73b4937fb31005a51 |
The 1-Jettiness DIS Spectrum: Factorization, Resummation, and Jet Algorthim Dependence
13 Mar 2015
Zhong-Bo Kang
Theoretical Division
Los Alamos National Laboratory
87545Los AlamosNMUSA
Xiaohui Liu
Maryland Center for Fundamental Physics
University of Maryland
20742College Park, MarylandUSA
Center for High Energy Physics
Peking University
100871BeijingChina
Sonny Mantry
Department of Physics
University of North Georgia
30597DahlonegaGAUSA
Jianwei Qiu
Physics Department
Brookhaven National Laboratory
Upton11973NYUSA
C.N. Yang Institute for Theoretical Physics
Stony Brook University
11794Stony BrookNYUSA
The 1-Jettiness DIS Spectrum: Factorization, Resummation, and Jet Algorthim Dependence
13 Mar 20151 *Speaker at PAVI 14: From Parity Violation to Hadron Structure, Skaneateles, NY.
The 1-Jettiness (τ1) event shape for Deep Inelastic Scattering (DIS), allows for a quantitative and global description of the pattern of QCD radiation for single jet (J) production in electron-nucleus (NA) collisions e − + NA → e − + J + X. It allows for precision studies of QCD and is a sensitive probe of nuclear structure and dynamics. The large transverse momentum (PJ T ) of the final state jet J, characterizes the hard scale in the problem. The region of phase space where τ1 PJ T , corresponds to configurations where energetic radiation (E ∼ PJ T ) is only along either the single jet direction or the beam direction with only soft radiation (E ∼ τ1 PJ T ) in between. Thus, the restriction τ1 PJ T corresponds to a veto on additional jets and leads to large Sudakov logarithms of τ1/PJ T that must be resummed. Based on a factorization framework, derived using the Soft Collinear Effective Theory (SCET), we provide resummation results at the NNLL level of accuracy and match them onto the NLO result in fixed order perturbation theory, appropriate in the τ1 ∼ PJ T region where additional jets and hard radiation are allowed. The τ1-distribution depends on the jet algorithm used to find the leading jet in the region τ1 ∼ PJ T , unlike the resummation region where this dependence is power suppressed in τ1/PJ T 1. We give results for the entire τ1 spectrum, with a smooth matching between the resummation region and fixed-order region, where we make use of the anti-kt jet algorithm. The 1-Jettiness event shape can be a powerful probe of nuclear and QCD dynamics at future electron-ion colliders and by analyzing existing HERA data.Event shapes probe QCD dynamics, providing a quantitative global measure to characterize QCD radiation in the final state and allowing good analytic control over the corresponding calculations. Various event shapes for Deep Inelastic Scattering (DIS) processes were first studied in Refs.[1,2,3,4]. Thrust [1] and Broadening [3] distributions were studied at the next-toleading logarithmic (NLL) level of accuracy and matched at O(α s ) to fixed-order results. A numerical comparison was also done against O(α 2 s ) results[5,6]. Thrust distributions were measured at HERA by the H1 [7, 8] and ZEUS [9, 10] collaborations.In Refs. [11,12,13,14], single jet (J) production in the DIS process1
1
Abstract. The 1-Jettiness (τ1) event shape for Deep Inelastic Scattering (DIS), allows for a quantitative and global description of the pattern of QCD radiation for single jet (J) production in electron-nucleus (NA) collisions e − + NA → e − + J + X. It allows for precision studies of QCD and is a sensitive probe of nuclear structure and dynamics. The large transverse momentum (PJ T ) of the final state jet J, characterizes the hard scale in the problem. The region of phase space where τ1 PJ T , corresponds to configurations where energetic radiation (E ∼ PJ T ) is only along either the single jet direction or the beam direction with only soft radiation (E ∼ τ1 PJ T ) in between. Thus, the restriction τ1 PJ T corresponds to a veto on additional jets and leads to large Sudakov logarithms of τ1/PJ T that must be resummed. Based on a factorization framework, derived using the Soft Collinear Effective Theory (SCET), we provide resummation results at the NNLL level of accuracy and match them onto the NLO result in fixed order perturbation theory, appropriate in the τ1 ∼ PJ T region where additional jets and hard radiation are allowed. The τ1-distribution depends on the jet algorithm used to find the leading jet in the region τ1 ∼ PJ T , unlike the resummation region where this dependence is power suppressed in τ1/PJ T 1. We give results for the entire τ1 spectrum, with a smooth matching between the resummation region and fixed-order region, where we make use of the anti-kt jet algorithm. The 1-Jettiness event shape can be a powerful probe of nuclear and QCD dynamics at future electron-ion colliders and by analyzing existing HERA data.
Event shapes probe QCD dynamics, providing a quantitative global measure to characterize QCD radiation in the final state and allowing good analytic control over the corresponding calculations. Various event shapes for Deep Inelastic Scattering (DIS) processes were first studied in Refs. [1,2,3,4]. Thrust [1] and Broadening [3] distributions were studied at the next-toleading logarithmic (NLL) level of accuracy and matched at O(α s ) to fixed-order results. A numerical comparison was also done against O(α 2 s ) results [5,6]. Thrust distributions were measured at HERA by the H1 [7,8] and ZEUS [9,10] collaborations.
In Refs. [11,12,13,14], single jet (J) production in the DIS process
e − + N A → J + X,(1)
e e Beam remnants Soft Radiation p nce parton correlation functions. Alternatively, one can consider jet production where cattered electron is unobserved. In this case, it is the large transverse momentum of et that plays the role of the hard scale in the process. Such a process has been studied e past in the context of spin-dependent observables [41]. this work, we consider the process in Eq.(1) with an additional constraint imposed by -jettiness event shape ⌧ 1 . The use of 1-jettiness as a global DIS event shape was first osed in Ref. [42]. In particular, we are interested in the di↵erential cross-section
d A ⌘ d 3 (e + N A ! J + X) dy dP J T d⌧ 1 ,(2)
e P J T and y are the transverse momentum and rapidity of the jet J, respectively. The t shape ⌧ 1 restricts the radiation between the final state jet and the nuclear beam tions. In the limit ⌧ 1 ! 0, the final state jet becomes infinitely narrow and only soft tion (of energy E ⇠ ⌧ 1 ) is allowed between the nuclear beam and jet directions. Any etic radiation must be closely aligned with either the beam or jet directions. This is atically illustrated in Fig. 1. We restrict ourselves to such configurations by imposing hase space condition
⌧ 1 ⌧ P J T .(3)
ctorization and resummation framework for the 1-jettiness DIS event shape, in this n of phase space, was first derived in Ref.
[42] he detailed properties of the radiation illustrated in Fig. 1 will be a↵ected by the nuclear t in the process. For example, for larger nuclei one typically expects enhanced hadronic ity between the jet and beam directions. The soft radiation between the beam and jet tions can be a↵ected by jet quenching or energy loss as the jet emerges from the nuclear um. This is because partons produced in the hard collisions could undergo multiple ering inside the large nucleus and thus lead to induced gluon radiation [14,43,44] when ng through the nucleus to form the observed hadron or jet. While such e↵ects can be ed by varying jet shape parameters, the information about soft radiation at wide angles the jet is often lost. The main idea advocated in this paper is to study the properties e observed radiation in Fig. 1, quantified by distributions in the configuration space
P J = X k p k ✓( 2q A · p k Q a 2q J · p k Q J ),(7)2 ⇠ ⌧ 1 P J T .(8)⌧ 1 ⇠ P J T(9)
II. BASIC IDEA
To perferm a numerical evaluation of the integration, we have to be able to extract the infrared poles. At the NLO level, the idea is very straight forward: we parametrize the phase space using variables x i 's where x i 2 [0, 1], to make the phase space integration has the form
Z dPS F = Z Y dx i x 1 ai✏ i ⇥ [x bi i ⇥ F ] ,(10)
where we demand that x bi i ⇥ F is finite when x i ! 0. Given that all the observables are infrared safe, all the infrared poles can be extracted by expanding
x 1 ai✏ i = 1 a i ✏ (x i ) + X ( ✏a i ) n n! ✓ log n (x) x ◆ + .(11)
And therefore in
Z dPS F = A ✏ 2 + B ✏ + C ,(12)
all A, B and C can obtained at least numerically. Since the physical nature of infrared divergence is related only to soft E ! 1 and collinear ✓ ! 0, the parametrization is very easy to figure out at NLO or even NNLO level. To achieve this, in most cases we need to partition the phase space into di↵erent sectors. In each sector, only one parton can reach its soft singularity and only one pair of partons can have collinear singularity. For instance, for e i q i ! e f q f g case, we have to introduce partitioning to isolate the cases where g is parallel to q i or q f , while e i g ! e f qq no partitioning is needed, as long as we demand at least one high p T jet. where N A denotes a nucleus with atomic weight A, was first studied using the 1-Jettiness event shape (τ 1 ), a specific application of the N -Jettiness event shape [15,16] first introduced to study exclusive N -jet production at the LHC. A factorization and resummation framework was derived [11,12] for the observable
dσ A ≡ d 3 σ(e − + N A → J + X) dy dP J T dτ 1 ,(2)
in the limit τ 1 P J T , where P J T and y denote the transverse momentum and the rapidity of the jet (J). The 1-jettiness global event shape τ 1 is defined as
τ 1 = k min 2q A · p k Q a , 2q J · p k Q J ,(3)
where the sum is over all final state particles (except the final state lepton) with momenta p k . The light-like four-vectors q A and q J denote reference vectors along the nuclear beam and final state jet directions respectively. In general, an external jet algorithm is used to determine leading jet and the light-like vector q J is aligned with it. The constants Q a and Q J are of the order of the hard scale and their choices are not unique; different choices correspond to different definitions of τ 1 . The 1-Jettiness algorithm associates all final state particles either with the beam region or with the jet region according to the minimization condition in Eq. (3). The momentum of the final state jet P J , defined in the 1-Jettiness framework, is then given by the sum of the momenta of all particles in the jet region
P J = k p k θ( 2q A · p k Q a − 2q J · p k Q J ).(4)
Note that the external jet algorithm is only used to determine the light-like reference vector q J and that the 1-Jettiness jet momentum P J is in general distinct from that of the leading jet found by the external algorithm, as explained in detail in Ref. [14]. The limit τ 1 P J T corresponds to configurations that typically look like that shown in Fig. 1; any energetic radiation (E ∼ P J T ) in the final state is closely aligned either along the jet direction or along the beam direction. At wide angles from these directions, the restriction τ 1 P J T only allows for soft radiation (E ∼ τ 1 P J T ). In effect, the restriction τ 1 P J T acts as a veto on additional jets or hard radiation at wide angles from the beam or leading jet directions. This restriction on final state radiation gives rise to large Sudakov logarithms of the form α n s ln 2m (τ 1 /P J T ) for m ≤ n, that require resummation. Since the dynamics in the τ 1 P J T region is dominated by radiation collinear with either the jet or beam directions and soft radiation in all directions, the Soft Collinear Effective Theory (SCET) [17,18,19,20,21,22] is the appropriate effective theory to derive a factorization and resummation framework. This factorization and resummation framework was first developed in Refs. [11,12] and has the schematic form
dσ resum ≡ d 3 σ resum dydP J T dτ 1 ∼ H ⊗ B ⊗ J ⊗ S,(5)
where H is the hard function, B is the beam function [23,15] that describes the dynamics of the initial state PDF and the perturbative initial state radiation collinear with the beam direction, J is the jet function describing the dynamics of the collinear radiation in the final state jet, and S is the soft function describing the dynamics of soft radiation (E ∼ τ 1 ) throughout the event.
The beam function is matched onto the standard PDF B ∼ I ⊗ f, where I is perturbatively calculable and describes perturbative collinear radiation along the beam direction. The hard, jet, beam, and soft functions have renormalization scales with respective scalings
µ H ∼ P J T , µ J ∼ µ B ∼ τ 1 P J T , µ S ∼ τ 1 ,(6)
that minimize any large logarithms in the respective functions. All objects in Eqs. (5) are evaluated at a common scale µ after using renormalization group equations in SCET to evolve them from their natural scalings in Eq. (6), thereby resuming large logarithms. We refer the reader to Refs. [12,13] for a detailed version of the factorization formula, including the derivation and field-theoretic definitions of the hard, jet, beam, and soft functions. Numerical results with a resummation of Sudakov logarithms at the next-to-next-leading logarithmic (NNLL) level of accuracy were first presented in Ref. [12]. These results also included a wide range of nuclear targets: proton, Carbon, Calcium, Iron, Gold, and Uranium. Shortly thereafter, NNLL resummation results for a proton target were presented in Ref. [24] and they also introduced two new definitions of 1-jettiness, corresponding to different choices of the jet reference vector q J used in the definition of the 1-jettiness event shape.
The situation is quite different in the region
τ 1 ∼ P J T ,(7)
for which the configurations typically look like that shown in Fig. 2. The large value of τ 1 corresponds to easing the veto on hard radiation at wide angles from the beam and leading jet directions. The Sudakov logarithms of τ 1 /P J T are now small and the use of standard perturbation theory is appropriate. In order to obtain the full τ 1 -spectrum, one must smoothly match the resummation (τ 1 P J T ) and fixed-order (τ 1 ∼ P J T ) regions. This was done in Ref. [14], where the resummation region was smoothly matched onto the fixed-order region, at the NNLL + NLO(∼ α s ). More recently, results at NNLL+NLO were also obtained in Ref. [25] for a different definition of 1-Jettiness, which was shown to be equivalent to the DIS thrust [1] event shape and does not use a jet algorithm in its implementation. Schematically, the differential cross-section for the full spectrum can be written as
dσ = [dσ resum − dσ F O resum ] + dσ F O .(8)
Here dσ resum denotes the resummed cross section computed in the region τ 1 P J T . The dσ F O resum is this resummed cross section expanded to fixed-order perturbation theory and is given by setting all scales in the factorization formula equal to each other
dσ F O resum = dσ resum (µ = µ H = µ J = µ B = µ S ),(9)
thereby turning off resummation and leaving only the contributions of fixed-order SCET matrix elements. The dσ F O is the full cross section at the same order in perturbation theory. The dσ F O differs from dσ F O resum by terms that are non-singular in the limit τ 1 → 0. In the resummation region τ 1 P J T , dσ is dominated by dσ resum due to a cancellation between dσ F O resum and dσ F O , up to suppressed non-singular terms. Similarly, in the fixed-order region τ 1 ∼ P J T , dσ is dominated by dσ F O due to a cancellation between dσ resum and dσ F O resum , up to terms suppressed in perturbation theory. Furthermore, in order to smoothly match the τ 1 -spectrum in the resummation and fixedorder regions, one must make use of profile functions [26,27,28] so that the scales µ H , µ B , µ J , and µ S appearing in dσ resum smoothly converge to the single scale µ ∼ P J T that appears in dσ F O . This is essential for important cancellations to occur between the various terms Eq. (8). The full τ 1 -spectrum has three distinct regions
τ 1 ∼ Λ QCD , Λ QCD τ 1 P J T , τ 1 ∼ P J T ,(10)
that must be smoothly connected by matching and the use of profile functions. In the region τ 1 ∼ Λ QCD the soft radiation (E ∼ τ 1 ) becomes non-perturbative and must be modeled. The model must smoothly converge to the perturbative soft function in the region τ 1 Λ QCD . This is accomplished by writing the soft function as a convolution [26,29] between a model function and the partonic soft function.
Note that in the resummation region τ 1 P J T , the dependence on the jet algorithm used to determine q J in Eqs. (3) and (4) power suppressed [16] in τ 1 /P J T . This can be understood by noting that τ 1 P J T corresponds to the configuration in Fig. 1 with a narrow collimnated jet of energetic radiation well-separated from the beam direction with only soft radiation in between. For such configurations, different jet algorithms will find the same direction q J for the leading jet momentum, up to power corrections in τ 1 /P J T corresponding to differences in how soft radiation is grouped into the leading jet. On the other hand, for τ 1 ∼ P J T as in Fig. 2, different jet algorithms can give rise to different directions q J for the leading jet momentum corresponding to differences in how hard radiation at wide angles is grouped into the leading jet.
Thus, in the schematic formula in Eq. (9), dσ resum and dσ F O resum are independent of the jet algorithm of used to find the leading jet, up to power corrections. This allowed the works in Refs. [12,13,24] to provide NNLL results in the resummation region τ 1 P J T without reference to an explicit jet algorithm. On the other hand, in the region τ 1 ∼ P J T an explicit jet algorithm must be used and the reference vector q J will strongly depend on the algorithm used. In particular, the computation of dσ F O in Eq. (8) requires an explicit jet algorithm. In our work we make use of the anti-kt [30] jet algorithm, although our numerical code is flexible enough to use other algorithms. The dσ F O has the schematic form
dσ F O ∼ dP SF meas. ([P S]) |M| 2 ⊗ f,(11)
where dP S is final state phase space measure, |M| 2 is the UV renormalized amplitude squared for the partonic process, f denotes the initial state PDF, andF meas. is the measurement function that imposes restrictions on the final state. In particular, for the observable in Eq.(2) it restricts the final state jet to have a transverse momentum and rapidity of P J T and y respectively and the final state radiation to have the value τ 1 for the 1-jettiness event shape.These final state restrictions along with the anti-kt jet algorithm are implemented numerically using Vegas [31]. For each phase space point, a jet algorithm is implemented to cluster final state particles and find the leading jet. The transverse momentum (K J T ) and rapidity (y K ) of the leading jet are then used to construct the light-like jet reference vector q J = (K J T cosh y K , K J T , K J T sinh y K ). A set of values τ 1 , P J T , y is returned for each phase space point. Numerical integrations are then performed by restricting the phase space to be within specified bin sizes around specified values for τ 1 , P J T , and y.
The partonic channels LO are
e − + q i → e − + q i , e − +q i → e − +q i(12)
where the index i runs over the quark and antiquark flavors. The NLO contribution has three types of partonic channels with the real emission of an extra parton in the final state
e − + q i → e − + q i + g, e − +q i → e − +q i + g e − + g → e − + q i +q i ,(13)
and virtual corrections to the leading order channels in Eq. (12). Infrared (IR) singularities arise in these NLO calculations from the real emission of an extra parton in the final state as well as from the virtual corrections to the leading order process. In order to numerically evaluate dσ F O , it becomes necessary to analytically isolate these IR singularities. We use dimensional regularization, working in d = 4 − 2 dimensions, to isolate the IR divergences as poles in . We also implement the sector decomposition technique [32,33,34,35] in order to break up the phase space into sectors where only single parton or a single pair of partons becomes unresolved, corresponding to the soft and collinear IR divergences. This facilitates the isolation of IR poles after which numerical integration techniques can be used straightforwardly. We refer the reader to Ref. [14] for more details. We now give numerical results for the full 1-Jettiness spectrum for the case of a proton target. We work at an electron-nucleus center of mass energy of 90 GeV and integrate over the jet transverse momentum and rapidity over the ranges [P low J T , P high J T ]=[20 GeV, 30 GeV] and |y| < 2.5 respectively. In Fig. 3 we show the perturbative results for dσ F O /dτ 1 (solid red curve) and |dσ F O resum /dτ 1 | (dashed blade curve). We see that in the limit τ 1 → 0, the dσ F O converges to dσ F O resum as expected since in this limit dσ F O is dominated by terms singular in this limit. This is one of several cross-checks performed [14] on the NLO calculation.
We also note that around τ 1 ∼ 5 GeV, dσ F O resum becomes negative. This corresponds to the region where the non-singular terms, not contained in dσ F O resum , become important. A smooth matching of the resummation and fixed-order regions, as in Eq. (8) is required to properly describe the spectrum over its full range. The result of this matching which also incorporates a non-perturbative soft function in the region τ 1 ∼ Λ QCD is shown in Fig. 4. The various bands in Fig. 4 correspond to the τ 1 -distribution at the NLL (widest blue band), the NLL'+NLO (red band), and the NNLL+NLO (green band) levels of accuracy. The width of the bands indicate the perturbative uncertainty obtained via scale variation. We see that resummation tames the singular behavior in the τ 1 → 0 limit and that the perturbative uncertainty reduces as we go to higher orders in resummation.
Figure 1 .
1Typical
Figure 2 .
2Typical configuration for τ 1 ∼ P J T .
Figure 3 . τ 1
31-distributions for the full NLO (solid red line) and the magnitude of the resummed SCET result expanded to NLO for kinematics given in the text.
Figure 4 .
4The NLL (widest blue band), NLL'+NLO (red band), and the NNLL+NLO (green band) τ 1distributions.
*Speaker at PAVI 14: From Parity Violation to Hadron Structure, Skaneateles, NY.
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. S Frixione, Z Kunszt, A Signer, hep-ph/9512328Nucl.Phys. 467Frixione S, Kunszt Z and Signer A 1996 Nucl.Phys. B467 399-442 (Preprint hep-ph/9512328)
. M Czakon, Phys.Lett. 693Preprint 1005.0274Czakon M 2010 Phys.Lett. B693 259-268 (Preprint 1005.0274)
. R Boughezal, K Melnikov, F Petriello, 1111.7041Phys.Rev. 8534025Boughezal R, Melnikov K and Petriello F 2012 Phys.Rev. D85 034025 (Preprint 1111.7041)
. R Boughezal, F Caola, K Melnikov, F Petriello, M Schulze, JHEP. 130672Preprint 1302.6216Boughezal R, Caola F, Melnikov K, Petriello F and Schulze M 2013 JHEP 1306 072 (Preprint 1302.6216)
| []
|
[
"Observation of a dissipative phase transition in a one-dimensional circuit QED lattice",
"Observation of a dissipative phase transition in a one-dimensional circuit QED lattice"
]
| [
"Mattias Fitzpatrick \nDepartment of Electrical Engineering\nPrinceton University\n08540PrincetonNJUSA\n",
"Neereja M Sundaresan \nDepartment of Electrical Engineering\nPrinceton University\n08540PrincetonNJUSA\n",
"Andy C Y Li \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonILUSA\n",
"Jens Koch \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonILUSA\n",
"A A Houck \nDepartment of Electrical Engineering\nPrinceton University\n08540PrincetonNJUSA\n"
]
| [
"Department of Electrical Engineering\nPrinceton University\n08540PrincetonNJUSA",
"Department of Electrical Engineering\nPrinceton University\n08540PrincetonNJUSA",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonILUSA",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonILUSA",
"Department of Electrical Engineering\nPrinceton University\n08540PrincetonNJUSA"
]
| []
| Condensed matter physics has been driven forward by significant experimental and theoretical progress in the study and understanding of equilibrium phase transitions based on symmetry and topology. However, nonequilibrium phase transitions have remained a challenge, in part due to their complexity in theoretical descriptions and the additional experimental difficulties in systematically controlling systems out of equilibrium. Here, we study a one-dimensional chain of 72 microwave cavities, each coupled to a superconducting qubit, and coherently drive the system into a nonequilibrium steady state. We find experimental evidence for a dissipative phase transition in the system in which the steady state changes dramatically as the mean photon number is increased. Near the boundary between the two observed phases, the system demonstrates bistability, with characteristic switching times as long as 60 ms -far longer than any of the intrinsic rates known for the system. This experiment demonstrates the power of circuit QED systems for studying nonequilibrium condensed matter physics and paves the way for future experiments exploring nonequilbrium physics with many-body quantum optics.Over the past decades, there has been remarkable progress in studying both real and synthetic quantum materials. Advances in nanoscale fabrication and cryogenics have allowed for exquisite control of electronic systems -unlocking strongly correlated electronic states and topological materials 1 . Simultaneously, the ability to model desired Hamiltonians with ultra-cold Fermi and Bose gases has allowed unprecedented access to synthetic material properties 2 . As a whole, much of the development of condensed matter physics has focused on the study of (quasi-)equilibrium physics, which is more accessible both experimentally and theoretically. However, the constant presence of dissipation, noise, and decoherence belie the fact that, ultimately, the world is nonequilibrium.A phase transition indicates a sometimes sudden change in the physical properties of a system as a function of some external system parameter. Thermal phase transitions are well-understood in the context of statistical mechanics and occur when the free energy becomes nonanalytic. At zero temperature, the role of quantum fluctuations gives rise to a new set of quantum phase transitions which involve a sudden change in the ground state of a Hamiltonian H; a phase transition occurs when the gap between the first excited state and the ground state closes. These concepts need to be extended to consider nonequilibrium steady states, as the system is no longer in its ground state but rather in a state that balances drive and dissipation. In a dissipative phase transition, the steady state abruptly changes as a system parameter is varied 26 . Whenever the system is describable in terms of a Lindblad master equation 3 ,ρ = Lρ, then such a transition is signalled by the closing of the lowest excitation gap in the spectrum of the Liouvillian superoperator L.In recent years, interacting photons have emerged as an excellent candidate for studying nonequilibrium condensed matter physics due to the lack of particle number conservation 4 . In cavity quantum electrodynamics, strong coupling between atoms and a cavity can mediate effective photon-photon interactions 5-7 . Arrays of coupled microwave 8 or optical 9 cavities can be fabricated by conventional lithographic techniques, and the competition between on-site interactions and hopping between neighboring cavities can give rise to quantum phase transitions of light 10-12 . A wide range of many-body effects have been predicted in these systems, including a Mott insulator-superfluid phase transition 10-12 and fractional quantum Hall-like states of light 13-15 . Experiments on small systems have demonstrated low-disorder lattices 8 , a dynamical quantum phase transition in a cavity dimer 16 , and chiral ground state currents in a cavity trimer 17 . Circuit QED lattices are inherently open systems, with dissipation an ever-present force that leads both to qubit relaxation as well as the inevitable loss of photons from microwave cavities. While dissipation presents an obstacle for quantum information processing, it is of fundamental interest in the study of nonequilibrium phase transitions. Just as excitations inevitably leak from the system, it is easy to add photons back and to drive into a steady state, making these systems particularly amenable to the study of dissipative phase transitions.In this paper we present experimental evidence for a dissipative phase transition in a circuit QED lattice. We observe that at drive frequencies between the low-power resonance frequencies of the system, there exists a region of hysteresis and bistability where the steady state of the system switches stochastically between two states ρ 1 and ρ 2 . By determining the corresponding switching rates, we can obtain the so-called asymptotic decay rate which characterizes the closing of the spectral gap of L. At the transition between the two states, the characteristic switching times become exceptionally long, a key characteristic of a dissipative phase transition. A similar observation has recently been made in a single cavity system with multiple qubits 18 . arXiv:1607.06895v1 [quant-ph] 23 Jul 2016 2 FIG. 1. 72-site circuit QED lattice. a, Coplanar waveguide resonators, each with a bare cavity frequency of 7.5 GHz, are capacitively coupled to form a linear chain on a 2.5 × 2.5 cm 2 chip. Each resonator is coupled to its neighboring resonators to yield a hopping rate t/2π ≈ 144 MHz and has an average photon loss rate of κ/2π ≈ 1.6 MHz. At three intermediate chain sites, three-way coupling capacitors provide ports for secondary input and output lines (arrows on sides). b-c, A transmon qubit is capacitively coupled to the center pin near the edge of each resonator in the lattice, ensuring coupling to the fundamental mode of each resonator with strength g. The coupled resonator-qubit system forms the fundamental unit cell of the lattice. d, The circuit can be modeled as a linear chain of coupled oscillators, each dispersively coupled to a weakly anharmonic multi-level system.Our device, shown inFig. 1a, consists of a linear chain of 72 lattice sites. Each site comprises a coplanarwaveguide resonator with fundamental-mode frequency ω/2π = 7.5 GHz, coupled to a transmon qubit 19(Fig. 1b)placed at one of the resonator's voltage antinodes. Resonators are capacitively coupled to neighboring resonators(Fig. 1c), so that photons can hop between nearest-neighbor sites. Variations in transmon qubit frequencies in fabrication are a likely source of uncontrolled disorder that is difficult to compensate for in our lattice. We therefore use an asymmetric SQUID-loop geometry allowing each qubit to be tuned over a finite frequency range via an applied magnetic flux. Because individually tuning 72 qubits is currently infeasible in our system, we instead employ a global magnetic field to simultaneously tune all qubit frequencies. Because each qubit is intentionally fabricated with a SQUID-loop of random area, this randomizes the frequency of all qubits within a band of frequencies near 8.5 GHz. In this way, we can ensure that features of interest are universal to the system rather than artifacts of a particular instance of disorder (see Supplementary Information I).To experimentally study the nonequilibrium behav-ior of the device, we monitor the homodyne transmission across the lattice while varying the drive frequency and scanning the drive power over more than five orders of magnitude(Fig. 2a). At low drive powers, we find the expected discrete transmission peaks associated with the interaction-shifted eigenmode frequencies of the resonator lattice. As we vary the mean photon number in the system by increasing the strength of the drive, we observe that a sudden change in system behavior occurs: transmission peaks split and then, at around −10 dB of drive power, abruptly give way to a region of strongly suppressed transmission. In this high-power region, peak-like features are completely absent.The transition between the low-and high-power phases can be more thoroughly explored by measuring the transmission at a single drive frequency while sweeping the drive power either from low to high (2c) or from high to low(Fig. 2d). Doing so reveals a significant region exhibiting hysteresis, which is located at the top of the low-power lobes where the transition to the high-power phase occurs. Subtracting the transmission signals for the two different sweep directions clearly marks the hysteretic regime, as shown inFig. 2e.To gain insight into this behavior, we model the system as a one-dimensional chain of identical circuit QED elements, as illustrated inFig. 1d. The corresponding Hamiltonian H = | 10.1103/physrevx.7.011016 | [
"https://arxiv.org/pdf/1607.06895v1.pdf"
]
| 3,550,701 | 1607.06895 | 37c376c8cec7959a164534f108a1de9a8ea4027a |
Observation of a dissipative phase transition in a one-dimensional circuit QED lattice
Mattias Fitzpatrick
Department of Electrical Engineering
Princeton University
08540PrincetonNJUSA
Neereja M Sundaresan
Department of Electrical Engineering
Princeton University
08540PrincetonNJUSA
Andy C Y Li
Department of Physics and Astronomy
Northwestern University
60208EvanstonILUSA
Jens Koch
Department of Physics and Astronomy
Northwestern University
60208EvanstonILUSA
A A Houck
Department of Electrical Engineering
Princeton University
08540PrincetonNJUSA
Observation of a dissipative phase transition in a one-dimensional circuit QED lattice
(Dated: July 22, 2016)j (H r j + H q j + H rq j ) + j,j
Condensed matter physics has been driven forward by significant experimental and theoretical progress in the study and understanding of equilibrium phase transitions based on symmetry and topology. However, nonequilibrium phase transitions have remained a challenge, in part due to their complexity in theoretical descriptions and the additional experimental difficulties in systematically controlling systems out of equilibrium. Here, we study a one-dimensional chain of 72 microwave cavities, each coupled to a superconducting qubit, and coherently drive the system into a nonequilibrium steady state. We find experimental evidence for a dissipative phase transition in the system in which the steady state changes dramatically as the mean photon number is increased. Near the boundary between the two observed phases, the system demonstrates bistability, with characteristic switching times as long as 60 ms -far longer than any of the intrinsic rates known for the system. This experiment demonstrates the power of circuit QED systems for studying nonequilibrium condensed matter physics and paves the way for future experiments exploring nonequilbrium physics with many-body quantum optics.Over the past decades, there has been remarkable progress in studying both real and synthetic quantum materials. Advances in nanoscale fabrication and cryogenics have allowed for exquisite control of electronic systems -unlocking strongly correlated electronic states and topological materials 1 . Simultaneously, the ability to model desired Hamiltonians with ultra-cold Fermi and Bose gases has allowed unprecedented access to synthetic material properties 2 . As a whole, much of the development of condensed matter physics has focused on the study of (quasi-)equilibrium physics, which is more accessible both experimentally and theoretically. However, the constant presence of dissipation, noise, and decoherence belie the fact that, ultimately, the world is nonequilibrium.A phase transition indicates a sometimes sudden change in the physical properties of a system as a function of some external system parameter. Thermal phase transitions are well-understood in the context of statistical mechanics and occur when the free energy becomes nonanalytic. At zero temperature, the role of quantum fluctuations gives rise to a new set of quantum phase transitions which involve a sudden change in the ground state of a Hamiltonian H; a phase transition occurs when the gap between the first excited state and the ground state closes. These concepts need to be extended to consider nonequilibrium steady states, as the system is no longer in its ground state but rather in a state that balances drive and dissipation. In a dissipative phase transition, the steady state abruptly changes as a system parameter is varied 26 . Whenever the system is describable in terms of a Lindblad master equation 3 ,ρ = Lρ, then such a transition is signalled by the closing of the lowest excitation gap in the spectrum of the Liouvillian superoperator L.In recent years, interacting photons have emerged as an excellent candidate for studying nonequilibrium condensed matter physics due to the lack of particle number conservation 4 . In cavity quantum electrodynamics, strong coupling between atoms and a cavity can mediate effective photon-photon interactions 5-7 . Arrays of coupled microwave 8 or optical 9 cavities can be fabricated by conventional lithographic techniques, and the competition between on-site interactions and hopping between neighboring cavities can give rise to quantum phase transitions of light 10-12 . A wide range of many-body effects have been predicted in these systems, including a Mott insulator-superfluid phase transition 10-12 and fractional quantum Hall-like states of light 13-15 . Experiments on small systems have demonstrated low-disorder lattices 8 , a dynamical quantum phase transition in a cavity dimer 16 , and chiral ground state currents in a cavity trimer 17 . Circuit QED lattices are inherently open systems, with dissipation an ever-present force that leads both to qubit relaxation as well as the inevitable loss of photons from microwave cavities. While dissipation presents an obstacle for quantum information processing, it is of fundamental interest in the study of nonequilibrium phase transitions. Just as excitations inevitably leak from the system, it is easy to add photons back and to drive into a steady state, making these systems particularly amenable to the study of dissipative phase transitions.In this paper we present experimental evidence for a dissipative phase transition in a circuit QED lattice. We observe that at drive frequencies between the low-power resonance frequencies of the system, there exists a region of hysteresis and bistability where the steady state of the system switches stochastically between two states ρ 1 and ρ 2 . By determining the corresponding switching rates, we can obtain the so-called asymptotic decay rate which characterizes the closing of the spectral gap of L. At the transition between the two states, the characteristic switching times become exceptionally long, a key characteristic of a dissipative phase transition. A similar observation has recently been made in a single cavity system with multiple qubits 18 . arXiv:1607.06895v1 [quant-ph] 23 Jul 2016 2 FIG. 1. 72-site circuit QED lattice. a, Coplanar waveguide resonators, each with a bare cavity frequency of 7.5 GHz, are capacitively coupled to form a linear chain on a 2.5 × 2.5 cm 2 chip. Each resonator is coupled to its neighboring resonators to yield a hopping rate t/2π ≈ 144 MHz and has an average photon loss rate of κ/2π ≈ 1.6 MHz. At three intermediate chain sites, three-way coupling capacitors provide ports for secondary input and output lines (arrows on sides). b-c, A transmon qubit is capacitively coupled to the center pin near the edge of each resonator in the lattice, ensuring coupling to the fundamental mode of each resonator with strength g. The coupled resonator-qubit system forms the fundamental unit cell of the lattice. d, The circuit can be modeled as a linear chain of coupled oscillators, each dispersively coupled to a weakly anharmonic multi-level system.Our device, shown inFig. 1a, consists of a linear chain of 72 lattice sites. Each site comprises a coplanarwaveguide resonator with fundamental-mode frequency ω/2π = 7.5 GHz, coupled to a transmon qubit 19(Fig. 1b)placed at one of the resonator's voltage antinodes. Resonators are capacitively coupled to neighboring resonators(Fig. 1c), so that photons can hop between nearest-neighbor sites. Variations in transmon qubit frequencies in fabrication are a likely source of uncontrolled disorder that is difficult to compensate for in our lattice. We therefore use an asymmetric SQUID-loop geometry allowing each qubit to be tuned over a finite frequency range via an applied magnetic flux. Because individually tuning 72 qubits is currently infeasible in our system, we instead employ a global magnetic field to simultaneously tune all qubit frequencies. Because each qubit is intentionally fabricated with a SQUID-loop of random area, this randomizes the frequency of all qubits within a band of frequencies near 8.5 GHz. In this way, we can ensure that features of interest are universal to the system rather than artifacts of a particular instance of disorder (see Supplementary Information I).To experimentally study the nonequilibrium behav-ior of the device, we monitor the homodyne transmission across the lattice while varying the drive frequency and scanning the drive power over more than five orders of magnitude(Fig. 2a). At low drive powers, we find the expected discrete transmission peaks associated with the interaction-shifted eigenmode frequencies of the resonator lattice. As we vary the mean photon number in the system by increasing the strength of the drive, we observe that a sudden change in system behavior occurs: transmission peaks split and then, at around −10 dB of drive power, abruptly give way to a region of strongly suppressed transmission. In this high-power region, peak-like features are completely absent.The transition between the low-and high-power phases can be more thoroughly explored by measuring the transmission at a single drive frequency while sweeping the drive power either from low to high (2c) or from high to low(Fig. 2d). Doing so reveals a significant region exhibiting hysteresis, which is located at the top of the low-power lobes where the transition to the high-power phase occurs. Subtracting the transmission signals for the two different sweep directions clearly marks the hysteretic regime, as shown inFig. 2e.To gain insight into this behavior, we model the system as a one-dimensional chain of identical circuit QED elements, as illustrated inFig. 1d. The corresponding Hamiltonian H =
Condensed matter physics has been driven forward by significant experimental and theoretical progress in the study and understanding of equilibrium phase transitions based on symmetry and topology. However, nonequilibrium phase transitions have remained a challenge, in part due to their complexity in theoretical descriptions and the additional experimental difficulties in systematically controlling systems out of equilibrium. Here, we study a one-dimensional chain of 72 microwave cavities, each coupled to a superconducting qubit, and coherently drive the system into a nonequilibrium steady state. We find experimental evidence for a dissipative phase transition in the system in which the steady state changes dramatically as the mean photon number is increased. Near the boundary between the two observed phases, the system demonstrates bistability, with characteristic switching times as long as 60 ms -far longer than any of the intrinsic rates known for the system. This experiment demonstrates the power of circuit QED systems for studying nonequilibrium condensed matter physics and paves the way for future experiments exploring nonequilbrium physics with many-body quantum optics.
Over the past decades, there has been remarkable progress in studying both real and synthetic quantum materials. Advances in nanoscale fabrication and cryogenics have allowed for exquisite control of electronic systems -unlocking strongly correlated electronic states and topological materials 1 . Simultaneously, the ability to model desired Hamiltonians with ultra-cold Fermi and Bose gases has allowed unprecedented access to synthetic material properties 2 . As a whole, much of the development of condensed matter physics has focused on the study of (quasi-)equilibrium physics, which is more accessible both experimentally and theoretically. However, the constant presence of dissipation, noise, and decoherence belie the fact that, ultimately, the world is nonequilibrium.
A phase transition indicates a sometimes sudden change in the physical properties of a system as a function of some external system parameter. Thermal phase transitions are well-understood in the context of statistical mechanics and occur when the free energy becomes nonanalytic. At zero temperature, the role of quantum fluctuations gives rise to a new set of quantum phase transitions which involve a sudden change in the ground state of a Hamiltonian H; a phase transition occurs when the gap between the first excited state and the ground state closes. These concepts need to be extended to consider nonequilibrium steady states, as the system is no longer in its ground state but rather in a state that balances drive and dissipation. In a dissipative phase transition, the steady state abruptly changes as a system parameter is varied 26 . Whenever the system is describable in terms of a Lindblad master equation 3 ,ρ = Lρ, then such a transition is signalled by the closing of the lowest excitation gap in the spectrum of the Liouvillian superoperator L.
In recent years, interacting photons have emerged as an excellent candidate for studying nonequilibrium condensed matter physics due to the lack of particle number conservation 4 . In cavity quantum electrodynamics, strong coupling between atoms and a cavity can mediate effective photon-photon interactions [5][6][7] . Arrays of coupled microwave 8 or optical 9 cavities can be fabricated by conventional lithographic techniques, and the competition between on-site interactions and hopping between neighboring cavities can give rise to quantum phase transitions of light [10][11][12] . A wide range of many-body effects have been predicted in these systems, including a Mott insulator-superfluid phase transition 10-12 and fractional quantum Hall-like states of light [13][14][15] . Experiments on small systems have demonstrated low-disorder lattices 8 , a dynamical quantum phase transition in a cavity dimer 16 , and chiral ground state currents in a cavity trimer 17 . Circuit QED lattices are inherently open systems, with dissipation an ever-present force that leads both to qubit relaxation as well as the inevitable loss of photons from microwave cavities. While dissipation presents an obstacle for quantum information processing, it is of fundamental interest in the study of nonequilibrium phase transitions. Just as excitations inevitably leak from the system, it is easy to add photons back and to drive into a steady state, making these systems particularly amenable to the study of dissipative phase transitions.
In this paper we present experimental evidence for a dissipative phase transition in a circuit QED lattice. We observe that at drive frequencies between the low-power resonance frequencies of the system, there exists a region of hysteresis and bistability where the steady state of the system switches stochastically between two states ρ 1 and ρ 2 . By determining the corresponding switching rates, we can obtain the so-called asymptotic decay rate which characterizes the closing of the spectral gap of L. At the transition between the two states, the characteristic switching times become exceptionally long, a key characteristic of a dissipative phase transition. A similar observation has recently been made in a single cavity system with multiple qubits 18 .
FIG. 1. 72-site circuit QED lattice. a, Coplanar waveguide resonators, each with a bare cavity frequency of 7.5 GHz, are capacitively coupled to form a linear chain on a 2.5 × 2.5 cm 2 chip. Each resonator is coupled to its neighboring resonators to yield a hopping rate t/2π ≈ 144 MHz and has an average photon loss rate of κ/2π ≈ 1.6 MHz. At three intermediate chain sites, three-way coupling capacitors provide ports for secondary input and output lines (arrows on sides). b-c, A transmon qubit is capacitively coupled to the center pin near the edge of each resonator in the lattice, ensuring coupling to the fundamental mode of each resonator with strength g. The coupled resonator-qubit system forms the fundamental unit cell of the lattice. d, The circuit can be modeled as a linear chain of coupled oscillators, each dispersively coupled to a weakly anharmonic multi-level system.
Our device, shown in Fig. 1a, consists of a linear chain of 72 lattice sites. Each site comprises a coplanarwaveguide resonator with fundamental-mode frequency ω/2π = 7.5 GHz, coupled to a transmon qubit 19 (Fig. 1b) placed at one of the resonator's voltage antinodes. Resonators are capacitively coupled to neighboring resonators (Fig. 1c), so that photons can hop between nearest-neighbor sites. Variations in transmon qubit frequencies in fabrication are a likely source of uncontrolled disorder that is difficult to compensate for in our lattice. We therefore use an asymmetric SQUID-loop geometry allowing each qubit to be tuned over a finite frequency range via an applied magnetic flux. Because individually tuning 72 qubits is currently infeasible in our system, we instead employ a global magnetic field to simultaneously tune all qubit frequencies. Because each qubit is intentionally fabricated with a SQUID-loop of random area, this randomizes the frequency of all qubits within a band of frequencies near 8.5 GHz. In this way, we can ensure that features of interest are universal to the system rather than artifacts of a particular instance of disorder (see Supplementary Information I).
To experimentally study the nonequilibrium behav-ior of the device, we monitor the homodyne transmission across the lattice while varying the drive frequency and scanning the drive power over more than five orders of magnitude ( Fig. 2a). At low drive powers, we find the expected discrete transmission peaks associated with the interaction-shifted eigenmode frequencies of the resonator lattice. As we vary the mean photon number in the system by increasing the strength of the drive, we observe that a sudden change in system behavior occurs: transmission peaks split and then, at around −10 dB of drive power, abruptly give way to a region of strongly suppressed transmission. In this high-power region, peak-like features are completely absent.
The transition between the low-and high-power phases can be more thoroughly explored by measuring the transmission at a single drive frequency while sweeping the drive power either from low to high (2c) or from high to low (Fig. 2d). Doing so reveals a significant region exhibiting hysteresis, which is located at the top of the low-power lobes where the transition to the high-power phase occurs. Subtracting the transmission signals for the two different sweep directions clearly marks the hysteretic regime, as shown in Fig. 2e.
To gain insight into this behavior, we model the system as a one-dimensional chain of identical circuit QED elements, as illustrated in Fig. 1d. The corresponding Hamiltonian
H = j (H r j + H q j + H rq j ) + j,j H hop j,j + H d ,(1)
includes terms for the resonator, qubit, and the resonator-qubit coupling on each site j, hopping of photons between nearest-neighbor resonators, and a coherent drive (acting on site 1 only). Each resonator contributes a single harmonic mode, H r j = ω a † j a j , where a † j and a j are the creation and annihilation operators for photons on site j. The low-lying transmon levels are described as an anharmonic oscillator H q
j = P N (Ω b † j b j + 1 2 U b † j b † j b j b j )P N with negative Hubbard/Kerr interaction U = −E C ,
and projectors P N that truncate the Hilbert space to levels N E J /2E C within the transmon's cosine well. (Interestingly, the sign of U only affects the system dynamics, but not the steady state given that the qubit and drive frequencies are tuned accordingly -see Supplementary Information II.A.) The operators b † j and b create and annihilate qubit excitations, E C is the single electron charging energy and E J the effective Josephson energy of the transmon qubit. Note that we neglect disorder effects within this model. Qubit-resonator coupling and photon hopping take the simple forms H rq j = g(a j b † j + h.c.) and H hop j,j = t(a j a † j + h.c.). Within rotating-wave approximation, the microwave drive acting on site 1 is given by H d = (t)a 1 e iω d t +h.c. In our model, we account for qubit relaxation and intrinsic photon loss (at rates Γ and κ, respectively) by employing the standard Lindblad master
ρ = −i[H, ρ] + κ j D[a j ]ρ + Γ j D[b j ]ρ (2) where D[L]ρ = LρL † − 1 2 {L † L,
ρ} is the usual action of the Lindblad damping operator.
The experimentally observed transition is remarkably well captured by simple, quasi-classical mean-field theory that decouples the sites, but allows for mean-field parameters to differ from site to site. Allowing for sitedependent parameters is particularly relevant for our case in which the drive only acts on one end of the resonator chain, rather than on every site.) Within the quasi-classical treatment 20 , the quadrature amplitudes α j = a j and β j = b j play the role of mean-field parameters and obey the equations
iα j = (ω − ω p − i κ 2 )α j + g β j + t(α j−1 + α j+1 ) + δ j,1 iβ j = (Ω − ω p − i Γ 2 )β j + U |β j | 2 β j + g α j .(3)
From these equations, we obtain the steady-state transmission signal S 21 ∼ a j and the second-order coherence function g (2) (0) = a † j a † j a j a j /| a † j a j | 2 , choosing j as the label of the output port resonator (see Methods).
The steady-state transmission (Fig. 2b) reproduces all of the qualitative features of the experimental data. We find that the transition occurs beyond the point where the dispersive approximation holds, and further observe that the mean-field solution predicts an increasing accu-mulation of transmon excitations. Higher transmon levels are a crucial model ingredient, as calculations based on the simpler Jaynes-Cummings lattice do not yield results consistent with experiment. Quasi-classically, the drop in transmission in the high-power phase is associated with chaotic dynamics with parallels to results previously obtained for a driven, dissipative Bose-Hubbard chain 20 .
While bistability obtained within nonequilibrium mean-field theory generally has to be considered with care, it is interesting to note that the mean-field solution reveals a region of bistability and hysteresis consistent with that detected experimentally (see Supplementary Information, Fig. 2). Finding multiple steady states appears at odds with Spohn's theorem 21 : the steady-state solution of the Lindblad master equation is unique as long as Hilbert space is finite (or can be safely truncated), and minimal conditions for nature and number of relaxation channels are satisfied. Thus, dissipative phase transitions and stationary bistability can only occur in the thermodynamic limit -when the number of lattice sites tends to infinity and/or when truncation fails due to accumulation of excitations in the strong drive limit, such as in the breakdown of photon blockade on a single Jaynes-Cummings site 18,22 .
Bistability and hysteresis can, however, be produced dynamically 23 as recently discussed by Casteels and coworkers in the context of a Bose-Hubbard dimer 24 . As shown in their paper, hysteresis arises from parameter sweeps across a point where the spectral gap of the Liouvillian superoperator L (nearly) closes, in a manner 7 GHz is plotted as a function of power. κ and γ are included for reference to indicate that ADR can be as large as five orders of magnitude slower than relevant timescales of the device. When either γ1→2 or γ2→1 are slower than the duration of the measurement pulse, τm, we cannot reliable extract a characteristic switch rate. In these cases we select the smallest extracted switching time which is larger than 1/τm. Upward (downward) pointing triangles indicate when γ1→2 (γ2→1) are less than 1/τm, circles indicate when both rates are used.
analogous to the Kibble-Zurek mechanism 25 . Similar to the case studied by Casteels et al., we find that meanfield theory can capture certain qualitative aspects of the bistability and hysteresis, but necessarily fails in aspects related to sweep times and quantum fluctuations. The dramatic suppression of transmission and loss of all resonance peaks beyond a certain drive power threshold are indicative of a dissipative phase transition, arising from the intricate interplay of dissipation, driving, and nonlinearity of the system. The crucial quantity for such a transition is the gap in the spectrum of the Lindblad superoperator L. If the real part of one of its eigenval-ues approaches zero, then deviations of the steady state along the "direction" of the corresponding L-eigenstate become increasingly long-lived and ultimately allow for a new steady state to emerge. The negative real part of the eigenvalue λ closest to zero, − Re λ, is known as the asymptotic decay rate (ADR) 26 . An approximation for the ADR can be extracted by single-shot measurements of the dynamics in the bistable region as follows.
We apply a drive with constant frequency and amplitude, and record single-shot time traces of the homodyne amplitude and phase. Our measurements show that the system undergoes switching between two metastable states on timescales large compared to system-intrinsic timescales (Fig. 3a). The state of the system at each point along a single-shot trajectory is classified as either ρ 1 or ρ 2 , and characteristic dwell times are extracted. The statistics acquired from many single-shot trajectories allow us to extract average rates γ 1→2 , γ 2→1 for the switching between the two metastable states ρ 1 and ρ 2 observed at low power and high power, respectively (labeled as 1 and 2 in the figure).
The extracted switching rates allow us to estimate the asymptotic decay rate by adopting a simplified rateequation model 27 describing the probabilities p 1 and p 2 for the system to be in metastable state ρ 1 or ρ 2 (see Methods section for details):
d dt p 1 p 2 = −γ 1→2 γ 2→1 γ 1→2 −γ 2→1 p 1 p 2 .(4)
Diagonalization of this system yields the stationary and purely decaying eigenmodes ρ s = (γ 2→1 ρ 1 + γ 1→2 ρ 2 )/γ Σ and ρ ADR = γ 2→1 ρ 1 − γ 1→2 ρ 2 with corresponding eigenvalues zero and λ ADR = −γ Σ = −(γ 1→2 + γ 2→1 ). Hence, this simplified model predicts an asymptotic decay rate of − Re λ ADR = γ Σ . Remarkably, the asymptotic decay rate, shown in Fig. 3b-c, reaches a minimum value as low as ∼ 10 Hz, which is five orders of magnitudes lower than the rates set by photon decay and transmon relaxation in our system. This vast timescale discrepancy delivers strong evidence for the onset of a dissipative phase transition. Similar to the situation of equilibrium phase transitions, it is only in the thermodynamic limit that the the spectral gap can fully close and turn the crossover between two steadystate phases into a phase transition in the strict sense 28 .
We gather additional evidence for the approach to a dissipative phase transition by measuring fluorescence power spectra and second-order coherence functions in our system. To this end, two different driving pulse shapes, Fig. 4a-b, are used to access the distinct states of the system. Within the region of bistability, we can perform state initialization either in the low-power phase 1 or the high-power phase 2 by approaching the final drive amplitude ξ either from a lower or a higher drive amplitude. After this initialization period, the two pulses maintain the constant drive amplitude ξ, during which time, the transmitted signal is detected using heterodyne detection with a 32 MHz intermediate frequency. The power spectrum is then obtained by performing a Fourier transform on the heterodyne signal. The second-order correlation function g (2) (0) is measured using techniques outlined in Ref. 29 . Figure 4 indicates that the low-power state can be characterized by a single, coherent drive tone (g (2) = 1) and that the high-power state can be characterized by broadband and multimode (see supplement) emission and bunching (g (2) ≈ 2). In addition, the onset of the high-power state has a stark linewidth broadening of the drive tone and has a region of strong bunching g (2) ≈ 5 for the down pulse as the system transitions to the high-power state. Experimental measurements of g (2) in Fig. 4e agree well with theory results shown Fig. 4f, barring the strong bunching observed with the 'down pulse' (Fig. 4b) at the high ξ side of the bistability region. Based on our modeling, the experiment involves both large numbers of photons and excitations of higher trans-mon levels, and hence may indeed approach the thermodynamic limit necessary for the observation of a dissipative phase transition. This work demonstrates the potential for circuit QED lattices as a controllable platform that can guide a deeper theoretical and experimental understanding of nonequilibrium condensed matter physics.
METHODS
Experimental methods. The cavities of the circuit QED lattice were etched using standard optical lithography and plasma etching techniques from a 200 nm thick Nb film on a 25 × 25 cm 2 sapphire substrate. Transmon qubits were designed to have Josephson junctions with dimensions 200 × 180 nm 2 and 450 × 450 nm 2 and were fabricated according to the "Manhattan" technique outlined by Potts et al. 30 , using electron beam lithography and aluminum evaporation. Similar transmon qubits have coherence times T 1 = 1 µs and coupling constant of g/2π = 265 MHz. Measurements were performed at a temperature of 7.5 mK in a dilution refrigerator, and inside a superconducting solenoid magnet controlled by a room-temperature DC voltage source. Transmission measurements are performed using a network analyzer, switching-rate measurements using standard homodyne detection techniques. All power-spectrum measurements were done by taking the Fourier transform of a heterodyne signal, and g (2) measurements were implemented using the homodyne techniques described by Eichler et al. 29 (see Supplementary Information for further details).
Numerical solution of the mean-field equations.
We solve for the stationary state of the mean-field equations (3) by time evolution and extracting the long-time limit, since root-finding methods are difficult to handle for the large system of nonlinear equations 20 . In the high-power phase the dynamics is chaotic, so that additional time averaging in the long-time limit is required. For instance, the second-order coherence function is obtained by evaluating g (2) (0) = |α(t)| 2 t /| |α(t)| t | 2 , where the time average · t is carried out over a time interval that excludes any initial transient behavior.
Model underlying the ADR estimate. First, consider stochastic switching between two pure states |1 and |2 . The simplest description is based on a two-level Hamiltonian H = E 21 |2 2| where E 21 is the energy difference between the two states, and the master equatioṅ (5) with D[L] denoting the usual Lindblad damping superoperator for jump operator L. The resulting 4 × 4 Liouvillian L is block-diagonal, where one of the two blocks fully captures the dynamics of density matrices of the form ρ(t) = p 1 (t)|1 1| + p 2 (t)|2 2|, where the probabilities p 1,2 obey the rate equation (4). This model can be extended and made more realistic by considering subsets of pure states that make up the two metastable states ρ 1 and ρ 2 , which are likely to be mixed states rather than pure states.
ρ = −i [H, ρ] + γ 1→2 D [|2 1|] ρ + γ 2→1 D [|1 2|] ρ,
Supplemental Information
I. DEVICE PARAMETERS
The device consists of 72-coplanar waveguide cavities coupled by capacitors which are formed using gaps in the center pin of the resonator. Each tunneling capacitor is designed to have a capacitance of 20.7 pF, resulting in a tunneling matrix element t/2π = 144 MHz. The threeway couplers are integrated in the lattice to maintain the cavity-cavity hopping rate while introducing a weak coupling to an input/output transmission line to weakly probe the internal behavior of the lattice.
One qubit is coupled to each cavity. From finiteelement simulations, we predict a charging energy E c /h = 180 MHz. Similar qubits measured in other devices have measured coupling rates g/2π = 265 MHz. Each transmon SQUID has a fixed width of 9 µm but the heights are chosen to be a random number between 8 µm and 20 µm, ensuring that a global magnetic field can continuously change the qubit frequencies but will never return to the exact same qubit frequency distribution. Together with previous measurements of E J and simulated values for E c , we expect qubit frequencies to be between 8 and 8.8 GHz. This is confirmed in Fig. 5 which shows a measurement of transmission as a function of external magnetic field. Each mode exhibits frequency shifts of different flux periodicity, indicating the existence of numerous qubits. Based on optical microscope inspection of the sample, it is expected that after the experiment was complete, a minimum 60% of our qubits are fully functional, with another 20% for which the smaller Josephson junction of the SQUID did not make a physical connection; it is unknown if this damage occured due to handling the device after the experiment was complete.
II. THEORY
A. Mapping of positive -negative U The sign of the Hubbard interaction U has a significant impact on the system's energy spectrum. However, if we are merely interested in the steady-state behavior, the sign of U is relatively less important as long as the system remains in the transmon's cosine well. We begin our discussion first with the quasi-classical treatment. For the steady state, the time derivative of any expectation value vanishes. Hence, the quasi-classical dynamical equations reduce to
0 = (ω − ω p − i γ 2 )α j + g β j + t(α j−1 + α j+1 ) + δ j,1 , 0 = (ω + ∆ − ω p − i Γ 2 )β j + U |β j | 2 β j + g α j ,(6)
with the qubit-resonator detuning ∆ = Ω − ω. Now, we consider a mapping such that
(ω p , ∆, g, t, α j , β j ) → 2ω − ω p , −∆, −g, −t, α * j , β * j .
This mapping physically corresponds to tuning both the drive and qubit frequencies in a specific way.
(For our open chain, the sign flip of g and t is allowed by a local gauge transformation.) By taking the complex conjugate on both sides of the dynamical equations together with the mapping, we find:
0 = (ω − ω p − i γ 2 )α j + g β j + t(α j−1 + α j+1 ) + δ j,1 , 0 = (ω + ∆ − ω p − i Γ 2 )β j + (−U )|β j | 2 β j + g α j .(7)
The mapping thus effectively flips the sign of U . In other words, if both the drive and qubit frequencies are tunable, the sign of U is not a concern and the truncation to low-lying transmon levels is justified. The above discussion can be easily generalized to the Lindblad master equation. In that case, instead of mapping α j and β j to their complex conjugates, we map the steady-state density matrix ρ s to its complex conjugate ρ * s . Note that any steady-state expectation value can be calculated using ρ * s through the equation
O = tr (Oρ s ) = [tr (O * ρ * s )] * .(8)
B. Simulation of hysteresis
Hysteresis appears in the simulation if the quasiclassical dynamical equations have more than one attractor. Hence, we can numerically investigate the region of hysteresis by picking different initial states and checking whether the system evolves to the same or different attractors in the long-time limit. We illustrate this by simulating the transmission for a 20-site chain in Figure 6. (The region of hysteresis is qualitative the same for Transmission as a function of external magnetic field. By changing the strength of the external magnetic field, individual qubit frequencies can be continuously changed. The non-periodic behaviour of the transmission peaks is due to the random areas of the transmons in the cQED lattice. both 20-site and 72-site chains.) Subtraction of the simulated transmission with two different initial states shows a region with more than one attractor. The shape of this hysteresis region is consistent with the experimental data. Readers may notice that the result becomes 'noisy' in the high power regime. This is the result of the chaotic dynamics which makes the time average strongly sensitive to the initial state and the averaging window. However, the hysteresis is reassuring by direct inspection of the system time evolution. In the region of hysteresis, one initial state gives stationary behavior (stable fixed point) in the long time limit while the other gives chaotic dynamics (strange attractor).
III. SWITCHING RATE EXTRACTION
In order to extract switching rates necessary for the calculation of the asymptotic decay rate (ADR), it is necessary to gather statistics on the lifetimes of each state in the region of bistability. To do this, a pulse shown in Fig. 4 (CW) microwave source. After the initialization phase of the pulse, the homodyne amplitude and phase are extracted from the output signal by first passing the raw signal through a series of amplifiers before being mixed down to using a local oscillator (LO) at the drive frequency using an IQ mixer. The I and Q outputs of the IQ mixer are then sent through a 1.9 MHz low-pass filter (LPF) before being amplified, digitized, and sent to the measurement computer (see Fig. 7). The amplitude and phase of the homodyne signal are then extracted by taking
A = I 2 + Q 2 (9) θ = tan −1 (I/Q)(10)
The 1.9 MHz filter was chosen to reduce noise in the single-shot signal in an effort to reduce the number of false-counts in the steady state transitions. This filter frequency is well above observed transition rates. The digitizer sampling rate was chosen to be 50 MS/s and the data was down-sampled to include only every tenth data point to avoid memory constraints. After all processing, each trajectory consists of 1.5 × 10 6 data points. Figure 7 outlines the thresholding algorithm used to determine switching rates γ 1→2 and γ 2→1 . For each drive amplitude and frequency, seven 0.3 s trajectories are acquired and the data is placed in histograms for amplitude and phase. It is then determined whether the resultant distributions are gaussian, meaning there is no bistability, or bimodal, meaning that the there is bistability. In the case of a bimodal distribution, data are classified as either being in the high or low-power state with a threshold given by the mean of the peak locations. The measured quantity (amplitude or phase) which has the fewest his- togram counts at the threshold is then used as the measured quantity for the state lifetime determination. New data is then acquired and categorized according to the threshold. Once the new data has been categorized, state lifetimes are extracted. The lifetimes are then binned in a nonuniform histogram bins shown in Fig. 8.
Depending on the population of the histogram bins, the bins can be summed to create histograms with larger time intervals. This scheme ensures that histograms are sufficiently populated for situations involving short or long lifetimes. After the binning procedure has been performed, the resultant distribution is fit to an exponential to extract a characteristic switching time, τ c . From this, the switching rate for the state of interest is γ = 1/τ c .
The physical measurement setup for the jump rate extraction is shown in Fig. 10.
IV. EMISSION PROPERTIES
The transmission shown in Figure 2 of the main text exhibits an abrupt change as the system crosses into the high power regime. As illustrated in Figure 4 of the main text, the high power state can be characterized by broadband emission around the drive tone. When a power spectrum measurement is performed (in this case using a spectrum analyzer) observing the entire range of frequencies with low-power transmission peaks, the system emits at nearly all the low-power transmission peaks of the system. This can be understood qualitatively by con- 9. Multimode Emission. By driving a single mode of the system, in this case at 7.535 GHz, emission is observed both below (a) and above (b) the drive tone. Emission peaks (shown in red) qualitatively match low-power transmission peaks (shown in blue). Emission is plotted in units of dB above the background.
sidering the system Hamiltonian in the eigenmode basis of the cavity chain.
We can rewrite the Hamiltonian in the cavity eigenmode basis by the substitution: a j = N µ=1 W jµãµ , where W jµ is the weight of the µth eigenmode at site j. This gives
H = N µ=1ω µã † µãµ + N j=1 Ωb † j b j + U 2 b † j b † j b j b j + (11) g N j,µ=1 W * jµã † µ b j + h.c. + N µ=1 W * 1µã † µ + h.c. ,
whereω µ is the frequency of the µth eigenmode. This equation can be thought of as a collection of modes coupled to a bath of transmon qubits. In this basis, each cavity eigenmode interacts with all other cavity eigenmodes through the communal qubit bath. This effective interaction explains the presence of multimode emission shown in Figure 9. Fig. 2 are performed using the same experimental setup but using a network analyzer (Keysight PNA-X N5241A) to source the RF signal and the returning signal after the two Miteq amplifiers is sent directly back into the network analyzer.
FIG. 2 .
2Microwave transmission spectra as a function of power, exhibiting an abrupt transition to a suppressed transmission regime and a region of bistability. a, Dispersively shifted transmission peaks show nonlinear splitting at increased power and give rise to a region of strongly suppressed transmission without resonance peaks. Here data is acquired using constant power, frequency sweeps. b, Corresponding mean-field result for transmission through a 72-site lattice, showing features qualitatively consistent with the experiment. c, Zoom into one lobe, showing the sharp transition to a state of suppressed transmission as the drive power is swept from low to high power using constant-frequency, linear power sweep over a 31.95 ms period. d, Same region as in c, but now sweeping power in the opposite direction (high to low) over the same time period as c. The transition now occurs along a down-shifted curve. e, Subtraction of the data shown in c and d uncovers the large region of hysteresis. equation formalism for the reduced density matrix,
FIG. 3 .
3Asymptotic decay rate in the transition region. a, Single-shot time trace of the homodyne phase in the hysteretic region for constant drive amplitude. Data show stochastic switching between two distinct metastable states 1 and 2 on timescales vastly exceeding those intrinsic to the system. b, Asymptotic decay rate obtained from the sum of the characteristic switching times, γ1→2 + γ2→1, as a function of drive frequency and power. c, ADR for a drive frequency of 7.5059
FIG. 4 .
4State characterization. To probe properties of the states within the region of bistability, two pulse sequences are used to initialize the system: a, an 'up pulse' for initialization in the high-power state, and b, a 'down pulse' for initialization in the low-power state. Due to the long timescales, the power spectra shown in c, d and the second-order correlation function g(2) (0) in e can be obtained for each state independently, enabling state characterization within the region of bistability. f, Mean-field result for the second-order correlation function for comparison (note: x axes cannot be compared directly).
FIG. 5 .
5FIG. 5. Transmission as a function of external magnetic field. By changing the strength of the external magnetic field, individual qubit frequencies can be continuously changed. The non-periodic behaviour of the transmission peaks is due to the random areas of the transmons in the cQED lattice.
FIG. 6 .
6Simulation of the region of hysteresis. Transmission of a 20-site chain is simulated with parameters motivated by the experiment, and initial state being the vacuum state in a and a highly excited state in b. c, Subtraction of the simulated transmission shown in a and b uncovers the region of hysteresis.
FIG. 9. Multimode Emission. By driving a single mode of the system, in this case at 7.535 GHz, emission is observed both below (a) and above (b) the drive tone. Emission peaks (shown in red) qualitatively match low-power transmission peaks (shown in blue). Emission is plotted in units of dB above the background.
FIG. 10 .
10Measurement schematic for extraction of switching times. Measurements are performed inside a superconducting magnet at the base of a dilution refrigerator. Transmission measurements shown in
between two distinct steady states of the system. From the raw data acquired in (a) and (b), a histogram is compiled and, if the resulting distribution is bimodal, the mean of the peak locations is used as a discriminating threshold. Each point along a trajectory is categorized based on the threshold, resulting in (c) and (d). Ultimately, the state lifetimes are determined using the threshold which has the fewest histogram counts at the threshold (phase in this case)., lasting 0.3 s is used to modulate a continuous-wave
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FIG. 8. Switching time histogram bins. When transition times are short, the short time bins provide a precise determination of short characteristic switching times. When the switching times are long, however, the shorter time interval bins can be summed to form larger bins, creating wellpopulated bins of larger time intervals, enabling a precise determination of longer characteristic switching times.100 bins
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AcknowledgmentsThe authors thank Iacopo Carusotto and Cristiano Ciuti for helpful discussions. This work was supported by the Army Research Office through grant W911NF-15-1-0397 and the National Science Foundation through Grants No. DMR-0953475 and No. PHY-1055993. NS was supported by an NDSEG fellowship.
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"A NEW CONTRIBUTION TO THE FLAVOUR-CHANGING LEPTON-PHOTON VERTEX",
"A NEW CONTRIBUTION TO THE FLAVOUR-CHANGING LEPTON-PHOTON VERTEX"
]
| [
"D Palle \nDepartment of Theoretical Physics\nRugjer Bošković Institute\nP.O.Box 1016ZagrebCROATIA\n"
]
| [
"Department of Theoretical Physics\nRugjer Bošković Institute\nP.O.Box 1016ZagrebCROATIA"
]
| []
| We show that the correct perturbation theory for mixed fermion states leads to nonvanishing contributions of the dimension-four vertex operators for flavour-changing transitions. Their contributions to the amplitude are of the same order of magnitude as the dimension-five vertex operators. Considerations are valid irrespective of the electroweak model. PACS 11.10.Gh Renormalization 12.15.Ff Quark and lepton masses and mixing 13.40 Hq Electromagnetic decays | 10.1007/bf03185553 | [
"https://arxiv.org/pdf/hep-ph/9706295v1.pdf"
]
| 18,935,943 | hep-ph/9706295 | 1e8808dc8046efa63c36d5884a9054a06769031f |
A NEW CONTRIBUTION TO THE FLAVOUR-CHANGING LEPTON-PHOTON VERTEX
Jun 1997
D Palle
Department of Theoretical Physics
Rugjer Bošković Institute
P.O.Box 1016ZagrebCROATIA
A NEW CONTRIBUTION TO THE FLAVOUR-CHANGING LEPTON-PHOTON VERTEX
Jun 1997arXiv:hep-ph/9706295v1 9
We show that the correct perturbation theory for mixed fermion states leads to nonvanishing contributions of the dimension-four vertex operators for flavour-changing transitions. Their contributions to the amplitude are of the same order of magnitude as the dimension-five vertex operators. Considerations are valid irrespective of the electroweak model. PACS 11.10.Gh Renormalization 12.15.Ff Quark and lepton masses and mixing 13.40 Hq Electromagnetic decays
It seems that current measurements in astro-and particle physics (solar and atmospheric neutrinos, COBE data, LSND data, observed ionization of the Universe, etc.) strongly suggest that neutrinos should be massive particles. In this paper we show that the correct treatment of the flavour mixing of massive leptons in perturbation theory results in new terms in the decay amplitude due to the dimension-four operators.
Contributions to flavour-changing radiative decays of leptons in the perturbative calculations of the majority of electroweak models appear through quantum loops owing to the existence of flavour-changing charged weak currents. We can write the general form of the f 1 → f 2 + γ amplitude (vertex) with the following Lorentz structures [1]:
M µ (f 1 (p 1 ) → f 2 (p 2 ) + γ(q)) = −ıū(p 2 )Γ µ;f 1 f 2 u(p 1 ) = −ıū(p 2 )[γ µ (F L 1 (q 2 )P L + F R 1 (q 2 )P R ) + ıσ µν q ν (F L 2 (q 2 )P L + F R 2 (q 2 )P R ) +q µ (F L 3 (q 2 )P L + F R 3 (q 2 )P R )]u(p 1 ), (1) where : P L,R ≡ 1 2 (1 ∓ γ 5 ) .
Previous calculations [1] of the decay amplitude were focused on the F L,R 2 form factors because the F L,R 3 form factors gave zero contribution to f 1 → f 2 + γ. However, the F L,R 1 form factors were claimed to give contributions that should vanish because of the conservation of the electromagnetic current. We show that the latter claim is incorrect and that the F L,R 1 form factors give contributions to the amplitude comparable with that of the F L,R 2 [1]. The most natural choice for the renormalization scheme in electroweak theory is the on-shell renormalization scheme [2]. In this scheme, we repeat the most important ingredients concerning mixed fermion states. The onshell renormalization conditions for mixed fermions (propagators) are
S ren ij [pole] = δ ij m i − p ,(2)
where : S ij ≡ F ourier T ransf orm ı 0|T ψ i (x)ψ j (y)|0 .
These renormalization conditions ensure a correct input for fermion masses and a correct form of renormalized propagators. These conditions can be written in a more transparent form by introducing on-shell spinors [2]:
K ren ij u(m j ) = 0, u(m i )K ren ij = 0, { 1 m i − p K ren ii }u(m i ) = u(m i ), (3)ū (m i ){K ren ii 1 m i − p } =ū(m i ), def inition : K ren ij (p) ≡ {S ren (p)} −1 ij , i, j = f lavour indices.
For Majorana fields, one obtains the same formulae for on-shell renormalization conditions, but with a different number of conditions in comparison with Dirac fermions [2].
Furthermore, any correctly quantized electroweak theory preserves the BRST symmetry [2]. Thus, the generalized Ward-Takahashi identity for the flavour-changing lepton-photon vertex can be written as
q µ Γ ren µ;ij (p + q, p|q) = −eΣ ren ij (p) + eΣ ren ij (p + q).(4)
From the general Lorentz structure of the lepton-photon amplitude one can see that only the F L,R 1 form factors are related to flavour off-diagonal selfenergies through Ward-Takahashi identities [2,3]. These identities are valid for renormalized Green functions, and the on-shell renormalization conditions (3) should be used to uniquely fix finite terms of self-energies:
Σ ren ij (p)u(p, m j ) = 0, u(p, m i )Σ ren ij (p) = 0.(5)
It is now evident that the electromagnetic current remains conserved because of the on-shell conditions:
q µū (p, m i )Γ ren µ;ij (p + q, p|q)u(p + q, m j ) = 0.
In addition, we can evaluate the F L,R 1 (q 2 ) form factors from the on-shell conditions at q 2 = 0. Let us write the most general form of the renormalized self-energy (flavour indices suppressed):
Σ ren (p) ≡ (σ 1 (p 2 ) + δZ 1 ) pP L + (σ 2 (p 2 ) + δZ 2 ) pP R +(σ 3 (p 2 ) + δZ 3 )P L + (σ 4 (p 2 ) + δZ 4 )P R .(6)
Inserting the above into the Ward-Takahashi identity and setting p µ = 0 and q 2 = 0, we obtain the following expressions for the form factors:
F L 1 (0) f 1 =f 2 = e(σ 1 (0) + δZ 1 ), F R 1 (0) f 1 =f 2 = e(σ 2 (0) + δZ 2 ).(7)
The four renormalization constants are defined by the on-shell renormalization conditions (5) (f i f ermion has a mass m i ):
δZ 1 = 1 m 2 1 − m 2 2 [−m 2 1 σ 1 (m 2 1 ) + m 2 2 σ 1 (m 2 2 ) + m 1 m 2 (σ 2 (m 2 1 ) − σ 2 (m 2 2 )) −m 2 (σ 3 (m 2 1 ) − σ 3 (m 2 2 )) + m 1 (σ 4 (m 2 1 ) − σ 4 (m 2 2 ))], δZ 2 = 1 m 2 1 − m 2 2 [m 1 m 2 (σ 1 (m 2 1 ) − σ 1 (m 2 2 )) − m 2 1 σ 2 (m 2 1 ) + m 2 2 σ 2 (m 2 2 ) +m 1 (σ 3 (m 2 1 ) − σ 3 (m 2 2 )) − m 2 (σ 4 (m 2 1 ) − σ 4 (m 2 2 ))],(8)δZ 3 = 1 m 2 1 − m 2 2 [m 2 m 2 1 (σ 1 (m 2 1 ) − σ 1 (m 2 2 )) − m 1 m 2 2 (σ 2 (m 2 1 ) − σ 2 (m 2 2 )) +m 2 2 σ 3 (m 2 1 ) − m 2 1 σ 3 (m 2 2 ) − m 1 m 2 (σ 4 (m 2 1 ) − σ 4 (m 2 2 ))], δZ 4 = 1 m 2 1 − m 2 2 [−m 1 m 2 2 (σ 1 (m 2 1 ) − σ 1 (m 2 2 )) + m 2 m 2 1 (σ 2 (m 2 1 ) − σ 2 (m 2 2 )) −m 1 m 2 (σ 3 (m 2 1 ) − σ 3 (m 2 2 )) + m 2 2 σ 4 (m 2 1 ) − m 2 1 σ 4 (m 2 2 )].
For definiteness, let us write the interaction Lagrangian with flavourchanging lepton charged currents with Dirac neutrinos, in a form that is valid irrespective of the symmetry-breaking mechanism :
L I = g √ 2 ij W µν i U ij γ µ P L l j + g √ 2M W ij φ +ν i [m l j U ij P R − m ν i U ij P L ]l j + h.c.,
ν = neutrino; l = charged lepton; φ + = Nambu − Goldstone scalar.
Then we have to find renormalized neutrino self-energies with the above interaction Lagrangian and the renormalization conditions (5).
In the 't Hooft-Feynman gauge, one can easily verify that
σ 1;2;3;4 (p 2 ) = −ı g 2 32π 2 l U il U * ml × {(2 + m 2 l M 2 W )B 1 (p 2 ; m 2 l , M 2 W ); m ν i m νm M 2 W B 1 (p 2 ; m 2 l , M 2 W ); m 2 l M 2 W m ν i B 0 (p 2 ; m 2 l , M 2 W ); m 2 l M 2 W m νm B 0 (p 2 ; m 2 l , M 2 W )}, {m, i, l} are f lavours of {ν 1 , ν 2 , charged lepton l}.
The scalar functions B 0 (p 2 ) and B 1 (p 2 ) [3] have to be evaluated for p 2 ≪ (M 1 − M 2 ) 2 , so it would be useful to make an expansion in the vicinity of p 2 = 0:
ı 16π 2 {1; p µ }B 0;1 (p 2 ; M 1 , M 2 ) = d 4 k (2π) 4 {1; k µ } (k 2 − M 2 1 + ıǫ)((k + p) 2 − M 2 2 + ıǫ) , B 0 (p 2 ; M 1 , M 2 ) = θ(M 1 , M 2 ) + b 2 p 2 + b 4 p 4 + O(p 6 ), B 1 (p 2 ; M 1 , M 2 ) = η(M 1 , M 2 ) + 1 2 (b 4 M 2 2 − M 2 1 2 − b 2 )p 2 + O(p 4 ), b 2 = 1 2 M 2 1 + M 2 2 (M 2 1 − M 2 2 ) 2 + 2M 2 1 M 2 2 (M 2 1 − M 2 2 ) 3 ln M 2 M 1 , b 4 = M 4 1 + 10M 2 1 M 2 2 + M 4 2 6(M 2 1 − M 2 2 ) 4 − 2 M 2 1 M 2 2 (M 2 1 + M 2 2 ) (M 2 1 − M 2 2 ) 5 ln M 2 M 1 ,
θ, η f unctions contain ultraviolet inf inity.
From the above we can evaluate the leading terms of the F L,R
1 (0) f 1 =f 2 form factors (m 2 ≪ m 1 and m l ≪ M W ) F L 1 (0) i =m ≃ −ı eG F 4 √ 2π 2 m 2 1 l U il U * ml ( 9 8 m 2 l M 2 W − 3 m 2 l M 2 W ln M W m l ), F R 1 (0) i =m ≃ −ı eG F 4 √ 2π 2 m 1 m 2 l U il U * ml (− 5 8 m 2 l M 2 W + 3 m 2 l M 2 W ln M W m l ),(9)
and at the same time (see Eq.(10.27) in the book of Ref. [1])
F L 2 (0) i =m ≃ −ı eG F 4 √ 2π 2 m 2 l U il U * ml ( 3 4 m 2 l M 2 W ), F R 2 (0) i =m ≃ −ı eG F 4 √ 2π 2 m 1 l U il U * ml ( 3 4 m 2 l M 2 W ).(10)
It is straightforward to evaluate the rate:
Γ(f 1 (m 1 ) → f 2 (m 2 ) + γ) = m 2 1 − m 2 2 4πm 2 1 [2p 2 (p + p 2 + m 2 2 )(|F V 2 | 2 + |F A 2 | 2 ) +|F V 1 | 2 ( p 2 + m 2 2 − m 2 ) + |F A 1 | 2 ( p 2 + m 2 2 + m 2 ) − m 2 p(F V 1 F V * 2 + F V * 1 F V 2 −F A 1 F A * 2 − F A * 1 F A 2 ) + p(p + p 2 + m 2 2 )(F V 1 F V * 2 + F V * 1 F V 2 + F A 1 F A * 2 + F A * 1 F A 2 )], where p = m 2 1 − m 2 2 2m 1 , F V,A i = F R i ± F L i 2 .
Thus, the contributions from the F L,R 1 (ν 1 = ν 2 ) and F L,R 2 (ν 1 = ν 2 ) form factors to the rate of ν 1 → ν 2 + γ (or µ → e + γ) are of the same order of magnitude [1](however the neutrino flavour diagonal charges vanish F L,R 1 (0) ν i = 0). To conclude, one can say that if perturbation theory is correctly applied to mixed fermion states, one has to calculate all F L,R 1,2 (f 1 = f 2 ) form factors to evaluate decay amplitudes in any electroweak model with lepton mixing. In our calculation we respect Lorentz, gauge and BRST symmetries, as well as the renormalization conditions for mixed fermion states. A correct evaluation of the neutrino lifetime for certain electroweak models could be of great importance in the theoretical cosmology and astrophysics [1,4], with Sciama's decaying neutrino hypothesis as an example [5].
. B W Lee, R E Shrock, Phys. Rev. D. 161444Lee B. W. and Shrock R. E., Phys. Rev. D, 16 (1977) 1444;
R N Mohapatra, P B Pal, Massive neutrinos in physics and astrophysics. SingaporeWorld Scientificand references thereinMohapatra R. N. and Pal P. B., Massive neutrinos in physics and astrophysics (World Scientific, Singapore) 1991, and references therein.
. K Aoki, Suppl. of the Prog. Th. Phys. 731Aoki K. et al, Suppl. of the Prog. Th. Phys., 73 (1982) 1.
. M Böhm, H Spiesberger, W Hollik, Fort. der Phys. 34687Böhm M., Spiesberger H. and Hollik W., Fort. der Phys., 34 (1986) 687.
E W Kolb, M S Turner, The Early Universe. California, USAddison-Wesley Pub. CoKolb E. W. and Turner M. S., The Early Universe (Addison-Wesley Pub. Co., California, US) 1990.
. D W Sciama, Nature. 40Sciama D. W., Nature, 346 (1990) 40;
D W Sciama, Modern Cosmology and the Dark Matter Problem. UKCambridge Univ. PressSciama D. W., Modern Cos- mology and the Dark Matter Problem (Cambridge Univ. Press, UK) 1995.
| []
|
[
"STABILITY AND INSTABILITY OF BREATHERS IN THE U (1) SASA-SATUSUMA AND NONLINEAR SCHRÖDINGER MODELS",
"STABILITY AND INSTABILITY OF BREATHERS IN THE U (1) SASA-SATUSUMA AND NONLINEAR SCHRÖDINGER MODELS"
]
| [
"Miguel A Alejo ",
"ANDLuca Fanelli ",
"Claudio Muñoz "
]
| []
| []
| We consider the Sasa-Satsuma (SS) and Nonlinear Schrödinger (NLS) equations posed along the line, in 1+1 dimensions. Both equations are canonical integrable U (1) models, with solitons, multi-solitons and breather solutions[43]. For these two equations, we recognize four distinct localized breather modes: the Sasa-Satsuma for SS, and for NLS the Satsuma-Yajima, Kuznetsov-Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior[17]. In this paper we find the natural H 2 variational characterization for each of them, and prove that Sasa-Satsuma breathers are H 2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang[36]. Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H 2 based Lyapunov functional, in the spirit of [4], extended this time to the vectorvalued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma-Yajima, Peregrine y Kuznetsov-Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in[32].For this model, we will assume two boundary value conditions (BC) at infinity: 1 This equation is similar to the well-known linear Schödinger i∂tw + ∂ 2 x w = 0, but instead of dealing with the additional term 2 Re w only as a perturbative term, we will consider all linear terms as a whole for later purposes (not considered in this paper), in particular, long time existence and decay issues, see e.g.[18,19].2 Another model corresponds to the Gross-Pitaevskii equation: i∂tu + ∂ 2 x u + u(1 − |u| 2 ) = 0, for which the Stokes wave is modulationally stable. | null | [
"https://arxiv.org/pdf/1901.10381v1.pdf"
]
| 119,705,951 | 1901.10381 | 18479d447adee1c0d4ffe99b4c5fe83473f25bdb |
STABILITY AND INSTABILITY OF BREATHERS IN THE U (1) SASA-SATUSUMA AND NONLINEAR SCHRÖDINGER MODELS
Miguel A Alejo
ANDLuca Fanelli
Claudio Muñoz
STABILITY AND INSTABILITY OF BREATHERS IN THE U (1) SASA-SATUSUMA AND NONLINEAR SCHRÖDINGER MODELS
We consider the Sasa-Satsuma (SS) and Nonlinear Schrödinger (NLS) equations posed along the line, in 1+1 dimensions. Both equations are canonical integrable U (1) models, with solitons, multi-solitons and breather solutions[43]. For these two equations, we recognize four distinct localized breather modes: the Sasa-Satsuma for SS, and for NLS the Satsuma-Yajima, Kuznetsov-Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior[17]. In this paper we find the natural H 2 variational characterization for each of them, and prove that Sasa-Satsuma breathers are H 2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang[36]. Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H 2 based Lyapunov functional, in the spirit of [4], extended this time to the vectorvalued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma-Yajima, Peregrine y Kuznetsov-Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in[32].For this model, we will assume two boundary value conditions (BC) at infinity: 1 This equation is similar to the well-known linear Schödinger i∂tw + ∂ 2 x w = 0, but instead of dealing with the additional term 2 Re w only as a perturbative term, we will consider all linear terms as a whole for later purposes (not considered in this paper), in particular, long time existence and decay issues, see e.g.[18,19].2 Another model corresponds to the Gross-Pitaevskii equation: i∂tu + ∂ 2 x u + u(1 − |u| 2 ) = 0, for which the Stokes wave is modulationally stable.
1. Introduction 1.1. Setting. In this paper our main purpose is to deal with the variational stability of complex soliton-like solutions for Schrödinger-type, U (1) invariant models appearing in nonlinear Physics and integrability theory. By U (1) symmetry, we refer to the classical invariance of the equation under the transformation u → ue iγ , with γ ∈ R and u complex-valued solution.
The first model that we shall consider is the cubic focusing Nonlinear Schrödinger (NLS) equation posed on the real line iu t + u xx + |u| 2 u = 0, u(t, x) ∈ C, (t, x) ∈ R 2 .
(1. Additionally, we will consider the Sasa-Satsuma (SS) equation for a function q = q(T, X) posed on the line [38] iq T + 1 2 q XX + |q| 2 q + i q XXX + 6|q| 2 q X + 3q(|q| 2 ) X = 0, q = q(T, X) ∈ C, T, X ∈ R. Note that in this equation (and after a suitable rescaling) is the parameter of bifurcation from (the integrable) cubic NLS (1.1). However, it is important to notice that, unless = 0, (1.3) represents a third order complex-valued model for the unknown q, with important differences with respect to (1.1).
Following Sasa and Satsuma [38], we have that under the change of variables u(t, x) = q(T, X)e −i(X−T /(18 ))/ (6 ) , t = T, x = X − T /(12 ), and assuming = 1, equation (1.3) reads now [43, p. 114] u t + u xxx + 6|u| 2 u x + 3u(|u| 2 ) x = 0, u = u(t, x) ∈ C, t, x ∈ R.
(1.4)
In this paper we will focus on this third order, complex-valued, modified KdV (mKdV) model. In particular, this equation will retain several properties of the standard, scalar valued mKdV equation.
Both equations, (1.1) and (1.3), are well-known integrable models, see [45] and [38] respectively. NLS describes propagation pulses in nonlinear media and gravity waves in the ocean [13], and was proved integrable by Zakharov and Shabat [45]. NLS (1.1) with nonzero BC (1.2) is believed to describe the emergence of rogue or freak waves in deep sea [35], and also it is a well-known example of the mechanism known as modulational instability [35,1]. On the other hand, SS was introduced by Sasa and Satsuma [38] as an integrable model for which the Lax pair is 3 × 3 matrix valued, and it is closely related to another integrable model, the Hirota equation (see e.g. [43] for additional details).
Finally, in the case of (1.1) with nonzero boundary conditions at infinity, note that the Stokes wave e it is a particular, non localized solution of (1.1). A complete family of standing waves can be obtained by using the scaling, phase and Galilean invariances of (1.1):
u c,v,γ (t, x) := √ c exp ict + i 2 xv − i 4 v 2 t + iγ . (1.5)
This wave is another solution to (1.1), for any scaling c > 0, velocity v ∈ R, and phase γ ∈ R. However, since all these symmetries represent invariances of the equation, they will not be essential in our proofs, and we will assume in this paper c = 1, v = γ = 0.
Consequently, we will seek for solutions in the form of a Stoke wave, which means that we set u(t, x) = e it (1 + w(t, x)).
(1.6)
We will deal with solutions to (1.7) for which the modulational instability phenomenon is present. Indeed, note that w now solves [32] iw t + w xx + 2 Re w + w 2 + 2|w| 2 + |w| 2 w = 0, (1. 7) with initial data in a certain Sobolev space. The associated linearized equation for (1.7) is just 1 i∂ t w + ∂ 2 x w + 2 Re w = 0. (1.8) Written only in terms of φ = Re w, we have the wave-like equation (compare with [14] in the periodic setting)
∂ 2 t φ + ∂ 4 x φ + 2∂ 2 x φ = 0. (1.9)
This problem has some instability issues, as reveal a standard frequency analysis: looking for a formal standing wave φ = e i(kx−ωt) solution to (1.9), one has ω(k) = ±|k| k 2 − 2, which reveals that for small wave numbers (|k| < √ 2) the linear equation behaves in an "elliptic" fashion, and exponentially (in time) growing modes are present from small perturbations of the vacuum solution. A completely similar conclusion is obtained working in the Fourier variable. This singular behavior is not present if now the equation is defocusing, that is (1.7) with nonlinearity −|u| 2 u. 2 Summarizing, in this paper we will focus on models (1.1) and (1.4) with zero boundary values at infinity, and on the model (1.7), which represents (1.1) with nonzero boundary conditions, in the form of a Stoke wave (1.2). Additionally, and appealing to physical considerations, we will only consider solutions to these models with finite energy, in a sense to be described below.
Concerning the well-posedness theory for the three models (1.1)-(1.1a), (1.4), and (1.7), we have the following result. The proof of this result in the case of Sasa-Satsuma (1.4) follows easily from the arguments in Kenig-Ponce-Vega [22], and for (1.7) it was recently proved in [32]. The proof of (1.1) is standard, and is due to Ginibre and Velo [15], Tsutsumi [41] and Cazenave and Weissler [12]. See Cazenave [9] for a complete account on the different NLS equations.
1.2. U (1) invariant Breathers. In this paper we are interested in variational stability properties associated to particular but not less important exact solutions to (1.1)-(1.1a), (1.4) and (1.7), usually referred as breathers.
Definition 1.2. We will say that a particular smooth solution to (1.1)-(1.1a) or (1.1)-(1.1b), or (1.4), is a breather if modulo the invariances of the equation, it is periodic in time, but with nontrivial period.
This definition leaves outside of our paper standard solitons for (1.1):
√ ce i(ct+ 1 2 xv− 1 4 v 2 t+γ 0) cosh( √ c(x − vt − x 0 )) , c > 0, v, x 0 , γ 0 ∈ R,(1.10)
which are time periodic solutions of (1.1), thanks to scaling and Galilean transformations, but its time period is trivial (its infimum equals zero). This last soliton is a well-known orbitally stable solution of NLS, see Cazenave-Lions [10], Weinstein [42], and Grillakis-Shatah-Strauss [17].
It turns out that models (1.1)-(1.1a), (1.1)-(1.1b) and (1.4) possess explicit breather solutions, each one with a particular different behavior. More precisely, these are the breather solutions that we will study in this paper:
(i) The Sasa-Satsuma (SS) breather. Let α, β > 0 be arbitrary but fixed parameters.
Following [38, eqns. where the phase Θ and the scaled Q β obey Θ := α(x + δt + x 1 ), Q β (x) := βQ(βx), and the speeds γ and δ are given by (compare with [4] for instance) γ := 3α 2 − β 2 , δ := α 2 − 3β 2 .
(1.12)
Above, Q is complex-valued, exponentially decaying:
Q(x) := Q η (x) := 2(e x + ηe −x ) e 2x + 2 + |η| 2 e −2x , (1.13) and η := α α + iβ .
(1.14)
It is well-known that the real-valued function |Q| is single humped when |η| > 1/2 (i.e. |α| > 1 2 α 2 + β 2 ), and double humped when 0 < |η| ≤ 1/2 (or |α| ≤ 1 2 α 2 + β 2 ), see [43,36]. This mixed shape is in strong contrast with the standard NLS soliton (1.1) given in (1.10), which is only single humped. Moreover, from the formula in (1.11)-(1.13)-(1.14), one can clearly see that an increasingly small NLS soliton (1.10) is recovered in the limit η → 1 (or β → 0). See Fig. 1 for more details.
Another important fact in the SS breather is the fact that the single humped condition |α| > 1 2 α 2 + β 2 leads to 3α 2 > β 2 , which is nothing but having γ > 0 (i.e., a SS breather of negative speed). Similarly, the double-humped condition |α| ≤ 1 2 α 2 + β 2 means that γ ≤ 0, that is to say, the SS breather moves to the right.
The B SS solution is usually referred in the literature (see e.g. [43,36] and references therein) as an embedded soliton, because it is embedded in the continuous spectrum of the associated linear operator (see Remark 3.1 for more details on this concept). From the techniques exposed in this paper, we will see that B SS fits perfectly the description associated to a breather solution, including its stability characterization.
The stability of the SS breather has been studied by Pelinovsky and Yang in [36]. It was proved in this work that in the η → 1 limit, the SS breather is linearly stable (single humped case). No other regime seems to be rigorously described in the literature, as far as we understand. Also, the nonlinear stability/instability of the SS breather seems a completely open question.
(ii) The Satsuma-Yajima (SY) breather. Let c 1 , c 2 > 0, and γ ± := c 2 ± c 1 . The NLS equation with zero background (1.1) has the standing, exponentially decaying breather [39]
B SY (t, x) := 2 √ 2γ + γ − e ic 2 1 t (c 1 cosh(c 2 x) + c 2 e iγ + γ − t cosh(c 1 x)) γ 2 − cosh(γ + x) + γ 2 + cosh(γ − x) + 4c 1 c 2 cos(γ + γ − t)
, (1.15) as solution which is a perturbation of the zero state, see Fig . 2. By invariances of the equation under time-space shifts, it is possible to give a more general form for (1.15) involving shifts x 1 , x 2 ∈ R in the t and x variables, respectively. Note that by choosing c 1 = 1 and c 2 = 3, we recover the original breather discovered by Satsuma-Yajima [39]:
4 √ 2e it (cosh(3x) + 3e 8it cosh x) cosh(4x) + 4 cosh(2x) + 3 cos(8t)
.
(1.16)
The SY breather has been observed in nonlinear optics as well as in quantum mechanics, and plays a key role in the description of the precise dynamics of optical and matter waves in nonlinear and non autonomous dispersive physical systems, driven by nonautonomous NLS and Gross-Pitaevskii (GP) models. For instance, two matter wave soliton solutions in a Bose-Einstein condensate reduce to the SY breather with a suitable constant selection (see [37] for further details). Moreover, in a hydrodynamical context, it has been reported the observation of the SY breather from a precise initial condition for exciting the two soliton solution, which gives rise to this SY breather, from the mechanical instruments generating the waves ( [11]).
It is also well-known that SY breathers are unstable [43]. Their instability is simply based in the fact that there are explicit 2-solitons solutions (see (7.1) for example) arbitrarily close to the SY breather, but with completely different long-time behavior at infinity in time. This instability property is motivated, in terms of inverse scattering data, as the understanding of the 2-soliton and SY breather as objects described by 2-parameter "complex-valued eigenvalues", with no restriction at all, see [43] for more details. On the contrary, the 2-soliton and mKdV breather are defined by using realvalued and complex-valued eigenvalues respectively, a distinction that avoids arbitrary closeness in any standard metric. Left below: |SS| with η = 0.51, and right below: |SS| with η = 0.9. Note that for η close to 1, one recovers the NLS soliton, and for η close to zero, the breather decouples and two clearly defined humps, at equal distance for all time (of order O(| log η|)), emerge in the dynamics. and grow quickly. This unstable growth leads to a nontrivial competition with the (focusing) nonlinearity, time at which the solution is apparently stabilized.
(iii.1) The Peregrine (P) breather [35]. Given by
B P (t, x) := e it 1 − 4(1 + 2it) 1 + 4t 2 + 2x 2 , (1.17)
which is a polynomially decaying (in space and time) perturbation of the nonzero background given by the Stokes wave e it , which appears and disappears from nowhere [1]. See Fig. 3 left for details. Some interesting connections have been made between the Peregrine soliton (1.17) and the intensely studied subject of rogue waves in ocean [44,40,1,23] (see also [8] for an alternative explanation to the rogue wave phenomenon). Very recently, Biondini and Mantzavinos [7] showed, using inverse scattering techniques, the existence and long-time behavior of a global solution to (1.7) in the integrable case (p = 3), but under certain exponential decay assumptions at infinity, and a no-soliton spectral condition (which, as far as we understand, does not define an open subset of the space of initial data).
Note that, because of time and space invariances in NLS, for any t 0 ,
x 0 ∈ R, B P (t − t 0 , x − x 0 )
is also a Peregrine breather.
(iii.
2) The Kuznetsov-Ma (KM) breather. The final object that we will consider in this paper is the Kuznetsov-Ma (KM) breather [26,27], given by the compact expression [2] B KM (t, x) := e it 1 − √ 2β (β 2 cos(αt) + iα sin(αt)) α cosh(βx) − √ 2β cos(αt) , α := (8a(2a − 1)) 1/2 , β := (2(2a − 1)) 1/2 , a > 1 2 .
(1.18) Notice that in the formal limit a ↓ 1 2 one recovers the Peregrine breather. See Fig. 3 right for details. Note that B KM is a Schwartz perturbation of the Stokes wave, and therefore a smooth classical solution of (1.7). It has been also observed in optical fibre experiments, see Kliber et al. [24] and references therein for a complete background on the mathematical problem and its physical applications.
Using a simple argument coming from the modulational instability of the equation (1.7), in [32] it was proved for the first time, and in a rigorous form, that both B KM and B P are unstable with respect to perturbations in Sobolev spaces H s , s > 1 2 . Previously, Haragus and Klein [25] showed numerical instability of the Peregrine breather, giving a first hint of its unstable character. The proof of this result uses the fact that Peregrine and Kuznetsov-Ma breathers are in some sense converging to the background final state (i.e. they are asymptotically stable) in the whole space norm H s (R), a fact forbidden in Hamiltonian systems with conserved quantities and stable solitary waves. A further extension of this result, valid for periodic perturbations of the Akhmediev breather, was proved in [3]. Please see more details on the Akhmediev breather in [3]. The left axis represents the x variable, and the right one the t variable.
Main results
The results in this paper can be characterized in two principal guidelines: a first one concerning a variational characterization for each breather above considered, and a second one related to stability and instability properties associated to that characterization. We will also identify each dispersive model in this paper with its respective breather solution. Indeed, let . Our first result is the following variational characterization of all these breather solutions. We will prove that, essentially, all of them satisfy the same nonlinear fourth order ODE, up to particular constants.
B (4x) + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B − 2(β 2 − α 2 )(B xx + 4|B| 2 B) + (α 2 + β 2 ) 2 B = 0. (2.1) (2) If X = SY and B = B SY , B (4x) + 3B 2 xB + 4|B| 2 B xx + B 2B xx + 2|B x | 2 B + 3 2 |B| 4 B − (c 2 2 + c 2 1 )(B xx + |B| 2 B) + c 2 2 c 2 1 B = 0.B (4x) + 3B 2 xB + (4|B| 2 − 3)B xx + B 2B xx + 2|B x | 2 B + 3 2 (|B| 2 − 1) 2 B − β 2 (B xx + (|B| 2 − 1)B) = 0. (2.3)
In particular, for X = P one has that B = B P satisfies the limiting case Theorem 2.1 will be a particular consequence of the following variational characterization of each breather above mentioned. Recall that for m ∈ N, the vector space H m (R; C) corresponds to the Hilbert space of complex-valued functions f : R → C, with m derivatives in L 2 (R; C), endowed with the standard norm. where (1) F X , E X and M X are respective H 2 , H 1 and L 2 based conserved quantities for the dispersive model X around the zero background or the Stokes wave e it , depending on the particular limit value of the breather at infinity. Here, E X and M X corresponds to suitable energy and mass, respectively; (2) H X is well-defined for u ∈ B X + H 2 (R; C); (3) This functional is conserved for H 2 perturbations of the respective dispersive model X. (4) m X , n X ∈ R are well-chosen parameters, depending only on the nontrivial internal parameters of the breather B X ; in particular: (a) For X = SS, one has m X = −2(β 2 − α 2 ) and n X = (α 2 + β 2 ) 2 .
B (4x) + 3B 2 xB + (4|B| 2 − 3)B xx + B 2B xx + 2|B x | 2 B + 3 2 (|B| 2 − 1) 2 B =
(b) For X = SY , one has m X = (c 2 2 + c 2 1 ) and n X = c 2 2 c 2 1 . (c) For X = KM , one has m X = −β 2 and n X = 0. (d) For X = P , one has m X = n X = 0. 2 states that all U (1) breathers considered in this paper (and possibly several others not considered here by lenght considerations) satisfy the same variational characterization. This property exactly coincides in the SS case with the classical mKdV characterization [4]; however, in the remaining SY , KM and P cases, it certainly differs in the choice of respective constants for the construction of H.
Remark 2.4. Theorem 2.2 reveals that KM and P breathers obey, in some sense, degenerate variational characterizations. More precisely, the KM breather characterization do not require the use of the L 2 based mass term M KM , and even worse, the P breather does not require the mass and the energy M P and E P , respectively. The absence of these two quantities may be related to the fact that
M P [B P ] = E P [B P ] = 0, (see Remark 3.2),
meaning a particular form of instability (recall that mass and energy terms are somehow convex terms aiding to the stability of solitonic structures). We would like to further stress the fact that the variational characterization of the famous Peregrine breather is in H 2 , since mass and energy are useless. See also Remark 3.3 for more about the zero character of KM and P conservation laws.
Remark 2.5. Theorem 2.1 will be a (not so direct) consequence of the critical point character of each breather in Theorem 2.2, identity (2.6). Section 5 is devoted to the proof of this fact.
The proof of Theorem 2.1 is simple, variational and follows previous ideas presented in [4] for the case of mKdV breathers, and [6] for the case of the Sine-Gordon breather (see also [33] for a recent improvement of this last result, based in [5]). The main differences are in the complex-valued nature of the involved breathers, and the nonlocal character of the KM and P breathers. Some special attention must be put to find the constants m X and n X above, a task that required some time and a large amount of computations, but finally we have found each of them.
2.2. Stability and instability results. Next, we establish some stability and instability properties for the considered breathers. As usual, we start out with the SS case. In this paper, we show nonlinear stability of this breather. A more precise statement of stability is given in Theorem 6.7. The proof of Theorem 2.3 follows the ideas in [4], but the proofs are considerably harder, because of the complexvalued character of the involved linearized operator around the breather solution. After some nontrivial preliminary results, we prove that this linear operator is nondegenerate and has only a unique negative eigenvalue, a property shared by the mKdV breather. Recall that the mKdV breather is real-valued, and proofs are considerably simpler in that case. Theorem 2.3 is, as far as we understand, the first rigorous nonlinear stability result for a U (1) symmetry breather.
Our proof does work even in the double humped case, despite the fact that in this case the linearized operator H SS [B SS ] has a more complex structure. No such nonlinear stability result was known in the literature, even in the single humped case. Now we consider the SY breather. Recall that it is well-known that the SY breather is unstable, see e.g. [43]. However, this lack of stability is only mild, in the sense that the SY breather (1.15) is instead part of a larger family of 2-soliton states B SY,gen , given by a complicated formula, see (7.1). This larger family is indeed, stable, as it was proved by Kapitula [21]. On the other hand, the construction of N -solitons in the nonintegrable NLS cases has been carried out for the first time by Martel and Merle [29], and more recently by Nguyen [34]. Note that in this last reference, a breather like solution such as the SY breather (1.15) has not yet been constructed. The stability of these nonintegrable N -soliton solutions has been addressed in H 1 and for some particular nonlinearities (essentially supercritical), see [30]. Finally, nonexistence of NLS breathers with the oddness parity property and any nonlinearity has been recently proved in [31].
H SY [B SY ](D, D) = 0, in addition to the standard kernel ∂ x B SY , ∂ t B SY and negative direction ∂ c B SY , c = max{c 1 , c 2 }.
Remark 2.6. Theorem 2.4 states that SY breathers are unstable at the linear level because among the three natural symmetries associated to the SY breather, there is no symmetry capable to control two additional zero and negative directions. This unstable character is certainly not present in the case of the general SY 2-soliton, which is stable [21].
Remark 2.7. We believe that in Theorem 2.4 there is an additional direction of instability, which is in this case of negative character, and not part of a generalized kernel. The proof of this result will be probably published elsewhere.
Finally, we consider the case of KM and P breathers. Recall that both are unstable, see [32]. In this paper we further improve the results in [32] by showing the following nonlinear instability property: Theorem 2.5 (Direction of instability of the Peregrine breather). Let B = B P be a Peregrine breather, critical point of the functional H P defined in (2.5). Then the following is satisfied. Let z 0 ∈ H 2 be any sufficiently small perturbation. Then, as t → −∞,
H P [B P ](z 0 ) = 0, but H P [B P ](z 0 , z 0 ) = 1 2 (|w x | 2 − |w| 2 − w 2 )(t) + O( z 0 3 H 1 ) + o t→+∞ (1), (2.7) where w = w(t) := e −it ∂ x z 0 ∈ H 1 .
Remark 2.8. The previous result gives a precise expression for the lack of stability in Peregrine breathers. Essentially, the continuous spectrum of the second derivative of the Lyapunov functional H P stays below zero, a phenomenon that induces exponential growth in time for arbitrary perturbations of the associated linear dynamics.
Remark 2.9. Theorem 2.5 can be recast as an absence of spectral gap for the linearized dynamics; we will not pursue this fact in the Peregrine case, but instead we will exemplify this fact using the Kuznetsov-Ma breather KM.
In the case of the KM breather, things are more complicated, and the previous result is not valid, since B KM does not decay to the Stokes wave at time infinity (recall that KM breather oscillates around a Schwartz perturbation of the Stokes wave). Instead, we will prove the following
H KM [B KM ](∂ x B KM ) = 0, inf σ c (H KM [B KM ]) < 0. (2.8)
Here σ c stands for the continuum spectrum of the linear operator associated to
H KM [B KM ].
Remark 2.10. The above theorem shows that the KM linearized operator H KM has at least one embedded eigenvalue. This is not true in the case of linear, real-valued operators with fast decaying potentials, but since H KM is a matrix operator, this is perfectly possible. Additionally, a similar result for the Peregrine case could be proved, but the polynomial decay in space of the Peregrine breather makes this result more complicated to establish for the moment. Organization of this paper. This paper is organized as follows. In Section 3 we establish some preliminary results needed for the proof of Theorems 2.1 and 2.2. Section 4 deals with the proof of Theorem 2.2, needed for the proof of Theorem 2.1. Section 5 is devoted to the proof of Theorem 2.1. In Section 6 we prove Theorem 2.3. Section 7 attacks Theorem 2.4 and Section 8 is concerned with the proof of Theorem 2.5. Finally, Section 9 deals with Theorem 2.6.
Preliminaries
The purpose of this section is to gather several results present in the literature, needed below. We first present a result for the Sasa-Satsuma breather.
3.1. Non variational PDE in the SS case. The following results are essentially contained in [36]. From (1.11) and (1.4), it is not difficult to see that the soliton profile Q β satisfies the ODE
Q β + 3iαQ β + 6|Q β | 2 Q β + 6iα|Q β | 2 Q β + 3Q β (|Q β | 2 ) − β 2 Q β − 3iαβ 2 Q β = 0.
This equation can be rewritten as
Q β + 9Q βQβ Q β + 3Q 2 βQ β − β 2 Q β + 3iα Q β − β 2 Q β + 2Q 2 βQβ = 0. (3.1)
Note that this is a third order equation, and it seems that it cannot be integrated one more time. This exact equation will be used to prove (2.1).
Remark 3.1. Note that the term embedded soliton comes from (3.1). Unlike the standard
NLS ODE Q − Q + Q 3 = 0, in its linear form Q β − β 2 Q β + 3iα Q β − β 2 Q β = 0 (3.1) has
"continuous spectrum" solutions of the form e iax , a ∈ R; see [36] for more details about this concept.
3.2. Conserved quantities. In this subsection we consider the conserved quantities needed for the proof of Theorem 2.1 and the definition of H in (2.5). In what follows, we adopt the subscript X ∈ {SS, SY, KM, P } to denote the conservation laws needed according to the respective breather B X . the energy
E SS [u] := |u x | 2 − 2|u| 4 dx,(3.3)
and the H 2 based energy
F SS [u] := |u xx | 2 − 8|u| 2 |u x | 2 − 3((|u| 2 ) x ) 2 + 8|u| 6 dx. (3.4)
Satsuma-Yajima. It is known that the NLS (1.7) with zero boundary condition at infinity possesses the following formally conserved quantities: the classical mass 5) and the focusing energy
M SY [u] := |u| 2 ,(3.E SY [u] := |u x | 2 − 1 2 |u| 4 . (3.6)
The additional H 2 based energy is given by the expression
F SY [u] := |u xx | 2 − 3|u| 2 |u x | 2 − 2(Re(ūu x )) 2 + 1 2 |u| 6 . (3.7)
Peregrine and Kuznetsov-Ma. For simplicity in the computations, it is convenient to write (1.7) for w in terms of the function u in (1.6). With this choice, both for X = KM and P , one has the mass
M X [u] := (|u| 2 − 1), (3.8)
the energy 9) and the Stokes wave + H 2 perturbations conserved energy:
E X [u] := |u x | 2 − 1 2 (|u| 2 − 1) 2 ,(3.F X [u] := |u xx | 2 − 3(|u| 2 − 1)|u x | 2 − 1 2 ((|u| 2 ) x ) 2 + 1 2 (|u| 2 − 1) 3 . (3.10)
Remark 3.2. In [32], it was computed the mass and energy (3.8)-(3.9) of the Peregrine (1.17) and breathers. Indeed, one has
M P [B P ] = E P [B P ] = 0,
(however, the L 2 -norm of B P (t) is never zero, but converges to zero as t → +∞), and
M KM [B KM ] = 4β, E KM [B KM ] = − 8 3 β 3 .
Note that P has same energy and mass as the Stokes wave solution (the nonzero background), a property not satisfied by the standard soliton on zero background. Also, compare the mass and energy of the Kuznetsov-Ma breather with the ones obtained in [4] for the mKdV breather. valid in the X = SS, SY cases, and
P X [u] := Im (ū − e −it )u x ,(3.12)
for the X = P, KM cases. Note that both quantities are well-defined and finite in the case of a breather B X , and essentially measure the speed of each breather. It is not difficult to show (or using a symbolic calculation program) that
P SS [B SS ] = −α α 2 + β 2 log 1 α 2 2β 2 + α 2 + 2β α 2 + β 2 ,(3.13)
and
P SY [B SY ] = P P [B P ] = P KM [B KM ] = 0. (3.14)
We can then conclude that, except for SS breathers, which have nonzero momentum, SY , KM and P breathers are zero speed solutions. This is in concordance with the characterization of periodic in time breathers, for which
d dt M SY [u] = const.P SY [u].
Therefor, breathers must have zero momentum. See [31] for another point of view about this fact. Note instead that, under a suitable Galilean transformation, they must have nonzero momentum.
Higher energy expansions: Proof of Theorem 2.2
This section is devoted to the the proof of Theorem 2.2. In what follows, we consider real-valued parameters m X , n X , for each X ∈ {SS, SY, KM, P } as follows:
(1) For X = SS, one has m X = −2(β 2 − α 2 ) and n X = (α 2 + β 2 ) 2 (see (1.11)).
(2) For X = SY , one has m X = (c 2 2 + c 2 1 ) and n X = c 2 2 c 2 1 . (3) For X = KM , one has m X = β 2 and n X = 0 (see (1.18)). (4) For X = P , one has m X = n X = 0 (see (1.17)).
These are the parameters previously mentioned in Theorem 2.2, item (4).
Consider the Lyapunov functional H X defined by
H X [u] = F X [u] + m X E X [u] + n X M X [u],
where F X , E X and M X were introduced in Subsection 3.2. This is exactly the functional considered in Theorem 2.2, and more specifically, (2.5). Note that this functional is a linear combination of conserved quantities mass (3.2)-(3.8), energy (3.3)-(3.9), and the second energy in F X (3.4)-(3.10).
Consequently, items (1)-(4) in Theorem 2.2 are easily proved.
It remains to prove item (5) in Theorem 2.2, and the fact that breathers B X are critical points for H X . These last facts will be a consequence of the following Proposition, and Theorem 2.1.
H X [B X + z] = H X [B X ] + G X [z] + Q X [z] + N X [z], (4.1) where • H X [B X ] does not depend on time. Moreover, H X [B P ] = 0.
• The linear term in z is given as
G X [z] = 2 Re zG[B X ], (4.2) with G[B SS ] := B (4x) + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B − 2(β 2 − α 2 )(B xx + 4|B| 2 B) + (α 2 + β 2 ) 2 B; (4.3) G[B SY ] := B (4x) + 3B 2 xB + 4|B| 2 B xx + 2|B x | 2 B + B 2B xx + 3 2 |B| 4 B − (c 2 2 + c 2 1 )(B xx + |B| 2 B) + c 2 2 c 2 1 B; (4.4) G[B KM ] := B (4x) + 3B 2 xB + (4|B| 2 − 3)B xx + B 2B xx + 2|B x | 2 B + 3 2 (|B| 2 − 1) 2 B − β 2 (B xx + (|B| 2 − 1)B); (4.5) and G[B P ] := B (4x) + 3B 2 xB + (4|B| 2 − 3)B xx + B 2B xx + 2|B x | 2 B + 3 2 (|B| 2 − 1) 2 B.
(4.6)
• The quadratic functional is given as
Q X [z] := Re zL X [z]dx (4.7) where L SS [z] := z 4x + (14|B| 2 + m SS )z xx + 6B 2z xx + (12BB x + 16BB x )z x + 12BB xzx + (14BB xx + 12|B x | 2 + 12BB xx + 72|B| 4 + 8m SS |B| 2 + n SS )z + (14BB xx + 8B 2 x + 48|B| 2 B 2 + 4m SS B 2 )z, (4.8) L SY [z] := z 4x + 4|B| 2 z xx + 3B 2 xz + 6BB x z x + 4BB xxz + 4BB xxz + 2BB xx z + B 2z xx + 2BB x z x + 2BB xzx + 2|B x | 2 z + 9 2 |B| 4 z + 3|B| 2 B 2z − m SY [z xx + B 2z + 2|B| 2 z] + n SY z. (4.9) L KM [z] := z 4x + 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) + 3(|B| 2 − 1)z xx − 4|B x | 2 z − 6BB xzx − 4BB x z x − B 2 xz + B 2z xx + |B| 2 z xx − m KM [z xx + B 2z + (2|B| 2 − 1)z], (4.10) and L P [z] := z 4x + 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) + 3(|B| 2 − 1)z xx − 4|B x | 2 z − 6BB xzx − 4BB x z x − B 2 xz + B 2z xx + |B| 2 z xx .
(4.11)
• Finally, assuming z H 1 small enough, we have the nonlinear estimate
|N X [z]| z 3 H 1 . (4.12)
Remark 4.1. Note that terms (4.3)-(4.6) precisely correspond to the nonlinear elliptic equations presented in Theorem 2.1; in that sense, once Theorem 2.1 is proved, Theorem 2.2 is also completely proved.
Remark 4.2. All linearized operators appearing from (4.7) contain terms in z andz. Consequently, these are 2 × 2 matrix valued operators with fourth order components each, more demanding that the ones found in [6] for the Sine-Gordon case, which was composed by fourth and second order mixed terms only.
Proof of Proposition 4.1. We proceed following standard steps. We will prove (4.1) decomposing H X [B X + z] into zeroth, first (linear in z), second (quadratic in z) and higher order terms (cubic or higher in z). The convention that we will use below is the following:
• Zeroth order terms will have the subscript "0". • First order terms will have the subscript lin.
• Second order terms will have the subscript quad.
• Higher order terms will have the subscript non.
Step 1. Contribution of the mass terms. Recall the masses (3.2), (3.5) and (3.8). We have for X = SS, SY and B = B X ,
M X [B + z] = |B + z| 2 = |B| 2 + 2 Re Bz + |z| 2 .
Similarly, for X = KM, P , (4.14)
M X [B + z] = |B + z| 2 − 1 = (|B| 2 − 1) + 2 Re Bz + |z| 2 .
Step 2. Contribution of the energy terms. Recall the energies (3.3) and (3.6). If X = SS and B = B X ,
E SS [B + z] = |B x + z x | 2 − 2 |B + z| 4 = |B x | 2 + 2 Re B xzx + |z x | 2 − 2 |B| 2 + 2 Re(Bz) + |z| 2 2 .
Therefore, we have
E SS [B + z] = E SS [B] + 2 Re z(−B xx ) + |z x | 2 − 2 (2 Re(Bz)) 2 + |z| 4 + 4|B| 2 Re(Bz) + 2|B| 2 |z| 2 + 4|z| 2 Re(Bz)
.
Clearly E SS,0 = E SS [B]
. The linear contribution here is 15) and the quadratic contribution is
E SS,lin := 2 Re z(−B xx − 4|B| 2 B),(4.E SS,quad = |z x | 2 − 2 2(Re(Bz)) 2 + 2|B| 2 |z| 2 . (4.16)
Finally, the higher order contribution is given by and the quadratic contribution is given by
E SS,non = −2 |z| 4 + 4|z| 2 Re(Bz) .E SY,quad := |z x | 2 − 1 2 (2 Re(Bz)) 2 + 2|B| 2 |z| 2 . (4.19)
Finally, the higher order contributions are
E SY,non := − 1 2 |z| 4 + 4|z| 2 Re(Bz) . (4.20)
Consider now the NLS case. The energy is given by (3.9), and if X = KM or P , and B = B X , we have
E X [B + z] = |B x + z x | 2 − 1 2 (|B + z| 2 − 1) 2 = |B x | 2 + 2 Re B xzx + |z x | 2 − 1 2 |B| 2 − 1 + 2 Re(Bz) + |z| 2 2 .
Therefore, we have
E X [B + z] = E X [B] + 2 Re z(−B xx ) + |z x | 2 − 1 2
(2 Re(Bz)) 2 + |z| 4 + 4(|B| 2 − 1) Re(Bz) + 2(|B| 2 − 1)|z| 2 + 4|z| 2 Re(Bz) .
Consequently, E X,0 := E X [B]. The linear contribution here is 21) and the quadratic contribution is Step 3. Contribution of the second energy terms. The SS case. We start by considering the case X = SS. Note that from (3.4),
E X,lin := 2 Re z(−B xx − (|B| 2 − 1)B),(4.E X,quad := |z x | 2 − 1 2 (2 Re(Bz)) 2 + 2(|B| 2 − 1)|z| 2 .F SS [B + z] = |B xx + z xx | 2 − 8|B + z| 2 |B x + z x | 2 − 3((|B + z| 2 ) x ) 2 + 8|B + z| 6 = |B xx | 2 + |z xx | 2 + 2 Re(B xxzxx ) F SS,1 −8 |B| 2 + |z| 2 + 2 Re(Bz) |B x | 2 + |z x | 2 + 2 Re(B xzx ) F SS,2 −3 (B x + z x )(B +z) + (B + z)(B x +z x ) 2 F SS,3 +8 |B| 2 + 2 Re(Bz) + |z| 2 3 F SS,4 (4.24)
We have
F SS,1 = (|B xx | 2 + |z xx | 2 ) + 2 Re zB xxxx , hence F SS,1,0 = |B xx | 2 ,F SS,2 = −8 |B| 2 + |z| 2 + 2 Re(Bz) |B x | 2 + |z x | 2 + 2 Re(B xzx ) We have F SS,2,0 = −8 |B| 2 |B x | 2 .
The linear terms are 28) and the quadratic terms are
F SS,2,lin = − 8 2|B| 2 Re(B xzx ) + 2|B x | 2 Re(Bz) = − 16 Re z − (|B| 2 B x ) x + |B x | 2 B = 16 Re z(B 2 xB + |B| 2 B xx ),(4.F SS,2,quad = − 8 |B| 2 |z x | 2 + |B x | 2 |z| 2 + 2 Re(Bz)2 Re(B xzx ) = − 2 Re 4|B| 2 |z x | 2 + 4|B x | 2 |z| 2 + 8Bz2 Re(B xzx ) = − 2 Re z − 4(|B| 2 ) x z x − 4|B| 2 z xx + 4|B x | 2 z + 8BB xzx + 8BB x z x .
(4.29)
Finally, the higher order terms are
F SS,2,non = − 8 |z| 2 + 2 Re(Bz) |z x | 2 − 16 |z| 2 Re(B xzx ) = − 8 |z x | 2 |z| 2 + 2|z x | 2 Re(Bz) − 2|z| 2 Re(B xzx ) .
(4.30)
Now, we deal with F SS,3 :
F SS,3 = − 3 (B x + z x )(B +z) + (B + z)(B x +z x ) 2 = − 3 B xB + B xz + z xB + z xz +B x B +B x z +z x B +z x z 2 .
(4.31)
The linear terms are
F SS,3,lin = − 3 B xB B xz + z xB +B x z +z x B − 3 B xz + z xB +B x z +z x B B xB +B x B − 3 B x B B xz + z xB +B x z +z x B = − 6 Re B xB B xz + z xB +B x z +z x B − 12 Re B xz + z xB Re(B xB ).
Therefore,
F SS,3,lin = − 6 Re B xB B xz + z xB +B x z +z x B − 12 Re z B x Re(B xB ) − (Re(B xB )B) x = − 12 Re B xB Re(B xz ) + Re(z x B) − 12 Re z B x Re(B xB ) − (Re(B xB )B) x = − 12 Re Re(B xB ) B xz +z x B − 12 Re z B x Re(B xB ) − (Re(B xB )B) x .
Collecting similar terms, we get
F SS,3,lin = − 12 Re z Re(B xB )B x − (Re(B xB )B) x + Re(B xB )B x − (Re(B xB )B) x = − 24 Re z Re(B xB )B x − (Re(B xB )B) x = 24 Re z(Re(B xB )) x B = 12 Re z(B xxB + B xBx +B xx B + B xBx )B, so that F SS,3,lin = 12 Re z(|B| 2 B xx + B 2B xx + 2B|B x | 2 ). (4.32)
The quadratic terms are F SS,3,quad
= − 3 B 2 xz 2 +B 2 x z 2 +B 2 z 2 x + B 2z2 x + 4BB x z xz + 4BB x zz x + 2|B x | 2 |z| 2 + 2|B| 2 |z x | 2 + 2(2 Re(Bz))(2 Re(B xzx )) = − 6 Re B 2 xz 2 + B 2z2 x + 4BB x z xz + |B x | 2 |z| 2 + |B| 2 |z x | 2 + 4Bz Re(B xzx ) = − 6 Re z B 2 xz − 2BB xzx − B 2z xx + 4BB x z x + |B x | 2 z − (|B| 2 ) x z x − |B| 2 |z xx + 4B Re(B xzx ) = − 6 Re z B 2 xz − B 2z xx + 4BB x z x + |B x | 2 z − (|B| 2 ) x z x − |B| 2 z xx + 2BB x z x .
(4.33)
Finally,
F SS,3,non = − 3 (z xz ) 2 + (z x z) 2 + 2B xz 2 z x + 2B xzx |z| 2 + 2z 2 xBz + 2|z x | 2B z + 2z x |z| 2B x + 2|z x | 2z B + 2|z x | 2 |z| 2 + 2B x z 2z x + 2z 2 x Bz .
(4.34)
As for F SS,4 , we have
F SS,4 = 8 |B| 2 + 2 Re(Bz) + |z| 2 3 = 8 |B| 4 + 4(Re(Bz)) 2 + |z| 4 + 4|B| 2 Re(Bz) + 2|B| 2 |z| 2 + 4|z| 2 Re(Bz) × |B| 2 + 2 Re(Bz) + |z| 2 .
Expanding terms, we have that the linear terms are given by On the other hand, the quadratic terms are given by Step 4. Gathering terms. We conclude from (4.25), (4.28), (4.32) and (4.35) that the linear part F SS,lin of F SS is given by
F SS,4,quad = 8 3|B| 2 B 2z2 + 3|B| 2B2 z 2 + 3|B| 4 |z| 2 = 2 Re 24|B| 2 B 2z2 + 12|B| 4 |z| 2 .+ 6 z(|B| 2 B xx + B 2B xx + 2B|B x | 2 ) + 24 z|B| 4 B = 2 Re z B xxxx + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B .
On the other hand, collecting terms in (4.26), (4.29), (4.33) and (4.36), the quadratic part of F SS is given by
F SS,quad := F SS,1,quad + F SS,2,quad + F SS,3,quad + F SS,4,quad = 2 Re z 1 2 z xxxx + 4(|B| 2 ) x z x + 4|B| 2 z xx − 4|B x | 2 z − 8BB xzx − 8BB x z x − 3B 2 xz + 3B 2z xx − 12BB x z x − 3|B x | 2 z + 3(|B| 2 ) x z x + 3|B| 2 z xx − 6BB x z x + 24|B| 2 B 2z + 12|B| 4 z = 2 Re z 1 2 z xxxx + 7|B| 2 z xx − 7(|B 2 |) x z x − 7|B x | 2 z − 4(B 2 ) xzx − 3B 2 xz + 3B 2z xx + 2BB x z x + 24|B| 4 B 2 z + 12|B| 4 z .= − 8 |z x | 2 |z| 2 + 2|z x | 2 Re(Bz) − 2|z| 2 Re(B xzx ) − 3 (z xz ) 2 + (z x z) 2 + 2B xz 2 z x + 2B xzx |z| 2 + 2z 2 xBz + 2|z x | 2B z + 2z x |z| 2B x + 2|z x | 2z B + 2|z x | 2 |z| 2 + 2B x z 2z x + 2z 2 x Bz + 8 4(Re(Bz)) 2 (2 Re(Bz) + |z| 2 ) + |z| 4 (|B| 2 + 2 Re(Bz) + |z| 2 ) + 4|B| 2 |z| 2 Re(Bz) + 2|B| 2 |z| 2 (2 Re(Bz) + |z| 2 ) + 4|z| 2 Re(Bz)(|B| 2 + 2 Re(Bz) + |z| 2 ) .
(4.38)
We can also collect higher order terms in the Lyapunov expansion. Specifically we have that from (4.14), (4.17) and (4.38),
N SS [z] := F SS,non + m SS E SS,non + n SS M SS,non = − 8 Re(Bz)|z x | 2 − 8 Re(B xzx )|z 2 | − 8|z| 2 |z x | 2 − 12 Re(B x z xz 2 ) − 12 Re(Bzz 2 x ) − 6 Re(z 2 xz 2 ) − 12 Re(B xzx )|z| 2 − 12 Re(Bz)|z x | 2
− 6|z| 2 |z x | 2 + 24|B| 2 (|z| 2 + 2 Re(Bz)) 2 + 8(|z| 2 + 2 Re(Bz)) 3 − 2m SS (|z| 4 + 2(2 Re(Bz))|z| 2 ) .
(4.39)
Clearly, in the case z H 2 small, one has |N SS [z]| z 3 H 1 . Summarizing, we have the following expansion for the Lyapunov functional H SS : End of proof in the SS case. This finally proves (4.1), (4.2)-(4.3), (4.7)-(4.8) and (4.12) in the SS case.
H SS [B + z] = F SS [B + z] + m SS E SS [B + z] + n SS M SS [B + z] = H SS [B] + 2 Re z B xxxx + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B − m SS (B xx + 4|B| 2 B) + n SS B + 2 Re z 2 z 4x − 14(|B| 2 ) x z x − 8(B 2 ) xzx + 14|B| 2 z xx − 14|B x | 2 z + 4BB x z x + 24|B| 4 z + 48|B| 2 B 2 z − 6B 2 xz + 6B 2z xx − m SS [z xx + 4B 2z + 8|B| 2 z] + n SS z + N
Since the SY case is somehow standard and close to SS, we will prefer to prove in full detail the more complicated case of KM and P breathers; the remaining SY case will be at the end of the proof.
Step 5. Contribution of the second energy terms. The case of Kuznetsov-Ma and Peregrine. Let X = KM or X = P. Now we deal with the contribution in F X , given in (3.10). Compared with F SS , there are minor differences, that we explain below. First of all, we also have the decomposition
F X [B + z] = |B xx + z xx | 2 − 3(|B + z| 2 − 1)|B x + z x | 2 − 1 2 ((|B + z| 2 ) x ) 2 + 1 2 (|B + z| 2 − 1) 3 = |B xx | 2 + |z xx | 2 + 2 Re(B xxzxx ) F X,1 −3 (|B| 2 − 1) + |z| 2 + 2 Re(Bz) |B x | 2 + |z x | 2 + 2 Re(B xzx ) F X,2 − 1 2 (B x + z x )(B +z) + (B + z)(B x +z x ) 2 F X,3 + 1 2 (|B| 2 − 1) + 2 Re(Bz) + |z| 2 (|B| 2 − 1) + 2 Re(Bz) + |z| 2 2 F X,4 .
Consequently the zeroth, linear, quadratic and nonlinear parts F X,1,0 , F X,1,lin , F X,1,quad and F X,1,non described above, compared with (4.25), (4.26) and (4.27), rest unchanged and we have F X,1,0 = |B xx | 2 , F X,1,lin = 2 Re zB xxxx , F X,1,quad = |z xx | 2 , F X,1,non = 0. (4.40)
The term F X,2,lin is analogous to F SS,2,lin in (4.28), except by a constant 3 (instead of 8) in front of it, and also the asymptotic constant equals 1. In fact, we have Also, the term F X,2,quad is analogous to F SS,2,quad in (4.29), except by a constant 3 (instead of 8) in front of it and the asymptotic constant 1. In fact, we have
F X,2,lin = − 3 2(|B| 2 − 1) Re(B xzx ) + 2|B x | 2 Re(Bz) = − 6 Re z − ((|B| 2 − 1)B x ) x + |B x | 2 B = 6 Re z(B 2 xB + (|B| 2 − 1)B xx ).F X,2,quad = − 3 (|B| 2 − 1)|z x | 2 + |B x | 2 |z| 2 + 2 Re(Bz) × 2 Re(B xzx ) = − 2 Re z − 3 2 (|B| 2 − 1)z xx − 3 2 (|B| 2 ) x z x + 3 2 |B x | 2 z + 3BB xzx + 3BB x z x .
(4.42)
Finally, the nonlinear term F X,2,non is given by
F X,2,non = − 3 |z| 2 |z x | 2 + 2 Re(B xzx ) − 6 Re(Bz)|z x | 2 . (4.43)
Similarly, the term F X,3,lin is analogous to F SS,3,lin in (4.32), except by a constant 1 2 (instead of 3) in front of it. We have first 44) and the linear contribution is given by
F X,3 = − 1 2 (B x + z x )(B +z) + (B + z)(B x +z x ) 2 = − 1 2 B xB + B xz + z xB + z xz +B x B +B x z +z x B +z x z 2 ,(4.F X,3,lin = 2 Re z(|B| 2 B xx + B 2B xx + 2B|B x | 2 ). (4.45)
On the other hand, the quadratic contribution is analogous to F SS,3,quad in (4.33), except by a constant 1 2 (instead of 3) in front of it. Therefore, the quadratic term is given by
F X,3,quad = − 1 2 B 2 xz 2 +B 2 z 2 x +B 2 x z 2 + B 2z2 x + 2B xB z xz + 2B xBzx z + 2B xBz z x + 2|B x | 2 |z| 2 + 2B x Bzz x + 2BB x z x z + 2|B| 2 |z x | 2 + 2B x Bz xz + 2B x Bzz x + 2B x Bzz x = − 1 2 (B 2 xz 2 +B 2 x z 2 ) + (B 2 z 2 x + B 2z2 x ) + 2|B x | 2 |z| 2 + 2|B| 2 |z x | 2 + 2(B xB z xz +B x Bzz x ) + 2(B xBzx z +B x Bz xz ) + 2(B xBz z x +B x Bzz x ) + 2(B x Bzz x +BB x z x z) .
Rearranging terms,
F X,3,quad = − Re(B 2 xz 2 ) + Re(B 2 z 2 x ) + |B x | 2 |z| 2 + |B| 2 |z x | 2 + 2 Re(B xB z xz ) + 2 Re(B xBzx z) + 2 Re(B xBz z x ) + 2 Re(B x Bzz x ) = − Re B 2 xz 2 +B 2 z 2 x + |B x | 2 |z| 2 + |B| 2 |z x | 2 + 4B xB z xz + 2B xBzx z + 2B x Bzz x .
(4.46)
The term F X,3,non is given now by (Compare with F SS,4 in (4.24).) First of all, we have F X,4 = 1 2 (|B| 2 − 1) 2 + 4(Re(Bz)) 2 + |z| 4 + 4(|B| 2 − 1) Re(Bz) + 2(|B| 2 − 1)|z| 2 + 4|z| 2 Re(Bz) × (|B| 2 − 1) + 2 Re(Bz) + |z| 2 .
F X,3,non = − 1 2 (z xz ) 2 + (z x z) 2 − B x z xz 2 +z x |z| 2 − B z 2 xz + |z x | 2 z − B x z x |z| 2 + B|z x | 2z + |z x | 2 |z| 2 − B xzx z 2 + Bz 2 x z .
Therefore, the linear terms are given by
F X,4,lin = 1 2 3(|B| 2 − 1) 2 (2 Re(Bz)) = 2 Re 3 2 (|B| 2 − 1) 2 Bz.+ 2 Re z(|B| 2 B xx + B 2B xx + 2B|B x | 2 ) + 2 Re 3 2 (|B| 2 − 1) 2 Bz = 2 Re z B xxxx + (4|B| 2 − 3)B xx + 3B 2 xB + 2B|B x | 2 + B 2B xx + 3 2 (|B| 2 − 1) 2 B .
On the other hand, collecting the terms in (4.40), (4.42), (4.46) and (4.49), the quadratic part of F N LS is given by F X,quad := F X,1,quad + F X,2,quad + F X,3,quad + F X,4,quad
= |z xx | 2 − 2 Re z − 3 2 (|B| 2 − 1)z xx − 3 2 (|B| 2 ) x z x + 3 2 |B x | 2 z + 3BB xzx + 3BB x z x − Re B 2 xz 2 +B 2 z 2 x + |B x | 2 |z| 2 + |B| 2 |z x | 2 + (4B xB z x + 2B x Bz x )z + 2B xBzx z + Re z 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) .
Arranging terms F X,quad = Re z z xxxx + 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz)
+ 3(|B| 2 − 1)z xx + 3(|B| 2 ) x z x − 3|B x | 2 z − 6BB xzx − 6BB x z x − Re B 2 xz 2 + B 2z xzx + |B x | 2 zz + |B| 2 z xzx + (4B xB z x + 2B x Bz x )z + 2B x Bz xz Re z z xxxx + 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) + 3(|B| 2 − 1)z xx + 3(|B| 2 ) x z x − 3|B x | 2 z − 6BB xzx − 6BB x z x + Re z − B 2 xz + (B 2z x ) x − |B x | 2 z + (|B| 2 z x ) x − (4B xB z x + 2B x Bz x ) − 2B x Bz x Therefore, F X,quad = Re z z xxxx + 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) + 3(|B| 2 − 1)z xx − 4|B x | 2 z − 6BB xzx − 4BB x z x − B 2 xz + B 2z xx + |B| 2 z xx .
(4.51)
Finally, we also collect the higher order terms in the Lyapunov expansion. Specifically we have that (4.40), (4.43), (4.47) and (4.50) leads to N X [z] := F X,1,non + F X,2,non + F X,3,non + F X,4,non
= − 6 Re(Bz)|z x | 2 − 6 Re(B xzx )|z 2 | − 3|z| 2 |z x | 2 − 2 Re(B x z xz 2 ) − 2 Re(Bzz 2 x ) − Re(z 2 xz 2 ) − 2 Re(B xzx )|z| 2 − 2 Re(Bz)|z x | 2 − |z| 2 |z x | 2 + 3 2 (|B| 2 − 1)(|z| 2 + 2 Re(Bz)) 2 + 1 2 (|z| 2 + 2 Re(Bz)) 3 − 1 2
m X (|z| 4 + 2(2 Re(Bz))|z| 2 ) .
(4.52)
We conclude that Proposition 4.1 in the KM and P cases (except for the proof of G X [z] = 0) is deduced from the above representation. Indeed, we have (4.1) by gathering
H X [B + z] = F X [B + z] + m X E X [B + z] + n X M X [B + z] = H X [B] + G X [z] + Q X [z] + N X [z],
as desired, selecting X = KM for the KM breather, m KM = β 2 , n KM = 0; and selecting X = P for the Peregrine breather, and m P = n P = 0.
Step 7. The case of Satsuma-Yajima. This case is very similar to the previous KM/P cases, with some minor differences in constants. Let m SY = (c 2 2 +c 2 1 ), n SY = c 2 2 c 2 1 as in the beginning of Section 4. Let also X = SY , B = B X and consider F SY [B + z] as in (3.7). First of all, note that the linear and quadratic contributions F SY,lin and F SY,quad from F SY [B + z] are as in the KM/P cases, but removing the asymptotic constant 1. Additionally, the higher order terms are given by for the choices of m X and n X given at the beginning of Section 4. Although these proofs are straightforward and painful, we present them in some detail to further checking by the reader.
N SY [z] := − 6 Re(Bz)|z x | 2 − 6 Re(B xzx )|z 2 | − 3|z| 2 |z x | 2 − 2 Re(B x z xz 2 ) − 2 Re(Bzz 2 x ) − Re(z 2 xz 2 ) − 2 Re(B xzx )|z| 2 − 2 Re(Bz)|z x | 2 − |z| 2 |z x | 2 + 3 2 |B| 2 (|z| 2 + 2 Re(Bz)) 2 + 1 2 (|z| 2 + 2 Re(Bz)) 3 − 1 2 m SY (|z| 4 + 2(2 Re(Bz))|z| 2 ) .+ 2 Re z B xxxx + 3B 2 xB + 4|B| 2 B xx + 2B|B x | 2 + B 2B xx + 3 2 |B| 4 B − m SY (B xx + |B| 2 B) + n SY B + 2 Re z 2 z 4x + 4|B| 2 z xx − 4|B x | 2 z − 3(B 2 ) xzx − 4BB x z x + B 2z xx − B 2 xz + 9 2 |B| 4 z + 3|B| 2 B 2z − m SY (z xx + B 2z + 2|B| 2 z) + n SY z + N SY [z] =: H SY [B] + G SS [z] + Q SS [z] + N SS [z],
5.1. Proof of (5.1) in the SS case. First we have
Q β + 4iαQ β − 6α 2 Q β − 4iα 3 Q β + α 4 Q β + 8Q β Q 2 β + 14Q βQβ Q β + 12Q βQ β Q β + 32iαQ βQβ Q β − 16α 2 Q 2 βQβ + 6Q 2 βQ β + 24Q 3 βQ 2 β − m SS Q β + 2iαQ β − α 2 Q β + 4Q 2 βQβ + n SS Q β = 0. (5.2)
Proof. See Appendix A for a proof of this result.
We continue with the proof of (5.1). Replacing m SS and n SS ,
Q β + 4iαQ β − 2(2α 2 + β 2 )Q β − 4iαβ 2 Q β + β 2 (4α 2 + β 2 )Q β + 8Q β Q 2 β + 32iαQ βQβ Q β + 14Q βQβ Q β − 8(α 2 + β 2 )Q 2 βQβ + 12Q βQ β Q β + 6Q 2 βQ β + 24Q 3 βQ 2 β = 0. (5.3)
From the third order ODE (3.1) satisfied by the profile Q β , we have
Q β + 9Q βQβ Q β + 3Q 2 βQ β − β 2 Q β + 3iα Q β − β 2 Q β + 2Q 2 βQβ = 0.
Therefore,
Q β + 4iαQ β − iαQ β + 9Q β Q 2 β + 9Q βQβ Q β + 15Q βQ β Q β + 3Q 2 βQ β − β 2 Q β + 3iαQ β − 3iαβ 2 Q β + 12iαQ β Q βQβ + 6iαQ 2
βQ β = 0. Using (3.1) and replacing above, we have
Q β + 4iαQ β + 9Q β Q 2 β + 9Q βQβ Q β + 15Q βQ β Q β + 3Q 2 βQ β − β 2 Q β − 3iαβ 2 Q β + 12iαQ β Q βQβ + 6iαQ 2 βQ β + iα 9Q βQβ Q β + 3Q 2 βQ β − β 2 Q β + 3iα Q β − β 2 Q β + 2Q 2 βQβ = 0. Namely Q β + 4iαQ β + 9Q β Q 2 β + 9Q βQβ Q β + 15Q βQ β Q β + 3Q 2 βQ β − (3α 2 + β 2 )Q β − 4iαβ 2 Q β + 21iαQ β Q βQβ + 9iαQ 2 βQ β + 3α 2 β 2 Q β − 6α 2 Q 2 βQβ = 0.
Comparing with (5.3), we just must show the following nonlinear identity satisfied by the soliton Q β (1.13):
− (α 2 + β 2 )Q β + β 2 (α 2 + β 2 )Q β − 2(α 2 + 4β 2 )Q 2 βQβ + 11iαQ β Q βQβ − 9iαQ 2 βQ β −Q β Q 2 β − 3Q βQ β Q β + 3Q 2 βQ β + 5Q βQβ Q β + 24Q 3 βQ 2 β = 0. (5.4)
For the proof of this nonlinear identity is direct but cumbersome: see Appendix B for a proof. This ends the proof of (5.1).
5.2.
Proof of (5.1) in the remaining cases. The rest of proofs in the cases SY , KM and P ((2.2), (2.3) and (2.4)) are similar to the above written, and add no new insights nor mathematical clues about the breathers themselves. For this reason, we have placed them in the Appendix C.
Stability of the SS breather. Proof of Theorem 2.3
This Section is devoted to the proof of Theorem 2.3. The proof requires several steps, that we represent in different subsections.
Without loss of generality, using the scaling and space invariances of the equation, we assume β = 1 and x 2 = 0.
6.1. Continuous spectrum and nondegeneracy of the kernel. Let B = B SS be a SS breather as in (1.11), and L SS be the linear operator in (4.8). By considering z andz as independent variables, as usual, and with a slight abuse of notation, we can write L SS as
L SS = L SS,1 L SS,2 L SS,3 L SS,4 , where L SS,1 := ∂ 4 x + (14|B| 2 + m SS )∂ 2 x + (12BB x + 16BB x )∂ x + (14BB xx + 12|B x | 2 + 12BB xx + 72|B| 4 + 8m SS |B| 2 + n SS ), L SS,2 := 6B 2 ∂ 2 x + 12BB x ∂ x + (14BB xx + 8B 2 x + 48|B| 2 B 2 + 4m SS B 2 ), L SS,3 := 6B 2 ∂ 2 x + 12BB x ∂ x + (14BB xx + 8B 2 x + 48|B| 2B2 + 4m SSB 2 ) = L SS,2 , and L SS,4 := ∂ 4 x + (14|B| 2 + m SS )∂ 2 x + (12BB x + 16BB x )∂ x + (14BB xx + 12|B x | 2 + 12BB xx + 72|B| 4 + 8m SS |B| 2 + n SS ) = L SS,1 .
Note that L SS is Hermitian as an operator defined in H 2 (R; C) with dense domain H 4 (R, C). Therefore, its spectrum is real-valued. We start with the following result, essentially proved in [4].
= ∂ 4 x + m SS ∂ 2 x + n SS 0 0 ∂ 4 x + m SS ∂ 2 x + n SS . (6.1)
In particular, the continuous spectrum of L is the closed interval [(α 2 + β 2 ) 2 , +∞) in the case β ≥ α, and [4α 2 β 2 , +∞) in the case β < α.
Now we study the kernel of L SS . We have directly from (2.1)
∂ x 1 B ∂ x 1B , ∂ x 2 B ∂ x 2B ∈ ker L SS .
Note that ∂ x 1 B = iαB, which is nothing but the instability direction associated to the U (1) invariance. Moreover, following the ideas in [4], based on the 1-D character of the ODEs involved, we have Lemma 6.2 (Nondegeneracy).
ker L SS = span ∂ x 1 B ∂ x 1B , ∂ x 2 B ∂ x 2B .
Remark 6.1. The proof of this result follows the ideas in [4], but not every vector valued linear operator around breathers will follow the same idea of proof. See [6] for a case were the argument in [4] does not apply. We will benefit here from the fact that the second component of L SS [z] = 0 corresponds to the complex conjugate of the first one.
Proof. Let z ∈ H 4 be such that L SS [z] = 0, such that {z, B 1 , B 2 } is linearly independent. For all large x we have that L SS behaves like L SS,0 in (6.1), which determines the large x behavior of solutions of L SS [z] = 0. Fortunately, L SS,0 is a diagonal operator with the same components, so we only need to consider the first one, the second one being identical since it corresponds to the complex conjugate. As in [4], we have that z must have the large x behavior
z(x) ∼ e ±x±iα .
Among these, there are only two linearly independent possible behaviors as x → ±∞ representing localized data: e −x±iαx , the same number as the dimension of ker L SS . This implies that dim ker L SS ≤ 2, proving the result.
L SS ∂ α B = −4α(B xx + 4|B| 2 B) − 4α(α 2 + β 2 )B,(6.
2)
and
L SS ∂ β B = 4β(B xx + 4|B| 2 B) − 4β(α 2 + β 2 )B. (6.3)
Additionally, we have
L SS (B 0 ) = −B, B 0 := β∂ α B + α∂ β B 8αβ(α 2 + β 2 ) ,(6.
4)
and
Re (β∂ α B + α∂ β B)L SS (β∂ α B + α∂ β B) = −4α 2 β(α 2 + β 2 ) |Q| 2 < 0. (6.5)
Remark 6.2. Lemma 6.3 shows that β∂ α B + α∂ β B is a negative direction for the functional L SS .
Proof. The proofs of (6.2) and (6.3) are direct from (2.1). The proof of (6.4) follows from (6.2) and (6.3). Finally, from (6.4), (1.11) and (1.13),
Re (β∂ α B + α∂ β B)L SS (β∂ α B + α∂ β B) = − 8αβ(α 2 + β 2 ) Re (β∂ α B + α∂ β B)B = − 4αβ(α 2 + β 2 ) β∂ α |B| 2 + α∂ β |B| 2 = − 4α 2 β(α 2 + β 2 ) |Q| 2 .
This proves (6.5).
It turns out that the most important consequence of the previous result is the fact that L SS possesses only one negative eigenvalue. Indeed, in order to prove that result, we follow the Greenberg and Maddocks-Sachs strategy [16,28], applied this time to the linear operator L SS . This time, we need some important changes. Lemma 6.4 (Uniqueness criterium, see also [16,28]). Let B = B SS be any SS breather (1.11), and ∂ x 1 B, ∂ x 2 B the corresponding kernel of the operator L SS . Then L SS has
x∈R dim (ker W [∂ x 1 B, ∂ x 2 B] ∩ ker W [∂ x ∂ x 1 B, ∂ x ∂ x 2 B]) (x)
negative eigenvalues, counting multiplicity. Here, W is the Wronskian matrix of the functions ∂ x 1 B and ∂ x 2 B,
W [A 1 , A 2 ](x) := A 1 A 2 A 1 A 2 (x). (6.6)
Proof. This result is essentially contained in [16,Theorem 2.2], where the finite interval case was considered. As shown in several articles (see e.g. [28,20]), the extension to the real line is direct. Here we need some changes, that sketch below.
Fix θ ∈ R. Let us consider the eigenvalue problem
L SS z = λ(θ)z, z ∈ H θ ,(6.7)
where
H θ := {z ∈ H 4 ((−∞, θ), C) : z(θ) = z x (θ) = 0}
. With a slight abuse of notation we will denote by L SS,θ the unbounded operator L SS with domain H θ and values in L 2 (R). Clearly for any θ ∈ R, L SS,θ is self-adjoint. Moreover, its continuous spectrum is given by σ c (L SS ). Also, for any θ ∈ R, L SS,θ is bounded below.
For any θ ∈ R, the number of eigenvalues of L SS,θ is nonempty. We define by n(θ) ≥ 1 (maybe infinite) the number of eigenvalues of L SS,θ . Notice that n(θ) is never zero, since λ 1 (θ) always exists.
Recall that λ j (+∞) represent the eigenvalues of L SS in R. Our objective is to determine the number of indices j such that λ j (+∞) < 0. We remark that we know that there is at least one and at most a finite number of negative eigenvalues for L SS .
Let θ ∈ R and λ 1 (θ) ≤ λ 2 (θ) ≤ · · · ≤ λ n(θ) , be the eigenvalues of L SS,θ , counted as many times according to their multiplicity. Note that n(θ) may vary but it is always finite and ≥ 1. Moreover, λ j (θ) are continuous and strictly decreasing functions of θ, with λ j (θ) ≥ λ j (+∞).
Fix now j ∈ {1, . . . , n(θ)}. There is at most one θ j such that λ j (θ j ) = 0, and such θ j exists if and only if λ j (+∞) < 0. Since the set of eigenvalues {λ k (+∞) : λ k (+∞) < 0} is finite and nonempty, we conclude the number of negative eigenvalues of L SS equals the number of points θ such that λ j (θ) = 0, where j ∈ {1, . . . , n(θ)}. And fixed θ ∈ R, the multiplicity of 0 as an eigenvalue of L SS,θ is equal to the number of indices j such that λ j (θ) = 0. Now, let us characterize 0 as an eigenvalue of L SS,θ . Indeed, we have that 0 is an eigenvalue of L SS,θ if and only if there are constants c 1 , c 2 ∈ C, not all zero, such that the vector-valued function
Z(x) := B 1 (x) B 2 (x) B 1 (x) B 2 (x)
c 1 c 2 is nontrivial and belongs to H θ (note that any other linearly independent element of the vector space L SS Z = 0 is exponentially increasing as x → −∞). Additionally, taking space derivative and using the definition of H θ we have
B 1 (θ) B 2 (θ) B 1 (θ) B 2 (θ) c 1 c 2 = 0 0 , (6.8)
as well as, for constants,c 1 ,c 2 ∈ C,
∂ x B 1 (θ) ∂ x B 2 (θ) ∂ x B 1 (θ) ∂ x B 2 (θ) c 1 c 2 = 0 0 . (6.9)
Summing on x ∈ R we conclude.
In what follows, we compute the double Wronskians (6.8) and (6.9) in the explicit SS case. We easily have
det W [∂ x 1 B, ∂ x 2 B](x) = det iαQ η e iΘ Q η e iΘ −iαQ η e −iΘ Q η e −iΘ = 2iα Re Q η Q η . (6.10)
We have, for η =: a + ib = α 2 α 2 +β 2 − αβi α 2 +β 2 , and u := e 2x > 0,
Re Q η Q η = u a 4 + 4a 3 u − 2a 2 u − b 2 − 4au u 2 − b 2 + b 4 − 2b 2 u + 2u 3 − u 4 (a 2 + b 2 + 2u + u 2 ) 3 .
Let us find a positive root u for the term in the numerator above. First of all, we have
a 4 + 4a 3 u − 2a 2 u − b 2 − 4au u 2 − b 2 + b 4 − 2b 2 u + 2u 3 − u 4 = α 4 (u − 1)(u + 1) 3 + 2α 2 β 2 u u 3 + 1 + β 4 (u − 2)u 3 (α 2 + β 2 ) 2 .
The solutions to this equation equals zero are u 1,± := ±α
α 2 + β 2 , u 2,± := β 2 − α 2 ± β β 2 − 3α 2 α 2 + β 2 .
Clearly u 1,− is not a valid solution. Now, if β 2 − 3α 2 < 0, the only valid positive root is u 1,+ = e 2x . It is not difficult to see in this case that
dim ker W [∂ x 1 B, ∂ x 2 B] 1 2 log u 1,+ = 1.
Assume now β 2 ≥ 3α 2 in (1.11). We have now at least a second root, u 2,+ , always positive. Additionally, u 2,− > 0 means
(β 2 − α 2 ) 2 > β 2 (β 2 − 3α 2 ) ⇐⇒ − 2β 2 α 2 + α 4 > −3α 2 β 2 ⇐⇒ β 2 + α 2 > 0,
so both u 2,± are positive, therefore, three roots are present in this case. In all these cases, dim ker W [∂ x 1 B, ∂ x 2 B] 1 2 log u 2,± = 1. Now we impose the second condition on the derivatives, i.e. (6.9). From (6.9) we have at
x = θ, iα(Q η + iαQ η )e iΘ (Q η + iαQ η )e iΘ −iα(Q η − iαQ η )e −iΘ (Q η − iαQ η )e −iΘ c 1 c 2 = 0 0 . (6.11)
A necessary condition to satisfy the previous equation with c 1 , c 2 not both zero is that at
x = θ we have (Q η + iαQ η )(Q η − iαQ η ) + (Q η + iαQ η )(Q η − iαQ η ) = 0.
The previous identity simplifies to
2 Re{Q η Q η } + 2α Im{Q η Q η } + 2α 2 Re{Q η Q η } = 0.
From (6.10) we have the last term in the previous identity equals zero. On the other hand, after some computations, one has
Re{Q η Q η } x= 1 2 log u 1,+ = Im{Q η Q η } x= 1 2 log u 1,+ = 0.
However, one can easily check (with numerics, for instance) that
Re{Q η Q η } x= 1 2 log u 2,± = 0, Im{Q η Q η } x= 1 2 log u 2,± = 0.
The following result summarizes our findings: In what follows, we define as B −1 the unique eigenfunction associated to the unique negative eigenvalue, such that B −1 L 2 = 1. We have Proposition 6.6 (Coercivity). Let B = B SS be a Sasa-Satsuma breather, and ∂ x 1 B, ∂ x 2 B the corresponding kernel of the associated operator L SS . There exists µ 0 > 0, depending on α, β only, such that, for any z ∈ H 2 (R) satisfying
Re ∂ x 1 Bz = Re ∂ x 2 Bz = 0, (6.12) one has
Re zL SS z ≥ µ 0 z 2 H 2 (R) − 1 µ 0 Re zB 2 . (6.13)
Proof. For the sake of simplicity, we denote B j := ∂ x j B. Indeed, it is enough to prove that, under the conditions (6.12) and the additional orthogonality condition Re zB = 0, one has
Re zL SS z ≥ µ 0 z 2 H 2 (R) .
Indeed, note that from (6.4), the function B 0 satisfies L SS [B 0 ] = −B, and from (6.5),
Re B 0 B = − Re B 0 L SS [B 0 ] > 0. (6.
14)
The next step is to decompose z and B 0 in span(B −1 , B 1 , B 2 ) and the corresponding orthogonal subspace. One has
z =z + mB −1 , B 0 = b 0 + nB −1 + p 1 B 1 + p 2 B 2 , m, n, p 1 , p 2 ∈ C,
where Re zB −1 = Re zB 1 = Re zB 2 = 0,
Re b 0 B −1 = Re b 0 B 1 = Re b 0 B 2 = 0. Note in addition that Re B −1 B 1 = Re B −1 B 2 = 0.
From here and the previous identities we have
Re zL SS z = Re (L SSz − mλ 2 0 B −1 )(z + mB −1 ) = Re zL SSz − m 2 λ 2 0 .]B 0 = Re (L SS [z] − mλ 2 0 B −1 )(b 0 + nB −1 + p 1 B 1 + p 2 B 2 ) = Re L SS [z]b 0 − mnλ 2 0 . (6.16)
On the other hand, In particular, ifz = λb 0 ,
Re B 0 B = − Re B 0 L SS [B 0 ] = − Re (b 0 + nB −1 )(L SS [b 0 ] − nλ 2 0 B −1 ) = − Re b 0 L SS b 0 + n 2 λ 2 0 .Re L SS [z]b 0 2 Re B 0 B + Re b 0 L SS b 0 ≤ a Re zL SSz , 0 < a < 1. (6.19)
In the general case, using the orthogonal decomposition induced by the scalar product (L SS ·, ·) L 2 on span(B −1 , B 1 , B 2 ), we get the same conclusion as before. Therefore, we have proved (6.19) for all possiblez.
Finally, replacing in (6.18) and (6.15), Re zL SS z ≥ (1−a) Re zL SSz ≥ 0, and Re zL SSz ≥ m 2 λ 2 0 . We have, for some C > 0,
Re zL SS z ≥ (1 − a) Re zL SSz ≥ 1 2 (1 − a) Re zL SSz + (1 − a)m 2 λ 2 0 ≥ 1 C (2 z 2 H 2 (R) + 2m 2 B −1 2 H 2 (R) ) ≥ 1 C z 2 H 2 (R) .
6.2. End of proof. We shall prove now the following explicit version of Theorem 2.3:
Theorem 6.7 (Explicit nonlinear stability of SS breathers). Let B = B SS be a SS breather with profile defined by a single hump, that is, γ = 3α 2 − β 2 > 0 in (1.12). Assume that u 0 ∈ H 2 (R; C) is such that
u 0 − B H 2 < η,
for some η sufficiently small. Then there exists K > 0 and shifts
x 1 (t), x 2 (t) ∈ R as in (1.11) such that sup t∈R u(t) − B(t; x 1 (t), x 2 (t)) H 2 < Kη.
Moreover, one has sup t∈R |x j (t)| Kη.
We prove the theorem only for positive times, since the negative time case is completely analogous. From the continuity of the SS flow for H 2 (R) data, there exists a time T 0 > 0 and continuous parameters x 1 (t), x 2 (t) ∈ R, defined for all t ∈ [0, T 0 ], and such that the solution u(t) of the Cauchy problem for the SS equation (1.4), with initial data u 0 , satisfies
sup t∈[0,T 0 ] u(t) − B SS (t; x 1 (t), x 2 (t)) H 2 (R) ≤ 2ν. (6.20)
The idea is to prove that T 0 = +∞. In order to do this, let K * > 2 be a constant, to be fixed later. Let us suppose, by contradiction, that the maximal time of stability T * , namely
T * := sup T > 0 for all t ∈ [0, T ], there existx 1 (t),x 2 (t) ∈ R such that sup t∈[0,T ] u(t) − B SS (t;x 1 (t),x 2 (t)) H 2 (R) ≤ K * ν ,(6.21)
is finite. It is clear from (6.20) that T * is a well-defined quantity. Our idea is to find a suitable contradiction to the assumption T * < +∞.
By taking η 0 smaller, if necessary, we can apply a well known theory of modulation for the solution u(t). There exists ν 0 > 0 such that, for all ν ∈ (0, ν 0 ), the following holds. There exist C 1 functions x 1 (t), x 2 (t) ∈ R, defined for all t ∈ [0, T * ], and such that
z(t) := u(t) − B(t), B(t, x) := B(t, x; x 1 (t), x 2 (t)) (6.22) satisfies, for t ∈ [0, T * ], Re ∂ x 1 B(t; x 1 (t), x 2 (t))z(t) = Re ∂ x 2 B(t; x 1 (t), x 2 (t))z(t) = 0. (6.23)
Moreover, one has
z(t) H 2 (R) + |x 1 (t)| + |x 2 (t)| ≤ KK * η, z(0) H 2 (R) ≤ Kη, (6.24)
for some constant K > 0, independent of K * .
Proof. For simplicity, we denote B j := ∂ x j B. The proof of this result is a classical application of the Implicit Function Theorem. Let
J j (u(t), x 1 , x 2 ) := Re (u(t, x) − B(t, x; x 1 , x 2 ))B j (t, x; x 1 , x 2 )dx, j = 1, 2.
It is clear that J j (B(t; x 1 , x 2 ), x 1 , x 2 ) ≡ 0, for all x 1 , x 2 ∈ R. On the other hand, one has for j, k = 1, 2,
∂ x k J j (u(t), x 1 , x 2 ) (B(t),0,0) = − Re B k (t, x; 0, 0)B j (t, x; 0, 0)dx.
Let J be the 2 × 2 matrix with components J j,k := (∂ x k J j ) j,k=1,2 . From the identity above, one has
det J = − |B 1 | 2 |B 2 | 2 − (Re B 1 B 2 ) 2 (t; 0, 0),
which is different from zero from the Cauchy-Schwarz inequality and the fact that B 1 and B 2 are not parallel for all time. Therefore, in a small H 2 neighborhood of B(t; 0, 0), t ∈ [0, T * ] (given by the definition of (6.21)), it is possible to write the decomposition (6.22)-(6.23).
Now we look at the bounds (6.24). The first bounds are consequence of the decomposition itself and the equations satisfied by the derivatives of the scaling parameters, after taking time derivative in (6.23) and using that det J = 0.
Proof of Theorem 2.4. The case of the SY breather
The proof is simple and is based in the fact that 2-soliton solutions can be arbitrarily close to the SY breather (1.15). Let α ∈ R. The SY 2-soliton solution is given by the expression
B SY,gen (t, x) := 2 √ 2ie −i(αx−(c 2 1 −α 2 )t) G(t, x) F (t, x) , with G(t, x) := c 1 A − cosh(y 2 ) + 4ic 2 α sinh(y 2 ) + e i(2αx+tγ + γ − ) c 2 A + cosh(y 1 ) + 4ic 1 α sinh(y 1 ) , F (t, x) := a − cosh(y + ) + a + cosh(y − ) + 4c 1 c 2 cos((2αx + γ + γ − t)),(7.1)
where
A + := γ + γ − + 4α 2 , A − := γ + γ − − 4α 2 , a + := γ 2 + + 4α 2 , a − := γ 2 − + 4α 2 , y 1 := c 1 (x + 2αt), y 2 := c 2 (x − 2αt), y ± := xγ ± − 2αγ ∓ t.
Note that (7.1) reduces to (1.15) when the frequency parameter α = 0: for each (t, x) ∈ R 2 , lim α→0 B SY,gen (t, x) = B SY (t, x). We start out with a simple lemma.
F [B KM ] = F e i π 2α 1 − i √ 2β cosh(βx) = F 1 − i √ 2β cosh(βx) = F KM √ 2β cosh(βx) = β 5 F KM √ 2 cosh x , where F KM [u] := u 2 xx − 5u 2 u 2 x + 1 2 u 6 .(8.H P [B + z] = Q P [z] + N P [z],
where N P satisfies (4.12). Recall that Q P is given by (4.7)-(4.11). More precisely, we have
Q P [z] = 1 2 z z 4x + 3 2 (|B| 2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) + 3(|B| 2 − 1)z xx − 4|B x | 2 z − 6BB xzx − 4BB x z x − B 2 xz + B 2z xx + |B| 2 z xx .
Write B P = e it + B P , where lim t→±∞ B P (t) L ∞ = 0. We claim
Q P [z] = Q e it [z] + o t→+∞ (1). (8.4)
Assuming this property, we can conclude (8.3), since
Q e it [z] = 1 2 z(z 4x + z xx + e 2itz xx ) = 1 2 (|z xx | 2 − |z x | 2 − (e itz x ) 2 ).
It only remains to prove (8.4).
In what follows, we make the change of variables w := e −it z x . We have from (8.3),
H P [B + z] = 1 2 (|w x | 2 − |w| 2 − w 2 ) + O( z 3 H 1 ) + o t→+∞ (1). (8.5)
From (8.5) we have that, no matter the orthogonality conditions posed on z,
d 2 ds 2 H P [B + sz] s=0 = 1 2 (|w x | 2 − |w| 2 − w 2 ) + o t→+∞ (1).
Then we conclude by choosing appropriate z.
9. Proof of Theorem 2.6. The Kuznetsov-Ma case
We start with the following result.
L KM,0 [z] := z 4x + z xx + e 2itz xx − β 2 (z xx + e 2itz + z). (9.1)
The proof of this result is direct in view of the spatial exponential decay of the KM breather to the Stokes wave, and the Weil's Theorem.
σ c (L KM,0 ) = [−2β 2 , ∞) β ≥ √ 2, [− 1 4 (2 − β 2 ) 2 − 2β 2 , ∞) β ∈ (0, √ 2). (9.2)
Proof. Let λ ∈ R be such that L KM,0 z = λz. In matrix form, we have
∂ 4 x − (β 2 − 1)∂ 2 x − β 2 −e 2it (−∂ 2 x + β 2 ) −e −2it (−∂ 2 x + β 2 ) ∂ 4 x − (β 2 − 1)∂ 2 x − β 2 z z = λ z z .
Let us diagonalize the matrix operator on the LHS. In Fourier variables we have
ξ 4 + (β 2 − 1)ξ 2 − β 2 −e 2it (ξ 2 + β 2 ) −e −2it (ξ 2 + β 2 ) ξ 4 + (β 2 − 1)ξ 2 − β 2 ,
for which the diagonal operators L KM,0,± are in Fourier variables
F (L KM,0,± ) := ξ 4 + (β 2 − 1)ξ 2 − β 2 ± (ξ 2 + β 2 ) = ξ 4 + β 2 ξ 2 ξ 4 + (β 2 − 2)ξ 2 − 2β 2 .
Consider now the operator L KM,0,− = ∂ 4 x − (β 2 − 2)∂ 2 x − 2β 2 . If β 2 ≥ 2, then σ c (L KM,0,− ) = [−2β 2 , ∞), proving the first part in (9.2). If now 0 < β 2 < 2, we have after a simple computation that σ c (L KM,0,− ) = [− 1 4 (2 − β 2 ) 2 − 2β 2 , ∞). The proof is complete. 9.1. End of proof of Theorem 2.6. We have that (2.8) is a direct consequence of (9.
B x = Q β e iΘ + iαB, B xx = Q β e iΘ + 2iαQ β e iΘ − α 2 B, B xxx = Q β e iΘ + 3iαQ β e iΘ − 3α 2 Q β e iΘ − iα 3 B, B xxxx = Q β e iΘ + 4iαQ β e iΘ − 6α 2 Q β e iΘ − 4iα 3 Q β e iΘ + α 4 B.
Now, substituting the above derivatives in LHS of (2.1), we have
B (4x) + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B − m SS (B xx + 4|B| 2 B) + n SS B = e iΘ Q β + 4iαQ β − 6α 2 Q β − 4iα 3 Q β + α 4 Q β + 8Q β (Q β + iαQ β ) 2 + 14Q βQβ (Q β + 2iαQ β − α 2 Q β ) + 12Q β (Q β + iαQ β )(Q β − iαQ β ) + 6Q 2 β (Q β − 2iαQ β − α 2Q β ) + 24Q 3 βQ 2 β − m SS (Q β + 2iαQ β − α 2 Q β + 4Q β Q 2
βQβ ) + n SS Q β Expanding and simplifying we get
B (4x) + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B − m SS (B xx + 4|B| 2 B) + n SS B = e iΘ Q β + 4iαQ β − 6α 2 Q β − 4iα 3 Q β + α 4 Q β + 8Q β (Q 2 β + 2iαQ β Q β − α 2 Q 2 β ) + 14Q βQβ (Q β + 2iαQ β − α 2 Q β ) + 12Q β (Q βQ β − iαQ βQβ + iαQ β Q β + α 2 Q βQβ ) + 6Q 2 β (Q β − 2iαQ β − α 2Q β ) + 24Q 3 βQ 2 β − m SS (Q β + 2iαQ β − α 2 Q β + 4Q β Q 2 βQβ ) + n SS Q β . This implies that B (4x) + 8B 2 xB + 14|B| 2 B xx + 6B 2B xx + 12|B x | 2 B + 24|B| 4 B − m SS (B xx + 4|B| 2 B) + n SS B = e iΘ Q β + 4iαQ β − 6α 2 Q β − 4iα 3 Q β + α 4 Q β + 8Q β Q 2 β + 32iαQ βQβ Q β − 16α 2 Q 2 βQβ + 14Q βQβ Q β + 12Q β Q βQ β + 6Q 2 βQ β + 24Q 3 βQ 2 β − m SS (Q β + 2iαQ β − α 2 Q β + 4Q β Q 2 βQβ ) + n SS Q β , which is nothing but (5.2).
Appendix B. Proof of (5.4) Denote Q β = 2β(e βx + ηe −βx ) D , D := 2 + e 2βx + |η| 2 e −2βx , η = α α + iβ . Now, substituting Q β , expanding and collecting similar terms, we rewrite the nonlinear identity (5.4) as follows:
(5.4) = 1 D 5 A 7 e 7βx + A 5 e 5βx + A 3 e 3βx + A 1 e βx + A −1 e −βx + A −3 e −3βx + A −5 e −5βx + A −7 e −7βx ,
where
A 7 := 2β 3 (8β + iα) α 2 + β 2 β − iα − 16β 3 α 2 + 4β 2 + 2β 3 α 2 + β 2 8 + α α + iβ − 16iαβ 4 + 32β 5 = 0, A 5 := − 32β 5 (β − iα) 3 5α 2 + 4iαβ + 20β 2 (α + iβ) 4 (β + iα) + 40β 3 α 2 + 2iαβ + 2β 2 + 8β 3 9α 2 − 2iαβ + 6β 2 − 16β 3 α 2 + 4β 2 7α 2 − iαβ + 4β 2 α 2 + β 2 − 16iα 2 β 4 5α 3 − 21iα 2 β + 5αβ 2 − 21iβ 3 (α 2 + β 2 ) 2 + 768β 5 = 0, A 3 := 8β 3 α 2 + β 2 −21iα 3 + 34α 2 β + 6iαβ 2 + 8β 3 (β − iα) 2 (β + iα) + 8β 3 α 2 + β 2 −21iα 3 + 14α 2 β − 14iαβ 2 + 8β 3 (β − iα) 2 (β + iα) − 16β 3 α 2 + 4β 2 −21iα 3 + 15α 2 β − 8iαβ 2 + 4β 3 (β − iα) 2 (β + iα) + 32β 5 α 2 + β 2 2 39iα 3 − 42α 2 β + 8iαβ 2 + 4β 3 (α + iβ) 4 (β + iα) 3 + 16iαβ 4 −9α 4 + 104iα 3 β + 19α 2 β 2 + 80iαβ 3 + 4β 4 (α 2 + β 2 ) 2 + 768αβ 5 (β + 5iα) (α + iβ)(β + iα) = 0, A 1 := 1536α 2 β 5 5α 2 − 2iαβ + β 2 (α 2 + β 2 ) 2 − 16αβ 3 α 2 + 4β 2 35α 3 − 15iα 2 β + 20αβ 2 − 4iβ 3 (α 2 + β 2 ) 2 − 16iα 2 β 4 5α 3 − 205iα 2 β − 84αβ 2 − 76iβ 3 (α 2 + β 2 ) 2 + 4β 3 63α 4 − 28iα 3 β + 56α 2 β 2 − 16iαβ 3 + 8β 4 α 2 + β 2 − 4β 3 −77α 4 + 12iα 3 β − 24α 2 β 2 − 16iαβ 3 + 8β 4 α 2 + β 2 + 32αβ 5 85iα 4 + 112α 3 β − 47iα 2 β 2 − 28αβ 3 + 8iβ 4 (α + iβ) 2 (β + iα) 3 = 0, A −1 := − 1536iα 3 β 5 5α 2 + 2iαβ + β 2 (β − iα) 3 (β + iα) 2 − 16α 2 β 3 α 2 + 4β 2 −35iα 3 + 15α 2 β − 20iαβ 2 + 4β 3 (β − iα) 3 (β + iα) 2 + 16iα 3 β 4 5α 3 + 205iα 2 β − 84αβ 2 + 76iβ 3
(α + iβ) (α 2 + β 2 ) 2 + 4αβ 3 α 2 + β 2 −63iα 4 + 28α 3 β − 56iα 2 β 2 + 16αβ 3 − 8iβ 4 (β − iα) 3 (β + iα) 2 − 4αβ 3 α 2 + β 2 77iα 4 − 12α 3 β + 24iα 2 β 2 + 16αβ 3 − 8iβ 4 (β − iα) 3 (β + iα) 2 + 32α 2 β 5 85iα 5 − 27α 4 β + 65iα 3 β 2 − 19α 2 β 3 − 20iαβ 4 + 8β 5 (α + iβ) 4 (β + iα) 3 = 0,
A −3 := 768α 4 β 5 (β − 5iα) (β − iα) 3 (β + iα) 2 + 8α 3 β 3 α 2 + β 2 21iα 3 + 14α 2 β + 14iαβ 2 + 8β 3 (α − iβ) 3 (β − iα) 3 − 16α 3 β 3 α 2 + 4β 2 21iα 3 + 15α 2 β + 8iαβ 2 + 4β 3 (α − iβ) 3 (β − iα) 3 − 8α 3 β 3 α 2 + β 2 21iα 3 + 34α 2 β − 6iαβ 2 + 8β 3 (α + iβ) 3 (β + iα) 3 − 16iα 4 β 4 −9iα 3 + 95α 2 β − 76iαβ 2 + 4β 3
(α + iβ)(β + iα) (α 2 + β 2 ) 2 + 32α 3 β 5 39iα 4 + 3α 3 β + 50iα 2 β 2 − 12αβ 3 − 4iβ 4 (α + iβ) 4 (β + iα) 3 = 0,
A −5 := 768α 5 β 5 (α − iβ) 2 (α + iβ) 3 − 16iα 7 β 4 (−21β + 5iα) (β − iα) 2 (β + iα) (α 2 + β 2 ) 2 +
32α 5 β 5 5iα 2 + 4αβ + 20iβ 2 (α + iβ) 4 (β + iα) 3 + 8α 5 β 3 α 2 + β 2 9α 2 + 2iαβ + 6β 2 (α − iβ) 3 (α + iβ) 4 − 40α 5 β 3 iα 2 + 2αβ + 2iβ 2 α 2 + β 2 (α + iβ) 4 (β + iα) 3 − 16α 5 β 3 α 2 + 4β 2 7α 2 + iαβ + 4β 2 (α − iβ) 3 (α + iβ) 4 = 0, and A −7 := − 32iα 7 β 5 (α + iβ) 4 (β + iα) 3 + 16iα 8 β 4 (α + iβ) (α 2 + β 2 ) 3 − 16α 7 β 3 α 2 + 4β 2 (α − iβ)(α + iβ) 2 (α 2 + β 2 ) 2 − 2α 7 β 3 (α + 8iβ) (α 2 + β 2 ) 3 + 2α 7 β 3 (9α − 8iβ) (α 2 + β 2 ) 3 = 0, and we conclude. C.1. Proof of (2.2). We will use, for the sake of simplicity, the following notation for the SY breather solution (1.15):
B SY = M N , with
M := 2 √ 2γ + γ − e ic 2 1 t (c 1 cosh(c 2 x) + c 2 e iγ + γ − t cosh(c 1 x)), N := γ 2 − cosh(γ + x) + γ 2 + cosh(γ − x) + 2c 1 c 2 (e iγ + γ − t + e −iγ + γ − t ).
(C.1) Now, we rewrite the identity (2.2) in terms of M, N in the following way
(2.2) = 1 N 5 5 i=1 S i , (C.2)
with S i given explicitly by:
S 1 = iN 6M N t N 2 x − 2N (N x (M t N x + 2M N xt ) + N t (2M x N x + M N xx )) − N 3 M xxt + N 2 (2N x M xt + 2M x N xt + M xx N t + M t N xx + M N xxt ) , (C.3) S 2 =M (N M x − M N x ) 2 , (C.4) S 3 = 2MM 2M N 2 x + N 2 M xx − N (2M x N x + M N xx ) , (C.5) S 4 = 2M (M N x − M x N )(NM x −M N x ), (C.6) and S 5 = 3 2 M 3M 2 + nN 4 M − mN 2 M 2M + N (N M xx − 2M x N x ) + M (2N 2 x − N N xx ) , (C.7)
where we skipped index SY in parameters m SY , n SY for simplicity. Now substituting the explicit functions M, N (C.1) in S i , i = 1, . . . , 5 and collecting terms, we get after lengthy manipulations that with S i given explicitly by: S i = a 1 cosh 2 (xβ) cos(tα) + a 2 cosh 4 (βx) cos(αt) + a 3 cosh(βx) cos 2 (αt) + a 4 cosh 3 (βx) cos 2 (αt) + a 5 cos 2 (αt) sin(αt) + a 6 cos 3 (αt) + a 7 cosh 2 (βx) cos 3 (αt) + a 8 cosh(βx) cos 3 (αt) sin(αt) + a 9 cosh(βx) cos 4 (αt) + a 10 cos(αt) 4 sin(αt) + a 11 cos 5 (αt), with coefficients a i , i = 1, . . . , 11 given as follows a 1 := 4 √ 2α 4 β 3 α 2 − β 2 β 2 + 2 , a 2 := − 1 2 a 1 , a 3 := − 7β √ 2α a 1 , a 4 := β √ 2α a 1 , a 5 := 3i β 2 α 2 a 1 , a 6 := β 2 α 2 (3β 2 + 5)a 1 , a 7 := 3 β 2 α 2 a 1 , a 8 := −4i β 2 α 2 a 1 , a 9 := 4αβ 4 3α 4 − 2α 2 β 2 5β 2 + 8 + β 4 7β 4 + 24β 2 + 20 , a 10 := −4i √ 2αβ 5 3α 4 − 2α 2 β 2 3β 2 + 5 + β 4 3β 4 + 10β 2 + 8 ,
S 1 := − N 6iM N t N 2 x − 2iN (N x (M t N x + M (iN x + 2N xt )) + N t (2M x N x + M N xx )) + N 3 (M xx − iM xxt ) + N 2 (−2M x (N x − iN xt ) + i(2N x M xt + N t M xx + iM N xx + M t N xx + M N xxt )) ,(a 11 := − i α (β 2 + 1)a 10 .
Finally, using that β = 2(2a − 1) and α = 8a(2a − 1), we have that all a i vanish, and we conclude.
C.3. Proof of (2.4). This identity follows in the same way that the proof of identity (2.3) above. We include it for the sake of completeness, but it can be formally obtained by a standard limiting procedure.
Let us use the following notation for the Peregrine breather solution (1. = 32(1 + 2it)(1 + 4t 2 − 6x 2 )(−3 − 8it + 4t 2 + 2x 2 )(−3 + 8it + 4t 2 + 2x 2 ), S 4 = (C.18) = −512(2t − i)(2t + i)x 2 4t 2 − 8it + 2x 2 − 3 , S 5 = (C.19) = 96 1 + 4t 2 − 2x 2 2 4t 2 − 8it + 2x 2 − 3 , and S 6 = −48i(2t − i)(1 + 4t 2 + 2x 2 ) 2 4t 2 − 6x 2 + 1 . Now collecting terms, it is easy to see that we get a polynomial
6 i=1 S i = b 0 + b 2 x 2 + b 4 x 4 + b 6 x 6 ,
where we have that b 0 = b 2 = b 4 = b 6 = 0.
Proposition 1 . 1 (
11Local and global well-posedness for (1.1)-(1.1a), (1.4), and (1.7)). The Sasa-Satsuma equation (1.4) is locally well-posed in H s , s > 1 4 , and globally well-posed if s ≥ 1. Similarly, NLS with zero background (1.1) is globally well-posed for s ≥ 0, while NLS with nonzero background (1.7) is locally well-posed in H s , s > 1 2 .
(38)-(39)], and [43, eqns. (3-250)-(3-252)], an exact breather solution of Sasa-Satsuma (1.4) is given by the expressionB SS (t, x) := Q β (x + γt + x 2 )e iΘ ,(1.11)
(
iii) The NLS case with nonzero background. Finally, NLS with nonzero boundary condition, represented in (1.6)-(1.7), possesses at least two important localized solutions characteristic of the modulational instability phenomenon, which -roughly speaking-says that small perturbations of the exact Stokes solution e it are unstable
Figure 1 .
1Absolute value of the SS breather (1.4), for different values of the parameter η. Left above: |SS| with η = 0.05; right above: |SS| with η = 0.19; note that these are cases where the double hump is clearly devised.
Figure 2 .
2Left: Absolute value of the SY breather (1.15). Note the periodic in time behavior of this solution. Right: Absolute value of the double soliton (7.1) close to (with α = 0.1) the SY breather (1.15). The left axis represents the x variable, and the right axis, the t variable.
Figure 3 .
3Left: Absolute value of the P breather (1.17). Note the localized character in space and time. Right: Absolute value of the KM breather (1.18).
2. 1 .
1Variational characterization. Our first result is the following variational characterization of B SS , B SY , B P and B KM in (1.11)-(1.15)-(1.17)-(1.18).
SS = Sasa-Satsuma (1.4), SY = Satsuma-Yajima (1.1), and KM = Kuznetsov-Ma (1.7), P = Peregrine (1.7)
Theorem 2 . 1 (
21Elliptic equations satisfied by U (1) breather solutions). Let B = B X be any of the solutions defined in (1.11)-(1.15)-(1.17)-(1.18), with X ∈ {SS, SY, KM, P }. Then we have (1) For X = SS, B = B X satisfies
)
For X = KM and β as in (1.18), B = B KM solves
Theorem 2 . 2 (
22Variational characterization). Each breather mentioned in Theorem 2.1 is critical point of a real-valued functional of the form H X [u] := F X [u] + m X E X [u] + n X M X [u], (2.5)
( 5 )
5Each breather B X is a critical point for the functional H X , in the sense that for X ∈ {SS, SY, KM, P }, H X [B X ](z) = 0, for all z ∈ H 2 (R; C). (2.6) Remark 2.3. Theorem 2.
Theorem 2 . 3 (
23Nonlinear stability of the SS breather). The the SS breather (1.11) is orbitally stable in H 2 (R; C).
Theorem 2 . 4 (
24Characterization of the linear instability of the SY breather). There exists at least one instability direction D in the Schwartz class, and associated to the SY breather, for which there is no invariance nor symmetry present in (1.15) allowing to control it. More precisely, one has for the linear operator H SY [B SY ] associated to B SY :
Theorem 2. 6 (
6Absence of spectral gap and instability of the KM breather). Let B = B KM be a Kuznetsov-Ma breather (1.18), critical point of the functional H KM defined in (2.5). Then for all a > 1 2 we have H P [B KM ] = 0,
Remark 2 . 11 .
211Note that classical stable solitons or solitary waves Q easily satisfy the estimate inf σ c (H Q [Q]) > 0, where H Q is the standard quadratic form associated to the energy-mass or energy-momentum variational characterization of Q. Even in the cases of the mKdV breather B mKdV [4] or Sine-Gordon breather B SG [6], one has inf σ c (H B mKdV [B mKdV ]) > 0 and also inf σ c (H B SG [B SG ]) > 0. The KM breather does not follow this property at all, another consequence of the modulational instability present in the NLS equation with nonzero boundary value at infinity. Consequently, to our knowledge, both the KM and P breathers cannot represent nor exemplify any stable process in Nature.
Sasa-Satsuma. Recall the Sasa-Satsuma equation (1.4). The following quantities are invariant of the motion, on sufficiently regular solutions: the mass M SS [u] := |u| 2 dx, (3.2)
Remark 3. 3 (
3Momentum laws). Another important conserved quantity here is the Momentum P X [u] := Im ūu x , (3.11)
Proposition 4 . 1 (
41Variational characterization of SS, SY, KM and P breathers). For each X ∈ {SS, SY, KM, P }, and for each z ∈ H 2 (R), we have
The linear and quadratic contributions here are the same for both equations. Therefore, if X = SS, SY, KM, P, M X,0 := M X [B], M X,lin := 2 Re Bz and M X,quad := |z| 2 .
M KM,lin and M P,lin may not be necessarily well-defined, without adding cancelling terms (see below). As for the mass terms, there are no higher order contributions to the expansion of H X [B X + z]: M SS,non = M SY,non = M KM,non = M P,non = 0.
consider the energy in the Satsuma-Yajima (SY) case(3.6). If X = SY and B = B X ,E SY [B + z] = E SY [B] + 2 Re z(−B xx ) + |z x | 2 −12 4(Re(Bz)) 2 + |z| 4 + 4|B| 2 Re(Bz) + 2|B| 2 |z| 2 + 4|z| 2 Re(Bz) , so that E SY,0 := E SY [B], and the linear contribution is E SY,lin := 2 Re z −B xx − |B| 2 B . (4.18)
the higher order contribution is E X,non := − 1 2 |z| 4 + 4|z| 2 Re(Bz) .
F
SS,4,lin = 8 2|B| 4 Re(Bz) + 4|B| 4 Re(Bz) = 48 |B| 4 Re(Bz) = 48 Re z|B| 4 B.
F
SS,4,non = 8 4(Re(Bz)) 2 (2 Re(Bz) + |z| 2 ) + |z| 4 (|B| 2 + 2 Re(Bz) + |z| 2 ) + 4|B| 2 |z| 2 Re(Bz) + 2|B| 2 |z| 2 (2 Re(Bz) + |z| 2 ) + 4|z| 2 Re(Bz)(|B| 2 + 2 Re(Bz) + |z| 2 ) .
F
SS,lin := F SS,1,lin + F SS,2,lin + F SS,3,lin + F SS,4,lin = 2 Re zB xxxx + 8 z(B 2 xB + |B| 2 B xx )
F
the term F X,4,lin requires more care than the others. We have this time (X = KM, P ) X2 − 1) + 2 Re(Bz) + |z| 2 (|B| 2 − 1) + 2 Re(Bz) + |z|2 2 .
2 − 1) 2 z + 6(|B| 2 − 1)B Re(Bz) .
2 |z| 2 + 4 Re(Bz) (|B| 2 − 1) + 2 Re(Bz) + |z| 2 .
Step 6 .
6Gathering terms. The case of Kuznetsov-Ma and Peregrine. From (4.40), (4.41), (4.45) and (4.48) we conclude that the linear part of F X , X = N LS is given by F X,lin := F X,1,lin + F X,2,lin + F X,3,lin + F X,4,lin = 2 Re zB xxxx + 6 Re z(B 2 xB + (|B| 2 − 1)B xx )
( 4 .
453) Clearly we have the estimate |N SY [z]| z 3 H 1 under small data assumptions. Finally, the expansion of the Lyapunov functional H SY [B + z] is given by: H SY [B + z] = F SY [B + z] + m SY E SY [B + z] + n SY M SS [B + z] = H SY [B]
Lemma 5. 1 (
1Alternative form for (2.1)). Let m SS = −2(β 2 − α 2 ) and n SS = (α 2 + β 2 ) 2 , and let B = B SS = Q β e iΘ be the breather solution (1.11) of (1.4). Then B satisfies (2.1) if and only if Q β solves
Lemma 6 . 1 .
61The operator L SS is a compact perturbation of the constant coefficients operator L SS,0 :
Lemma 6. 3 (
3Existence of negative directions). Let B = B SS be a SS breather as in (1.11), and L SS be the linear operator in (4.8). Then we have
Lemma 6. 5 (
5Negative eigenvalues of L SS ). Let B = B SS be a SS breather with parameters α, β > 0, and let L SS be the associated linearized operator (4.8). Then L SS has always only one negative eigenvalue.
since L SS [B 0 ] = −B (see (6.4)), one has 0 = Re zB = − Re zL SS [B 0 ] = Re L SS [z + mB −1
=
zL SS z = Re zL SSz − Re L SS [z]b 0 2 Re B 0 B + Re b 0 L SS b 0 . (6.18)Note that both quantities in the denominator are positive. Additionally, note that ifz = λb 0 Re zL SSz Re b 0 L SS b 0 .
Lemma 6 . 8 (
68Modulation and orthogonality). Let B = B SS be a SS breather as in (1.11).
not difficult to check that B SY,gen satisfies a nonlinear ODE which converges to the one satisfied by B SY as α → 0. Let B j := lim α→0 ∂ x j B SY,gen , j = 1, 2. Consequently, we have H SY [B SY ](B j ) = 0, H SY [B SY ](iB SY ) = 0. However, there is no way to control y 1 in (1.15). 8. Instability of the Peregrine bilinear form. Proof of Theorem 2.5
Lemma 9. 1 (
1Essential spectrum). Let L KM be the linear operator in (4.10) associated to the KM breather (1.18). Then L KM is a compact perturbation of the constant (in x) coefficients operator with dense domain H 4 (R; C)
Lemma 9 . 2 .
92Let a > 1 2 be any fixed parameter in (1.18), and β given in (1.18) as well. Then we have
2), and H KM [B KM ](∂ x B KM ) = 0 is also a consequence of Theorems 2.2 and 2.1. Appendix A. Proof of (5.2) Let B SS = B = Q β e iΘ be the soliton solution (1.11) of (1.4). Then we have
Appendix C .
.Proofs of (2.2), (2.3) and (2.4) This section continues and ends the proof mentioned in Subsection 5.2.
a
labeling r = sinh(c 1 x) sinh(c 2 x), s 1 = cosh(c 1 x), s 2 = cosh(c 2 x), s 3 = s 1 r, s 4 = s 3 1 s 5 = s 3 1 r, s 6 = s 5 1 , s 7 = s 1 rs 4 2 , s 8 = s 2 r, s 9 = s 2 1 s 2 , s 10 = rs 2 1 s 2 , s 11 = s 4 1 s 2 , s 12 = rs 4 1 s 2 , s 13 = s ij (c 1 , c 2 , m, n)e j·(2iγ + γ − t) , L i ∈ N. (C.10) For instance, we have for the first term in (C.8), i.e. p 1 s 1 = 4 j=0 a 1j e 2j·(2iγ + γ − t) s 1 , and analyzing the rest of polynomials p i , i = 2, . . . 29, in (C.8)-(C.10), it is easy to see that all coefficients a ij , i = 2, . . . 29 are proportional to the factor (c 2 1 + c 2 2 − m), i.e. a ij = b ij (c 1 , c 2 ) · (c 2 1 + c 2 2 − m), with b ij a polynomial in c 1 , c 2 . Therefore, selecting m = c 2 1 + c 2 2 , we get a ij = 0, ∀i = 2, . . . , 29, ∀j = 0, . . . , L i and we conclude. C.2. Proof of (2.3). The proof is similar to the one for (2.2). Let us use the following notation for the KM breather solution (1.18): B KM = e it 1 − M N , with M := √ 2β β 2 cos (αt) + iα sin (αt) , N := α cosh(βx) − √ 2β cos (αt) . (C.13) Now, we rewrite the identity (2.3) in terms of M, N in the following
C.15) S 2 := −(M − N )(N M x − M N x ) 2 , (C.16)S 3 := 2(M − N )(M − N ) − 2M N 2 x − N 2 M xx + N (2M x N x + M N xx ) , (C.17) S 4 := −2(M − N )(−N M x + M N x )(NM x −M N x − N )(M N + M (−M + N )) 2 , (C.19)andS 6 := N 2 β 2 M 2 (M − N ) + N (β 2M N − (3 + β 2 )(2M x N x − N M xx )) + M (−2β 2M N + β 2 N 2 + 2(3 + β 2 )N 2 x − (3 + β 2 )N N xx ) .(C.20)Now substituting the explicit functions M, N (C.13) in S i , i = 1, . . . ,
B
P = e it 1 − M N , with M := 4(1 + 2it), N := 1 + 4t 2 + 2x 2 . Now, we rewrite the identity (2.4) in terms of M, N in the2 + 2x 2 )(3 − 80t 4 + 32it 5 − 12x 2 − 36x 4 − 16it 3 (−5 + 2x 2 ) + 8t 2 (−1 + 34x 2 ) − 6it(−3 + 28x 2 + 4x 4 )), S 2 = (C.16) = −256(i − 2t) 2 x 2 (−3 + 8it + 4t 2 + 2x 2 ), S 3 = (C.17)
Nonzero BC, in the form of an Stoke wave: for all t ∈ R,1)
(1.1a) Zero BC: |u(t, x)| → 0 as x → ±∞, and
(1.1b) |u(t, x) − e it | → 0 as x → ±∞.
(1.2)
with N SY [z] as defined in(4.53). This proves in the SY case. The proof is complete.5. Existence of critical points: Proof of Theorem 2.1
In this section we prove Theorem 2.1. Recall that Theorem 2.1 is a fundamental part to
complete the proof of Theorem 2.2.
From Proposition 4.1 (more precisely, (4.3), (4.4), (4.5) and (4.6)), we see that (2.1), (2.2),
(2.3) and (2.4) are proved (and so Theorem 2.1) if we show in (4.2) that
G[B X ] ≡ 0,
(5.1)
Lemma 8.1. Let X = P or KM , B = B X be the Peregrine and Kuznetsov-Ma breathers from (1.17)-(1.18), and F = F X given by (3.10). Then we have Proof. (Justificar mejor) We deal first with the Peregrine case. Since F is a conserved quantity, we have from (1.17) that F [B P ] = lim t→+∞ F [B P ] = lim t→+∞ F [e it ]. Now, from (3.10) F [e it ] = 0 = 0.This proves the first identity in (8.1). Now we deal with F [B KM ]. Since F is conserved, we can assume t = π 2α . Then we have from (1.18) and (3.10),F [B P ] = 0,
F [B KM ] =
4
5
β 5 .
(8.1)
2 )
2After some lengthy computations, we see that F KMRemark 8.1. Note that from Remark 3.2 we also have M P [B P ] = E P [B P ] = 0. Additionally, M KM [B KM ] = 4β and E KM [B KM ] = − 8 3 β 3 . Consequently, H KM defined in (2.5) satisfies H KM [B KM ] = F KM [B KM ] + m KM E KM [B KM ] + n KM M KM [B KM ]which is strictly positive for β > 0.Lemma 8.2. Let B = B P be the Peregrine breather (1.17) and z ∈ H 2 (R; C) be a small perturbation. We have Proof. From Proposition 4.1 in the X = P case, we have H P [B + z] = H P [B] + G P [z] + Q P [z] + N P [z], and H P [B] = 0. From 5.1, we have G P [z] = 0. Therefore,√
2
cosh x = 4
5 , so that (8.1) is proved.
=
4
5
β 5 +
8
3
β 5 ,
H P [B + z] =
1
2
(|z xx | 2 − |z x | 2 − (e itz
x ) 2 ) + O( z 3
H 1 ) + o t→+∞ (1).
(8.3)
From the conservation laws for H SS and Proposition 4.1, H SS [u](t) = H SS [B](t) + Q SS [z](t) + N SS [z](t).(6.25)On the other hand, by the translation invariance in space,Indeed, from (1.11), we havefor some specific t 0 , x 0 . Since H involves integration in space of polynomial functions on B, B x and B xx , we haveis constant in time. Now we compare (6.25) at times t = 0 and t ≤ T * . We have. Additionally, from (6.12)-(6.13) applied this time to the time-dependent function z(t), which satisfies (6.23), we getConclusion of the proof. Using the conservation of mass M SS in (3.2), we have, after expanding u = B + z,Replacing this last identity in (6.26), we getby taking K * large enough. This last fact contradicts the definition of T * and therefore the stability property holds true.
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Existence of multi-solitary waves with logarithmic relative distances for the NLS equation. T V Nguyen, C. R. Acad. Sci. Paris, Ser. I. 357T. V. Nguyen, Existence of multi-solitary waves with logarithmic relative distances for the NLS equation, C. R. Acad. Sci. Paris, Ser. I 357 (2019) 13-58.
Water waves, nonlinear Schrödinger equations and their solutions. D H Peregrine, J. Austral. Math. Soc. Ser. B. 25D.H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. Ser. B 25, 16-43 (1983).
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| []
|
[
"Accelerating Strangelets via Penrose process in non-BPS fuzzballs",
"Accelerating Strangelets via Penrose process in non-BPS fuzzballs"
]
| [
"Massimo Bianchi \nDipartimento di Fisica\nI.N.F.N. Sezione di Roma Tor Vergata\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica1 -00133RomaITALY\n",
"Marco Casolino \nDipartimento di Fisica\nI.N.F.N. Sezione di Roma Tor Vergata\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica1 -00133RomaITALY\n",
"Gabriele Rizzo \nDipartimento di Fisica\nI.N.F.N. Sezione di Roma Tor Vergata\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica1 -00133RomaITALY\n"
]
| [
"Dipartimento di Fisica\nI.N.F.N. Sezione di Roma Tor Vergata\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica1 -00133RomaITALY",
"Dipartimento di Fisica\nI.N.F.N. Sezione di Roma Tor Vergata\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica1 -00133RomaITALY",
"Dipartimento di Fisica\nI.N.F.N. Sezione di Roma Tor Vergata\nUniversità di Roma Tor Vergata\nVia della Ricerca Scientifica1 -00133RomaITALY"
]
| []
| Ultra High Energy Cosmic Rays may include strangelets, a form of Strange Quark Matter, among their components. We briefly review their properties and discuss how they can be accelerated via Penrose process taking place in singular rotating Kerr black holes or in their smooth, horizonless counterparts in string theory, according to the fuzzball proposal. We focus on non-BPS solutions of the JMaRT kind and compute the efficiency of Penrose process that turns out not to be bounded unlike for Kerr BHs.Introduction and motivationsCosmic rays (CR) and in particular Ultra High Energy CR (UHECR) tend to play an important role in any progress of high-energy physics, from the identification of new elementary particles in the past to the recent confirmation of rare phenomena such as neutrino oscillations. Although the flux is tremendously suppressed at very high energies[1,2,3]and practically ends at the ZeV scale set by the GZK cutoff[4,5], the question remains as for how UHECR can be accelerated up to such high energies. Among the components of UHECR one can include the so-called "strangelets", a form of Strange Quark Matter (SQM)[22,28,30] that can be present in the dense core of a Neutron Star (NS) or in Quark Stars (QS)[23], where the temperature and density may be significantly higher than on the crust. When the mass of a NS exceeds the Chandrasekhar-Oppenheimer-Volkoff bound (around a few solar masses), it becomes unstable wrt gravitational collapse and produces a Black Hole (BH). In turn, BHs may provide both tidal tearing of captured astrophysical objects, including NS and QS, and a powerful acceleration mechanism of UHECR, including strangelets, aka Penrose process[6]. This can take place in rotating (Kerr) BHs surrounded by an 'ergo-region' where a time-like Killing vector becomes space-like. Thanks to Penrose mechanism Kerr/rotating BHs can be used as cosmic slings to accelerate UHECR and reach the GZK cutoff scale. BHs are the epitome of quantum gravity (QG), which is still poorly understood with quantum fieldtheory means. Luckily there is a leading contender: string theory. Based on the idea that point-like particles be replaced by one-dimensional objects, string theory can accommodate gravity (mediated by closed strings) together with gauge interactions (mediated by open strings) in an entirely consistent framework. General Relativity or higher dimensional extensions thereof, coupled to gauge fields and fermions, governs the dynamics at very low energies, compared to the string mass scale. The realization that string theory in addition to fundamental strings admits stable, extended solitons with p spatial dimensions, called p-branes, allows to represent BHs as bound states of strings and branes and to quantitatively address and partly solve some long-standing issues in the physics of BHs [7], including BH production in high-energy collisions[8]. In particular there are classes of charged BPS 1 BHs for which one can precisely count the micro-states responsible for the macroscopic entropy, which according to Beckenstein and Hawking is proportional to the area of the event horizon[9]. In the emerging 'fuzz-ball' proposal [10], BHs are described as ensembles of smooth horizonless geometries with the same asymptotic behavior at infinity, that have a non-trivial structure at the putative horizon. By replacing BHs with fuzzballs, dense, tangled balls of strings some of the subtle paradoxes can be avoided or clarified since they were generated by accepting the very presence of the singularity and of the horizon that must be only an approximation valid in the classical limit. The aim of the present paper is to use non-BPS fuzz-balls with an ergo-region as cosmic slings for the acceleration of UHECR. Due to a no-go theorem, that prevents the existence of non-trivial smooth horizonless solutions in D = 4 [11], we have to rely on fuzz-balls in higher dimensions[12]. In particular, D = 5 and D = 6 will be our starting point and represent a toy model for the physically interesting case. We will mostly use JMaRT (after Jejjalla, Madden, Ross and Titchener [14]) solitonic solution that is smooth and horizonless, yet with an ergo-region. As we will see, the asymptotic geometry, though free from pathological Closed Time-like Curves (CTC's), is over-rotating and cannot be as such strictly identified with the fuzz-ball of a BH, not even in D = 5 [15]. In fact JMaRT displays an instability that suggests that this kind of charged non-BPS solutions should decay into BPS ones with the same charges and lower angular momenta such as to satisfy the bound for BHs. The instability of JMaRT has been addressed by various groups, including [16] 2 . To the best of our knowledge however the role of the Penrose process in JMaRT or in similar smooth, horizonless solitonic geometries have not been addressed previously 3 . The plan of the paper is as follows.1 A BPS state (after Bogomolny, Prasad and Sommerfield) is an extremal state that saturates a (supersymmetric) bound between mass M , charges Q and angular momentum J.2 For a similar analysis in the BPS context see[17].3We thank G. Bossard and D. Turton for confirming this. | 10.1016/j.nuclphysb.2020.115010 | [
"https://arxiv.org/pdf/1904.01097v1.pdf"
]
| 91,184,427 | 1904.01097 | a0fb88b8c0dff4644f94237215fd1c09a342167f |
Accelerating Strangelets via Penrose process in non-BPS fuzzballs
April 3, 2019
Massimo Bianchi
Dipartimento di Fisica
I.N.F.N. Sezione di Roma Tor Vergata
Università di Roma Tor Vergata
Via della Ricerca Scientifica1 -00133RomaITALY
Marco Casolino
Dipartimento di Fisica
I.N.F.N. Sezione di Roma Tor Vergata
Università di Roma Tor Vergata
Via della Ricerca Scientifica1 -00133RomaITALY
Gabriele Rizzo
Dipartimento di Fisica
I.N.F.N. Sezione di Roma Tor Vergata
Università di Roma Tor Vergata
Via della Ricerca Scientifica1 -00133RomaITALY
Accelerating Strangelets via Penrose process in non-BPS fuzzballs
April 3, 2019
Ultra High Energy Cosmic Rays may include strangelets, a form of Strange Quark Matter, among their components. We briefly review their properties and discuss how they can be accelerated via Penrose process taking place in singular rotating Kerr black holes or in their smooth, horizonless counterparts in string theory, according to the fuzzball proposal. We focus on non-BPS solutions of the JMaRT kind and compute the efficiency of Penrose process that turns out not to be bounded unlike for Kerr BHs.Introduction and motivationsCosmic rays (CR) and in particular Ultra High Energy CR (UHECR) tend to play an important role in any progress of high-energy physics, from the identification of new elementary particles in the past to the recent confirmation of rare phenomena such as neutrino oscillations. Although the flux is tremendously suppressed at very high energies[1,2,3]and practically ends at the ZeV scale set by the GZK cutoff[4,5], the question remains as for how UHECR can be accelerated up to such high energies. Among the components of UHECR one can include the so-called "strangelets", a form of Strange Quark Matter (SQM)[22,28,30] that can be present in the dense core of a Neutron Star (NS) or in Quark Stars (QS)[23], where the temperature and density may be significantly higher than on the crust. When the mass of a NS exceeds the Chandrasekhar-Oppenheimer-Volkoff bound (around a few solar masses), it becomes unstable wrt gravitational collapse and produces a Black Hole (BH). In turn, BHs may provide both tidal tearing of captured astrophysical objects, including NS and QS, and a powerful acceleration mechanism of UHECR, including strangelets, aka Penrose process[6]. This can take place in rotating (Kerr) BHs surrounded by an 'ergo-region' where a time-like Killing vector becomes space-like. Thanks to Penrose mechanism Kerr/rotating BHs can be used as cosmic slings to accelerate UHECR and reach the GZK cutoff scale. BHs are the epitome of quantum gravity (QG), which is still poorly understood with quantum fieldtheory means. Luckily there is a leading contender: string theory. Based on the idea that point-like particles be replaced by one-dimensional objects, string theory can accommodate gravity (mediated by closed strings) together with gauge interactions (mediated by open strings) in an entirely consistent framework. General Relativity or higher dimensional extensions thereof, coupled to gauge fields and fermions, governs the dynamics at very low energies, compared to the string mass scale. The realization that string theory in addition to fundamental strings admits stable, extended solitons with p spatial dimensions, called p-branes, allows to represent BHs as bound states of strings and branes and to quantitatively address and partly solve some long-standing issues in the physics of BHs [7], including BH production in high-energy collisions[8]. In particular there are classes of charged BPS 1 BHs for which one can precisely count the micro-states responsible for the macroscopic entropy, which according to Beckenstein and Hawking is proportional to the area of the event horizon[9]. In the emerging 'fuzz-ball' proposal [10], BHs are described as ensembles of smooth horizonless geometries with the same asymptotic behavior at infinity, that have a non-trivial structure at the putative horizon. By replacing BHs with fuzzballs, dense, tangled balls of strings some of the subtle paradoxes can be avoided or clarified since they were generated by accepting the very presence of the singularity and of the horizon that must be only an approximation valid in the classical limit. The aim of the present paper is to use non-BPS fuzz-balls with an ergo-region as cosmic slings for the acceleration of UHECR. Due to a no-go theorem, that prevents the existence of non-trivial smooth horizonless solutions in D = 4 [11], we have to rely on fuzz-balls in higher dimensions[12]. In particular, D = 5 and D = 6 will be our starting point and represent a toy model for the physically interesting case. We will mostly use JMaRT (after Jejjalla, Madden, Ross and Titchener [14]) solitonic solution that is smooth and horizonless, yet with an ergo-region. As we will see, the asymptotic geometry, though free from pathological Closed Time-like Curves (CTC's), is over-rotating and cannot be as such strictly identified with the fuzz-ball of a BH, not even in D = 5 [15]. In fact JMaRT displays an instability that suggests that this kind of charged non-BPS solutions should decay into BPS ones with the same charges and lower angular momenta such as to satisfy the bound for BHs. The instability of JMaRT has been addressed by various groups, including [16] 2 . To the best of our knowledge however the role of the Penrose process in JMaRT or in similar smooth, horizonless solitonic geometries have not been addressed previously 3 . The plan of the paper is as follows.1 A BPS state (after Bogomolny, Prasad and Sommerfield) is an extremal state that saturates a (supersymmetric) bound between mass M , charges Q and angular momentum J.2 For a similar analysis in the BPS context see[17].3We thank G. Bossard and D. Turton for confirming this.
In Section 2 we will briefly review the notion of Strange Quark Matter and strangelets, discuss how Penrose mechanism can have a role in accelerating UHECRs including strangelets and SQM and sketch the alternative acceleration mechanisms proposed so far. In Section 3 we will recall the JMaRT solution and its properties. We set the stage for the analysis of the Penrose process with the study of geodesics motion in JMaRT geometry. Thanks to the large amount of isometry it proves convenient to work in the Hamiltonian formulation. We focus on the geodesics in the θ = 0 hyperplane and compute the efficiency of the Penrose process for massive spin-less particles that in-fall in a counter-rotating way. We discuss the results in comparison with the analogous process for Kerr BHs, reviewed in an Appendix. Section 4 contains our conclusions and an outlook for future investigation in this subject. In particular we will comment on upper limits that MINI-EUSO can set on the flux of strangelets and on the Penrose mechanism for their acceleration derived in Section 3 for non-BPS fuzz-balls and reviewed in the Appendix. In the Appendix, for the sake of convenience and for comparison with our analysis for non-BPS fuzzballs, we briefly review the rotating BH solution, originally found by Kerr, and its properties and discuss the Penrose process for both massive particles decaying into a massless pair (photons).
Strangelets and their acceleration via Penrose process
Neutron stars (NS) are probably the most compact and dense form of 'ordinary' matter in the Universe [18,19]. When thermo-nuclear reactions have exhausted their fuel, protons tend to recombine with electrons and form neutrons via emission of neutrinos. Matter becomes so dense that in a radius of a few km one can package masses of the order of the Sun's. In fact modelling the system as a perfect fluid with spherical symmetry Tolman, Oppenheimer and Volkoff (TOV) wrote down relativistic equations that allow determining quantitative bounds on the masses of these compact stars [18,19]. Assuming very low temperatures so much so that NS be supported only by the pressure of the degenerate Fermi gas of neutrons, one can set an upper limit on the mass of NS around the mass of the Sun. Mutatis mutandis i.e. replacing electrons with neutrons and using a 'reasonable' equation of state (EoS), relating pressure and energy density, the result is strikingly similar to Chandrasekar bound on the mass of white dwarves [20,18,13]. More recently, stimulated by the observation of candidate NS's violating the bound through direct GW detection [21], more elaborate equations of states have been proposed that amount to slice NS radially with different layers satisfying different EoS's that are glued at the interface [18,19]. In particular it has been suggested that Strange Quark Matter (SQM) [22], composed of up u, down d and strange s quarks, may play a role in internal layers, whereby temperature can raise and deconfinement can take place thus giving rise to a quark-gluon plasma. In fact the concept of a Quark Star (QS) has been put forward [23].
Strange Quark Matter and strangelets
The existence of SQM as a different state of hadronic matter other than ordinary nuclear matter was proposed for the first time in 1984 [22]. SQM would be composed by roughly an equal number of u, d and s quarks, with the presence of a third quark lowering the nucleon Fermi level with respect to a system with only two quark flavours [?]. In this case SQM may constitute the true ground state of hadronic matter and be stable. Quarks would be lumped together and not separated in nucleons, resulting in quark matter being much denser than ordinary matter. SQM could have been produced in the Big Bang [?], be present in the core of neutron stars or in "Strange Quark Stars" (SQS) [23] and be a candidate for baryonic dark matter [24]. Portions of SQM could be ejected as a consequence of collisions of these stars in binary systems [25]. Such collisions can inject a small fraction of this matter (also called strangelets) in the galactic radiation where it could be identifiable with cosmic ray detectors or mass spectrometers. Various experiments have tried to produce or search for SQM in various environments, on the ground, on balloons and in satellites, both of active and passive nature. A review on strangelet search and models can be found in [26,27]. SQM should be neutral (uncharged), if an exactly equal number of u, d, and s quarks is dynamically favoured, however the neutrality condition may be approximate, allowing strangelets to have a small residual electrical charge. In the light mass range, these objects could be identified as having an anomalous A/Z >> 2 ratio. A search with the PAMELA space-borne magnetic spectrometer has yielded upper limits 2 · 10 3 p/(m 2 sr yr) in the mass range 2 < A < 10 5 [83]. In [28] it has been suggested that heavier objects could interact with the atmosphere through an adiabatic compression mechanism similar to that of meteors. The higher density of SQM would result in longer and more uniform tracks than in case of meteorites, which tend to break up and flash during atmospheric entry. Furthermore since SQM is expected to be of interstellar origin, the speed of the track would be around 220 km/s (galactic velocity), higher than that of meteors that have an average speed of 40 km/s (solar system velocity), although interstellar meteors have also been observed [29]. Strangelets [30] may represent a fraction of UHECR. The first 'evidence' of such form of SQM could be provided by Price's event (balloon) that was initially proposed as a magnetic monopole candidate and subsequently rebutted and interpreted instead as a strangelet, having penetrated the Earth atmosphere.
Penrose process
After gravitational collapse of a NS a (rotating) black hole (BH) can form that is the only astrophysical object that could carry out a tidal tearing of a Quark Star (QS) -thus providing a source of SQM to be scattered throughout the Universe 4 . Rotating (Kerr) BHs are special in that in addition to a horizon that hides the curvature singularity they are surrounded by an 'ergo-region' external to the horizon where a time-like Killing vector becomes space-like [31]. Although curvature is finite in the ergo-region, tidal forces are much stronger than outside the ergo-sphere and a compact object captured by the rotating BH may be torn into pieces with different energies and angular momenta. As a result Penrose process can take place in Kerr BHs. This will be reviewed in the Appendix for the reader's convenience and for comparison with the case of BPS fuzz balls. According to Penrose [6] a particle with positive energy (wrt to flat infinity) can enter the ergo-region and split into two (or more) particles one of which has negative energy (as seen from infinity) and crosses the horizon to finally fall into the singularity. The rest of the products can escape back to infinity carrying out more energy than their 'mother'. The extra energy is provided by the BH that loses energy and angular momentum, since the initial particle should be counter-rotating, ie have opposite angular momentum wrt to the BH, for the very process to take place. The efficiency of the process is defined as the energy gained wrt to the initial energy
η = E f − E i E i
As we will see momentarily, the efficiency depends on the mass, energy and spin of the initial particle as well as of the products and of the BH and the place where the splitting takes place. In the simple case when the initial counter-rotating object with energy equal to its rest mass (E 0 = µ 0 ) decomposes into two massless products at the turning point of its geodesics in the equatorial plane (θ = π/2), it is simple to express η in terms of the 'radial' position r * where the process takes place. We have reproduced the text-book analysis in the Appendix for the interested reader and for comparison with the similar process in non-BPS fuzz balls. Though almost obvious η is positive -in fact with an upper bound η ≤ (1/2)( √ 2 − 1) -only when r * lies inside the ergo-region and the particle is massive and counter-rotating. The analogue process for massless particles or waves is called super-radiance [32].
Acceleration mechanisms for UHECR
Thanks to Penrose mechanism Kerr/rotating BHs can be used as cosmic slings thus allowing one to reach the peak (GZK cutoff) of the UHECRs mountain. Sling-shot by other small magnetized objects such as white dwarves, neutron stars and quark stars has been proposed by Blandford & Znajek, Berti, Brito & Cardoso, Banados and West. Contrary to BHs the efficiency η is not bounded from above. As we will momentarily see, this will turn out to be the case for non-BPS 'fuzz-balls', too. Before concluding this section, let us briefly recall the broad features of the two classes of acceleration mechanisms for CR proposed so far. According to the first "one-shot" mechanism, CR are accelerated by an extended/intense electric field directly to the ZeV scale [33]. The original idea put forward by Swann [34] has been elaborated on and the necessary electric field is usually related to the fast rotation of small, highly magnetized objects such as white dwarfs [35,36], neutron stars (pulsars) [37,38,39,40,41], or black holes [42,43,44]. While electric field acceleration has the advantage of being fast, it suffers from the drawback of occurring in astrophysical sites with extremely high energy density, where many energy-loss phenomena can take place at the same time. According to the second "stochastic" mechanism of acceleration, instead, particles gain energy gradually through multiple interactions with moving magnetized plasmas. The idea, pioneered by Fermi [45,46], can be realised in a variety of astrophysical environments, including the interplanetary medium [47,48], supernova remnants (SNRs) [49,50,51,52,53,54], the galactic disk and halo [55,56,57,58], AGN's [59,60,61], large-scale jets and lobes of giant radio-galaxies (RG) [62,63,64], blazars [65,66,67,68], gammaray bursts (GRBs) [69,70], starburst superwinds [71,72], galactic microquasar systems [73,74], and clusters of galaxies [75,76,77]. Contrary to the previous case, stochastic acceleration tends to be slow. Furthermore it poses the issue of how to keep relativistic particles confined within the Fermi 'engine'.
Penrose mechanism for smooth non-BPS fuzzballs
In this Section, we analyze the Penrose process for neutral massive scalar particles in smooth non-BPS geometries such as JMaRT. In order to set the stage for the computation we will first recall JMaRT soliton solution and its properties and then study the geodesics motion in this geometry. Thanks to the large amount of isometry the problem is integrable very much as for Kerr BHs 5 as well as for some BPS fuzz-balls [78]. To exploit this property it is convenient to work in the Hamiltonian formulation that requires the determination of the canonical momenta P µ , conjugate to the generalised velocitiesẋ µ . We will restrict our attention on the case where the conserved KK momentum P y of the infalling particle is zero. Moreover, we focus on geodesics in the hyperplane θ = 0 whereby an effective dimensional reduction takes place since the radius of one of the angular directions (φ) shrinks to zero and one has to set the corresponding conserved (angular) momentum P φ to zero. One ends up with only three variables t, r, ψ and the dynamics looks remarkably similar to the one in Kerr BHs 6 . Despite the relatively compact and elegant form of JMaRT, explicit formulae for the 'effective potentials' E ↑↑/↑↓ ± and for the efficiency η tend to become unwieldy. We will express the results in compact form in terms of the coefficient functions that appear as components of the inverse metric. We refrain from displaying cumbersome formulae that cannot illuminate the understanding. To illustrate the results for various values of the parameters we present different plots of E ↑↑/↑↓ ± and η as well as for other relevant coefficient functions.
JMaRT solution and its properties
In string theory, the objects colloquially called black-holes (BHs) are bound states of strings and p-branes, i.e. p-dimensional extended solitons. This description allows reproducing the micro-states necessary to explain the origin of BH entropy that scales with the area of the event horizon, at least for charged BPS black-holes [9]. In the fuzzball proposal [10] classical BHs can be thought of as ensembles of smooth, horizon-less geometries with the same asymptotic behaviour as the would-be BH, i.e. same mass, charge and angular momenta. BPS systems with two charges give rise to small BHs with string-size horizon. In order to have a large BH with a finite (possibly large) area of the event horizon, one has to consider 5 We thank P. Fré for stressing this property. 6 The same happens for θ = π/2 after replacing ψ with φ and the parameters a1 and a2 with one another. systems with at least three charges in D = 5 or four charges in D = 4. One of the grand successes of string theory is the precise micro-state counting for charged BHs, mostly in a BPS context. Extension to non-BPS and un-charged BHs has proven much harder. For our purposes, as a toy model of the Penrose mechanism for non-BPS fuzz-balls, we will consider a non-BPS 3-charge solution in D = 5 originally found by Jejjalla, Madden, Ross and Titchener (JMaRT). JMaRT solutions 7 in Type IIB superstring theory depend on five parameters associated to charges: D1brane Q 1 and D5-brane Q 5 charge, the asymptotic radius R of the Kaluza-Klein circle and two additional integer parameters m and n. For m = n+1 the solutions turn out to be BPS. Imposing appropriate conditions on the parameters, that determine the mass and angular momenta, JMaRT has neither singularity nor event horizon and is free from CTC's. The reason why we are interested in JMaRT is the presence of an ergo-region, whereby particles with negative energy can propagate. It has been argued that an ergo-region that does not enclose a horizon and a singularity should lead to an instability: JMaRT should decay to an extremal BPS solution with the same charges. This ergo region or similar instabilities has been studied by various groups [16,17]. We will assume that the decay process would take a long time so much so that JMaRT could behave as a cosmic sling thanks to Penrose process, that in turn can also play a role in the relaxation of JMaRT to a stable BPS configuration. Having in mind SQM and strangelets, we focus on massive neutral scalar particles rather than on waves. The analogous process for waves is called 'super-radiance' and has been studied for JMaRT in [32]. Hawking process has also been considered for JMaRT in [79]. In order to construct JMaRT one starts fromType IIB supergravity in D = 10 and considers 3-charge micro-state geometries of the D1-D5-P system [80]. The D1-branes wrap a circle S 1 y , along which KKmomentum is added, while the D5-branes wrap a five-torus S 1 y × T 4 . The original solution depends on 8 parameters that determine the mass M ADM (related to M ), two independent angular momenta J φ and J ψ (parameterised in terms of a 1 and a 2 ), the three charges Q 1 , Q 5 and Q p (expressible in terms of the 'boost' parameters δ 1 , δ 5 and δ p ), the radius R of the S 1 y and the volume V 4 of the four-torus T 4 . Safely neglecting T 4 , whose volume can be taken to be very small, the six-dimensional geometry is parameterized in terms of t (time), r (radial coordinate), y (for S 1 y ) and three angular coordinates θ, φ and ψ and reads
ds 2 = M (s p dy − c p dt) 2 √ H 1 H 5 − f dt 2 − dy 2 √ H 1 H 5 + H 1 H 5 r 2 dr 2 (r 2 + a 1 2 ) (r 2 + a 2 2 ) − M r 2 + dθ 2 + s θ 2 dφ 2 H 1 H 5 + a 2 2 −a 1 2 s θ 2 (H 1 +H 5 −f ) √ H 1 H 5 + c θ 2 dψ 2 H 1 H 5 + a 1 2 −a 2 2 c θ 2 (H 1 +H 5 −f ) √ H 1 H 5 + 2M s θ 2 dφ[dt(a 2 c 1 c 5 c p − a 1 s 1 s 5 s p ) + dy(a 1 c p s 1 s 5 − a 2 c 1 c 5 s p )] √ H 1 H 5 + M a 1 c θ 2 dψ + a 2 s θ 2 dφ 2 √ H 1 H 5 + 2M c θ 2 dψ[dt(a 1 c 1 c 5 c p − a 2 s 1 s 5 s p ) + dy(a 2 c p s 1 s 5 − a 1 c 1 c 5 s p )] √ H 1 H 5 (3.1) where 8 H i = f + M sinh 2 δ i , f = r 2 + a 2 1 sin 2 θ + a 2 2 cos 2 θ,(3.2)
with c i = cosh δ i , s i = sinh δ i , for short henceforth, as well as c θ = cos θ, s θ = sin θ, c φ = cos φ, s φ = sin φ, c ψ = cos ψ, s ψ = sin ψ.
We have not displayed the profiles of the other Type IIB supergravity fields that are present in JMaRT since they play no role in our later analysis of the Penrose process. The 6-dimensional metric can be written in the form
ds 2 = −A dt 2 + B dr 2 + C ψ dψ 2 + C φ dφ 2 + U dθ 2 + F dy 2 + 2Ω ψ dtdψ + 2Ω φ dtdφ + 2K dtdy + 2Λ ψ dydψ + 2Λ φ dydφ + 2Γ dψdφ (3.3) 7
Henceforth we call it JMaRT for short. 8 Our Hi are denoted by Hi in JMaRT [14].
with −A = −f + c p 2 M √ H 1 H 5 , B = r 2 √ H 1 H 5 (a 1 2 + r 2 ) (a 2 2 + r 2 ) − M r 2 , U = H 1 H 5 , F = f + M s p 2 √ H 1 H 5 (3.4) C ψ = a 1 2 c θ 4 (−f + H 1 + H 5 + M ) + a 2 2 c θ 4 (f − H 1 − H 5 ) + c θ 2 H 1 H 5 √ H 1 H 5 (3.5) C φ = s θ 4 (a 1 − a 2 )(a 1 + a 2 )(f − H 1 − H 5 ) + a 2 2 M + s θ 2 H 1 H 5 √ H 1 H 5 (3.6) Γ = a 1 a 2 c θ 2 M s θ 2 √ H 1 H 5 , Ω ψ = c θ 2 M (a 1 c 1 c 5 c p − a 2 s 1 s 5 s p ) √ H 1 H 5 , Ω φ = s θ 2 M (a 2 c 1 c 5 c p − a 1 s 1 s 5 s p ) √ H 1 H 5 (3.7) K = − c p s p M √ H 1 H 5 , Λ ψ = c θ 2 M (a 2 c p s 1 s 5 − a 1 c 1 c 5 s p ) √ H 1 H 5 , Λ φ = s θ 2 M (a 1 c p s 1 s 5 − a 2 c 1 c 5 s p ) √ H 1 H 5 (3.8)
The ADM mass and angular momenta are given by
M ADM = M 2 i cosh 2δ i , J φ = M (a 1 s 1 s 2 s p − a 2 c 1 c 5 c p ) , J ψ = M (a 2 s 1 s 2 s p − a 1 c 1 c 5 c p ) (3.9)
where δ i ≥ 0, without loss of generality, and c i = cosh δ i and s i = sinh δ i , as before. Note that J φ and J ψ get exchanged under the exchange of a 1 and a 2 . Potential singularities appear when H 1 = 0 or H 5 = 0 (curvature singularities) and when det g = 0, where | det g| = r 2 H 1 H 5 cos θ 2 sin θ 2 (3.10)
that is for r 2 = 0 (coordinate singularity) or for θ = 0, π or θ = π/2 (degeneration of the polar coordinates on the 'poles' of S 3 ). The vanishing of G(r) = (r 2 + a 2 1 )(r 2 + a 2 2 ) − M r 2 , the denominator of g rr , at
r 2 ± = 1 2 (M − a 2 1 − a 2 2 ) ± (M − a 2 1 − a 2 2 ) 2 − 4a 2 1 a 2 2 (3.11)
require a detailed analysis. In order shows that r = 0 is a removable coordinate singularity it proves convenient to introduce the adimensional variable
x = r 2 − r 2 + r 2 + − r 2 − so that dx = 2rdr r 2 + − r 2 − (3.12)
Moreover, if one could smoothly shrink a circle to zero at the origin (x = 0), the space is capped at x = 0 i.e. at r 2 = r 2 + > r 2 − and the 'true' curvature singularity at x = −1 i.e. at r 2 = r 2 − is excised. Absence of singularities, horizons and closed-time-like curves imposes conditions on the parameters that can be satisfied in the low mass (parameter) regime
M ≤ (a 1 − a 2 ) 2 (3.13)
and fixes M and R to be given by
M = a 2 1 + a 2 2 − a 1 a 2 c 2 1 c 2 5 c 2 p + s 2 1 s 2 5 s 2 p c 1 c 5 c p s 1 s 5 s p , R = M c 1 c 5 s 1 s 5 √ c 1 c 5 c p s 1 s 5 s p √ a 1 a 2 (c 2 1 c 2 5 c 2 p − s 2 1 s 2 5 s 2 p ) (3.14)
As a result one gets
r 2 − < r 2 + = −a 1 a 2 s 1 s 5 s p c 1 c 5 c p < 0 (3.15)
Two quantization conditions (needed to have closed orbits for φ = φ + α(s i , c i )y and ψ = ψ + β(s i , c i )y as y → y + 2πR) constrain the remaining parameters in terms of two integers m and n
j + j −1 s + s −1 = m − n , j − j −1 s − s −1 = m + n (3.16)
where j = a 2 /a 1 ≤ 1 and s = s 1 s 5 s p /c 1 c 5 c p ≤ 1, indeed one can take a 1 ≥ a 2 ≥ 0 without loss of generality, thus getting m ≥ n+1 ≥ 1. In terms of j, s and a 1 the expression for M reads
M (j, s) = a 1 2 j 4 − j 2 s 2 − j 2 s 2 + 1 (3.17)
replacing j, s in terms of the integers m, n one finds
M (m, n) = a 1 2 2m 2 n 2 [m 2 − (n + 1) 2 ][m 2 − (n − 1) 2 ] {(m 2 − n 2 ) 2 − (m 2 − n 2 ) [m 2 − (n + 1) 2 ][m 2 − (n − 1) 2 ] − m 2 − n 2 } (3.18) that vanishes in the BPS case m = n + 1 whereby M → 0, δ i → ∞ with Q i = M s i c i fixed.
The remaining five independent parameters correspond to Q 1 , Q 5 , R, m and n that determine the KK charge Q p and the angular momenta J φ , J ψ
Q p = nm Q 1 Q 5 R 2 , J φ = −m Q 1 Q 5 R , J ψ = n Q 1 Q 5 R (3.19)
The Penrose process can take place in JMaRT thanks to the presence of an ergoregion, that can be identified as the region where the norm of the time-like Killing vector V t = ∂ t becomes positive. Using JMaRT one finds
||V t || 2 = g µν V µ t V ν t = g tt = M c 2 p − f √ H 1 H 5 (3.20)
where f (r, θ) = r 2 +a 2 1 sin θ 2 +a 2 2 cos θ 2 and H i = f (r, θ)+M s 2 i . An ergo-sphere appears at f (r, θ) = M c 2
p r 2 e = M c 2 p − a 2 1 sin θ 2 − a 2 2 cos θ 2 (3.21)
where V t becomes space-like. In the BPS limit the norm of V t is always negative: ||V t || 2 = −f / √ H 1 H 5 and no ergo-region appears.
Geodetic motion in JMaRT
As a preliminary step to investigate the Penrose process in JMaRT, we study the geodesics for massive or massless neutral particles. Probes of this kind only feel the presence of the curved metric but are unaffected by the other Type IIB fields present in JMaRT. The Lagrangian that governs geodetic motion is given by
L = 1 2 g µνẋ µẋν (3.22)
where g µν denotes the six-dimensional metric tensor 9 . Recall that dr and dθ appear diagonally in ds 2 , while dt, dy, dφ, dψ form a four-dimensional block. As in Kerr BH or in BPS fuzz balls, in order to take advantage of all the symmetries, i.e. time translation, KK shifts U (1) y and rotations U (1) φ × U (1) ψ , it is better to switch to the Hamiltonian formalism. The generalized momenta are given by
P µ = ∂L ∂ẋ µ = g µνẋ ν (3.23)
whereẋ ν = dx µ /dτ and the Hamiltonian reads
H = P µẋ µ − L = 1 2 g µν P µ P ν (3.24)
where g µν is the inverse six-dimensional metric. For JMaRT the explicit expressions for the generalized momenta read P r =ṙ r 2 √ H 1 H 5 (a 1 2 + r 2 ) (a 2 2 + r 2 ) − M r 2 (3.25) 9 The extra four directions compactified on T 4 play no role in our analysis.
P θ =θ H 1 H 5 (3.26) P t = −ṫ f − c p 2 M √ H 1 H 5 −ẏ c p M s p √ H 1 H 5 +ψ c θ 2 M (a 1 c 1 c 5 c p − a 2 s 1 s 5 s p ) √ H 1 H 5 +φ M s θ 2 (a 2 c 1 c 5 c p − a 1 s 1 s 5 s p ) √ H 1 H 5 (3.27) P y =ẏ f + M s p 2 √ H 1 H 5 −ṫ c p M s p √ H 1 H 5 +ψ c θ 2 M (a 2 c p s 1 s 5 − a 1 c 1 c 5 s p ) √ H 1 H 5 +φ M s θ 2 (a 1 c p s 1 s 5 − a 2 c 1 c 5 s p ) √ H 1 H 5 (3.28) P φ =φ s θ 2 {s θ 2 [(a 2 1 − a 2 2 )(f − H 1 − H 5 ) + a 2 2 M ] + H 1 H 5 } √ H 1 H 5 +ṫ s θ 2 (a 2 c 1 c 5 c p − a 1 s 1 s 5 s p )M √ H 1 H 5 +ẏ s θ 2 (a 1 c p s 1 s 5 − a 2 c 1 c 5 s p )M √ H 1 H 5 +ψ s θ 2 c θ 2 a 1 a 2 M √ H 1 H 5 (3.29) P ψ =ψ c θ 2 {c θ 2 [(a 2 2 − a 2 1 )(f − H 1 − H 5 ) + a 2 1 M ] + H 1 H 5 } √ H 1 H 5 +ṫ c θ 2 (a 1 c 1 c 5 c p − a 2 s 1 s 5 s p )M √ H 1 H 5 +ẏ c θ 2 (a 2 c p s 1 s 5 − a 1 c 1 c 5 s p )M √ H 1 H 5 +φ c θ 2 s θ 2 a 1 a 2 M √ H 1 H 5 (3.30)
and the Hamiltonian for JMaRT can be written as
H = 1 2 − AP 2 t + BP 2 r + U P 2 θ + C ψ P 2 ψ + C φ P 2 φ + F P 2 y + P t P y K + P t P ψ Ω ψ + P t P φ Ω φ + P y P ψ Λ ψ + P y P φ Λ φ + P ψ P φ Γ (3.31)
where the coefficient functions A, B, U , C φ , C ψ , F , Ω φ , Ω ψ , Γ, K, Λ φ , Λ ψ are the non-zero components of the inverse metric g µν , whose explicit expressions are quite cumbersome and will not be displayed, except for the special case of θ = 0. The generalized velocities can be expressed in terms of the momenta using the above functionṡ t = −P t A + P y K + P ψ Ω ψ + P φ Ω φ (3.32)
r = P r B (3.33) θ = P θ U (3.34) φ = P φ C φ + P t Ω φ + P ψ Γ + P y Λ φ (3.35) ψ = P ψ C ψ + P t Ω ψ + P y Λ ψ + P φ Γ (3.36) y = P y F + P t K + P ψ Λ ψ + P φ Λ φ (3.37)
Very much as for Kerr BH and for BPS fuzz-balls, the system is integrable in that the dynamics in the r and θ coordinates can be separated in principle. In practice the geodesics are non-planar and their explicit form is not very illuminating for our purposes. Following similar analysis in BPS fuzz-balls [78] and without losing any significant feature of the result, one can focus on the hyper-planes θ = 0 and θ = π/2. Indeed it is consistent to setθ = 0 and P θ = 0 in these two cases since dP θ dτ = − ∂H ∂θ = 0 for θ = 0 and θ = π/2 (3.38)
For both choices an effective dimensional reduction takes place. For θ = 0 all terms in dφ drop, being proportional to sin θ 2 , and one can safely set P φ = 0; while for θ = π/2 all terms in dψ drop, being proportional to cos θ 2 , and one can safely set P ψ = 0. The two cases are perfectly equivalent and one can get one from the other by simply exchanging a 1 and a 2 in any relevant formula. For definiteness we will focus on the θ = 0 hyperplane in the following.
Moreover, we are not interested in the motion along the compact y direction. In order to simplify the analysis, one can set P y = 0. This is consistent since P y = 0 is conserved:Ṗ y = 0. As a consequenceẏ is completely determined by the other velocities and conserved momenta, so much so that we will not consider it later on. If we fix θ = 0 and consequently P θ = 0, all the terms in dφ drop and the metric becomes
ds 2 θ=0 = dt 2 c p 2 M − f √ H 1 H 5 + dr 2 r 2 √ H 1 H 5 (a 1 2 + r 2 ) (a 2 2 + r 2 ) − M r 2 + dψ 2 [(a 1 2 −a 2 2 )(H 1 +H 5 −f ) + a 1 2 M + H 1 H 5 ] √ H 1 H 5 + dt 2dψM (a 1 c 1 c 5 c p − a 2 s 1 s 5 s p ) √ H 1 H 5 − 2c p dyM s p √ H 1 H 5 + 2dψdyM (a 2 c p s 1 s 5 − a 1 c 1 c 5 s p ) √ H 1 H 5 + dy 2 f + M s p 2 √ H 1 H 5 (3.39)
which takes the form
ds 2 θ=0 = −Adt 2 + Bdr 2 + C ψ dψ 2 + 2Ω ψ dtdψ + 2Kdtdy + F dy 2 + 2Λ ψ dψdy =ĝ µν dx µ dx ν (3.40) where −A = c p 2 M − f √ H 1 H 5 (3.41) B = r 2 √ H 1 H 5 (a 1 2 + r 2 ) (a 2 2 + r 2 ) − M r 2 (3.42) C ψ = a 1 2 (−f + H 1 + H 5 + M ) + a 2 2 (f − H 1 − H 5 ) + H 1 H 5 √ H 1 H 5 (3.43) Ω ψ = M (a 1 c 1 c 5 c p − a 2 s 1 s 5 s p ) √ H 1 H 5 (3.44) F = f + M s p 2 √ H 1 H 5 (3.45) K = − c p M s p √ H 1 H 5 (3.46) Λ ψ = M (a 2 c p s 1 s 5 − a 1 c 1 c 5 s p ) √ H 1 H 5 (3.47) with f = r 2 + a 2 2 , H 1 = f + M s 1 2 , H 5 = f + M s 5 2 (3.48)
Recall that a 1 and a 2 switch their role under a change of θ from 0 to π/2 and an exchange ψ ↔ φ. The reduced Hamiltonian expressed in terms of the components of the reduced inverse metric reads
H = 1 2 −P t 2 A + P 2 r B + P ψ 2 C ψ + P y 2 F + P t P y K + P t P ψ Ω ψ + P ψ P y Λ ψ = − µ 2 2 (3.49)
where µ is the mass of the probe and the coefficient functions A, B, C ψ , F , Ω ψ , K, Λ ψ are the non-zero components of the reduced inverse metric and µ is the rest mass of the probe spin-less particle. At θ = 0 one has
B = 1 B = a 1 2 + r 2 a 2 2 + r 2 − M r 2 r 2 √ H 1 H 5 (3.50) and A = H 1 H 5 a 1 2 M s P 2 −c 1 2 c 5 2 M + H 1 + H 5 + M − f 2 + f H 1 + H 5 − M s P 2 + M + 2a 1 a 2 c 1 c 5 c P M 2 s 1 s 5 s P + a 2 2 f 2 − f H 1 + H 5 − M s P 2 − M s P 2 (H 1 + H 5 ) + M s 1 2 s 5 2 s P 2 + 1 + H 1 H 5 f + M s P 2 (f − M ) a 2 2 f 2 − f (H 1 + H 5 ) − M 2 s 1 2 s 5 2 + f H 1 H 5 − a 1 2 f M −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f (H 1 + H 5 + 2M ) −1 (3.51) C ψ = f (f − M ) H 1 H 5 (f − M ) a 2 2 f 2 − f (H 1 + H 5 ) − M 2 s 1 2 s 5 2 + f H 1 H 5 − a 1 2 f M −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f (H 1 + H 5 + 2M ) −1 (3.52) F = H 1 H 5 a 1 2 − M s P 2 + 1 −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f H 1 + H 5 + M s P 2 + 2 − 2a 1 a 2 c 1 c 5 c P M 2 s 1 s 5 s P + a 2 2 f 2 − f H 1 + H 5 + M s P 2 + M + M H 1 s P 2 + H 1 + H 5 s P 2 + H 5 + M s 1 2 s 5 2 s P 2 + H 1 H 5 f − M s P 2 + 1 a 1 2 f M −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f (H 1 + H 5 + 2M ) − (f − M ) a 2 2 f 2 − f (H 1 + H 5 ) − M 2 s 1 2 s 5 2 + f H 1 H 5 −1 (3.53) Ω ψ = M H 1 H 5 (a 1 c 1 c 5 c P f + a 2 s 1 s 5 s P (M − f )) a 1 2 f M −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f (H 1 + H 5 + 2M ) − (f − M ) a 2 2 f 2 − f (H 1 + H 5 ) − M 2 s 1 2 s 5 2 + f H 1 H 5 −1 (3.54) K = M H 1 H 5 a 1 2 c P s P −c 1 2 c 5 2 M − f + H 1 + H 5 + M + a 1 a 2 c 1 c 5 M s 1 s 5 2s P 2 + 1 + c P s P a 2 2 f − H 1 − H 5 − M s 1 2 s 5 2 + H 1 H 5 a 1 2 f M −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f (H 1 + H 5 + 2M ) − (f − M ) a 2 2 f 2 − f (H 1 + H 5 ) − M 2 s 1 2 s 5 2 + f H 1 H 5 −1 (3.55) Λ ψ = M H 1 H 5 (a 1 c 1 c 5 f s P + a 2 c P s 1 s 5 (M − f )) a 1 2 f M −c 1 2 c 5 2 M + H 1 + H 5 + M + f 2 − f (H 1 + H 5 + 2M ) − (f − M ) a 2 2 f 2 − f (H 1 + H 5 ) − M 2 s 1 2 s 5 2 + f H 1 H 5 −1 (3.56)
Since we are not interested in motion along the compact circle direction we can safely set P y = 0 (3.57) and, for convenience of the notation,
P t = −E , P ψ = J (3.58)
so that we get
P 2 r = 1 B E 2 A − J 2 C ψ − 2EJ Ω ψ − µ 2 = A B (E − E + )(E − E − ) ≥ 0 (3.59)
where the 'effective potentials' read
E ± = J Ω ψ ± J 2 ( Ω 2 ψ + A C ψ ) + µ 2 A A (3.60)
Since A/ B ≥ 0 always, one has either E > E + > E − or E < E − < E + where, depending on whether the particle is co-rotating (↑↑), ie J J ψ > 0, or counter-rotating (↑↓), ie J J ψ < 0, one has
E ↑↑ + ≥ µ , E ↑↑ − ≤ −µ or E ↑↓ + ≤ 0
for r ≤ r e , E ↑↓ − ≤ −µ As we will see momentarily, the Penrose process can only take place in the latter case. In Figs. 1, 2, 3 we have plotted E ↑↓ ± as a function of r for some 'reasonable' choice of the parameters M, a 1 , a 2 of JMaRT (determined by the choice of m, n, δ 1 , δ 5 and a 1 or, equivalently, R) and of the angular momentum J of the (massive µ = 0 or mass-less µ = 0) probe particle. Despite their complexity, thanks to the existence of a frame-dragging term dtdψ in Eq. 4, these solutions expose the expected presence of regions with negative energy inside the ergo-sphere that can be studied computationally and plotted. For comparison we also plot E ↑↑ ± in Fig. 4. As evident from the plots of E ± there are two kinds of geodesics in the θ = 0 plane: unbounded ones for E ≥ µ and trapped ones E ≤ −µ. In the former case the massive probe impinges from infinity, reaches a turning point r * where P r = 0 and gets deflected back to infinity, possibly after making several turns around the 'center' (x = 0 ie r 2 = r 2 + < 0) of the fuzzball. In the latter case, the particle cannot escape to infinity and remains forever inside the fuzzball. The relevant equations can be integrated in terms of non-elementary functions and we will not attempt to present a detailed analysis here. Instead we turn our attention on the Penrose process in JMaRT.
Penrose process in JMaRT and its efficiency
The presence of an ergo-region in JMaRT allows the Penrose process to take place, whereby a counterrotating particle acquires negative energy after crossing the ergo-sphere and if it splits into two or more fragments, one of the product may escape to infinity with an energy larger than the initial particle, while the other fragment(s) get trapped in the fuzzball for a 'long' time. Following [20,13], mutatis mutandis we will derive the efficiency of Penrose process in JMaRT metric. Let us consider a spin-less probe with rest mass µ 0 , energy E 0 ≥ µ 0 (positive branch) and orbital angular momentum J 0 opposite to the angular momentum of JMaRT (counter-rotating ↑↓). Very much as in Penrose original analysis, it seems reasonable and computationally convenient to assume that the probe splits exactly at the turning point r = r * where P r (r * ) = 0. At this point the angular velocity reaches a maximum and the tidal tearing of the probe is more likely to take place. This has the additional advantage of simplifying the analysis since one gets a relation between r, E 0 and J 0 of the form
E 2 0 A − J 2 0 C ψ − 2E 0 J 0 Ω ψ = µ 2 0 (3.61)
which can be (implicitly) solved for r = r * as a function of E 0 (positive counter-rotating branch) and J 0 . Denoting by E 1 , E 2 and J 1 , J 2 the energies and (orbital) angular momenta of the two (spin-less) fragments with rest masses µ 1 and µ 2 , conservation of energy and angular momentum yield
E 1 + E 2 = E 0 J 1 + J 2 = J 0 (3.62)
which has two solutions, symmetric under the exchange 1 ↔ 2. Assuming that particle 1 escapes to infinity (positive branch of the energy) while particle 2 gets trapped in the fuzzball (negative branch of the energy), one can first express J 1 and J 2 in terms of E 1 = E + (µ 1 ) and E 2 = E − (µ 2 ) and get 10
J 1,2 = −E 1,2 Ω ψ ± E 2 1,2 ( Ω 2 ψ + A C ψ ) − µ 2 1,2 C ψ C ψ (3.63)
Plugging these in the second equation and solving the system for E 1,2 yields
E 1,2 = 1 2µ 0 2 E 0 (µ 0 2 ± µ 1 2 ∓ µ 2 2 ) ± F(µ 0 2 , µ 1 2 , µ 2 2 ) E 2 0 − C ψ Ω 2 ψ + C ψ A µ 0 2 (3.64)
Note the role of the "fake square"
F(µ 0 2 , µ 1 2 , µ 2 2 ) = µ 0 4 + µ 1 4 + µ 2 4 − 2µ 0 2 µ 1 2 − 2µ 1 2 µ 2 2 − 2µ 2 2 µ 0 2 (3.65)
that is ubiquitous in 3-body phase space. Note that F(µ 0 2 , µ 1 2 , µ 2 2 ) ≥ 0 for µ 0 ≥ µ 1 + µ 2 as required by standard kinematics considerations. In the symmetric case
µ 1 = µ 2 = µ ≤ µ 0 /2 one finds F(µ 0 2 , µ 2 , µ 2 ) = µ 4 0 − 4µ 2 0 µ 2 = µ 2 0 (µ 2 0 − 4µ 2 ) ≥ 0.
Note also that A > 0 and Ω ψ > 0 while C ψ > 0 for r > r e and C ψ < 0 for r < r e . The efficiency η of Penrose process for JMaRT is given by the energy E 1 − E 0 gained by the 'probe' particle escaping to infinity with respect to energy of the incoming particle E 0 . As a function of the radial decay point, that we have identified with the radial turning point r * , implicitly determined by the choice of E 0 and J 0 , viz.
η(r * ) = E 1 − E 0 E 0 = − E 2 E 0 = − 1 2µ 0 2 (µ 0 2 − µ 1 2 + µ 2 2 ) + F(µ 0 2 , µ 1 2 , µ 2 2 ) 1 − C ψ Ω 2 ψ + C ψ A µ 0 2 E 2 0 (3.66)
The efficiency is negative when r * > r e as evident from the plot in Fig. 5. For some choice of the parameters, η is larger than one (Figs. 6, 7). In general, contrary to what happens for rotating BHs, reviewed in the Appendix, there is no upper bound on η. This looks particularly promising for the acceleration of UHECR, including strangelets, by non-BPS rotating fuzz-balls that should replace putative rotating BHs of the kind found by Kerr. The case of JMaRT should be taken as a toy model in many respects. First of all the relevant dynamics is at least five-dimensional. Second, though non-BPS, the charges play a crucial role in the very existence of the solution that should be thought of as some excited state of a BPS configuration. Last but not least, achieving phenomenologically reasonable values for the mass and angular momenta require extrapolation to very large charges that may look rather unnatural. We have shown that the Penrose process can take place not only in singular rotating (Kerr) BHs but also in smooth horizonless geometries that are expected to represent the micro-states of (charged) rotating BHs. The common and crucial feature being the presence of an ergoregion. We have considered the case of JMaRT, which is in a loose sense a non-BPS fuzzball in D = 5. Actually it is over-rotating wrt to classical BHs with the same mass and charges so it is only a gravitational soliton. Anyway, we took it as a toy model for our analysis and computed the efficiency η of a non-collisional Penrose process in rotating geometries of this kind. Contrary to the case of Kerr BHs, reviewed in the Appendix, η is not bounded from above and depends in a highly non-trivial fashion of the parameters of the fuzz-ball as well as on the masses of the probe and of the fragments and above all on the 'radial' position where the decay occurs. As expected the efficiency is positive only if the in-falling particle is counter-rotating and the splitting happens inside the ergo-region. In order to make quantitative predictions on the relevance of such a mechanism for the acceleration of UHECR and in particular strangelets one should estimate the distribution of such or similar objects in our galaxy / universe as well as of rotating fuzz-balls with large enough mass and angular momentum to be useful as cosmic slings thus allowing one to reach the GZK cutoff energy of the UHECRs. Sling-shot by other small magnetized objects such as white dwarves, neutron stars (and quark stars) has been proposed by Blandford & Znajek, Berti, Brito & Cardoso, Banados and West. We plan to address these and related issues in the near future [81]. For the time being, we would like to comment on upper limits that MINI-EUSO can set on the flux of strangelets and on the Penrose mechanism for their acceleration derived in Section 3 for non-BPS fuzz-balls and reviewed in the Appendix for Kerr BH. MINI-EUSO is an instrument to be placed inside the International Space Station (ISS), looking toward the Earth from a nadir-facing window in the Russian Zvezda module [84]. The main telescope employs a Fresnel optics with a Multi-Anode-photomultiplier (MAPMT) focal surface (48 × 48 pixels), with ancillary cameras in the Near-Infrared and Visible regions. Mini-EUSO will map the earth in the UV range (300-400 nm) with a spatial resolution of 6.11 km and a temporal resolution of 2.5 µs, searching for Ultra-High Energy Cosmic rays E > 5 · 10 20 eV and studying a variety of atmospheric events such as transient luminous events (TLEs), bioluminescence and meteors. The spatial and temporal sampling of the detector allows searching for strange quark matter tracks in the atmosphere, discriminating them from meteors from the light curve (intensity and speed). An estimation of the upper limit of flux which can be posed by one month of night observations of MINI-EUSO is about 10 −21 cm −2 s −1 sr −1 in the mass range above 5 · 10 24 GeV/c 2 [82]. that means 11 r e (θ) = M + M 2 − a 2 cos 2 θ (4.6)
The ergo-sphere lies outside the horizon except at the poles, θ = 0, π, where they touch each other. In the ergo-region, inside the ergo-sphere, all particles, including photons, must rotate with the hole since g tt > 0. The presence of the ergo-region allows Penrose process to take place as we will see momentarily.
Focussing for simplicity on geodesics in the equatorial plane θ = π/2 allows to write the restricted metric in the compact form ds 2 = −A dt 2 + C dφ 2 + 2Ω dtdφ + B dr 2 (4.7)
where
A = 1 − 2M r , B = 1 − 2M r + a 2 r 2 −1
, C = r 2 + a 2 + 2M r a 2 , Ω = − 2M a r (4.8)
Computing the conjugate momenta P µ = g µνẋ ν and setting 12 P t = −E , P φ = J , P r = P (4.9)
one finds
H = 1 2B P 2 + 1 2 1 AC + Ω 2 −CE 2 − 2ΩEJ + AJ 2 = − µ 2 2 (4.10)
where µ is the mass of the probe particle. The geodesic is null for µ = 0. Resolving for the radial momentum P in terms of E and J one finds
P 2 = B AC + Ω 2 CE 2 + 2ΩEJ − AJ 2 + µ 2 AC + Ω 2 = BC AC + Ω 2 (E − E + )(E − E − ) ≥ 0 (4.11)
where the 'effective potentials' read E ± = −ΩJ ± (Ω 2 + AC)(J 2 − Cµ 2 ) C (4.12)
Since BC/AC + Ω 2 ≥ 0 outside the horizon either E ≥ E + > E − or E ≤ E − < E + . E ± determine allowed negative-energy regions. For co-rotating particles (J a ≥ 0) E ↑↑ + is always positive, while while for counter-rotating particles (J a ≤ 0) E ↑↓ + becomes negative inside the ergo-sphere (r e = 2M ). As mentioned above, if a positive energy counter-rotating particle enters the ergo-sphere it acquires negative energy and 'decays' into two or more products, one of which has negative energy and falls into the horizon, then the particle that escapes may have more energy than the initial particle. In this way Kerr BH loses mass angular momentum. Following [20,13], we now review the efficiency of the Penrose process in Kerr BH. For simplicity we will assume that the in-falling massive particle has E = µ ('rest mass'), that the products are massless scalars (no spin) µ 1 = µ 2 = 0 and that the decay takes place at the turning point r = r * (with r H < r * < r e = 2M ) where P r = 0. Since energy and angular momentum are conserved, we have
E 1 + E 2 = E , J 1 + J 2 = J (4.13)
For massless particles J i = αE i , while J = βE for the massive one with α and β depending on the turning point r * where the decay/splitting takes place. The second equation then simplifies drastically to α 1 E 1 + α 2 E 2 = βE (4.14)
Solving the 'linear' system one has
E 1 = β − α 2 α 1 − α 2 E , E 2 = β − α 1 α 2 − α 1 E (4.15)
Taking the positive branch for E = E + and E 1 = E 1,+ and the negative branch for E 2 = E 2,− < 0, one gets E 1 > E The efficiency of the process can be estimated in the following way. Since ∆E = E 1 − E = −E 2 is the gained energy, the efficiency of Penrose process as a function of r * is given by
η(r * ) = E 1 − E E = −E 2 E = β − α 1 α 1 − α 2 = 1 2 2M r * − 1 ≤ 1 2 ( √ 2 − 1) (4.16)
since r H ≤ r * ≤ 2M = r e (θ = π/2) for the very process to take place. In fact the maximum is reached when r * = r H = M + √ M 2 − a 2 and a = M (extremal Kerr BH).
Figure 1 :
1E + , E − regions for JMaRT with "quantum numbers" shown on the left and a counter-rotatingprobe; x = (r 2 − r + 2 )/(r + 2 − r − 2 ). Ergosphere boundary is denoted by the dashed (grey online) vertical line on the right, end of geometry is in x = 0.
Figure 2 :
2A close-up of the negative energy region for the same quantum numbers inFig.1. Note the infinite wall exploding in the region close to x = 0.
Figure 3 :
3Asymptotic behaviour for the same quantum numbers as in Figs.1, 2, but with probe mass µ = 0.
Figure 4 :
4E + , E − regions for JMaRT with "quantum numbers" shown on the left and a co-rotating probe;x = (r 2 − r + 2 )/(r + 2 − r − 2 ). Ergosphere boundary is denoted by the dashed (grey online) vertical line on the right, end of geometry is in x = 0. There is no ergoregion.
Figure 5 :
5Efficiency for a choice of quantum numbers as in Figs.1, 2.
Figure 6 :
6Efficiency for another choice of quantum numbers, shown on the right. It peaks at η max 1.3.
Figure 7 :
7A close-up of the peak in η max as claimed inFig.6.
One should however keep in mind the objections in[30], since there is no impact with stellar protons and no disintegration of SQM in the process.
Conclusions and outlookAfter summarising the results of our present analysis, we would like to draw our conclusions and identify directions for future investigation on the subject.10 We are implicitly assuming that the fragments are produced with zero radial momentum and continue to move in the θ = 0 plane with Py = 0 and P φ = 0. This means that r * is a turning point for the fragments, too.
For later use, note that re = 2M for θ = π/2.12 We denote the angular momentum of the probe by J in order to avoid confusion with the angular momentum of the Kerr BH, denoted by J = M a.
AcknowledgmentsWe would like to thank Guillaume Bossard, Dario Consoli, Pietro Fré, Giorgio Di Russo, Alfredo Grillo, Maurizio Firrotta for useful discussions and comments. M. B. also acknowledges CERN for hospitality during completion of this work. M. B. was partially supported by the MIUR-PRIN contract 2015MP2CX4002 "Non-perturbative aspects of gauge theories and strings".Appendix: Penrose process for rotating Kerr BHsFor comparison with the more laborious case of non-BPS fuzz-balls, represented by JMaRT solutions, let us review how Penrose process can take place in the Kerr metric. The Kerr black hole is axially symmetric and is characterized by two parameters: mass M and angular momentum J = M a, with a ≤ M . Setting G N = 1, the line element in Boyer-Lindquist coordinates readswhere x = √ r 2 + a 2 sin θ cos φ, y = √ r 2 + a 2 sin θ sin φ, z = r cos θ andIn this coordinate system, surfaces with constant t and r are deformed two-spheres. The metric for a = 0 coincides with Schwarzchild metric. In contrast to the latter, however, there is an off-diagonal termthat is responsible for the 'gravitational' dragging of inertial frames caused by the rotation of the source. In practice a particle dropped 'straight' in from infinity, i.e. with J ≡ P φ = 0 is 'dragged' just by the influence of gravity so that it acquires an angular velocity ω in the same sense as that of the source. For the Kerr metric, ω has the same sign as a = J/M . This effect weakens with the distance as 1/r 3 . Kerr metric presents a singularity, a horizon and an ergo-sphere. The singularity is a ring located in the equatorial plane θ = π/2, at r = 0 i.e. z = 0 and x 2 + y 2 = a 2 . The singularity is cloaked by a horizon where g rr = ∞, i.e. ∆ = 0 that corresponds to the radiusThe 'ergo-sphere' can be identified as the surface where the norm of the time-like Killing vector V t = ∂ t vanishes. It is also called the 'static limit', since inside it no particle can remain at fixed r, θ, φ. From (4) one finds ||V t || 2 = g µν V µ t V ν t = g tt = − ∆ − a 2 sin 2 θ ρ 2 = 0 for ∆ = a 2 sin 2 θ (4.5)
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[
"ON A DIFFUSIVE SIS EPIDEMIC MODEL WITH MASS ACTION MECHANISM AND BIRTH-DEATH EFFECT: ANALYSIS, SIMULATIONS AND COMPARISON WITH OTHER MECHANISMS *",
"ON A DIFFUSIVE SIS EPIDEMIC MODEL WITH MASS ACTION MECHANISM AND BIRTH-DEATH EFFECT: ANALYSIS, SIMULATIONS AND COMPARISON WITH OTHER MECHANISMS *"
]
| [
"Huicong Li ",
"ANDRui Peng ",
"Zhi-An Wang "
]
| []
| []
| In the present paper, we are concerned with an SIS epidemic reaction-diffusion model governed by mass action infection mechanism and linear birth-death growth with no flux boundary condition. By performing qualitative analysis, we study the stability of the disease-free equilibrium, uniform persistence property in terms of the basic reproduction number and the global stability of the endemic equilibrium in homogeneous environment, and investigate the asymptotic profile of endemic equilibria (when exist) in heterogeneous environment as one of the movement rate of the susceptible and infected populations is small. Our results, together with those in previous works on three other closely related modeling systems, suggest that the factors such as infection mechanism, variation of total population and population movement play vital but subtle roles in the transmission dynamics of diseases and hence provide useful insights into the strategies designed for disease control and prevention.2000 Mathematics Subject Classification. 35K57, 35J57, 35B40, 92D25. | null | [
"https://arxiv.org/pdf/1807.03451v1.pdf"
]
| 115,133,288 | 1807.03451 | 73b01f0885e74c15d26e2334697777bfd1ccd741 |
ON A DIFFUSIVE SIS EPIDEMIC MODEL WITH MASS ACTION MECHANISM AND BIRTH-DEATH EFFECT: ANALYSIS, SIMULATIONS AND COMPARISON WITH OTHER MECHANISMS *
10 Jul 2018
Huicong Li
ANDRui Peng
Zhi-An Wang
ON A DIFFUSIVE SIS EPIDEMIC MODEL WITH MASS ACTION MECHANISM AND BIRTH-DEATH EFFECT: ANALYSIS, SIMULATIONS AND COMPARISON WITH OTHER MECHANISMS *
10 Jul 2018
In the present paper, we are concerned with an SIS epidemic reaction-diffusion model governed by mass action infection mechanism and linear birth-death growth with no flux boundary condition. By performing qualitative analysis, we study the stability of the disease-free equilibrium, uniform persistence property in terms of the basic reproduction number and the global stability of the endemic equilibrium in homogeneous environment, and investigate the asymptotic profile of endemic equilibria (when exist) in heterogeneous environment as one of the movement rate of the susceptible and infected populations is small. Our results, together with those in previous works on three other closely related modeling systems, suggest that the factors such as infection mechanism, variation of total population and population movement play vital but subtle roles in the transmission dynamics of diseases and hence provide useful insights into the strategies designed for disease control and prevention.2000 Mathematics Subject Classification. 35K57, 35J57, 35B40, 92D25.
Introduction
The mathematical study of infectious diseases can be traced back to the classic work of Kermack and McKendrick [29] in 1927. In [29], the authors adopted the mass action infection mechanism (also called density-dependent infection mechanism) to study a deterministic SIR (susceptibleinfected-recovered) epidemic model, meaning that the infection (incidence) rate is proportional to the number of encounters between susceptible and infected individuals; mathematically, such infection rate is characterized by the bilinear function βSI, where β > 0 is the disease transmission rate and S(t) and I(t) represent the density of susceptible and infected populations respectively. The most significant achievement made in [29] is perhaps the epidemic threshold result that the density of susceptible individuals must exceed a critical value in order for the epidemic outbreak to occur. Due to the seminal importance of the Kermack-McKendrick theory to the field of theoretical epidemiology, their works were republished in 1991; see [30][31][32].
Employing the same infection mechanism and instead considering an SIS (susceptible-infectedsusceptible) model, one is led to the following ODE system (see, for instance, [43]):
S ′ = −βSI + γI, t > 0, I ′ = βSI − γI, t > 0,(1.1)
where γ > 0 is the disease recovery rate, together with initial data fulfilling S(0)+I(0) = N > 0 and I(0) > 0. As one of the simplest models in mathematical epidemiology, (1.1) still demonstrates the threshold result as Kermack and McKendrick [29] observed. In fact, it is clear that S(t) + I(t) = N for all t ≥ 0, and hence (1.1) can be reduced to the following logistic type equation:
I ′ = βI N − γ β − I .
Simple analysis shows that if N ≤ γ/β, then I(t) → 0 and in turn S(t) = N − I(t) → N as t → ∞, while if N > γ/β, it holds I(t) → N − γ/β > 0, and S(t) → γ/β > 0 as t → ∞. Defining the basic reproduction number R 0 = N β/γ, then the disease-free equilibrium (DFE) (N, 0) is globally attractive if R 0 ≤ 1, while the endemic equilibrium (EE) (γ/β, N − γ/β) is globally attractive if R 0 > 1. We also refer interested readers to the review paper [24] for various ODE models describing infectious diseases. Nowadays it is widely recognized that spatial spread of an infection is closely related to the heterogeneity of the environment and the spatial-temporal movement of the hosts. This is well supported by numerous research on diseases including malaria [38,39], rabies [27,28,45], dengue fever [52], West Nile virus [34,53], hantavirus [1,2], Asian longhorned beetle [22,23], etc; see [51] and references therein. A popular way to incorporate spatial movement of hosts into epidemic models is to assume host random movements, leading to coupled reaction-diffusion equations. Taking into account spatial diffusion and environmental heterogeneity, we obtain the PDE version of (1.1):
S t − d S ∆S = −β(x)SI + γ(x)I, x ∈ Ω, t > 0, I t − d I ∆I = β(x)SI − γ(x)I,
x ∈ Ω, t > 0, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, t > 0, (1.2) where the spatial domain Ω ⊂ R m (m ≥ 1) is bounded and has smooth boundary ∂Ω; positive constants d S and d I represent the diffusion rate of susceptible and infected individuals respectively; β(x) and γ(x) are positive Hölder continuous functions on Ω accounting for the disease transmission rate and recovery rate respectively; the Neumann boundary condition means that no population flux crosses the boundary ∂Ω. For this model, Deng and Wu [14] studied the global dynamics and existence of EE, while [56,57] investigated the asymptotic profile of EE (when exists) as the diffusion rate of susceptible or infected population is small or large, which consequently suggests interesting implication in terms of epidemiology; see the last section of our paper for further discussion. System (1.2) does not take into consideration birth/death effect of susceptible or infected individuals and thus the total population is conserved in the sense that However, it is quite natural to consider the situation that susceptible individuals are subject to a recruitment (source) term modeling their birth and death rate, especially a linear one [6,24]. Therefore, in this paper we are motivated to study the following reaction-diffusion epidemic system with varying total population and environmental heterogeneity:
S t − d S ∆S = Λ(x) − S − β(x)SI + γ(x)I, x ∈ Ω, t > 0, I t − d I ∆I = β(x)SI − [γ(x) + µ(x)] I,
x ∈ Ω, t > 0, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, t > 0, S(x, 0) = S 0 (x) ≥ 0, I(x, 0) = I 0 (x) ≥, ≡ 0, x ∈ Ω.
(1.
3)
The recruitment term Λ(x) − S represents that the susceptible population is subject to linear growth and µ(x) accounts for the death rate of the infected, with Λ and µ being assumed to be positive Hölder functions on Ω. All the other parameters have the same interpretation as before. Throughout the paper, the initial data S 0 and I 0 are nonnegative continuous functions on Ω, and there is a positive number of infected individuals initially, i.e., Ω I 0 (x)dx > 0.
Another widely accepted type of infection mechanism is the so-called frequency-dependent transmission (also called as standard incidence infection mechanism) of the form βSI/(S + I), initiated by de Jong, Diekmann and Heesterbeek [13] in 1995. In this scenario, (1.2) becomes (1.4) and its counterpart with linear recruitment reads
S t − d S ∆S = −β(x) SI S + I + γ(x)I, x ∈ Ω, t > 0, I t − d I ∆I = β(x) SI S + I − γ(x)I, x ∈ Ω, t > 0, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, t > 0, S(x, 0) = S 0 (x), I(x, 0) = I 0 (x), x ∈ Ω, S t − d S ∆S = Λ(x) − S − β(x) SI S + I + γ(x)I, x ∈ Ω, t > 0, I t − d I ∆I = β(x) SI S + I − γ(x)I, x ∈ Ω, t > 0, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, t > 0, S(x, 0) = S 0 (x), I(x, 0) = I 0 (x), x ∈ Ω.
(1.5)
We note that (1.4) was first proposed by Allen et al. [4] and then it (and its variants) was (were) studied extensively by many researchers [11, 12, 16, 18-20, 25, 33, 46, 47, 49, 50] while (1.5) was analyzed by Li et al. [36]; see also [35] for the case of logistic source instead of the linear one. One also observes that the total population in (1.4) is conserved and that in (1.5) varies.
In [42], by comparing the outcomes of models with density-dependent and frequency-dependent transmission rates to the observed epidemiology of certain diseases, McCallum, Barlow and Hone concluded that both density-dependent and frequency-dependent mechanisms have their own advantages in modeling disease spread, depending on the transmission mode of the disease under consideration. They further pointed out that the transmission mode could be in general decided by estimating the force of infection.
On the other hand, epidemic theory for many ODE models has demonstrated that the basic reproduction number, which may be considered as the fitness of a pathogen in a given population, must be greater than unity for the pathogen to invade a susceptible population; see [7,8,15,26,44,54] and references therein. For the PDE models (1.2)-(1.5), we can also find their respective basic reproduction number R 0 and show that R 0 serves as the threshold value to determine the transmission dynamics of disease; that is, if R 0 > 1 the disease persists whereas it becomes extinct in the long run if R 0 < 1. However, the total population N , and the movement (migration) rates d I and d S may affect R 0 of (1.2)-(1.5) in different manners. As a result, each of the parameters N, d I , d S plays a subtle role in disease control; more detailed description will be made in the last discussion section.
The main goal of the current paper is twofold. The first one is to rigorously investigate qualitative properties of (1.3) and the asymptotic profile of EE (when exists) with respect to the small movement rate d I or d S . Theorem 2.4 below tells us that once R 0 > 1, the infectious disease will uniformly persist in space. Thus it becomes important to understand how the mobility of population migration affects the spatial distribution of disease, because this will help decision-makers to predict the pattern of disease occurrence and henceforth to conduct effective/optimal control strategies of disease eradication. Our result in Theorem 3.1 indicates that restricting the motility rate of susceptible individuals cannot eradicate the disease for (1.3), while this strategy works perfectly for (1.2) with small total population size ([57, Corollary 2.4]). Similar phenomenon was also observed in models (1.4) and (1.5). Therefore, this suggests that varying total population tends to enhance the persistence of infectious disease. The second goal is to compare our main results on the model (1.3) with those on models (1.2), (1.4) and (1.5), so as to understand the influence of the factors such as infection mechanism, movement rate and source term on the eradication of epidemics, and to discuss possible applications in disease control. Numerical simulations are also carried out to reinforce the theoretical findings and illustrate possible outcomes for those unknown situations and hence provide clues for further analytical pursues. We refer to Section 4 for detailed discussion on the implications of analytical results and comparisons between four related SIS epidemic models mentioned above.
The remainder of this paper is organized as follows. In Section 2, we first obtain the global existence and boundedness of solutions to the parabolic problem (1.3), then discuss the stability of equilibrium and the uniform persistence property via the basic reproduction number R 0 , and finally we consider the global attractivity of DFE and EE in spatially homogeneous environment. Section 3 is devoted to the study of asymptotic profile of EE when the diffusion rate of susceptible population or infected population approaches zero. In the last section, we perform numerical simulations, compare our results for (1.3) with those of the other three models, and discuss the implication of our findings in detail from the viewpoint of disease control.
In the rest of the paper, for notational convenience, we denote g * = max x∈Ω g(x) and g * = min x∈Ω g(x), for g = Λ, β, γ and µ.
Properties of solutions to (1.3)
In this section, we consider the parabolic system (1.3) by first establishing the global existence and uniform boundedness of solutions, and then show the local stability of DFE and uniform persistence via the basic reproduction number. Lastly we investigate the global attractivity of DFE and EE in homogeneous environment. for some large T > 0.
Proof. From the standard theory for semilinear parabolic systems [5], it follows that (1.3) admits a unique solution (S(x, t), I(x, t)) for x ∈ Ω and t ∈ [0, T max ) with T max being the maximal existence time. Moreover, the strong maximum principle for parabolic equations yields that the solution is positive on Ω × (0, T max ). Integrating both PDEs of (1.3) and adding the resulting two identities, we are led to
d dt Ω (S(x, t) + I(x, t))dx = Ω Λ(x)dx − Ω (S(x, t) + µ(x)I(x, t))dx ≤ Ω Λ(x)dx − θ Ω (S(x, t) + I(x, t))dx,(2.3)
where θ = min{1, µ * } > 0. Then the well-known Gronwall's inequality applied to (2.3) asserts that there exists some constant M 1 > 0, such that Ω (S(x, t) + I(x, t))dx ≤ M 1 , ∀t ∈ (0, T max ).
(2.4) We now consider S t − d S ∆S = Λ(x) − S + [γ(x) − β(x)S] I, x ∈ Ω, t ∈ (0, T max ), ∂S ∂ν = 0, x ∈ ∂Ω, t ∈ (0, T max ), S(x, 0) = S 0 (x),
x ∈ Ω.
(2.5)
For any nonnegative I, it is straightforward to verify that the positive constant
M 2 := max Λ L ∞ (Ω) , S 0 L ∞ (Ω) , γ β L ∞ (Ω)
is an upper solution of (2.5). The comparison principle for parabolic equations gives
S(x, t) ≤ M 2 , ∀x ∈ Ω, t ∈ (0, T max ).
Since S is uniformly bounded and the L 1 -norm of I(·, t) is also bounded for t ∈ (0, T max ) thanks to (2.4), in view of [3,Theorem 3.1] or [50, Lemma 3.1] and using the I-equation, we deduce that I is also uniformly bounded in Ω × (0, T max ). As a result, we must have T max = ∞ and (2.1) is proved. We next show (2.2). To the aim, we need to construct a more accurate upper solution of problem (2.5), which is independent of S 0 for all large time. In fact, let u(t) be the unique solution of the following ODE.
u ′ (t) = Λ * + γ/β L ∞ (Ω) − u(t), t > 0; u(0) = S 0 L ∞ (Ω) + γ/β L ∞ (Ω) .
It is clear that
u(t) = S 0 L ∞ (Ω) + γ β L ∞ (Ω) e −t + Λ * + γ β L ∞ (Ω) 1 − e −t ≥ γ β L ∞ (Ω) , which implies γ(x) − β(x)u(t) ≤ 0, ∀x ∈ Ω, t > 0.
It can be easily checked that u(t) is an upper solution of (2.5) and consequently,
S(x, t) ≤ u(t) → Λ * + γ β L ∞ (Ω) as t → ∞, ∀x ∈ Ω.
That is, we obtain an upper bound of S(·, t) L ∞ (Ω) which is independent of initial data for all large time. Now applying [50, Lemma 3.1] to the I-equation, we deduce that I(·, t) L ∞ (Ω) can also be bounded by a positive constant independent of (S 0 , I 0 ) for large t > 0.
2.2.
Basic reproduction number and uniform persistence. It is easily seen that the following elliptic problem
− d S ∆S = Λ(x) − S, x ∈ Ω; ∂S ∂ν = 0, x ∈ ∂Ω (2.6)
admits a unique positive solutionS, which is globally asymptotically stable for the corresponding parabolic equation with nonnegative initial data. Then (S, 0) is an equilibrium of (1.3), which we call the disease-free equilibrium (DFE). Clearly, it is the unique DFE. We define the basic reproduction number R 0 as follows:
R 0 = sup 0 =ϕ∈H 1 (Ω) Ω βSϕ 2 dx Ω [d I |∇ϕ| 2 + (γ + µ)ϕ 2 ] dx .
(2.7)
Indeed, one can follow the idea of next generation operators as in [50] to introduce the basic reproduction number, which coincides with the value R 0 . It is worth mentioning that the basic reproduction number R 0 defined here is qualitatively different from that in [4] and [14] in that it also depends implicitly on the diffusion rate d S of the susceptible individuals. Let (λ * , ψ * ) be the principal eigenpair of the eigenvalue problem
d I ∆u + (βS − γ − µ)u + λu = 0, x ∈ Ω; ∂u ∂ν = 0, x ∈ ∂Ω. (2.8)
Then, we have the following properties of R 0 , the proof of which resembles that of [4, Lemma 2.3] and hence is omitted.
Proposition 2.2. The following assertions hold.
(a) R 0 is a monotone decreasing function of d I with R 0 → max Ω βS/(γ + µ) as d I → 0 and
R 0 → Ω βSdx/ Ω (γ + µ)dx as d I → ∞; (b) If Ω β(x)S(x)dx < Ω [γ(x) + µ(x)]dx, and βS − (γ + µ) changes sign, then there exists a threshold value d * I ∈ (0, ∞) such that R 0 > 1 for d I < d * I and R 0 < 1 for d I > d * I ; (c) If Ω β(x)S(x)dx > Ω [γ(x) + µ(x)]dx, then R 0 > 1 for all d I > 0. (d) R 0 > 1 when λ * < 0, R 0 = 1 when λ * = 0, and R 0 < 1 when λ * > 0.
It turns out that the stability of the DFE (S, 0) is completely determined by the size of R 0 .
Proposition 2.3. The DFE (S, 0) is linearly stable if R 0 < 1, and it is linearly unstable if R 0 > 1.
Proof. The linearization of (1.3) around the DFE (S, 0) reads
η t − d S ∆η = −η + (−βS + γ)ξ, x ∈ Ω, t > 0, ξ t − d I ∆ξ = (βS − γ − µ)ξ, x ∈ Ω, t > 0, ∂η ∂ν = ∂ξ ∂ν = 0, x ∈ ∂Ω, t > 0, with η(x, t) = S(x, t)−S(x) and ξ(x, t) = I(x, t). Now suppose that (η(x, t), ξ(x, t)) = (e −λt φ(x), e −λt ψ(x))
is a solution of the above linear system with λ being a complex number. Then simple calculations show that
d S ∆φ − φ + (−βS + γ)ψ + λφ = 0, x ∈ Ω, d I ∆ψ + (βS − γ − µ)ψ + λψ = 0, x ∈ Ω, ∂φ ∂ν = ∂ψ ∂ν = 0, x ∈ ∂Ω.
(2.9)
We first assume that R 0 < 1 and shall show that (S, 0) is linearly stable; that is, if (λ, φ, ψ) is any solution of (2.9) with φ or ψ not identically zero, then Re(λ) > 0. There are two cases to consider: ψ ≡ 0 and φ ≡ 0; ψ ≡ 0.
In the former case, clearly (λ, φ) is an eigenpair of the eigenvalue problem
d S ∆u − u + λu = 0, x ∈ Ω; ∂u ∂ν = 0, x ∈ ∂Ω. (2.10)
It is obvious that λ must be real due to the self-adjoint property of the operator involved in (2.10) and hence λ ≥ 1, as we wanted. If the latter case happens, it follows that (λ, ψ) is an eigenpair of the eigenvalue problem (2.8) and hence λ is real and λ ≥ λ * > 0 due to Proposition 2.2 (d). Thus, the linear stability of (S, 0) is proved. We now suppose R 0 > 1 and show the instability of (S, 0). Proposition 2.2(d) yields that λ * < 0. It is well known that the following linear problem
d S ∆φ − φ + λ * φ = (βS − γ)ψ * , x ∈ Ω, ∂φ ∂ν = 0, x ∈ ∂Ω
admits a solution φ * . Consequently, (λ * , φ * , ψ * ) becomes a solution of (2.9) with λ * < 0 and ψ * > 0 and so (S, 0) is linearly unstable.
Based on the "ultimately uniformly boundedness" (2.2), we are able to establish the uniform persistence property of (1.3) when the basic reproduction number R 0 > 1. In fact, one can easily adapt the arguments of [50,Theorem 3.3], developed by Magal and Zhao (see [41,Theorem 4.5] and [58,Chapter 13]), to conclude the following assertion. Furthermore, (1.3) admits at least one EE provided that R 0 > 1.
2.3.
Global stability in homogeneous environment. In this subsection, we consider the global stability of the DFE and EE of (1.3) in homogeneous environment, i.e., all of parameters Λ, β, γ and µ are positive constants. In view of (2.7), we now have an explicit expression for the basic reproduction number R 0 = Λβ γ+µ and the unique DFE is given by (S, 0) = (Λ, 0). On the other hand, there exists a unique constant EE (Ŝ,Î) if and only if R 0 > 1, wherê
S = γ + µ β = Λ R 0 andÎ = Λ µ 1 − 1 R 0 = γ + µ µβ (R 0 − 1) .
For later purpose, we recall a simple fact which can be found in [55, Lemma 2.5.1]:
Lemma 2.1. Let a and b be positive constants. Assume that ϕ, ψ ∈ C 1 ([a, ∞)), ψ(t) ≥ 0 in [a, ∞) and ϕ is bounded from below. If ϕ ′ (t) ≤ −bψ(t) and ψ ′ (t) ≤ K in [a, ∞) for some constant K, then lim t→∞ ψ(t) = 0.
By constructing suitable Lyapunov functionals, we can show
Theorem 2.5. Assume that d S = d I .
Then the following assertions hold.
(i) If R 0 ≤ 1, then the DFE is globally attractive; (ii) If R 0 > 1, then the EE is globally attractive. Proof. Set d S = d I = d.
To verify (i), for any solution (S, I) of (1.3), we define
V (t) = 1 2 Ω [(S − Λ) + I] 2 dx + µ + 1 β Ω Idx.
Then, for all t > 0, direct calculations show that
V ′ (t) = Ω [(S − Λ) + I] (S t + I t )dx + µ + 1 β Ω I t dx = Ω [(S − Λ) + I] (d S ∆S + Λ − S − µI + d I ∆I)dx + µ + 1 β Ω (d I ∆I + βSI − γI − µI)dx = −d Ω |∇(S + I)| 2 dx − Ω (S − Λ) 2 dx − µ Ω I(S − Λ)dx + Ω I(Λ − S)dx − µ Ω I 2 dx + µ + 1 β Ω (βSI − γI − µI)dx ≤ − Ω (S − Λ) 2 dx − µ Ω I 2 dx + µ + 1 β [βΛ − (γ + µ)] Ω Idx ≤ 0, due to the assumption that R 0 = βΛ/(γ + µ) ≤ 1. Define ψ(t) = Ω (S − Λ) 2 dx + µ Ω I 2 dx ≥ 0.
Recall that Theorem 2.1 tells us that both S(·, t) L ∞ (Ω) and I(·, t) L ∞ (Ω) are bounded. Hence, by [10, Theorem A2], we have
S(·, t) C 2+α (Ω) + I(·, t) C 2+α (Ω) ≤ C 0 , ∀t ≥ 1, (2.11)
for some positive constant C 0 . Furthermore, using both PDEs of (1.3), one can easily see that ψ ′ (t) is bounded from above for t ∈ [1, ∞). We deduce from Lemma 2.1 (by taking ϕ(t) = V (t)) that
(S(x, t), I(x, t)) → (Λ, 0) = (S, 0) in L 2 (Ω) 2 , as t → ∞.
Furthermore, (2.11) indicates that (S(·, t), I(·, t)) is compact in C 2 (Ω) for t ≥ 1. This, together with the above L 2 -convergence, yields that
(S(x, t), I(x, t)) → (S, 0) in C 2 (Ω) 2 , as t → ∞;
that is, (S, 0) attracts all solutions of (1.3).
We next prove (ii). Define
W (t) = 1 2 Ω S −Ŝ + I −Î 2 dx + µ + 1 β Ω I −Î −Î ln Î I dx ≥ 0, ∀t > 0.
By straightforward computations, we have
W ′ (t) = Ω S −Ŝ + I −Î (S t + I t )dx + µ + 1 β Ω 1 −Î I I t dx = Ω S −Ŝ + I −Î (d∆S + Λ − S − µI + d∆I)dx + µ + 1 β Ω 1 −Î I (d∆I + βSI − γI − µI)dx = −d Ω |∇(S + I)| 2 dx − µ + 1 β dÎ Ω |∇I| 2 I 2 dx + µ + 1 β Ω (I −Î)(βS − βŜ)dx + Ω (S −Ŝ) + (I −Î) Ŝ + µÎ − S − µI dx ≤ − Ω (S −Ŝ) 2 dx − µ Ω (I −Î) 2 dx ≤ 0,
where we have used the fact that Λ =Ŝ + µÎ and γ + µ = βŜ. In Lemma 2.1, let
φ(t) = W (t), ψ(t) = Ω (S −Ŝ) 2 dx + µ Ω (I −Î) 2 dx, ∀t > 0.
Then arguing similarly as before, we eventually conclude that
(S(x, t), I(x, t)) → (Ŝ,Î) in C 2 (Ω) 2 , as t → ∞.
The proof is complete.
The above theorem tells us that system (1.3) is uniformly persistent in homogeneous environment provided R 0 > 1, at least in the equal diffusion rate case.
Remark 2.1. For general positive functions Λ, β, γ, µ and constants d S , d I > 0, we suspect that (1.3) has a unique EE which is globally attractive if R 0 > 1, and the DFE is globally attractive if R 0 ≤ 1. However the justification of this suspicion is highly nontrivial and has to be left open in the current paper.
Asymptotic profile of EE
In this section, we are concerned with the asymptotic behavior of EE of (1.3), which is a positive solution to the elliptic system:
−d S ∆S = Λ(x) − S − β(x)SI + γ(x)I, x ∈ Ω, −d I ∆I = β(x)SI − [γ(x) + µ(x)]I, x ∈ Ω, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω (3.1)
as one of the diffusion rates d S , d I goes to zero.
3.1. The case of d S → 0. Using a singular perturbation argument, one can easily show thatS, being the unique positive solution of (2.6), converges uniformly to Λ as d S → 0 (see [48,Lemma 3.2]). Therefore, according to the continuity of eigenvalues with respect to the potential function, we see that the principal eigenvalue λ * of (2.8) converges to the principal eigenvalue of the following eigenvalue problem
d I ∆u + (βΛ − γ − µ) u + λu = 0, x ∈ Ω; ∂u ∂ν = 0, x ∈ ∂Ω, (3.2)
which is denoted by λ 0 . To ensure the existence of EE for all small d S , one has to assume λ 0 < 0. Now we are ready to establish the main result of this subsection.
Theorem 3.1. Assume that λ 0 < 0. Fix d I > 0, and let d S → 0, then every positive solution (S, I) of (3.1) satisfies (up to a subsequence of d S → 0)
(S, I) → (S, I) uniformly on Ω, where S(x) = Λ(x)+γI(x) 1+βI(x)
, and I is a positive solution to
− d I ∆I = β(x)SI − (γ(x) + µ(x))I, x ∈ Ω; ∂I ∂ν = 0, x ∈ ∂Ω. (3.3)
Proof. Mentioned as before, (3.1) has at least one EE for all small d S > 0 when λ 0 < 0. In the following, we divide our argument into three steps for sake of clarity.
Step 1: A priori bounds for S and I. Assume S(x 0 ) = max x∈Ω S(x). We apply the maximum principle [40, Proposition 2.2] to the first equation of (3.1) to derive Λ(
x 0 ) − S(x 0 ) − β(x 0 )S(x 0 )I(x 0 ) + γ(x 0 )I(x 0 ) ≥ 0, or, Λ * ≥ Λ(x 0 ) ≥ S(x 0 ) + I(x 0 ) (β(x 0 )S(x 0 ) − γ(x 0 )) . (3.4) If β(x 0 )S(x 0 ) − γ(x 0 ) ≤ 0, then max Ω S = S(x 0 ) ≤ γ(x 0 )/β(x 0 ) ≤ γ/β L ∞ (Ω) . If β(x 0 )S(x 0 ) − γ(x 0 ) > 0, it follows from (3.4) that max Ω S = S(x 0 ) ≤ Λ * . Thus, for any d S , d I > 0, we have max Ω S ≤ max Λ * , γ β L ∞ (Ω) . (3.5)
On the other hand, set S(x 1 ) = min x∈Ω S(x). Then an application of the maximum principle [40, Proposition 2.2] implies that Λ(
x 1 ) − S(x 1 ) − β(x 1 )S(x 1 )I(x 1 ) + γ(x 1 )I(x 1 ) ≤ 0, equivalently, Λ(x 1 ) + γ(x 1 )I(x 1 ) 1 + β(x 1 )I(x 1 ) ≤ S(x 1 ).
Obviously, there exists a positive constant c * , independent of d S , d I > 0, such that
c * ≤ Λ(x 1 ) + γ(x 1 )I(x 1 ) 1 + β(x 1 )I(x 1 ) .
Hence, for any d S , d I > 0, it holds
c * ≤ S(x), ∀x ∈ Ω. (3.6)
Integrating both PDEs of (3.1) over Ω yields
Ω {Λ(x) − S − β(x)SI + γ(x)I} dx = 0, Ω {β(x)SI − [γ(x) + µ(x)]I} dx = 0, from which it immediately follows that µ * Ω Idx ≤ Ω µIdx + Ω Sdx = Ω Λdx ≤ |Ω|Λ * (3.7) and β * Ω SIdx ≤ Ω βSIdx ≤ (γ * + µ * ) Ω Idx ≤ |Ω|Λ * (γ * + µ * ) µ * . (3.8)
We now write the I-equation as
− ∆I = 1 d I [βS − (γ + µ)] I, x ∈ Ω; ∂I ∂ν = 0, x ∈ ∂Ω. (3.9)
According to the Harnack-type inequality (see, e.g., [37]
I ≤ C min Ω I ≤ C 1 |Ω| Ω Idx ≤ C. (3.10)
Hereafter, C represents a positive constant independent of small d S > 0 which may vary from place to place.
Step 2: Convergence of I. Recall that I satisfies (3.9). By (3.5) and (3.10), we have
1 d I [βS − (γ + µ)] I L p (Ω) ≤ C, ∀ p > 1.
From the standard L p -estimate for elliptic equations (see, e.g., [21]), it follows that I W 2,p (Ω) ≤ C for any given p > 1. Taking p to be sufficiently large, we see from the Sobolev embedding that I C 1+α (Ω) ≤ C for some 0 < α < 1. As a result, there exists a subsequence of d S → 0, say d n := d S,n , satisfying d n → 0 as n → ∞, and a corresponding positive solution (S n , I n ) of (3.1) with d S = d n , such that I n → I uniformly on Ω, as n → ∞, (3.11) where 0 ≤ I ∈ C 1 (Ω). In view of (3.10), either I ≡ 0 on Ω or I > 0 on Ω.
(3.12)
Suppose the former holds in (3.12); that is, I n → 0 uniformly on Ω, as n → ∞.
(3.13)
Then for sufficiently small ǫ > 0, we have 0 ≤ I n (x) ≤ ǫ, ∀x ∈ Ω, for all large n. This fact, together with the first equation of (3.1), implies that for all large n, S n satisfies
−d n ∆S n ≤ Λ − S n + γ * ǫ, x ∈ Ω; ∂S n ∂ν = 0, x ∈ ∂Ω and −d n ∆S n ≥ Λ − S n − β * ǫS n , x ∈ Ω; ∂S n ∂ν = 0, x ∈ ∂Ω.
We consider the following two auxiliary problems:
− d n ∆u = Λ − u + γ * ǫ, x ∈ Ω; ∂u ∂ν = 0, x ∈ ∂Ω,(3.14)
and
− d n ∆v = Λ − v − β * ǫv, x ∈ Ω; ∂v ∂ν = 0, x ∈ ∂Ω. (3.15)
It is clear that systems (3.14) and (3.15) admit a unique positive solution, denoted by u n and v n , respectively. A simple sub-supsolution argument, combined with the uniqueness, guarantees that v n ≤ S n ≤ u n on Ω for all large n. Using a singular perturbation argument as in [17,Lemma 2.4], it can be shown that
u n → Λ + γ * ǫ, v n → Λ 1 + β * ǫ
uniformly on Ω, as n → ∞.
Sending n → ∞, we find
Λ(x) 1 + β * ǫ ≤ lim inf n→∞ S n (x) ≤ lim sup n→∞ S n (x) ≤ Λ(x) + γ * ǫ.
Thanks to the arbitrariness of small ǫ > 0, we obtain that S n → Λ uniformly on Ω, as n → ∞. . Then Ĩ n L ∞ (Ω) = 1 for all n ≥ 1, andĨ n solves
− d I ∆Ĩ n = [β(x)S n − (γ + µ)]Ĩ n , x ∈ Ω; ∂Ĩ n ∂ν = 0, x ∈ ∂Ω. (3.18)
As before, through a standard compactness argument for elliptic equations, after passing to a further subsequence if necessary, we may assume that
I n →Ĩ in C 1 (Ω), as n → ∞,
where 0 ≤Ĩ ∈ C 1 (Ω) with Ĩ L ∞ (Ω) = 1. By (3.16) and (3.18),Ĩ satisfies
− d I ∆Ĩ = [βΛ − (γ + µ)]Ĩ, x ∈ Ω; ∂Ĩ ∂ν = 0, x ∈ ∂Ω. (3.19)
The Harnack-type inequality (see, [37] or [48, Lemma 2.2]) applied to (3.19) yieldsĨ > 0 on Ω. However, the positiveness ofĨ indicates that the principal eigenvalue λ 0 of the eigenvalue problem (3.2) must be zero (withĨ being a corresponding eigenfunction), contradicting our assumption that λ 0 < 0. Thus, (3.13) cannot occur, and we must have I > 0 on Ω. That is, I n → I > 0 uniformly on Ω, as n → ∞.
(3.20)
Step 3: Convergence of S. Notice that S n solves − d n ∆S n = Λ − S n − βS n I n + γI n , x ∈ Ω; ∂S n ∂ν = 0, x ∈ ∂Ω.
(3.21)
In view of (3.20), we see that for any small ǫ > 0, it holds
0 < I(x) − ǫ ≤ I n (x) ≤ I(x) + ǫ, ∀x ∈ Ω (3.22)
for all large n. Thus, for all sufficiently large n, we have Λ − S n − βS n (I + ǫ) + γ(I − ǫ) ≤ Λ − S n − βS n I n + γI n ≤ Λ − S n − βS n (I − ǫ) + γ(I + ǫ).
Given large n, we consider the following auxiliary problem Because of (3.17), it can be easily seen that I satisfies (3.3). The proof is complete.
3.2.
The case of d I → 0. This subsection is devoted to the investigation of the asymptotic behavior of positive solutions of (3.1) with d S > 0 being fixed and d I → 0. Because of mathematical difficulty, we can only deal with one space dimension case, that is, the habitat Ω is an interval. Without loss of generality, we take Ω = (0, 1). In light of 2.2 (a) and 2.4, we assume that {β(x)S(x) > γ(x) + µ(x) : x ∈ [0, 1]} is non-empty so that R 0 > 1 and thus (3.1) admits positive solutions for all small d I > 0. Our main result reads as follows. we deduce from the elliptic L 1 -theory in [9] that, for any p > 1, S W 1,p (0,1) ≤ C, where C is a positive constant independent of d I but allows to be different below. Then for sufficiently large p, the Sobolev embedding theorem guarantees that S C α ([0,1]) ≤ C for some α ∈ (0, 1). Moreover, up to a sequence of d I → 0, say d n := d I,n → 0 with d n → 0 as n → ∞, the corresponding positive solution sequence (S n , I n ) of (3.1) with d I = d n satisfies S n → S 0 > 0 in C([0, 1]), as n → ∞ due to (3.6). In light of (3.7), by passing a subsequence of d n if necessary, we may assume that As S n (x) − S n (0) = x 0 S ′ n (y)dy for any n ≥ 1, we find that S 0 solves
S 0 (x) − S 0 (0) = − 1 d S x 0 y 0 [Λ(z) − S 0 (z)]dz dy, which in turn implies that − d S S ′′ 0 (x) = Λ(x) − S 0 (x), x ∈ (0, 1); S ′ 0 (0) = 0. (3.27)
If integrating (3.26) from x to 1, one can use the analysis similar as above to know that S ′ 0 (1) = 0. Therefore, this and (3.27) give that S 0 =S, that is, S n →S uniformly on [0, 1] as n → ∞.
On the other hand, observe that
λ 1 (d n , γ(x) + µ(x) − β(x)S n (x)) = 0, ∀n ≥ 1,
where λ 1 (d n , γ(x)+µ(x)−β(x)S n (x)) stands for the principal eigenvalue of the following eigenvalue problem:
d n ∆u + [β(x)S n (x) − γ(x) − µ(x)] u + λu = 0, x ∈ (0, 1); u ′ (0) = u ′ (1) = 0.
Combined with the fact that the principal eigenvalue continuously depends on the parameters, the argument as in [4, Lemma 2.3] yields 4) and (1.5) to model the SIS epidemic dynamics based on different infection mechanisms and modeling ideas, it will be helpful to summarize their results and make a comparison so as to understand the influence of the factors such as infection mechanism, movement rate and source term on the eradication of epidemics. Numerical simulations will be performed to validate theoretical results and to predict possible outcomes for those cases that remain unknown analytically. Then we discuss the implication of these theoretical and numerical findings from the disease control viewpoint. Since the results of the model (1.3) have been summarized above, below we shall briefly recall the results for the SIS models (1. Hereafter, N is a fixed positive constant, representing the total number of the susceptible and infected populations. That is N = Ω (S(x) + I(x))dx is a constant. As in [4,49], we introduce the notion of low/high/moderate risk site/domain. We say that x is a low (or high or moderate) risk site if the local disease transmission rate β(x) is lower (or higher or equal to) than the local disease recovery rate γ(x). Let
0 = λ 1 (d n , γ(x) + µ(x) − β(x)S n (x)) → min x∈[0,1] {γ(x) + µ(x) − β(x)S(x)}, as n → ∞,H − = {x ∈ Ω : β(x) < γ(x)} and H + = {x ∈ Ω : β(x) > γ(x)}
denote the set of low-risk sites and high-risk sites, respectively.
Assume that both H − and H + are nonempty. The authors in [4] defined the basic reproduction numberR 0 = sup 0 =ϕ∈H 1 (Ω) Ω βϕ 2 dx Ω (d I |∇ϕ| 2 + γϕ 2 )dx and showed that the unique DFE (N/|Ω|, 0) is globally stable ifR 0 < 1, while it is unstable and a unique EE exists ifR 0 > 1. Indeed, following the argument similar to [12], one can show that the uniform persistence property holds onceR 0 > 1.
The asymptotic profile of the EE was also investigated in [4] when the diffusivity of the susceptible individuals tends to zero. In particular, the result of [4] shows that • As d S → 0, the unique positive solution (S, I) (which exists ifR 0 > 1) of (4.1) fulfills (S, I) → (Ŝ, 0) uniformly on Ω, whereŜ satisfies a free boundary problem, is positive at all low-risk sites and is also positive at some (but not all) high-risk sites.
This result indicates that it may be possible to entirely eliminate the infectious disease by restricting the motility rate of the susceptible to be small. Further asymptotics of the EE in other cases were obtained by Peng [46] wherein it was shown that if d I → 0 and d := d I /d S → d 0 ∈ [0, ∞], then the unique positive solution (S, I) of (4.1) satisfies the following:
• If d 0 = 0, then S → N Ω [1 + (β − γ) + γ −1 ] and I → N (β − γ) + γ −1 Ω [1 + (β − γ) + γ −1 ] uniformly on Ω.
In what follows, (s) + = max{s, 0}. 4), we may conclude that the optimal strategy of eliminating the infectious disease is to restrict the motility rate of the susceptible population, while restricting the motility of infected population can only eradicate the disease in low-risk and moderate-risk sites. Of course, another strategy is to set d I → 0 and d S → 0 while the susceptible moves relatively slower than the infected.
• If d 0 ∈ (0, ∞), then S → N d 0 [1 − A(d 0 ; x)] Ω [A(d 0 ; x) + d 0 (1 − A(d 0 ; x))] , I → N A(d 0 ; x) Ω [A(d 0 ; x) + d 0 (1 − A(d 0 ; x))] uniformly on Ω, where A(d 0 ; x) = d 0 (β−γ) + d 0 (β−γ)+γ . • If d 0 = ∞,
4.1.2.
Result on (1.5). Now we consider the scenario that the susceptible individuals are allowed to have birth and death, and look at the SIS reaction-diffusion system (1.5) with a linear external source. One of the main results in [36] states that (1.5) admits at least one EE (S, I) ifR 0 > 1, which is in fact a positive steady state of (1.5) satisfying 2). In [14,56,57], the authors treated the SIS system (1.2) with mass action and its steady state problem: For the mass action system (1.2), the basic reproduction number depends on the total population size N and is defined asR
−d S ∆S = Λ(x) − S − β(x) SI S+I + γ(x)I, x ∈ Ω, −d I ∆I = β(x) SI S+I − γ(x)I, x ∈ Ω, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω. −d S ∆S = −β(x)SI + γ(x)I, x ∈ Ω, −d I ∆I = β(x)SI − γ(x)I, x ∈ Ω, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, Ω [S(x) + I(x)]dx = N.0 = sup 0 =ϕ∈H 1 (Ω) (N/|Ω|) Ω βϕ 2 Ω (d I |∇ϕ| 2 + γϕ 2 ) = N |Ω|R 0 .
It is shown that a positive solution (S, I) of (4.3) exists wheneverR 0 > 1. Indeed, following the argument similar to [12], one can show the uniform persistence property holds onceR 0 > 1. Moreover, one can show thatR 0 > 1 when N > Ω γ(x)
β(x) dx, andR 0 > 1 is also possible when N ≤ Ω γ(x) β(x) dx depending on the parameters β, γ and d I . Furthermore, for fixed d I > 0, the following asymptotics as d S → 0 have been shown in [56,57]:
• If either N − Ω γ β > 1 4 Ω |∇β| 2 β 3 or N |Ω| > γ β on Ω, then (S, I) → γ(x) β(x) , N |Ω| − 1 |Ω| Ω γ(x) β(x) dx uniformly on Ω; • If N ≤ Ω γ(x) β(x) dx,S * * = N |Ω| − N |Ω| − γ β + and I * * = N |Ω| − γ β + .
On the other hand, in the case of one-dimensional domain, say Ω = (0, 1), if γ < N β on [0, 1], then for fixed d S > 0, as d I → 0, the authors of [56] proved that 5) and their corresponding EE problem (when no confusion is caused) as MO, MW, SO and SW, respectively, in order that each label of models can bear a meaning (see Table 1). For convenience, we also list the basic reproduction number for each of models MO, MW, SO, SW in Table 1, where three observations are worth mentioning as follows.
(a) MO is the only one whose basic reproduction number depends on N via N/|Ω| which measures the number of population per unit space. This implies that the total population plays a role in the eradication of diseases only for MO, and also explains why a disease is easier to become endemic in a more crowded population than a sparse population as mentioned in [57]. (b) If the birth-death effect is considered, then MO becomes MW whose basic reproduction number no longer depends on total population N . This indicates that the birth and death effects could be an important factor for the eradication of diseases in SIS models with mass-action infection mechanism. However, the birth-death effect is not important for models SO and SW any more, since both have the same basic reproduction number. (c) MW is the only model whose basic reproduction number depends (implicitly) on the diffusivity d S of the susceptibles.
Model
Infection Table 1. Basic reproduction numbers for SIS epidemic models, whereS in the basic reproduction number for MW is the unique solution of (2.6).
Asymptotic behavior of EE.
From the disease control point of view, one is mainly concerned with whether the infectious disease can be eradicated (namely whether I(x) can go extinction either throughout the entire domain Ω or partially). One of the strategies as recalled above is to control the motility of susceptible and/or infected populations. Below in Table 2 and Table 3 we capsulize the asymptotic behavior of EE (S(x), I(x)) as d S → 0 or d I → 0 or both. Furthermore we use numerical simulations to illustrate known results and predict possible outcomes for unknown cases. In the following, we shall use (S * , I * ) to represent the asymptotic behavior of EE for all models for simplicity. We remark that the parameter values chosen in all simulations are sufficient to guarantee the existence of EE in models under consideration. For example, in Table 2. Asymptotic behavior of (S(x), I(x)) as d S → 0 or d I → 0.
When the movement rate d S of the susceptibles tends to zero, the asymptotics of solutions have been well understood to a large extent as seen in Table 2, and the asymptotic profiles of EE illustrated in Fig.2 are consistent with analytical results. It is worth mentioning that for MO, since the parameter values are taken so that 1 = N < 1 0 γ(x) β(x) dx, we have the convergence I → 0 as d S → 0 according to the results of [57] which our numerical simulations fit well.
With the same parameter values as in Fig.2, we illustrate the asymptotic profiles of EE as d I → 0 in Fig.3. For the two standard incidence infection models SO and SW, our simulations show that the limiting profile of I for both models is positive only at high-risk sites which match well with the analytical results. The limiting profile of S for model SW is constant because of the special choice of Λ (see [36,Theorem 5.2]). For models MW and MO, the exact limiting behavior of I(x) remains open except knowing that its total population is positive (see Table 2). Our numerical simulations Model Limit of (S(x), I(x)) as d S → 0 and d I → 0 MO For the case d I /d S → d ∈ (0, ∞), S * (x) > 0 and I * (x) ≥ 0 but Table 3. Asymptotic behavior of (S(x), I(x)) as both d S → 0 and d I → 0.
I * (x) ≡ 0 MW Unknown SO S * (x) > 0 and I * (x) ≡ 0 iff x ∈ H − when d I /d S → [0, ∞), S * (x) ≥ 0 and I * (x) ≡ 0 when d I /d S → ∞ SW Unknown
in Fig.3 demonstrate that the infectious disease tends to aggregate in a narrow region and is eradicated outside this region, where model MO has a narrower aggregation region than model MW. We remark that in our simulation the condition γ < N β required in [56] is not satisfied on [0, 1], and we observe that S tends to a positive constant though its rigorous proof still remains open.
The asymptotic behavior of EE as d S → 0 and d I → 0 is only partially understood (see results in Table 3). The numerical simulations shown in Fig.4 verify the known results on models SO and MO where the asymptotic profiles of (S, I) coincide because of our choice of the parameter values. However, the asymptotic behavior of EE as d S → 0 and d I → 0 for models MW and SW entirely remains open and our numerical simulations have the following predictions. Firstly, for MW, the simulation implies that S * (x) > 0 and I * (x) ≥ 0 but I * (x) ≡ 0 as d S → 0 and d I → 0 with d I /d S → d ∈ (0, ∞), which is analogous to the asymptotic behavior of EE for MO. In other words, the birth-death effect seems to be not important for SIS models with mass-action infection mechanisms if both diffusion rates of the susceptible and infectious are small with the same order. Secondly, for model SW, the numerical simulation shows that S * (x) is a positive constant and I * (x) ≥ 0 where I * (x) ≡ 0 if and only if x ∈ H − . These simulations suggest possible asymptotic behavior of models MW and SW as d S → 0 and d I → 0 for further analytical pursues.
Finally, to see whether the inclusion of a moderate-risk region will affect the asymptotic profiles of EE as considered in [49], we choose appropriate functions for β(x) and γ(x) as
β(x) = 1, x ∈ [0, 0.75] 2x − 0.5, x ∈ [0.75, 1] , γ(x) = −2x + 1.5, x ∈ [0, 0.25] 1, x ∈ [0.25, 1] (4.4)
such that β(x) = γ(x) on the interval [0.25, 0.75] (moderate-risk region), see a plot in Fig.1(b), and perform numerical simulations with small d I . For model MW, Fig.5(a) indicates that the infected population tends to aggregate on two narrow regions instead of one, compared to the case without a moderate-risk region as illustrated in Fig.3. Moreover, the simulation in Fig.5(b) illustrates that the limiting profile of I of SO, SW and MO is positive only at high-risk sites. This is in sharp contrast with Fig.3 where there is no moderate-risk region and the limiting profile of I for model MO is positive only on a narrow part within the high-risk region.
4.2.3.
Implication on disease control. We now discuss numerous implications/comments on disease control based on analytical and numerical results summarized in the preceding subsections. First consider models SO and MO which have conserved total population but subject to different infection mechanism. For model SO with any magnitude of total population, it is possible to eliminate the disease entirely by restricting d S while the disease cannot be eradicated on high-risk sites by limiting d I (see Table 2, Fig.2 and Fig.3). As for model MO, restricting d S can eliminate the disease only if the total population is small (see Table 2), whereas the infected individuals tend to aggregate on a narrow region if d I is small by the observation from Fig.3. Thus, if the total number of population remains unchanged, we may conclude that the disease described by standard incidence infection mechanism modeled by SO is easier to control by limiting the motility d S of susceptible population compared to the mass-action infection mechanism modeled by MO. Fig.1(a). Nevertheless, the disease subject to mass action infection mechanism can be eradicated to a larger extent (region) if the motility d I of infected individuals is restricted. Now consider models MW and SW that have the same linear recruitment but different infection mechanisms. From Table 2, Fig.3 and Fig.4, we see that the infectious disease cannot be eliminated at all by restricting d S for either models due to the source term of susceptible population, while restricting d I can eliminate the disease partially for both models but standard incidence infection mechanism seems to be more efficient than the mass-action one. Let us also consider the effect of linear recruitment on the same infection mechanism; that is, we compare model SO with SW, and MO with MW. Recall that restricting the motility of susceptible population (d S is small) yields the extinction of disease subject to standard incidence infection mechanism in SO, but this strategy fails for SW with linear recruitment subject to the same infection mechanism. Similar results hold between models MO and MW, but only with small total population. When d I is small, the infectious disease modeled by SO and SW is eradicated/persistent at the same region but the latter has a larger total mass, whereas the infectious disease modeled by MW is less condensed compared to its counterpart MO. Thus, if d S is small, whichever the infection mechanism is, a varying total population tends to enhance the persistence of disease, while this enhancement induced by standard incidence infection mechanics is not as strong as mass action one does. Nevertheless, for small d I , the disease subject to mass-action infection mechanism modelled by MO and MW seems to be less endemic since the infected population is more concentrated (see Fig.3).
If the environment is modified to include a moderate-risk region (see a graph in Fig.1(b)), then we see that for small d I , the disease modelled by SO, SW and MO can be eradicated precisely at low-risk and moderate-risk sites (see Fig.5(b)). This exhibits quite different behavior than that of model MW for which the infected disease may also persist in low-risk or moderate-risk sites but also be eradicated in part of high-risk sites (see Fig.5(a)). Compared to the profiles shown in Fig.3 for the case of small d I without moderate-risk site, from the standing point of disease control, this essentially implies that at least for model MO it is perhaps not a sound strategy to create a moderate-risk domain in the environment and restrict the motility of infected population at the same time.
We also would like to mention that due to the conservative property of the total population, the steady state problem of SO can be reduced to a single local elliptic equation while that of MO can be reduced to a single nonlocal elliptic equation. Hence, this property makes the corresponding system easier to attack, compared to the case of varying total population. Moreover, it is exactly because of this property that one can consider the asymptotic profiles of the positive solution for small d I and d I /d S → d 0 for some d 0 , as in [46,57]. This seems to be a rather challenging task for the steady state of models MW and SW due to lack of appropriate a prior estimates.
Finally, it is perhaps worth mentioning that one can also consider the effects of large motility rate of susceptible or infected population, as in [35,36,46]. In fact, one can easily follow the arguments there and conclude that when the motility of the susceptible population tends to infinity, the density of the susceptibles becomes positive and homogeneous and the density of the infected is also positive but inhomogeneous throughout the habitat; similar result holds if the movement rate of the infected population becomes large. Since these results are essentially the same as before and they indicate that large diffusion rate of the susceptibles or infected does not help to eradicate the disease, we do not present these results in this paper.
x, t) + I(x, t)] dx = Ω [S 0 (x) + I 0 (x)] dx =: N, ∀t ≥ 0.
2. 1 .
1Global existence and uniform boundedness. We now establish the global existence and boundedness of solutions to (1.3).
Theorem 2 . 1 .
21The solution (S(x, t), I(x, t)) of problem (1.3) exists uniquely and globally. Furthermore, there exists a positive constant M depending on initial data and the parameters d S , d I , Λ, β, γ and µ such that S(·, t) L ∞ (Ω) + I(·, t) L ∞ (Ω) ≤ M, ∀t ≥ 0. (2.1) Moreover, there exists some M ′ > 0 independent of initial data fulfilling S(·, t) L ∞ (Ω) + I(·, t) L ∞ (Ω) ≤ M ′ , ∀t ≥ T, (2.2)
Theorem 2 . 4 .
24Suppose that R 0 > 1. Then system (1.3) is uniformly persistent, i.e., there exists some η > 0 independent of the initial data (S 0 , I 0 ), such that lim inf t→∞ S(x, t) ≥ η and lim inf t→∞ I(x, t) ≥ η uniformly for x ∈ Ω.
I n fulfills − d I ∆I n = β(x)S n I n − (γ + µ)I n , x ∈ Ω; ∂I n ∂ν = 0, x ∈ ∂Ω. (3.17) DefineĨ n := In In L ∞ (Ω)
−
d n ∆w = Λ − w − βw(I + ǫ) + γ(I − ǫ), x ∈ Ω; ∂w ∂ν = 0, x ∈ ∂Ω. (3.23) It is clear that (3.23) admits a unique positive solution, denoted by w n . By similar arguments to those in the proof of [17, Lemma 2.4]), we notice that w n → Λ + γ(I − ǫ) 1 + β(I + ǫ) uniformly on Ω, as n → ∞. Since S n is an upper solution of (3.23), it then follows of (3.24) and (3.25), combined with the arbitrariness of small ǫ > 0, we have lim n→∞ S n (x) = S(x) := Λ(x) + γ(x)I(x) 1 + β(x)I(x) uniformly on Ω.
Theorem 3. 2 . 1 0
21Assume that the set {x ∈ [0, 1] : β(x)S(x) > γ(x) + µ(x)} is non-empty. Fix d S > 0 and let d I → 0, then every positive solution (S, I) of (3.1) satisfies (up to a subsequence of d I ) that S → S 0 uniformly on [0, 1], where S 0 ∈ C([0, 1]) and S 0 > 0 on [0, 1], and Idx → I 0 for some positive constant I 0 . Proof. Notice that (3.5), (3.6), (3.7) and (3.8) remain true in the current situation. Since the spatial domain is one dimensional and S satisfies − d S S ′′ (x) + S(x) = Λ − βS(x)I(x) + γI(x), x ∈ (0, 1); S ′ (0) = S ′ (1) = 0, (3.26)
1 0[
1I n dx → I 0 , as n → ∞, for some nonnegative constant I 0 . To show I 0 > 0, we proceed indirectly and suppose that I 0 = 0. By integrating (3.26) from 0 to x, y) − S n (y) − β(y)S n (y)I n (y) + γ(y)I n (y)}dy, ∀x ∈ [Λ(y) − S 0 (y)]dy uniformly on [0, 1].
.
Summary of analytical results. In this paper, we are concerned with the SIS epidemic model(1.3) with mass action infection mechanism and linear source. To study the parabolic problem (1.3), our first step is to establish the global existence and uniform boundedness of solutions. Then a basic reproduction number R 0 is defined via a variational characterization, which determines the local stability of the unique DFE. When the environment is spatially homogeneous and the diffusion rates of the susceptible and infected are equal, by constructing suitable Lyapunov functionals, we further prove the global attractivity of the DFE for R 0 ≤ 1 and that of the EE forR 0 > 1. We are mainly interested in the asymptotic behavior of positive steady states (S, I) of problem (1.3), which exist provided R 0 > 1 in general heterogeneous environment, as the diffusion rates of the susceptible or the infected tends to zero. For fixed d I > 0, Theorem 3.1 shows that the limiting functions of both S and I as d S → 0, are positive throughout the habitat. In the one dimensional interval, say [0, 1], for fixed d S > 0, Theorem 3.2 indicates that the limiting function of S as d I → 0 is positive in [0, 1] while the total infected population tends to a positive constant. Since there are four principle models (1.2), (1.3), (1.
.
Results on (1.4). The steady state problem corresponding to (1.S ∆S = −β(x) SI S+I + γ(x)I, x ∈ Ω, −d I ∆I = β(x) SI S+I − γ(x)I, x ∈ Ω, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, Ω [S(x) + I(x)]dx = N.
then I → 0 uniformly on Ω, and S → N [1−A(∞;x)] Ω [1−A(∞;x)] uniformly on any compact subset of H − and H + respectively, where A(∞; x) = 0, if x ∈ H − ; 1, if x ∈ H + .Clearly the limiting function of I when d I → 0 and d → d 0 ∈ [0, ∞) is positive on H + while zero on H − . In particular, if d I → 0 and d S > 0 is fixed, we are in the first scenario above. Thus, for model (1.
•
As d S → 0, both limiting functions of S and I are inhomogeneous and positive on the entire habitat Ω; • As d I → 0, the limiting function of S is positive on the entire habitat Ω and that of I is positive only on high-risk sites. 4.1.3. Result on (1.
I
then (S, I) → (S * , 0) uniformly on Ω, where S * is a positive function. Under the assumption that Ω + = x ∈ Ω : N |Ω| β(x) − γ(x) > 0 is nonempty, Wu and Zou [57] further proved the following: • If d I → 0 and d I /d S → d ∈ (0, ∞), then (S, I) → (S * * , I * * ) uniformly on Ω and I * * is the unique nonnegative solution of * * − dI * * . Therefore, the distribution of I * * depends critically on the magnitude of d. In fact, if d ∈ (0, 1), then {x ∈ Ω : I * * (x) > 0} is a proper subset of Ω + ; if d ∈ (1, ∞), then Ω + is a subset of {x ∈ Ω : I * * (x) > 0}; if d = 1, then
•
Any EE (S, I) satisfies S →Ŝ uniformly on [0, 1] with a positive functionŜ and 1 0 Idx converges to a positive constant. Biologically, this implies that the infectious disease still persists when the movement of the infected population is small. 4.2. Discussion and conclusions. 4.2.1. Comparison of the basic reproduction number. For readability, hereafter we call models (
. 2 )
2Mass-action incidence without birth-deathR 0 = N |Ω| sup 0 =ϕ∈H 1 (Ω)Ω βϕ 2 Ω (d I |∇ϕ| 2 +γϕ 2 ) MW=(1.3) Mass-action incidence with birth-death R 0 = sup 0 =ϕ∈H 1 (Ω) Ω βSϕ 2 Ω (d I |∇ϕ| 2 +(γ+µ)ϕ 2 ) SO = (1.4) Standard incidence without birth-deathR 0 = sup 0 =ϕ∈H 1 (Ω) Ω βϕ 2 Ω (d I |∇ϕ| 2 +γϕ 2 )SW=(1.5) Standard incidence with birth-deathR 0 = sup 0 =ϕ∈H 1 (Ω) Ω βϕ 2 Ω (d I |∇ϕ| 2 +γϕ 2 )
Fig. 2 ,
2for any d S > 0 and d I > 0, of (S(x), I(x)) as d S → 0 Limit of (S(x), I(x)) as d I → 0 MO S * (x) > 0 and I * (x) ≡ 0 (or > 0) for small (or large) N S * (x) > 0 and Ω I * (x) > 0 MW S * (x) > 0 and I * (x) > 0 S * (x) > 0 and Ω I * (x) > 0 SO S * (x) ≥ 0 and I * (x) ≡ 0 S * (x) > 0 and I * (x) ≡ 0 iff x ∈ H − SW S * (x) > 0 and I * (x) > 0 S * (x) > 0 and I * (x) ≡ 0 iff x ∈ H −
Figure 1 .Figure 2 .
12(a) Graphs of β(x) = 1.5 + sin(2πx), γ(x) = 1.2 + cos(2πx) for x ∈ [0, 1], and the set H − and H + (reproduction of Fig.1 in [4]); (b) Graphs of β(x) and γ(x) given by (4.4) with a moderate-risk region in [Numerical simulations of the profile of (S(x), I(x)) as d S → 0 for systems MO, MW, SO and SW, where parameters are chosen as: d S = 10 −6 , d I = 1, Λ(x) = 3, µ(x) = 0.5 + x and β(x) and γ(x) are as plotted in
Figure 3 .
3Numerical simulations of the profile of (S(x), I(x)) as d I → 0 for systems MO, MW, SO and SW, where d S = 1, d I = 10 −5 and other parameters are chosen same as those in Fig.2.
Figure 4 .Figure 5 .
45Numerical simulations of the profile of (S(x), I(x)) as d S , d I → 0 for systems MO, MW, SO and SW, where d S = d I = 10 −5 and other parameters are chosen same as those in Fig.Numerical simulations of the asymptotic profile of I(x) as d I → 0 for systems MO, MW, SO and SW with a moderate-risk site, where d S = 1, d I = 10 −5 , Λ(x) = 3, µ(x) = 0.5 + x, β(x) and γ(x) are given by (4.4) as plotted inFig.1(b).
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| []
|
[
"Visual Referring Expression Recognition: What Do Systems Actually Learn?",
"Visual Referring Expression Recognition: What Do Systems Actually Learn?"
]
| [
"Volkan Cirik [email protected] \nCarnegie Mellon University\n\n",
"Louis-Philippe Morency [email protected] \nCarnegie Mellon University\n\n",
"Taylor Berg-Kirkpatrick \nCarnegie Mellon University\n\n"
]
| [
"Carnegie Mellon University\n",
"Carnegie Mellon University\n",
"Carnegie Mellon University\n"
]
| [
"Proceedings of NAACL-HLT 2018"
]
| We present an empirical analysis of state-ofthe-art systems for referring expression recognition -the task of identifying the object in an image referred to by a natural language expression -with the goal of gaining insight into how these systems reason about language and vision. Surprisingly, we find strong evidence that even sophisticated and linguistically-motivated models for this task may ignore linguistic structure, instead relying on shallow correlations introduced by unintended biases in the data selection and annotation process. For example, we show that a system trained and tested on the input image without the input referring expression can achieve a precision of 71.2% in top-2 predictions. Furthermore, a system that predicts only the object category given the input can achieve a precision of 84.2% in top-2 predictions. These surprisingly positive results for what should be deficient prediction scenarios suggest that careful analysis of what our models are learning -and further, how our data is constructed -is critical as we seek to make substantive progress on grounded language tasks. | 10.18653/v1/n18-2123 | [
"https://www.aclweb.org/anthology/N18-2123.pdf"
]
| 44,096,203 | 1805.11818 | 8e2d8cee2fc474d0c08bfdfa707993d005067823 |
Visual Referring Expression Recognition: What Do Systems Actually Learn?
Association for Computational LinguisticsCopyright Association for Computational LinguisticsJune 1 -6. 2018. 2018
Volkan Cirik [email protected]
Carnegie Mellon University
Louis-Philippe Morency [email protected]
Carnegie Mellon University
Taylor Berg-Kirkpatrick
Carnegie Mellon University
Visual Referring Expression Recognition: What Do Systems Actually Learn?
Proceedings of NAACL-HLT 2018
NAACL-HLT 2018New Orleans, LouisianaAssociation for Computational LinguisticsJune 1 -6. 2018. 2018
We present an empirical analysis of state-ofthe-art systems for referring expression recognition -the task of identifying the object in an image referred to by a natural language expression -with the goal of gaining insight into how these systems reason about language and vision. Surprisingly, we find strong evidence that even sophisticated and linguistically-motivated models for this task may ignore linguistic structure, instead relying on shallow correlations introduced by unintended biases in the data selection and annotation process. For example, we show that a system trained and tested on the input image without the input referring expression can achieve a precision of 71.2% in top-2 predictions. Furthermore, a system that predicts only the object category given the input can achieve a precision of 84.2% in top-2 predictions. These surprisingly positive results for what should be deficient prediction scenarios suggest that careful analysis of what our models are learning -and further, how our data is constructed -is critical as we seek to make substantive progress on grounded language tasks.
Introduction
There has been increasing interest in modeling natural language in the context of a visual grounding. Several benchmark datasets have recently been introduced for describing a visual scene with natural language (Chen et al., 2015), describing or localizing specific objects in a scene (Kazemzadeh et al., 2014;Mao et al., 2016), answering natural language questions about the scenes (Antol et al., 2015), and performing visually grounded dialogue (Das et al., 2016). Here, we focus on referring expression recognition (RER) -the task of identifying the object in an image that is referred to by a natural language expression produced by a human (Kazemzadeh et al., 2014;Mao et al., 2016;Yu et al., 2016;Nagaraja et al., 2016;Hu et al., 2017).
Recent work on RER has sought to make progress by introducing models that are better capable of reasoning about linguistic structure (Hu et al., 2017;Nagaraja et al., 2016) -however, since most of the state-of-the-arts systems involve complex neural parameterizations, what these models actually learn has been difficult to interpret. This is concerning because several post-hoc analyses of related tasks (Zhou et al., 2015;Devlin et al., 2015;Agrawal et al., 2016;Jabri et al., 2016;Goyal et al., 2016) have revealed that some positive results are actually driven by superficial biases in datasets or shallow correlations without deeper visual or linguistic understanding. Evidently, it is hard to be completely sure if a model is performing well for the right reasons.
To increase our understanding of how RER systems function, we present several analyses inspired by approaches that probe systems with perturbed inputs (Jia and Liang, 2017) and employ simple models to exploit and reveal biases in datasets (Chen et al., 2016a). First, we investigate whether systems that were designed to incorporate linguistic structure actually require it and make use of it. To test this, we perform perturbation experiments on the input referring expressions. Surprisingly, we find that models are robust to shuffling the word order and limiting the word categories to nouns and adjectives. Second, we attempt to reveal shallower correlations that systems might instead be leveraging to do well on this task. We build two simple systems called Neural Sieves: one that completely ignores the input referring expression and another that only predicts the category of the referred object from the input expression. Again, surprisingly, both sieves are able to identify the correct object with surprising precision in top-2 and top-3 predictions. When these two simple systems are com-bined, the resulting system achieves precisions of 84.2% and 95.3% for top-2 and top-3 predictions, respectively. These results suggest that to make meaningful progress on grounded language tasks, we need to pay careful attention to what and how our models are learning, and whether our datasets contain exploitable bias.
Related Work
Referring expression recognition and generation is a well studied problem in intelligent user interfaces (Chai et al., 2004), human-robot interaction (Fang et al., 2012;Chai et al., 2014;Williams et al., 2016), and situated dialogue (Kennington and Schlangen, 2017). Kazemzadeh et al. (2014) and Mao et al. (2016) introduce two benchmark datasets for referring expression recognition. Several models that leverage linguistic structure have been proposed. Nagaraja et al. (2016) propose a model where target and supporting objects (i.e. objects that are mentioned in order to disambiguate the target object) are identified and scored jointly.
The resulting model is able to localize supporting objects without direct supervision. Hu et al. (2017) introduce a compositional approach for the RER task. They assume that the referring expression can be decomposed into a triplet consisting of the target object, the supporting object, and their spatial relationship. This structured model achieves state-of-the-art accuracy on the Google-Ref dataset. Cirik et al. (2018) propose a type of neural modular network (Andreas et al., 2016) where the computation graph is defined in terms of a constituency parse of the input referring expression.
Previous studies on other tasks have found that state-of-the-art systems may be successful for reasons different than originally assumed. For example, Chen et al. (2016b) show that a simple logistic regression baseline with carefully defined features can achieve competitive results for reading comprehension on CNN/Daily Mail datasets (Hermann et al., 2015), indicating that more sophisticated models may be learning realtively simple correlations. Similarly, Gururangan et al. (2018) reveal bias in a dataset for semantic inference by demonstrating a simple model that achieves competitive results without looking at the premise.
Analysis by Perturbation
In this section, we would like to analyze how the state-of-the-art referring expression recognition systems utilize linguistic structure. We conduct experiments with perturbed referring expressions where various aspects of linguistic structure are obscured. We perform three types of analyses: the first one studying syntactic structure (Section 3.2), the second focusing on the importance of word categories (Section 3.3), and the final one analyzing potential biases in the dataset (Section 3.4).
Analysis Methodology
To perform our analysis, we take two state-of-theart systems CNN+LSTM-MIL (Nagaraja et al., 2016) and CMN (Hu et al., 2017) and train them from scratch with perturbed referring expressions. We note that the perturbation experiments explained in next subsections are performed on all train and test instances. All experiments are done on the standard train/test splits for the Google-Ref dataset (Mao et al., 2016). Systems are evaluated using the precision@k metric, the fraction of test instances for which the target object is contained in the model's top-k predictions. We provide further details of our experimental methodology in Section 4.1.
Syntactic Analysis by Permuting Word Order
In English, word order is important for correctly understanding the syntactic structure of a sentence. Both models we analyze use Recurrent Neural Networks (RNN) (Elman, 1990) with Long Short-Term Memory (LSTM) cells (Hochreiter and Schmidhuber, 1997). Previous studies have shown that reccurrent architectures can perform well on tasks where word order and syntax are important: for example, tagging (Lample et al., 2016), parsing (Sutskever et al., 2014), and machine translation (Bahdanau et al., 2014). We seek to determine whether recurrent models for RER depend on syntactic structure. Premise 1: Randomly permuting the word order of an English referring expression will obscure its syntactic structure.
We train CMN and CNN+LSTM-MIL with shuffled referring expressions as input and evaluate their performance. Expressions. ∆ is the difference between no perturbation and shuffled version of the same system. Table 1 shows accuracies for models with and without shuffled referring expressions. The column with ∆ shows the difference in accuracy compared to the best performing model without shuffling. The drop in accuracy is surprisingly low. Thus, we conclude that these models do not stongly depend on the syntactic structure of the input expression and may instead leverage other, shallower, correlations.
Lexical Analysis by Discarding Words
Following the analysis presented in Section 3.2, we are curious to study what other aspects of the input referring expression may be essential for state-ofthe-art performance. If syntactic structure is largely unimportant, it may be that spatial relationships can be ignored. Spatial relationships between objects are usually represented by prepositional phrases and verb phrases. In contrast, simple descriptors (e.g. green) and object types (e.g. table) are most often represented by adjectives and nouns, respectively. By discarding all words in the input that are not nouns or adjectives, we hope to test whether spatial relationships are actually important to stateof-the-art models. Notably, both systems we test were specifically designed to model object relationships. Premise 2: Keeping only nouns and adjectives from the input expression will obscure the relationships between objects that the referring expression describes. Table 2 shows accuracies resulting from training and testing these models on only the nouns and adjectives in the input expression. Our first observation is that the accuracies of models drop the most when we discard the nouns (the rightmost column in Table 2). This is reasonable since nouns define the types of the objects referred to in the expression. Without nouns, it is extremely difficult to identify which objects are being described. Second, although both systems we analyze model the relationship between objects, discarding verbs and prepositions, which are essential in determining the relationship among objects, does not drastically effect their performance (the second column in Table 2). This may indicate the superior performance of these systems does not specifically come from their modeling approach for object relationships.
Bias Analysis by Discarding Referring Expressions
Goyal et al. (2016) show that some language and vision datasets have exploitable biases. Could there be a dataset bias that is exploited by the models for RER? Premise 3: Discarding the referring expression entirely and keeping only the input image creates a deficient prediction problem: achieving highpeformance on this task indicates dataset bias. We train CMN by removing all referring expressions from train and test. We call this model "image-only" since it ignores the referring expresion and will only use the input image. We compare the CMN "image-only" model with the state-of-theart configuration of CMN and a random baseline. only" model is able to surpass the random baseline by a large margin. This result indicates that the dataset is biased, likely as a result of the data selection and annotation process. During the construction of the dataset, Mao et al. (2016) annotate an object box only if there are at least 2 to 4 objects of the same type in the image. Thus, only a subset of object categories ever appear as targets because some object types rarely occur multiple times in an image. In fact, out of 90 object categories in MSCOCO, 43 of the object categories are selected as target objects less than 1% of the time they occur in images. This potentially explains the relative high performance of the "image-only" system.
Discussion
The previous analyses indicate that exploiting bias in the data selection process and leveraging shallow linguistic correlations with the input expression may go a long way towards achieving high performance on this dataset. First, it may be possible to simplify the decision of picking an object to a much smaller set of candidates without even considering the referring expression. Second, because removing all words except for nouns and adjectives only marginally hurt performance for the systems tested, it may be possible to further reduce the set of candidates by focusing only on simple properties like the category of the target object rather than its relations with the environment or with adjacent objects.
Neural Sieves
We introduce a simple pipeline of neural networks, Neural Sieves, that attempt to reduce the set of candidate objects down to a much smaller set that still contains the target object given an image, a set of objects, and the referring expression describing one of the objects.
Sieve I: Filtering Unlikely Objects. Inspired by the results from Section 3.4, we design an "imageonly" model as the first sieve for filtering unlikely objects. For example in Figure 1, Sieve I filters out the backpack and the bench from the list of bounding boxes since there is only one instance of these object types. We use a similar parameterization of one of the baselines (CMN LOC ) proposed by Hu et al. (2017) for Sieve I and train it by only providing spatial and visual features for the boxes, ignoring the referring expression. More specifically, for visual features r vis of a bounding boxes of an object, we use Faster- RCNN (Ren et al., 2015). We use 5-dimensional vectors for spatial features
r spat = [ x min W V , y min H V , xmax W V , ymax W V , Ar A V ]
where A r is the size and [x min , y min , x max , y max ] are coordinates for bounding box r and A V , W V , H V are the area, the width, and the height of the input image V . These two representations are concatenated as r vis,spat = [r vis r spat ] for a bounding box r.
We parameterize Sieve I with a list of bounding boxes R as the input with parameter set Θ I as follows:
s I = W score I r vis,spat
(1)
f I (R; Θ I ) = sof tmax(s I )(2)
Each bounding box is scored using a matrix W score I . Scores for all bounding boxes are then fed to softmax to get a probability distribution over boxes. The learned parameter Θ I is the scoring matrix W score I .
Sieve II: Filtering Based on Objects Categories
After filtering unlikely objects based only on the image, the second step is to determine which object category to keep as a candidate for prediction, filtering out the other categories. For instance, in Figure 1, only instances of suitcases are left as candidates after determining which type of object the input expression is talking about. To perform this step, Sieve II takes the list of object candidates from Sieve I and keeps objects having the same object category as the referred object. Unlike Sieve I, Sieve II uses the referring expression to filter bounding boxes of objects. We again use the baseline model of CMN LOC from the previous work (Hu et al., 2017) for the parametrization of Sieve II with a minor modification: instead of predicting the referred object, we make a binary decision for each box of whether the object in the box is the same category as the target object.
More specifically, we parameterize Sieve II as follows:r vis,spat = W vis,spat II r vis,spat
z II =r vis,spat f att (T ) (4)
z II = z II / || z II || 2 (5) s II = W score IIẑ s2 (6) f II (T, R; Θ II ) = sigmoid(s II )(7)
We encode the referring expression T into an embedding with f att (T ) which uses an attention mechanism (Bahdanau et al., 2014) on top of a 2-layer bidirectional LSTM (Schuster and Paliwal, 1997). We project bounding box features r vis,spat to the same dimension as the embedding of referring expression (Eq 3). Text and box representations are element-wise multiplied to get z II as a joint representation of the text and bounding box (Eq 4). We L2-normalize to produceẑ II (Eq 5, 6). Box scores Model , and parameters of the encoding module f att .
Filtering Experiments
We are interested in determining how accurate these simple nueral sieves can be. High accuracy here would give a possible explanation for the high performance of more complex models.
Dataset. For our experiments, we use Google-Ref (Mao et al., 2016) which is one of the standard benchmarks for referring expression recognition. It consists of around 26K images with 104K annotations. We use their Ground-Truth evaluation setup where the ground truth bounding box annotations from MSCOCO (Lin et al., 2014) are provided to the system as a part of the input. We used the split provided by Nagaraja et al. (2016) where splits have disjoint sets of images. We use precision@k for evaluating the performance of models. Implementation Details. To train our models, we used stochastic gradient descent for 6 epochs with an initial learning rate of 0.01 and multiplied by 0.4 after each epoch. Word embeddings were initialized using GloVe (Pennington et al., 2014) and finetuned during training. We extracted features for bounding boxes using the fc7 layer output of Faster-RCNN VGG-16 network (Ren et al., 2015) pre-trained on MSCOCO dataset (Lin et al., 2014). Hyperparameters such as hidden layer size of LSTM networks were picked based on the best validation score. For perturbation experiments, we did not perform any grid search for hyperparameters. We used hyperparameters of the previously reported best performing model in the literature. We released our code for public use 1 .
Baseline Models. We compare Neural Sieves to the state-of-the-art models from the literature. LSTM + CNN - MIL Nagaraja et al. (2016) score target object-context object pairs using LSTMs for processing the referring expression and CNN features for bounding boxes. The pair with the highest score is predicted as the referred object. They use Multi-Instance Learning for training the model. CMN (Hu et al., 2017) is a neural module network with a tuple of object-relationship-subject nodes. The text encoding of tuples is calculated with a two-layer bi-directional LSTM and an attention mechanism (Bahdanau et al., 2014) over the referring expression. Table 4 shows the precision scores. The referred object is in the top-2 candidates selected by Sieve I 71.2% of the time and in the top-3 predictions 86.6% of the time. Combining both sieves into a pipeline, these numbers further increase to 84.2% for top-2 predictions and to 95.3% for top-3 predictions. Considering the simplicity of Neural Sieve approach, these are surprising results: two simple neural network systems, the first one ignoring the referring expression, the second predicting only object type, are able to reduce the number of candidate boxes down to 2 on 84.2% of instances.
Results
Conclusion
We have analyzed two RER systems by variously perturbing aspects of the input referring expressions: shuffling, removing word categories, and finally, by removing the referring expression entirely. Based on this analysis, we proposed a pipeline of simple neural sieves that captures many of the easy correlations in the standard dataset. Our results suggest that careful analysis is important both while constructing new datasets and while constructing new models for grounded language tasks. The techniques used here may be applied more generally to other tasks to give better insight into what our models are learning and whether our datasets contain exploitable bias.
Figure 1 :
1Overview of Neural Sieves. Sieve I filters object types having multiple instances. Sieve II filters objects of one category mentioned in referring expression. Objects of the same category have the same color frames. Best seen in color.
687 (-.018) .642 (-.063) .585 (-.120) LSTM+CNN-MIL .644 (-.040) .597 (-.087) .533 (-.151)
Table 1: Results for Shuffling Word Order for ReferringModel
No Perturbation Shuffled
∆
CMN
.705
.675 -.030
LSTM+CNN-MIL
.684
.630 -.054
Table 2 :
2Results with discarded word categories. Numbers in parentheses are ∆, the difference between the best performing version of the original model.
Table 3
3shows precision@k results. The "image-Model
P@1 P@2 P@3 P@4 P@5
CMN
.705
.926
.979
.993
.998
CMN "image-only" .411
.731
.885
.948
.977
Random Baseline
.204
.403
.557
.669
.750
Table 3 :
3Results with discarded referring expressions. Sur-prisingly, the top-2 prediction (73.1%) of the "image-only"
model is better than the top prediction of the state-of-the-art
(70.5%).
Table 4 :
4Precision@k accuracies for Neural Sieves and state-
of-the-art systems. Note that even without using the referring
expression, Sieve I is able to reduce the number of candidate
boxes to 3 for 86.6% of the instances. When we further predict
the type of objects with Sieve II, the number of candidate
boxes is reduced to 2 for 84.2% of the instances.
s II are calculated with a linear projection of the
joint representation (Eq 6) and fed to the sigmoid
function for a binary prediction for each box. The
learned parameters Θ II are W vis,spat
II
,W score
II
https://github.com/volkancirik/neural-sieves-refexp
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A thorough examination of the cnn/daily mail reading comprehension task. Danqi Chen, Jason Bolton, Christopher D Manning, Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics. the 54th Annual Meeting of the Association for Computational LinguisticsBerlin, GermanyAssociation for Computational Linguistics1Long Papers)Danqi Chen, Jason Bolton, and Christopher D. Man- ning. 2016b. A thorough examination of the cnn/daily mail reading comprehension task. In Pro- ceedings of the 54th Annual Meeting of the Associa- tion for Computational Linguistics (Volume 1: Long Papers). Association for Computational Linguistics, Berlin, Germany, pages 2358-2367. http://www. aclweb.org/anthology/P16-1223.
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Making the v in vqa matter: Elevating the role of image understanding in visual question answering. Yash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, Devi Parikh, arXiv:1612.00837arXiv preprintYash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. 2016. Making the v in vqa matter: Elevating the role of image un- derstanding in visual question answering. arXiv preprint arXiv:1612.00837 .
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Modeling context in referring expressions. Licheng Yu, Patrick Poirson, Shan Yang, Alexander C Berg, Tamara L Berg, European Conference on Computer Vision. SpringerLicheng Yu, Patrick Poirson, Shan Yang, Alexander C Berg, and Tamara L Berg. 2016. Modeling context in referring expressions. In European Conference on Computer Vision. Springer, pages 69-85.
Simple baseline for visual question answering. Bolei Zhou, Yuandong Tian, Sainbayar Sukhbaatar, Arthur Szlam, Rob Fergus, arXiv:1512.02167arXiv preprintBolei Zhou, Yuandong Tian, Sainbayar Sukhbaatar, Arthur Szlam, and Rob Fergus. 2015. Simple base- line for visual question answering. arXiv preprint arXiv:1512.02167 .
| [
"https://github.com/volkancirik/neural-sieves-refexp"
]
|
[
"The relevance of random choice in tests of Bell inequalities with two atomic qubits",
"The relevance of random choice in tests of Bell inequalities with two atomic qubits"
]
| [
"Emilio Santos \nDepartamento de Física\nUniversidad de Cantabria. Santander\nSpain\n"
]
| [
"Departamento de Física\nUniversidad de Cantabria. Santander\nSpain"
]
| []
| It is pointed out that a loophole exists in experimental tests of Bell inequality using atomic qubits, due to possible errors in the rotation angles of the atomic states. A sufficient condition is derived for closing the loophole. | 10.1103/physreva.79.044104 | [
"https://arxiv.org/pdf/0901.0812v1.pdf"
]
| 14,148,549 | 0901.0812 | 220a6dbb38bbf78e9cd27464c0593eeea0d0c657 |
The relevance of random choice in tests of Bell inequalities with two atomic qubits
7 Jan 2009 November 4, 2018
Emilio Santos
Departamento de Física
Universidad de Cantabria. Santander
Spain
The relevance of random choice in tests of Bell inequalities with two atomic qubits
7 Jan 2009 November 4, 2018
It is pointed out that a loophole exists in experimental tests of Bell inequality using atomic qubits, due to possible errors in the rotation angles of the atomic states. A sufficient condition is derived for closing the loophole.
A recent experiment, by a group of Maryland, has measured the correlation between the quantum states of two Yb + ions separated by a distance of about 1 meter [1]. The authors claim that the experiment is relevant because it violates a CHSH [2] (Bell) inequality, modulo the locality loophole, closing the detection loophole. In my opinion that assertion does not make full justice to the relevance of the experiment. The truth is that it is the first experiment which has tested a genuine Bell inequality. Actually the results of previous experiments, in particular those involving optical photon pairs [3], did not test any genuine Bell inequality, that is an inequality which is a necessary condition for the existence of local hidden variables (LHV) models. The inequalities tested in those experiments should not be qualified as Bell´s because their derivation involves additional assumptions. Consequently their violation refutes only restricted families of LHV models, namely those fulfilling the additional assumption. ( For details see [4].)
The aim of the present letter is to point out the existence of a loophole in the Maryland experiment [1], or more generally in Bell tests with atomic qubits, in addition to the locality loophole. Blocking that loophole will be straightforward using random choice of the measurements, as is explained below.
In general I will consider experiments where a pair of atoms (or ions) is prepared in an entangled state. Then Alice performs a rotation of the state of her atom by an angle θ a and, after a short time, she may detect fluorescence of the atom illuminated by an appropriate laser. Similarly Bob performs a rotation of his atom by an angle θ b and, after that, he may detect fluorescence too. I shall label p ++ (θ a , θ b ) the probability of coincidence detection and p −− (θ a , θ b ) the probability that neither Alice nor Bob detect fluorescence. Similarly p −+ (θ a , θ b ) ( p +− (θ a , θ b )) will be the probability that only Bob (Alice) detects fluorescence. In the Maryland experiment [1] (see their eq.(6)), a function E (θ a , θ b ) is defined by
E (θ a , θ b ) = p ++ (θ a , θ b ) + p −− (θ a , θ b ) − p +− (θ a , θ b ) − p −+ (θ a , θ b ) .(1)
Then the authors define a parameter S by
S = |E (θ a , θ b ) + E (θ ′ a , θ b )| + |E (θ a , θ ′ b ) − E (θ ′ a , θ ′ b )| ,(2)
and claim that the CHSH [2] inequality S ≤ 2 is violated. The notation used by the authors is, however, somewhat misleading. Instead of eq.(1) they write
E (θ a , θ b ) = p (θ a , θ b ) + p (θ a + π, θ b + π) − p (θ a , θ b + π) − p (θ a + π, θ b ) ,(3)
where they label p (θ a , θ b ) the quantity which I have labeled p ++ (θ a , θ b ) . Definition eq.(3) , in place of eq.(1) , rests upon assuming the equalities
p −+ (θ a , θ b ) = p (θ a + π, θ b ) , p +− (θ a , θ b ) = p (θ a + π, θ b ) , p −− (θ a , θ b ) = p (θ a + π, θ b + π) ,
which are true according to quantum mechanics, but may not be true in LHV theories. In any case the authors measured E (θ a , θ b ) as defined in eq.(1) [5].
In order to show that there is a loophole in the experiment, in addition to the locality loophole, I begin remembering that, according to Bell [6], a LHV model will contain a set of hidden variables, λ, a positive normalized density function, ρ (λ) , and two functions M a (λ, θ a ) , M b (λ, θ b ), θ a and θ b being parameters which may be controlled by Alice and Bob respectively. The latter functions fulfil
M a (λ, θ a ) , M b (λ, θ b ) ∈ {0, 1}.(4)
In the Maryland experiment [1] the parameters θ a and θ b are angles defining the quantum states of the two ions. The probability, p ++ (θ a , θ b ) , that the coincidence measurement of two dichotomic variables, in two distant regions, gives a positive answer for both variables should be obtained in the LHV model by means of the integral
p ++ (θ a , θ b ) = ρ (λ) M a (λ, θ a ) M b (λ, θ b ) dλ.(5)
Similarly the probability, p +− (θ a , θ b ) , that Alice gets the answer "yes" and Bob the answer "no" is given by
p +− (θ a , θ b ) = ρ (λ) M a (λ, θ a ) [1 − M b (λ, θ b )] dλ,(6)
and analogous expressions for p −+ and p −− . A LHV model for an atomic experiment may be obtained by choosing
ρ (λ) = 1 2π , λ ∈ [0, 2π] , M a (λ, θ a ) = Θ π 2 − |λ − θ a | , M b (λ, θ b ) = Θ π 2 − |λ − θ b − π| , mod (2π) ,(7)
where Θ (x) = 1 if x > 0, Θ (x) = 0 if x < 0. It is easy to see, taking eqs. (5) and (6) into account, that model predictions are (assuming θ a , θ b ∈ [0, π])
p ++ (θ a , θ b ) = p −− (θ a , θ b ) = |θ a − θ b | 2π , p +− (θ a , θ b ) = p −+ (θ a , θ b ) = 1 2 − |θ a − θ b | 2π .(8)
Hence I get
E (θ a , θ b ) = 2 π |θ a − θ b | − 1,(9)
and it is not difficult to show that, for any choice of the angles θ a , θ b , θ ′ a , θ ′ b , the model predicts S ≤ 2 with S given by eq. (2) . Now let us assume that the experiment is performed so that Alice and Bob start measuring the quantity E (θ a , θ b ) in a sequence of runs of the experiment. After that they measure E (θ a , θ ′ b ) in another sequence, then they measure E (θ ′ a , θ b ) and, finally, they measure E (θ ′ a , θ ′ b ) . Let α be the error in the rotation performed by Bob on his atom in the first sequence of runs, so that the rotation angle is θ b + α rather than θ b in the measurement of E (θ a , θ b ) . Similarly I shall assume that the rotation angles are θ
′ b + β, θ b + γ and θ ′ b + δ in the measurements of E (θ a , θ ′ b ), E (θ ′ a , θ b ) and E (θ ′ a , θ ′ b )
, respectively. For simplicity I will assume that no error appears in Alice rotations. The errors are considered small, specifically |α| , |β| , |γ| , |δ| < π/4. I shall prove that, taking into account the errors in the measurement of the angles, the LHV model prediction for the parameter S, eq.(2) may apparently violate the CHSH [2] inequality S ≤ 2. To do that let us choose, as in the Maryland experiment [1],
θ a = π 2 , θ b = π 4 , θ ′ a = 0, θ ′ b = 3π 4 .(10)
The values predicted by the LHV model for the relevant quantities are
E (θ a , θ b + α) = −0.5 − 2α π , E (θ a , θ ′ b + β) = −0.5 + 2β π , E (θ ′ a , θ b + γ) = −0.5 + 2γ π , E (θ ′ a , θ ′ b + δ) = 0.5 + 2δ π .(11)
Then the parameter actually measured in the experiment is
S ′ = |E (θ a , θ b + α) + E (θ ′ a , θ b + γ)|+ |E (θ a , θ ′ b + β) − E (θ ′ a , θ ′ b + δ)| ,(12)
and the LHV prediction for that parameter is
S ′ == 2 + 2 π (α − β − γ + δ) ,
which may violate the inequality S ′ ≤ 2 for some values of the parameters α, β, γ and δ. In particular the results of the Maryland experiment [1] are reproduced by choosing 2α/π = 0.018, 2β/π = −0.046, 2γ/π = −0.081, 2δ/π = −0.073.
The errors in the angles are of order 7 o or less. It is plausible that errors as high as these may appear in experiments with atomic qubits but not in optical photon experiments. I stress that no violation of a Bell inequality by a LHV model is produced. Actually the parameter S ′ of eq.(12) is not a CHSH parameter as defined in eq.(2) . In the following I shall prove that the loophole may be closed by random choice of the angles to be measured. To begin with, it is easy to see that the LHV model predictions do not violate the inequality S ′ ≤ 2 if the error in the measurement, by Bob, of the angle θ b is the same in all measurements of that angle, and similarly for θ ′ b . In fact the inequality is fulfilled if α = β and γ = δ, as may be seen by looking at eq.(12) . In the following I derive a sufficient condition for the fulfillement of the inequality, S ′ ≤ 2, for the actually measurable quantity S ′ , by the predictions of any LHV model.
Let us assume that there is a (normalized) probability distribution, f a (x), for the errors when Alice rotates her atom by an angle θ a and another distribution, f ′ a (y), when she rotates her atom by an angle θ ′ a . Similarly I shall assume that there are similar disitribuions f b (u) and f ′ b (v) for the errors in the rotations, by Bob, of the angles θ b and θ ′ b . I shall show that a sufficient condition for the inequality S ′ ≤ 2 is that the distributions of errors, in the rotations made by Alice, are the same independently of what rotation performs Bob on the partner atom. And similarly for the rotations made by Bob. If this is the case the predictions of any LHV model for the quantity S ′ will be obtained from probabilities defined as follows (compare with eqs. (5) and (6))
p ++ (θ a , θ b ) = ρ (λ) M a (λ, θ a + x) M b (λ, θ b + u) dλf a (x)dxf b (u)du, (13) p +− (θ a , θ b ) = ρ (λ) M a (λ, θ a + x) [1 − M b (λ, θ b + u)] dλf a (x)dxf b (u)du,
and similarly for the other quantities p ij with i, j = +, −. Now we may define new quantities
Q a (λ, a) = M a (λ, θ a + x) f a (x)dx,(14)Q b (λ, b) = M b (λ, θ b + u) f b (u)du, Q a (λ, a ′ ) = M a (λ, θ ′ a + y) f ′ a (y)dy, Q b (λ, b ′ ) = M b (λ, θ ′ b + v) f ′ b (v)dv,
which fulfil the conditions (compare with eqs.(4))
0 ≤ Q a (λ, a) , Q a (λ, a ′ ) , Q b (λ, b) , Q b (λ, b ′ ) ≤ 1.(15)
The consequence is that we may obtain a new LHV model for the experiment with the quantities Q, eqs.(15) , in place of the quantities M, eqs. (4) . The existence of the model implies the fulfillement of the inequality S ′ ≤ 2. From our proof it is rather obvious that the essential condition required to block the loophole is that the probability distribution of errors made by Bob are independent of what rotation is performed by Alice in the partner atom, and similarly the errors made by Alice should be independent of the rotation performed by Bob. A simple method to insure that independence is that Alice makes at random the choice whether to rotate her atom by the angle θ a or by the angle θ ′ a , and similarly Bob. That is, after every preparation of the entangled state of the atom, Alice should make a random choice (with equal probabilities) between the rotation angles θ a and θ ′ a and similarly Bob should make a random choice, independently of Alice, between θ b and θ ′ b .
. D N Matsukevich, P Maunz, D L Moehring, S Olmschenk, C Monroe, Phys. Rev. Lett. 100150404D. N. Matsukevich, P. Maunz, D.L. Moehring, S. Olmschenk and C. Monroe, Phys. Rev. Lett. 100, 150404 (2008).
. J F Clauser, M A Horne, A Shimony, R A Holt, Phys. Rev. Lett. 23880J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
. M Genovese, Phys. Reports. 413319M. Genovese, Phys. Reports 413, 319 (2005).
. E Santos, Found. Phys. 341643E. Santos, Found. Phys. 34, 1643 (2004).
. D N Matsukevich, private communicationD. N. Matsukevich, private communication.
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| []
|
[
"Clustering Coefficients in Multiplex Networks",
"Clustering Coefficients in Multiplex Networks"
]
| [
"Emanuele Cozzo \nInstitute for Biocomputation and Physics of Complex Systems (BIFI)\nUniversity of Zaragoza\n50018ZaragozaSpain\n",
"Mikko Kivelä \nOxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBOxfordUK\n",
"Manlio De Domenico \nDepartament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain\n",
"Albert Solé \nDepartament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain\n",
"Alex Arenas \nDepartament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain\n",
"Sergio Gómez \nDepartament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain\n",
"Mason A Porter \nOxford Centre for Industrial and Applied Mathematics\nMathematical Institute and CABDyN Complexity Centre\nUniversity of Oxford\nOX1 3LBOxfordUK\n",
"Yamir Moreno \nInstitute for Biocomputation and Physics of Complex Systems (BIFI)\nUniversity of Zaragoza\n50018ZaragozaSpain\n\nDepartment of Theoretical Physics\nUniversity of Zaragoza\n50009ZaragozaSpain\n\nComplex Networks and Systems Lagrange Lab\nInstitute for Scientific Interchange\nTurinItaly\n"
]
| [
"Institute for Biocomputation and Physics of Complex Systems (BIFI)\nUniversity of Zaragoza\n50018ZaragozaSpain",
"Oxford Centre for Industrial and Applied Mathematics\nMathematical Institute\nUniversity of Oxford\nOX1 3LBOxfordUK",
"Departament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain",
"Departament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain",
"Departament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain",
"Departament d'Enginyeria Informática i Matemátiques\nUniversitat Rovira I Virgili\n43007TarragonaSpain",
"Oxford Centre for Industrial and Applied Mathematics\nMathematical Institute and CABDyN Complexity Centre\nUniversity of Oxford\nOX1 3LBOxfordUK",
"Institute for Biocomputation and Physics of Complex Systems (BIFI)\nUniversity of Zaragoza\n50018ZaragozaSpain",
"Department of Theoretical Physics\nUniversity of Zaragoza\n50009ZaragozaSpain",
"Complex Networks and Systems Lagrange Lab\nInstitute for Scientific Interchange\nTurinItaly"
]
| []
| Recent advances in the study of complex networked systems has highlighted that our interconnected world is made of networks that are coupled together through different layers that each stand for one type of interaction or system. Despite this situation, it is traditional to aggregate multiplex data into a single weighted network in order take advantage of existing tools. This is admittedly convenient, but it is also extremely problematic. In this paper, we generalize the concept of clustering coefficients for multiplex networks. We show how the layered structure of multiplex networks introduces a new degree of freedom that has a fundamental effect on transitivity. We compute our new multiplex clustering coefficients for several real multiplex networks and illustrate why generalizing monoplex concepts to multiplex networks must be done with great care. | 10.1088/1367-2630/17/7/073029 | [
"https://arxiv.org/pdf/1307.6780v2.pdf"
]
| 2,321,303 | 1307.6780 | d1ad7a452bf8932c1d4e286c534b22e8065e6e13 |
Clustering Coefficients in Multiplex Networks
25 Jul 2013
Emanuele Cozzo
Institute for Biocomputation and Physics of Complex Systems (BIFI)
University of Zaragoza
50018ZaragozaSpain
Mikko Kivelä
Oxford Centre for Industrial and Applied Mathematics
Mathematical Institute
University of Oxford
OX1 3LBOxfordUK
Manlio De Domenico
Departament d'Enginyeria Informática i Matemátiques
Universitat Rovira I Virgili
43007TarragonaSpain
Albert Solé
Departament d'Enginyeria Informática i Matemátiques
Universitat Rovira I Virgili
43007TarragonaSpain
Alex Arenas
Departament d'Enginyeria Informática i Matemátiques
Universitat Rovira I Virgili
43007TarragonaSpain
Sergio Gómez
Departament d'Enginyeria Informática i Matemátiques
Universitat Rovira I Virgili
43007TarragonaSpain
Mason A Porter
Oxford Centre for Industrial and Applied Mathematics
Mathematical Institute and CABDyN Complexity Centre
University of Oxford
OX1 3LBOxfordUK
Yamir Moreno
Institute for Biocomputation and Physics of Complex Systems (BIFI)
University of Zaragoza
50018ZaragozaSpain
Department of Theoretical Physics
University of Zaragoza
50009ZaragozaSpain
Complex Networks and Systems Lagrange Lab
Institute for Scientific Interchange
TurinItaly
Clustering Coefficients in Multiplex Networks
25 Jul 2013(Dated: May 11, 2014)arXiv:1307.6780v1 [physics.soc-ph]
Recent advances in the study of complex networked systems has highlighted that our interconnected world is made of networks that are coupled together through different layers that each stand for one type of interaction or system. Despite this situation, it is traditional to aggregate multiplex data into a single weighted network in order take advantage of existing tools. This is admittedly convenient, but it is also extremely problematic. In this paper, we generalize the concept of clustering coefficients for multiplex networks. We show how the layered structure of multiplex networks introduces a new degree of freedom that has a fundamental effect on transitivity. We compute our new multiplex clustering coefficients for several real multiplex networks and illustrate why generalizing monoplex concepts to multiplex networks must be done with great care.
The quantitative study of networks is fundamental for the study of complex systems throughout the biological, social, information, engineering, and physical sciences [1-3]. The broad applicability of networks, and their success in providing insights into the structure and function of both natural and designed systems, has generated considerable excitement across myriad scientific disciplines. Numerous tools have been developed to study networks, and the realization that several common features arise in a diverse variety of networks has facilitated the development of theoretical tools to study them. For example, many networks constructed from empirical data have heavy-tailed degree distributions, satisfy the small-world property, and/or possess modular structures; and such structural features can have important implications for information diffusion, robustness against component failure, and more Traditional studies of networks generally assume that nodes are connected to each other by a single type of static edge that encapsulates all connections between them. This assumption is almost always a gross oversimplification, and it can lead to misleading results and even the sheer inability to address certain problems. Most real and engineered networks are multiplex [1], as there are almost always multiple types of ties or interactions that can occur between nodes, and it is crucial to take them into account. For example, transportation systems include multiple modes of travel, biological systems include multiple signaling channels that operate in parallel, and social networks include multiple modes of communication.
The notion of multiplexity was introduced years ago in fields such as engineering [4,5] and sociology [1, 6,7], but such discussions included few analytical tools to accompany them. This situation arose for a simple reason: although many aspects of single-layer networks are well understood, it is very challenging to properly generalize even the simplest concepts for multiplex networks. Theoretical developments have gained steam only in the last few years [8][9][10][11][12][13][14][15][16][17], and even basic notions like centrality and diffusion have barely been studied in multiplex settings [18][19][20][21][22][23][24][25][26]. Moreover, new degrees of freedom arise from the multi-layer structured from multiplex networks, and this brings both new challenges [27] and new phenomena. This includes multiplexity-induced correlations [16], new dynamical feedbacks [23], and "costs" of inter-layer connections [28].
In this Letter, we develop multiplex generalizations of the clustering coefficient, which is one of the most important properties to calculate in monoplex networks. In constructing such a generalization, myriad definitions are possible, and the most appropriate one to use depends on the application under study. Such considerations are crucial when developing multiplex generalizations of any monoplex network diagnostic. Using the example of clustering coefficients, our Letter illustrates how the new degrees of freedom that result from the existence of an interlayer structure yield rich new phenomena and subtle differences in how one should compute key network diagnostics. We thereby demonstrate that in order to understand real complex systems, it is insufficient to generalize existing diagnostics in a naive manner and one must instead construct their generalizations from first principles.
Supra-Adjacency Matrices. We represent a multiplex network by a sequence of graphs
{G α } b α=1 , with G α = G α (V α , E α ), where α ∈ {1, .
. . , b} indexes the layers of the network. For simplicity, we will examine unweighted multiplex networks. We define the intra-layer supra-graph as
G(V, E), where the set of nodes is V = ⊔ α V α = α {(v, α) : v ∈ V α }, and the set of edges is E = α {((v, α), (u, α)) : (u, v) ∈ E α }.
We also define the coupling supra-graph G C (V, E C ) using the same sets of nodes, and edges as )) ∈ E C , we say that (v, α) and (v, β) are inter-connected. We say that a multiplex network is fully interconnected if all layers share the same set of nodes (i.e., if V α = V β for all α and β). The supra-graph isḠ(V,Ē), whereĒ = E ∪ E C . It is useful to define at this point what we call supra-nodes [45]. Supra-nodeũ is defined by the set l(ũ) = {(α, u), (β, u), . . . } of inter-connected nodes, i.e. nodes connected by edges in E C . Supra-nodes made up the aggregated multi-graph which results from the contraction of all edges in E C .
E C = α,β {((v, α), (v, β)) : v ∈ V α , v ∈ V β , α = β}. If ((v, α), (v, β
We let A, C, andĀ denote the corresponding adjacency matrices for G, G C , andḠ and we call them respectively intra-layer supra-adjacency matrix, coupling supra-adjacency matrix, and supra-adjacency matrix. It is easy to see that A = α A α , where A α is the adjacency matrix of G α and denotes the direct sum of the matrices. Thus, the supraadjacency matrices satisfy the propertyĀ = A + C. In this Letter, we will consider undirected networks, so A = A T . Additionally, C = C T follows from the definition of E C .
Intra-Layer (Monoplex) Clustering Coefficient. The local clustering coefficient c i of node i in an unweighted monoplex network is the number of triangles that include node i divided by the number of connected tuples for which the node is central [3,29]. The local clustering coefficient is a measure of transitivity [30], and it can be interpreted as the density of the focal node's neighborhood. For our purposes, it is convenient to define the clustering coefficient c i as the number of 3-cycles t i that start and end at the focal node i divided by the number of 3-cycles d i such that the second step occurs in a complete graph (i.e., assuming that the neighborhood of the focal node is as dense as possible).
Using the above definition, we calculate the local clustering coefficient by using the fact that the number of length-n walks that start at node i and end at node j is ( n t=1 A t ) ij , where the tth step occurs in the graph whose adjacency matrix is A t . We thus write t i = (A 3 ) ii and d i = (AF A) ii , where A is the adjacency matrix of the graph and F is the adjacency matrix of a complete network with no self-loops (i.e., F = J − I, where J is a complete square matrix of 1s and I is the identity matrix). The local clustering coefficient is then c i = t i /d i . This is equivalent to the usual definition of the clustering coefficient:
c i = t i /k i /(k i − 1)
, where k i is the degree of node i. One can obtain a single global clustering coefficient for a monoplex network either by averaging c i over all nodes or by computing c = i ti i di , the latter is what we will consider in the rest of the letter.
Cycles on Multiplex Networks. In addition to 3-cycles taking place inside a single layer, in multiplex networks, there are other cycles that can go through different additional layers but still have 3 intra-layer steps. For example, this is very important for both social networks and transportation networks: in the former, transitivity involves social ties across multiple media [1,31]; in the latter, there are typically several choices for how to return to one's starting location. All of these 3cycles can be important for dynamical processes on networks, so they need to be considered when defining a multiplex clus-tering coefficient. Let define a supra-walk as a walk on the supra graph in which, after or before each intra-layer step, a walk can either continue on the same layer or change to some adjacent layer. We represent this choice using the following matrix:
C = βI + γC ,(1)
where the parameter β is the "weight" of staying in the current layer and γ is the "weight" of stepping to another layer. Suppose, for example, that one wishes to consider only intra-layer steps or steps that change between a pair of layers before or after having an intra-layer step (In this scenario, we disallow two consecutive inter-layer steps.) The number of these cycles is
t W,i = ((A C) 3 + ( CA) 3 ) ii ,(2)
where the first term corresponds to cycles in which the interlayer step is taken after an intra-layer one and the second term to cycles in which the inter-layer step is taken before an intralayer one. We can simplify Eq. 2 by exploiting the fact that both A and C are symmetric. This yields
t W,i = 2((A C) 3 ) ii .(3)
If we relax the condition of disallowing two consecutive inter-layer steps, we can write
t SW,i = (( CA C) 3 ) ii ,(4)t SW ′ ,i = (( C ′ A + A C ′ ) 3 ) ii ,(5)
where C ′ = 1 2 βI + γC. Unlike the matrices in definition Eq. (2), the matrices CA C and CA + A C are symmetric and we can interpret them as weighted adjacency matrices of symmetric supra graphs, and we can thereby calculate cycles and clustering coefficients in these graphs. In the Supplementary Material (SM), we include additional discussion about these types of cycles, which include walks with two consecutive inter-layer steps.
Multiplex Clustering Coefficients Based on Counting Cycles. To define multiplex clustering coefficients, we need both the number of cycles t i and a normalization d i . For normalization, we can follow the same idea as with monoplex clustering coefficients and use a complete multiplex network F = α F α , where F α = J − I is the adjacency matrix for a complete graph on layer α. We can then proceed from any definition of t * ,i to d * ,i by replacing the second intra-layer step with a step in the complete multiplex network. For exam-
ple, we get d W,i = 2A CF CA C for t W,i = 2(A C) 3 .
We can now define local and global clustering coefficients for multiplex graphs analogously to monoplex networks. We can also define a clustering coefficient for the supra-nodesĩ.
This yields
c * ,i = t * ,i d * ,i ,(6)c * ,ĩ = i∈l(ĩ) t * ,i i∈l(ĩ) d * ,i ,(7)c * = i t * ,i i d * ,i ,(8)
For simplicity, we henceforth use c(β, γ) to indicate that we calculate c with parameter values β and γ.
We decompose the formula in Eq. (8) in terms of contributions from cycles that traverse exactly one, two, and three layers:
t * ,i = t * ,1,i β 3 + t * ,2,i βγ 2 + t * ,3,i γ 3 ,(9)d * ,i = d * ,1,i β 3 + d * ,2,i βγ 2 + d * ,3,i γ 3 ,(10)c * ,l = i t * ,l,i i d * ,l,i .(11)
Using this decomposition yields alternative way to average over contributions from the three types of cycles:
c * (ω 1 , ω 2 , ω 3 ) = 3 l ω l c * ,l ,(12)
where ω is a vector that gives the relative weights of the different contribution. Similar formulas hold for the other two multiplex clustering coefficients of Eqs. (6,7). Clustering Coefficients on Aggregated Networks. In the previous paragraph we have defined a local-clustering coefficient for the supra-nodes in terms of the local clustering coefficients of the nodes belonging to it. Now, we show another way to assign local clustering coefficient to supra nodes. A common way to study multiplex systems is to aggregate layers to obtain either multi-graphs or weighted networks. One can represent a weighted network using a weighted adjacency matrix W whose elements are the weights of the links. The weighted adjacency matrix associated to the aggregation of a multiplex has elements Wĩj = i∈l(ĩ),j∈l(j) A ij . There are numerous ways to define clustering coefficients on weighted networks [32] and one can use any of these after one has aggregated a multiplex network into a weighted monoplex network. For example, one can calculate the clustering defined in Refs. [33], [34], and [35], respectively, as
c Z,ĩ = j ,h WĩjWĩhWjh w max j =h WĩjWĩh = (W 3 )ĩĩ ((W (w max F )W )ĩĩ ,(13)c O,ĩ = 1 w max kĩ(kĩ − 1) j ,h (WĩjWĩhWjh) 1/3 ,(14)
c Ba,ĩ = 1 sĩ(kĩ − 1)
j ,h (Wĩj + Wĩh) 2 AĩjAĩhAjh ,(15)
where A is the unweighted network corresponding to W , the degree ofĩ is kĩ = j Aĩj, the strength ofĩ is sĩ = j Wĩj, the quantity w max = maxĩ ,j Wĩ ,j is the maximum weight in W , and F is the adjacency matrix of the complete unweighted graph. We can define the global version c Z of c Z,ĩ by summing over all the supra-nodes in the numerator and the denominator of Eq. 13 similar to Eq. 8. The coefficients c Z,ĩ and c Z are related to some of the multiplex coefficients for fully inter-connected multiplex networks. Letting β = γ = 1 and summing over all layers
yields: i∈l(ĩ) ((A C) 3 ) ii = ((W ) 3 )ĩĩ.
That is, in this special case the weighted clustering coefficients c Z,ĩ and c Z , and the multiplex clustering coefficients c W,ĩ and c W are equivalent: c W,ĩ (1/2, 1/2) = wmax b c Z,ĩ and c W (1/2, 1/2) = wmax b c Z . The term w max /b to match the normalizations arises because aggregation removes the information about the number of layers b, so the normalization must be based on the maximum weight instead of the number of layers. That is, a step in the complete weighted network is described by w max F in Eq. 13 instead of by bF .
Multiplex Clustering Coefficients in the Literature. Now consider the clustering coefficient proposed in [36] defined for fully interconnected multiplex networks as
c Be,ĩ = j ,h b α (A α )ĩj b β (A β )ĩh b γ (A γ )jh j ,h b β (A β )ĩj b α max((A α )ĩh, (A α )jh) ,(16)
which can be simplified if it's written in terms of the aggregated network as
c Be,ĩ = j ,h WĩjWĩhWjh j ,h Wĩj b α max((A α )ĩh, (A α )jh)(17)
The numerator of Eq. (17) is the same as the numerator of the weighted clustering coefficient c Z,ĩ , but the denominator is different. Because of the denominator in Eq. (17), the values of the clustering coefficient c Be,ĩ do not have to lie in the interval [0, 1]. For example, c Be = (n − 2)b/n for a complete multiplex graph, where n is the number of nodes in the multiplex graph and c Be > 1 when b > n n−2 . Refs. [37,38] define some further multiplex clustering coefficients, but we left out them of the further analysis in this Letter, since they don't return to the monoplex clustering coefficient for unweighted (i.e. networks with binary weights) and undirected networks.
Comparison of the Different Definitions. Next, we provide a comparison between the different formulations of multiplex clustering coefficients. In Table I, we give the values of some of the global and mean local clustering coefficients for multiplex networks (4 social networks and 2 transportation networks) constructed from real data. As we will now discuss, multiplex clustering coefficients give insights that are impossible to infer by calculating weighted clustering coefficients for aggregated networks or even by calculating clustering coefficients separately for each layer of a multiplex network.
For each of the social networks in Table I We symmetrized directed networks considering two nodes to be connected if there is at least one edge between them. The social networks above are fully inter-connected multiplex graphs, but the transport networks are not fully inter-connected. We use · ĩ to denote average overĩ.
if fewer layers are involved. That is, there is more intra-layer clustering than inter-layer clustering. The opposite is the case for the London Tube network: c * (0, 0, 1) > c * (0, 1, 0) > c * (1, 0, 0). This reflects the fact that lines in the Tube are designed to avoid redundant connections, and the lines roughly lie on geographically straight lines. A single-layer triangle would require a line to make a full loop within 3 stations. Two-layer triangles, which are a bit more frequent, entail that two lines run in almost parallel directions and that one line jumps over a single station. For 3-layer triangles, the geographical constraints do not matter because one can construct a triangle with three straight lines. The airline network is also a transportation network, but it is organized differently. Each layer encompasses flights from a single airline. The intra-airline clustering coefficients have small values since it is not in the interest of an airline to introduce new flights between two airports which can already be reached by with the same airline through some other airpot. The two-layer cycles correspond to cases where an airline has a connection from an airport to two other airports and a second airline has a direct connection between those two airports. Completing a three-layer cycle requires using three distinct airlines, and this type of congregation of airlines to the same area is not frequent in the data.
It is also relevant to examine how the various weighted clustering coefficients behave on aggregated multiplex networks. Note that for fully interconnected multiplex networks wmax b c Z gives c W ( 1 2 , 1 2 ). Both c O,i and c Ba,i are based on the unweighted clustering coefficient of the aggregated graph, where c Ba,i uses the weights to weight the importance of the different triangles, and c O,i is the unweighted clustering coefficient multiplied by the average intensity of the triangles. c Be is the only previously defined multiplex clustering coefficient, but it seems to be sensitive to the number of network layers, and even gets values larger than one for the Krackhardt cognitive social structure network with 21 layers. The transport networks are free from this sensitivity to the number of layers since although they have many of them, they are mostly empty. For example, most airlines only use a small subset of the total of 3108 airports.
Conclusions. We derived measurements of transitivity for multiplex networks by developing multiplex generalizations of the clustering coefficient. By using examples from empirical data in diverse settings, we showed that different notions of multiplex transitivity are important in different situations. For example, the balance between intra-layer versus inter-layer clustering is different in social versus transportation networks (and even in different types of networks within each category, as we illustrate explicitly for transportation networks), reflecting the fact that transitivity arises from different mechanisms in these cases. Such differences are rooted in the new degrees of freedom that arise from inter-layer connections and are invisibles to calculations of clustering coefficients on singlelayer networks obtained via aggregation. Generalizing clustering coefficients for multiplex networks thus makes it possible to explore such phenomena and to gain deeper insights into different types of transitivity in networks. Finally, the existence of multiple types of transitivity also has important implications for multiplex network motifs and multiplex community structure. In particular, our work on multiplex clustering coefficients demonstrates that definitions of all clustering notions for multiplex networks need to be able to handle such features.
Acknowledgments. All authors were supported by the European Commission FET-Proactive project PLEXMATH (Grant No. 317614). AA also acknowledges financial support from the ICREA Academia, Generalitat de Catalunya (2009-SGR-838) and the James S. McDonnell Foundation, and SG and AA were supported by FIS2012-38266. YM was also supported by MINECO through Grants FIS2011-25167 and by DGA (Spain). MAP acknowledges a grant (EP/J001759/1) from the EPSRC. We thank David Krackhardt for useful comments.
SUPPLEMENTARY MATERIAL Other Possible Definitions of Cycles
There are many possible ways to define cycles in multiplex networks. For example, one might want to disallow the option of staying inside a layer in the first step of the second term. We can then write
t W ′ ,i = [(A C) 3 + γCA( CA) 2 ] ii .(18)
With this restriction, cycles that traverse two adjacent edges to the focal node i are only calculated two times instead of four times. Similar to Eq. (3) in the main text, we can simplify Eq. (18) to obtain
t W ′ ,i = [(A C) 2 A C W ′ ] ii ,(19)
where C W ′ = (βI + 2γC). Table II shows the values of the previous clustering coefficient for the same networks studied in the main text.
Writing Clustering Coefficients Using "Elementary" 3-Cycles
It may be useful to decompose multilayer clustering coefficients defined in terms of multilayer cycles into so-called elementary cycles by expanding the formulas. Because we are only interested in the diagonal elements of the terms and the intra-layer supra-graph and coupling supra-graph that we consider are undirected, we can transpose the terms and still write them in terms of the matrices A and C rather than their transposes.
We adopt a convention in which all elementary cycles are transposed in a way that we select the one in which the first different element is A rather than C when comparing the two versions of the term from left to right. We then express all of the cycles in a standard form with terms AAA, AACAC, ACAAC, ACACA, ACACAC, CAAAC, CAACAC, and CACACAC.
Similarly, we write the normalization formulas using the same set of terms, except that the second A is replaced with
t i = [w AAA AAA + w AACAC AACAC + w ACAAC ACAAC + w ACACA ACACA + w ACACAC ACACAC + w CAAAC CAAAC + w CAACAC CAACAC + w CACACAC CACACAC] ii (20) d i = [w AAA AFA + w AACAC AFCAC + w ACAAC ACFAC + w ACACA ACFCA + w ACACAC ACFCAC + w CAAAC CAFAC + w CAACAC CAFCAC + w CACACAC CACFCAC] ii ,(21)
where w * are scalars that correspond to the weights for each type of elementary cycle. Note that we have absorbed the parameters β and γ into these coefficients. We illustrate the elementary cycles in Fig. 1,2.
In Table III, we show the coefficient values for expansions in terms of elementary cycles. In Table IV, we show the expansions for the case β = γ = 1. These cycle decompositions illuminate the difference between c W and c W ′ . The clustering coefficient c W gives equal weight to each elementary cycle, whereas c W ′ gives half of the weight to AAA and ACACA cycles; these are exactly cycles that include an implicit double-counting of cycles.
One can even express the cycles that include two consecutive inter-layer steps in a standard form for fully interconnected multiplex networks, because CC = (b − 1)I + (b − 2)C in this case. Without the assumption that β = γ = 1, the expansion for the coefficient c SW is cumbersome because it includes coefficients β k γ h with all possible combinations of k and h such that k + h = 6 and h = 1. Furthermore, it is no longer possible to infer the number of layers in which a walk traverses an intra-layer edge from the exponents of β and γ for c SW and c SW ′ . For example, in c SW ′ , the intra-layer elementary triangle AAA includes a contribution from both β 3 (i.e., the walk stays at the original layer) and βγ 2 (i.e., the walk visits some other layer but then comes back to the original layer without traversing any intra-layer edges while it was gone). Moreover, all of the terms with b arise from a walk moving to a new layer and then coming right back to the original layer in the next step. Because there are b − 1 other layers from which to choose, the influence cycles with this type of transient layer visits is amplified by the total number of layers in the network. That is, adding more layers (even ones that do not contain any edges) changes the relative importance of different types of elementary cycles.
Defining Multiplex Clustering Coefficients Using Auxiliary Networks
An elegant way to generalize clustering coefficients for multiplex networks is to define a new (possibly weighted) auxiliary supra-graph G M so that one can define cycles of interest as weighted 3-cycles in G M . Once we have a function that produces the auxiliary supra-adjacency matrix M = M(A, C), we can define the auxiliary complete supraadjacency matrix M F = M(F, C). One can then define a local clustering coefficient for node i with the formula
c i = (M 3 ) ii (MM F M) ii .(22)
As in the monoplex case, the denominator written in terms of the complete matrix is equivalent to that usual one written in terms of connectivity. In this case the connectivity of a node is considered in the supra-graph induced by the matrix M. We refers to the matrix M as multiplex walk matrix to remind that this matrix encode allowed types of steps in the multiplex. Anyway, in the case for example of A C and CA, the induced supra-graph is directed, so one should differentiate between in-and out-connectivity degree. A clear advantage in defining clustering coefficients using an auxiliary supra-graph is that one can then use the auxiliary graph to calculate other diagnostics (e.g., degree or strength) for nodes. One can then check for correlation between clustering-coefficient values and the size of the multiplex neighborhood of a node. The size of the neighborhood being the number of nodes that are reachable in a single step of the type defined by matrix M
The symmetric multiplex walk matrices of Eqs.(4) and (5) are
M SW ′ =(A C ′ + C ′ A)(23)M SW = CA C .(24)
To avoid double-counting intra-layer steps in the definition of M SW ′ , we need to rescale either the intra-layer weight parameter C (i.e., we can write C ′ = β ′ I + γC = 1 2 βI + γC) or the inter-layer weight parameter [i.e., we can write C ′ = βI+ γ ′ C = βI + 2βC and also define M SW ′ = 1 2 (A C ′ + C ′ )]. Let's consider supra graphs that are induced by multiplex walks matrices. The difference between the matrices M SW and M SW ′ is that M SW also includes terms of the form CAC that take into account walks that have an inter-layer step (C) followed by an intra-layer step (A)followed by another inter-layer step (C). Therefore, in the supra-graph that is induced by M SW , two nodes in the same layer that are not connected in that layer can be connected nodes in the same supra-nodes are connected in another layer.
When β = γ = 1, note that the matrix C sums the contributions of all supra nodes that share the same node. In other words, if we associate to each node i a vector of the canonical basis e i the application of C to e i
Ce i = j∈l(i) e j .(25)
produces a vector with entries equal to one in correspondence to the nodes that belong to it and zero otherwise. Consquently, M SW is of particular interest, as it is related to the weight matrix of the aggregated graph for β = γ = 1. That is,
( CA C) ij = wĩj, with i ∈ l(ĩ), j ∈ l(j).(26)
One can also write the multiplex clustering coefficient induced by Eq. (2) in terms of auxiliary supra-adjacency matrix by considering Eq.(3), which is a simplified version of the equation that counts cycles only in one direction:
M W = 3 √ 2A C .(27)
The matrix M W is not symmetric, which implies that the corresponding graph is a directed supra.graph. Nevertheless, the clustering coefficient that is induced by M W is the same as that induced by its transpose M † W . It is evident that different clustering-coefficients depends differently on the level of overlap (repeated edges among layers) in a multiplex network. The one that should change more is CA C, because it counts as triangles attached to a node all the triangles attached to other nodes in the same supra-node.
Main properties of the clustering coefficients defined in the main text
In Table V, we summarize the main properties for global clustering coefficients obtained by averaging the local multiplex clustering coefficients over all nodes (see main text for For all choices of relative weightings (ω 1 , ω 2 , ω 3 ) and in the limit as the number of supra nodes n → ∞, the coefficients c W ′ and c W have the value p for a fully inter-connected multiplex network that consists of an independent Erdős-Rényi (ER) graph with edge probability p for each layer.
C.C. β h γ k AAA AACAC ACAAC ACACA ACACAC CAAAC CAACAC CACACAC β 3 2 cW βγ 2 2 2 2 γ 3 2 β 3 1 c W ′ βγ 2 2 2 1 γ 3 2 β 3 1 c SW ′ βγ 2 2(b − 1) 2 4 3 1 γ 3 2(b − 2) 2(b − 2) 2 2 cSW β 6 1 β 4 γ 2 2(b − 1) 4 4 4 1 β 3 γ 3 2(b − 2) 2(b − 2) 4(b − 2) 8 4 β 2 γ 4 (b − 1) 2 4(b − 1) 4(b − 1) (b − 2) 2 8(b − 2) 2(b − 1) 2(b − 2) 4 β 1 γ 5 2(b − 2)(b − 1) 2(b − 2)(b − 1) 2(b − 2) 2 4(b − 1) 4(b − 2) γ 6 (b − 1) 2 2(b − 2)(b − 1) (b − 2) 22 0 0 0 c W ′ 1 2 2 1 2 0 0 0 cSW 1b 2 2b 2 2b 2 b 2 2b 2 b 2 2b 2 b 2 c SW ′ 2b − 1 2 2b 2b − 1 2 1 2 0
A Simple Example
We will now use a simple example (that of Fig. 1, panel a) to illustrate the differences between the different notions of a multiplex clustering coefficient. Consider a two-layer multiplex network with three nodes in layer 1 and two nodes in layer 2. The three nodes in layer 1 form a connected triple, and the two exterior nodes of this triple are connected to the two nodes in layer 2, which are connected to each other.
The adjacency matrix A for the intra-layer graph is
A = ,(28)
and the adjacency matrix C of the coupling supra-graph is
C =
Thus, the supra-adjacency matrix is
A = 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 .(30)
The multiplex walk matrix M W is
M W = 3 √ 2 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 ,(31)
and we note that it is not symmetric. For example, node 4 is reachable from 1, but node 1 is not reachable from 4. The Property c W (,ĩ) c W ′ (,ĩ) c Be,ĩ c Z(,ĩ) c Ba,ĩ c O,ĩ 1) Reduces to monoplex c 2) c * ≤ 1 3) c * = p in multiplex ER 4) Monoplex c for copied layers 5) Def. for supra-graph nodes TABLE V: Summary of the properties of the different multiplex clustering coefficients. The notation c * (,ĩ) means that the property holds for both the global version and the supra-node version of the clustering coefficient.
(1) The value of the clustering coefficient reduces to the values of the associated monoplex clustering coefficient for a single-layer graph.
(2) The value of the clustering coefficient is properly normalized; that is, it less or equal to 1 for all networks.
(3) The clustering coefficient has a value of p in a large (i.e., number of supra nodes n → ∞) fully interconnected multiplex network in which each layer is an independent Erdős-Rényi network with an edge probability of p. We tested this property numerically. (4) Suppose that we construct a multiplex network by replicating the same given monoplex network in each layer. The clustering coefficient for the multiplex network has the same value as for the monoplex network. (5) The clustering coefficient can be defined for each node in the supra-graph and Ci = Cĩ.
edge (1, 4) in this graph represents the walk {1, 2, 4} in the multiplex network. The symmetric walk matrix M SW ′ is
M SW ′ = 0 1 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 0 .(32)
It is evident that M SW ′ is the sum of M W and M † W with rescaled diagonal blocks in order to not double-count edges
That is, it is a complete graph without self-loops. We now calculate c * ,i using the different definitions of a multiplex clustering coefficient. To calculate c W,i , we need to compute F C, which we obtain using the equation
The multiplex network F C has a complete graph in each layer, 0 inter-layer entries in correspondence of node 1 in layer 2 because it has no coupling with this, and interchanged interlayer coupling of the rest of nodes because each inter-layer step is preceded by an intra-layer step in the complete graph. The clustering coefficient of node 1, which is attached to two triangles walkable along the directions of the edges, is
c W,1 = 1 2 .(36)c W,2 = 1 ,(37)
and this coefficient is equal to the clustering coefficient of remaining nodes.
To calculate c SW ′ ,i , we need to compute (F C + CF), which we obtain using the equation
In the supra-graph F C + CF, all nodes are connected to all other nodes except those that belong to the same supra-nodes. The clustering coefficient of node 1, which is attached to 6 triangles, is c SW ′ ,1 = 1 2 = c W,1 .
The clustering coefficient of supra node 2, which is attached to one triangle, is c SW ′ ,2 = 1 .
To calculate c SW,i , we need to compute CF C, which we obtain from the equation
The only difference between the graphs CF C and (F C+ CF) is the weight of the edges in CF C that take into account the fact that edges might be repeated in the two layers.
The clustering coefficient of node 1, which is attached to 8 triangles, is
c SW,1 = 8 12 = 2 3 .(42)
The clustering coefficient of node 2, which is attached to 4 triangles, is
c SW,2 = 4 6 = 2 3 .(43)
Because we are weighting edges with their multiplexity (the number of times an edge between two nodes is repeated in different layer among nodes in the same supra-node ) in the normalization, none of the nodes has a clustering coefficient equal to 1, while they have all the same value, in the aggregated network, where information about the layer is lost, all the nodes have clustering coefficient equal to 1 (independent of the definition of the clustering coefficient).
, c W satisfies c * (1, 0, 0) > c * (0, 1, 0) > c * (0, 0, 1), so it takes larger values coefficients (rows) calculated for various networks (columns). "Padgett": Padgett Florentine families social network (n = 16, b = 2)[39]."Wiring": Roethlisberger and Dickson bank wiring room social network (n = 14, b = 6) [40]. "Kapltail": Kapferer tailor shop network (n = 39, b = 4) [41]. "Krack": Krackhardt office cognitive social structures (n = 21, b = 21) [42]. "Tube": The London Underground transportation network (n = 314, b = 14) [43]. "Airline": Network of flights between cities, in which each layer corresponds to a single airline (n = 3108, b = 530) [44].
[ 1 ]
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two-layer elementary cycles. Blue node is the starting point of the cycle. Panel a)AACAC,b)ACAAC,c)ACACA,d)CAAAC FIG. 2: Sketch of three-layer elementary cycles. Blue node is the starting point of the cycle. Panel a)CAACAC,b)ACACAC,c)CACACA F. This yields
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TABLE II :
IIClustering coefficients (rows) calculated for the same networks used in the main text (columns).
TABLE III :
IIICoefficients of elementary multiplex 3-cycle terms for different multiplex clustering coefficients. For C SW ′ and CSW , we
calculate the expansions only for fully interconnected multiplex networks.
C. C. AAA AACAC ACAAC ACACA ACACAC CAAAC CAACAC CACACAC
cW
2
2
2
2
TABLE IV :
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Note that the prefix supra-refers to the fact that the object is made up of several elements of the same type. Note that the prefix supra-refers to the fact that the object is made up of several elements of the same type.
| []
|
[
"Tailoring dispersion of room temperature exciton-polaritons with perovskite-based subwavelength metasurfaces",
"Tailoring dispersion of room temperature exciton-polaritons with perovskite-based subwavelength metasurfaces"
]
| [
"Nguyen Ha ",
"My Dang ",
"Dario Gerace ",
"Emmanuel Drouard ",
"Gaëlle Trippé-Allard ",
"Ferdinand Lédée ",
"Radoslaw Mazurczyk ",
"Emmanuelle Deleporte ",
"Christian Seassal ",
"Hai Son Nguyen [email protected] ",
"De Lyon ",
"\nInstitut des Nanotechnologies de Lyon\n‡Dipartimento di Fisica\nINL-UMR5270\nCNRS\nEcole Centrale de Lyon\n36 avenue Guy de CollongueF-69134EcullyFrance\n",
"\nUniversità di Pavia\nvia Bassi 6I-27100PaviaItaly\n",
"\n¶Laboratoire Aimé Cotton\nCNRS\nUniv. Paris-Sud\nENS Paris-Saclay\nUniversité Paris-Saclay\n91405Orsay CedexFrance\n"
]
| [
"Institut des Nanotechnologies de Lyon\n‡Dipartimento di Fisica\nINL-UMR5270\nCNRS\nEcole Centrale de Lyon\n36 avenue Guy de CollongueF-69134EcullyFrance",
"Università di Pavia\nvia Bassi 6I-27100PaviaItaly",
"¶Laboratoire Aimé Cotton\nCNRS\nUniv. Paris-Sud\nENS Paris-Saclay\nUniversité Paris-Saclay\n91405Orsay CedexFrance"
]
| []
| Exciton-polaritons, elementary excitations arising from the strong coupling regime between photons and excitons in insulators or semiconductors, represent a promising platform for studying quantum fluids of light and realizing prospective all-optical devices. Among different materials for room temperature polaritonic devices, twodimensional (2D) layered perovskites have recently emerged as one of the promising candidates thanks to their prominent excitonic features at room temperature. Here we report on the experimental demonstration of exciton-polaritons at room temperature in resonant metasurfaces made from a subwavelength 2D lattice of perovskite pillars. These metasurfaces are obtained via spincoating, followed by crystallization of the perovskite solution in a pre-patterned glass backbone. The strong coupling regime is revealed by both angular-resolved reflectivity and photoluminescence measurements, showing anticrossing between photonic modes and the exciton resonance with a Rabi splitting in the 200 meV range. Moreover, we show that the polaritonic dispersion can be engineered by tailoring the photonic Bloch mode to which perovskite excitons are coupled. Linear, parabolic, and multi-valley polaritonic dispersions are experimentally demonstrated. All of our results are perfectly reproduced by both numerical simulations based on a rigorous coupled wave analysis and an elementary model based on a quantum theory of radiation-matter interaction. Our results suggest a new approach to study exciton-polaritons and pave the way towards large-scale and low-cost integrated polaritonic devices operating at room temperature. Article Exciton polaritons -half-light/half-matter elementary excitations arising from the strong coupling between excitons and photons 1 offer unprecedented insight into fundamental physical phenomena, as well as exciting technological prospects. On the one hand, thanks to their photonic component, polaritons can exhibit ballistic propagation over macroscopic distances. 2 Their photonic nature also provides optical means to generate, probe, and detect polaritons. On the other hand, polaritons exhibit strong nonlinearities inherited from exciton-exciton interactions, which are orders of magnitudes more efficient than the photonphoton nonlinear Kerr effect. 3,4 These unique features make polaritons an attractive playground to study fascinating physical phenomena such as out-of-equilibrium Bose-Einstein condensation, 5,6 superfluidity, 7 quantum vortices, 8 analog gravity, 9 and topological insulators. 10,11 Furthermore, from an application point of view, these quasiparticles represent a promising platform to make all-optical devices such as polariton lasers, 5,12,13 optical transistors, 14,15 resonant tunneling diodes, 16 optical switches, 17,18 and rooters. 19Pioneering work on exciton-polaritons was performed with GaAs and CdTe based quantum wells embedded within vertical Fabry Perot cavities. 1 Although these materials, especially GaAs, are still the most used excitonic materials for studying polaritonic physics, their operation is limited to cryogenic temperatures due to the modest excitonic binding energy of few meV. 20 In view of making polaritonic devices suited for practical applications, a quest has been set towards room temperature excitons, and therefore the use of materials of high excitonic binding energy such as GaN, 12,13 ZnO, 21,22 organic semiconductors 23-25 and, more recently, monolayers of transition metal dichalcogenides (TMDCs).26,27Other appealing candidates for room temperature polaritonic devices are hybrid organicinorganic perovskites (HOPs). These materials possess remarkable optical properties mostly determined by their inorganic component, such as bandgap tunability, high luminescence | 10.1021/acs.nanolett.0c00125 | [
"https://arxiv.org/pdf/2001.06801v1.pdf"
]
| 210,838,772 | 2001.06801 | db7fd40232e83a17e3f3f562c8b68dd77f5dd8d7 |
Tailoring dispersion of room temperature exciton-polaritons with perovskite-based subwavelength metasurfaces
arXiv:2001.06801v1 [physics.optics] 19 Jan 2020
Nguyen Ha
My Dang
Dario Gerace
Emmanuel Drouard
Gaëlle Trippé-Allard
Ferdinand Lédée
Radoslaw Mazurczyk
Emmanuelle Deleporte
Christian Seassal
Hai Son Nguyen [email protected]
De Lyon
Institut des Nanotechnologies de Lyon
‡Dipartimento di Fisica
INL-UMR5270
CNRS
Ecole Centrale de Lyon
36 avenue Guy de CollongueF-69134EcullyFrance
Università di Pavia
via Bassi 6I-27100PaviaItaly
¶Laboratoire Aimé Cotton
CNRS
Univ. Paris-Sud
ENS Paris-Saclay
Université Paris-Saclay
91405Orsay CedexFrance
Tailoring dispersion of room temperature exciton-polaritons with perovskite-based subwavelength metasurfaces
arXiv:2001.06801v1 [physics.optics] 19 Jan 20201Strong coupling regimeexciton-polaritonspolaritonic devices2D layered perovskitesmeta- surfacesnanophotonics 2
Exciton-polaritons, elementary excitations arising from the strong coupling regime between photons and excitons in insulators or semiconductors, represent a promising platform for studying quantum fluids of light and realizing prospective all-optical devices. Among different materials for room temperature polaritonic devices, twodimensional (2D) layered perovskites have recently emerged as one of the promising candidates thanks to their prominent excitonic features at room temperature. Here we report on the experimental demonstration of exciton-polaritons at room temperature in resonant metasurfaces made from a subwavelength 2D lattice of perovskite pillars. These metasurfaces are obtained via spincoating, followed by crystallization of the perovskite solution in a pre-patterned glass backbone. The strong coupling regime is revealed by both angular-resolved reflectivity and photoluminescence measurements, showing anticrossing between photonic modes and the exciton resonance with a Rabi splitting in the 200 meV range. Moreover, we show that the polaritonic dispersion can be engineered by tailoring the photonic Bloch mode to which perovskite excitons are coupled. Linear, parabolic, and multi-valley polaritonic dispersions are experimentally demonstrated. All of our results are perfectly reproduced by both numerical simulations based on a rigorous coupled wave analysis and an elementary model based on a quantum theory of radiation-matter interaction. Our results suggest a new approach to study exciton-polaritons and pave the way towards large-scale and low-cost integrated polaritonic devices operating at room temperature. Article Exciton polaritons -half-light/half-matter elementary excitations arising from the strong coupling between excitons and photons 1 offer unprecedented insight into fundamental physical phenomena, as well as exciting technological prospects. On the one hand, thanks to their photonic component, polaritons can exhibit ballistic propagation over macroscopic distances. 2 Their photonic nature also provides optical means to generate, probe, and detect polaritons. On the other hand, polaritons exhibit strong nonlinearities inherited from exciton-exciton interactions, which are orders of magnitudes more efficient than the photonphoton nonlinear Kerr effect. 3,4 These unique features make polaritons an attractive playground to study fascinating physical phenomena such as out-of-equilibrium Bose-Einstein condensation, 5,6 superfluidity, 7 quantum vortices, 8 analog gravity, 9 and topological insulators. 10,11 Furthermore, from an application point of view, these quasiparticles represent a promising platform to make all-optical devices such as polariton lasers, 5,12,13 optical transistors, 14,15 resonant tunneling diodes, 16 optical switches, 17,18 and rooters. 19Pioneering work on exciton-polaritons was performed with GaAs and CdTe based quantum wells embedded within vertical Fabry Perot cavities. 1 Although these materials, especially GaAs, are still the most used excitonic materials for studying polaritonic physics, their operation is limited to cryogenic temperatures due to the modest excitonic binding energy of few meV. 20 In view of making polaritonic devices suited for practical applications, a quest has been set towards room temperature excitons, and therefore the use of materials of high excitonic binding energy such as GaN, 12,13 ZnO, 21,22 organic semiconductors 23-25 and, more recently, monolayers of transition metal dichalcogenides (TMDCs).26,27Other appealing candidates for room temperature polaritonic devices are hybrid organicinorganic perovskites (HOPs). These materials possess remarkable optical properties mostly determined by their inorganic component, such as bandgap tunability, high luminescence
quantum yield, and narrow emission line-width. They also exhibit key advantages of organic semiconductors as solution processability and the strong exciton binding energy needed for high temperature operation. 28 In particular, two-dimensional (2D) layered HOPs offer excitons with binding energies up to hundreds meV and superior oscillator strength in comparison with conventional inorganic quantum well excitons. Thanks to this robust binding energy, the strong coupling regime could be observed at room temperature. Indeed, 2D layered HOPs were employed as a highly relevant material for room temperature polariton even before the "perovskite fever". Using these materials, observation of cavity-exciton polaritons and plasmon-exciton polaritons have already been reported by many groups. [29][30][31][32][33][34][35][36][37] Most recently, it has been shown that the nonlinearity of HOP polaritons is mostly due to the exciton-exciton interaction. 38 Such behavior, reserved so far only for GaAs polaritons at cryogenic temperature, 39 suggests that 2D layered HOP would be the ideal material to exploit polariton physics at room temperature.
In this letter, we report on the experimental demonstration of exciton polaritons at room temperature in HOP-based subwavelength metasurfaces textured in a periodic 2D lattice. This is achieved when coupling the HOP excitons with the photonic Bloch modes of a lattice of HOP pillars. The strong coupling regime for different lattice designs and polarizations is observed in both angular-resolved reflectivity and photoluminescence measurements, showing Rabi splitting in the range of 200 meV. Most interestingly, we show that the polaritonic dispersion can be engineered by tailoring the photonic Bloch mode to which HOP excitons are coupled. Polaritonic dispersion with linear, parabolic, and even multi-valley characteristics are observed. All of the experimental measurements are perfectly reproduced by both numerical simulations and a simplified quantum theory model, the latter suggesting that we can interpret these results in terms of elementary excitations previously defined as photonic crystal polaritons. 40,41 Our results suggest a new approach to study exciton-polaritons beyond the textbook Fabry Perot configuration, and pave the way toward large-scale and low-cost integrated polaritonic devices operating at room temperature.
The active material employed in this work is the 2D layered HOP, namely bi-(phenethylammonium) tetraiodoplumbate, known as PEPI, with chemical formula (C 6 H 5 C 2 H 4 NH 3 ) 2 PbI 4 . Its molecular structure is described in Fig 1(a). The composition of alternating organic/inorganic monolayers features the multi-quantum well structure in PEPI: the organic layers play the role of potential barriers that are able to confine the electronic excitation within the inorganic quantum wells. 37,42 The confinement effect is strengthened thanks to the high dielectric contrast between the organic and inorganic layers. 43 The combination of quantum confinement and dielectric confinement results in the high exciton binding energy, ranging up to hundred meV at room temperature. 44 Additionally, PEPI has been reported to have higher nonlinearity related to the delocalized Wannier excitons in comparison with Frenkel excitons in all-organic materials. 45 We first study the bare material: 50 nm−thick PEPI film on flat The fabrication process of PEPI metasurfaces is illustrated in Fig 1(d). A periodically patterned SiO 2 backbone is first prepared by electron-beam lithography, and then under- regime is perfectly adapted to study the ballistic propagation of polaritons for transmitting information and gating signals between polaritonic devices. 22 Figure 3(a) presents the normalized |v g |/c corresponding to P-polarized LP in the structure B. Contrary to the previous case, here polaritons undergo a slow-light regime with a maximum of |v g |/c ∼ 0.08 corresponding to the inflexion points of the dispersion at k x ≈ ±5 µm −1 . We further notice that this behavior is quite similar to microcavity polariton dispersion. Such slow-light regime, exhibiting high density of states, would be well suited to study polaritonic non-linear effects, as well as to trigger the Bose-Einstein condensation of polaritons. Finally, the group velocity corresponding to the S-polarized multi-valley band in the structure B is reported in Fig 3(b).
The two valleys correspond to the vanishing of group velocity at k x ≈ ±1.5 µm −1 . To the best of our knowledge, this is the first experimental observation of multi-valley polaritonic dispersion. It could be the building block of polariton valleytronic physics. 52,53 Indeed, recent theoretical works have predicted that Bose-Einstein condensation of polaritons with W-shaped dispersion would take place at the valley extrema, 53 thus paving the way to study Josephson oscillation in momentum space, 54 spontaneous symmetry breaking and two-mode squeezing. 52 Moreover, when not working at the extrema but at k x = 0, the W-shaped dispersion is also a perfect test bed for parametric scattering experiment. 55 In conclusion, we propose periodically patterned excitonic metasurfaces as a novel platform to study exciton-polariton physics. In constrast to the traditional microcavity design, the metasurface approach offers high flexibility for the tailoring of polaritonic properties (group velocity, quality factor of localized modes, emission pattern, etc.) and can be applied to a wide range of excitonic materials. As a proof-of-concept, the strong coupling regime at room temperature in perovskite-based metasurfaces is experimentally observed. is collected via the same objective and its image in the Fourier space is projected onto the entrance slit of a spectrometer. The sample orientation is aligned so that the ΓX direction is along the entrance slit. The output of the spectrometer is coupled to a CCD camera, and the image obtained from the camera leads directly to the energy-momentum dispersion diagram along ΓX.
PEPI model: The dielectric function of PEPI, used in numerical simulations and analyical calculations, is given by:
P EP I (E) = n 2 + A X E 2 X − E 2 − iγ X E(1)
where n = 2.4 is the refractive index of the passive structure, A X = 0.85 eV 2 is the oscillator strength of PEPI exciton, E X = 2.394 eV is its energy and γ X = 30 meV is its linewidth. 46
Numerical simulations: The RCWA simulations of angular-resolved reflectivity and absorption have been done with S 4 , a freely available software package provided by Fan Group in the Stanford Electrical Engineering Department. 47 Quantum theory of exciton-photon coupling: The radiation-matter coupling in periodically patterned multilayers is described by quantizing both the electromagnetic field and the exciton center of mass field in a periodic piecewise constant potential. Photonic modes, in particular, are obtained through a guided mode expansion method, 56 which allows to obtain both real and imaginary parts (losses) of photonic Bloch modes of the textured structure. The second-quantized hamiltonian of the interacting system is then diagonalized with a generalized Hopfield method, thus yielding the complex dispersion of mixed exciton-photon modes. 40 The main approximation with respect to the present work lies in the assumption that the whole excitonic oscillator strength is concentrated at the center of the patterned layer, while the full thickness of the HOP layer is taken into account with a non-dispersive average index (2.4). This way, the only 'fitting' parameter with respect to experimental results is the effective oscillator strength assumed for this single quantum well layer. On the other hand, we assumed a value for the oscillator strength per unit surface (f /S) that is actually obtained by multiplying the oscillator strength per quantum well in the HOP layer times the number of wells compatible with the overall HOP layer thickness. So, without adjustable parameters, our assumption proves to be reliable a posteriori, by the remarkable agreement with experimental results for the polariton dispersions reported in Fig. 2 and 3 of this manuscript.
Acknowledgement
The authors would like to thank the staff from the NanoLyon Technical Platform for helping and supporting in all nanofabrication processes. We thank José Penuelas for his technical support in the XRD characterization of perovskite. This work is partly supported by the
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The authors declare no competing financial interest. Graphical TOC Entry
SiO 2 substrate is prepared by spincoating, then immediately encapsulated with Poly-methyl methacrylate (PMMA) on top to avoid contact with humidity. X-ray diffraction (XRD) measurements [see Fig 1(b)] clearly evidence the critical peaks of PEPI crystalline structure, [0 0 2l], with l = 1 − 6. The photoluminescence (PL) spectrum, under excitation at 3.06 eV, exhibits a relatively sharp emission peak at 2.36 eV with full width at half maximum of 0.062 eV [see Fig 1(c)]. The absorption spectrum displays a strong absorption peak at 2.4 eV, which is superimposed to the PL spectrum. These spectral features clearly show an excitonic behavior at room temperature. The absorption continuum in UV range corresponds to the absorption above the bandgap of the semiconductors. The strong excitonic peak and continuum observed in the absorption spectrum at room temperature are in good agreement with the results reported in the literature. 46
goes ionic dry-etching with 150 nm of etching depth. PEPI in DMF (dimethylformamide) solution is infiltrated inside the air holes of this backbone via spincoating, then crystallized with the help of thermal annealing to form a pillar-lattice of PEPI. The Scanning Electron Microscope (SEM) images confirm that almost the entire PEPI fills inside the holes after its deposition, even though there are still residues of PEPI outside. A 200 nm thick layer of PMMA is immediately spincoated on top to encapsulate the whole structure. From the photonic point of view, the PMMA/PEPI nano-pillars/SiO 2 stack is equivalent to PEPI nano-pillars standing in a homogeneous optical medium, since the refractive index of PMMA (1.49) is closely matched with the SiO 2 one (1.47) in the whole spectral range of interest (400-600 nm). At difference with previous reports on inorganic exciton polaritons at low temperature 41 in which the periodic metasurface was not directly modifying the active material, here the excitonic medium is strongly modulated by the periodic pattern.The strong coupling regime in the PEPI metasurfaces is investigated by angle-resolved reflectivity (ARR) and angle-resolved PL (ARPL) measurements through a Fourier spectroscopy set up. These angular-resolved experiments are performed along ΓX direction of reciprocal space, using two distinct polarizations:the S-polarization, corresponding to nonzero E y electric field component, and the P-polarization, corresponding to non-zero H y mag-netic field component [see Fig 1(e)]. Numerical simulations employing the Rigorous Coupled Wave Analysis (RCWA) 47 are performed to predict the photonic Bloch mode response as a function of energy and wave vector, to be directly compared to the experimental results. Two designs of PEPI metasurface are considered in our study [see table in Fig 1(e)]: 1/Structure A: 80 nm-height pillar-lattice with a d/a aspect ratio of 0.8 and a lattice parameter of 250 nm; 2/Structure B: 50 nm-height pillar-lattice with a d/a aspect ratio of 0.9 and a lattice parameter of 350 nm. These designs are chosen to provide a rich variety of photonic Bloch modes and mode dispersions in the vicinity of PEPI exciton energy. In fact, struc-ture A only exhibits a S-polarized photonic mode in this spectral region, while structure B exhibits both S-and P-polarized modes. Their dispersions can be numerically predicted by simulating the ARR of passive structures, in which PEPI is replaced by a dielectric medium of refractive index n = 2.4. The photonic dispersions of the two structures are strikingly different: i) Structure A displays a single S-polarized mode with an almost linear dispersion, as evidenced in Fig 2(a); ii) Structure B shows a P-polarized mode with an almost parabolic dispersion close to normal incidence [see Fig 2(b)], and three S-polarized modes [Fig 2(c)], two of which exhibit parabolic dispersions with opposite and small curvatures corresponding to a mini-gap opening, while the third one appears a quite exotic mode with a multi-valley dispersion of W-shape with two off-Γ minima. The coupling between the previous Bloch modes with PEPI excitons is first investigated by reflectivity measurements. Figures 2(d-f) present the ARR experimental results (right panels) directly compared to the numerical simulations (left panels) performed on the active structures. From these results, the strong coupling regime induced by the presence of the strong excitonic response is clearly evidenced: all dispersion curves are bent when approaching the exciton energy, and undergo the typical anticrossing behavior as a function of the in-plane wave vector (scanned through the incidence angle). The dispersion of each mode splits into an upper polariton (UP) and a lower polariton (LP) branch, respectively, with a very good overall agreement between measurements and numerical simulations. Rabi splittings of approximately 200 meV are measured. We note that the PEPI model used for numerical simulations only takes into account the excitonic resonance, but not the highly absorptive continuum states above the PEPI bandgap (see Methods section for more details). This explains the much weaker measured signal of UP as compared to the simulated one[Fig 2(d,f)]. In the case of P-polarized measurements of structure B[Fig 2(e)], the UP is not visible. This is also in agreement with the related simulations in the left panel, in which the simulated signal of this UP is already very weak.Although the reflectivity experiment proves the existence of photonic crystal polaritonic modes in this PEPI metasurface, it is important to demonstrate that polariton states can be populated in these structures via optical pumping.Figures 2(g-i)present the ARPL experimental results (right panels) performed on the active structures. To get a qualitative comparison, simulations of angle-resolved absorption are also presented on equivalent color scale plots (left panels). Below the bare exciton, the LP emission is clearly observed in these PL measurements, confirming the existence of polariton states in our samples. Typical to other room temperature polaritonic systems using high bandgap materials, the UP is not observed in PL measurements.48,49 Regarding the signal from a non-dispersive band corresponding to uncoupled PEPI excitons, which is evident both from PL measurements and absorption simulations with a small spectral shift corresponding to the Stokes-shift, we notice that depending on the location within the PEPI pillar, an exciton can be at an anti-node or a node position of the photonic mode, thus can undergo strong coupling or weak coupling with these modes. This is different from text-book exciton quantum well polaritons, where all the excitons are equally coupled to the same Fabry-Perot mode.The strong coupling mechanism in periodically textured excitonic metasurfaces can be analyzed with a full quantum theory of radiation matter coupling taking into account the in-plane periodicity of photonic modes and excitonic wave functions on an equal footing. The full Hopfield matrix can be obtained from a classical solution of Maxwell equations in such periodically patterned multilayer structure, and solving for the corresponding Schrödinger equation for the exciton envelope function, then coupling them to obtain the coupling matrix elements, as detailed in a previous work.40 Here, we approximate the excitonic response as if it was concentrated in a single quantum well layer at the center of the periodically patterned region, with an effective oscillator strength taking into account the finite thicknesss of the 2D HOP film. Polaritonic modes obtained by numerically diagonalizing the Hopfield matrix thus are presented in Figs 2(j-l). Correspondingly, the experimental data extracted from the ARR measurements are also superimposed, showing a remarkably good agreement over the whole parameters range.We finally discuss on the engineering of polaritonic dispersion in these PEPI metasurfaces. The results from both ARR and ARPL measurements presented above show that the polariton dispersion shares the same shape as the one of uncoupled photonic modes whenworking out of the anticrossing region. Most remarkably, this leads to polaritonic modes that can be engineered to display linear [Figs 2(j)], slow-light [Figs 2(k)], or even multivalley [Figs 2(l)] characteristics. Notice that this is due to the strong periodic modulation of the active material, at difference with previous reports on photonic crystal modulated dispersions of guided polaritons with in-plane uniform active medium. 41 To quantitatively assess these different dispersions, we extracted the group velocity (normalized to the speed of light c = 3.10 8 m/s), |v g |/c from experimental data and calculations of polariton dispersions from Hopfield matrix diagonalization. In structure A, linear dispersion in LP corresponds to |v g |/c ∼ 0.42 when |k x | exceeds 5 µm −1 . This linear polaritonic dispersion is similar to the one of guided polaritons recently reported from several groups. 50,51 Such high velocity
Moreover, we demonstrate the dispersion engineering with a variety of polariton dispersion characteristics such as linear, slow light, and for the first time multi-valley polaritonic mode dispersion, respectively. Finally, our approach is naturally in the scheme of integrated optics, and perfectly adapted for large-scale fabrication methods such as nanoimprint and solution spincoating. It thus would pave the way for making low-cost integrated polaritonic devices operating at room temperature.MethodsFabrication of the patterned SiO 2 backbone: The negative pattern (i.e. hole lattices) is defined in PMMA A4 950 Microchem (resist) using electron-beam lithography system equipped with Raith Elphy pattern generator. Before exposure, a film of 10 nm Al is deposited on top of the photoresist via thermal evaporation to prevent surface charging effects. After exposure, the Al film is removed by chemical etch applying aluminum etchant type D solution. The sample is then developed using MIBK/IPA solution. Later, the pattern is transferred into the fused silica substrate by Reactive Ion Etching using CHF 3 gas under pressure of 50 mTorr. Infiltration and crystallization of perovskite: In preparation for perovskite deposition, the SiO 2 backbone has been through cleaning processes including cleaning with acetone, ethanol in ultrasonic bath and ozone treatment. The treatments play an important role for quality enhancement of thin film deposition. PEPI solution in DMF wt 10% is spincoated on top of pre-treated SiO 2 backbone at 2000-3000 rpm within 30 s. The crystallization of PEPI pillars is achieved by annealing at 95 o C during 90 s. At last, a thin film of 200 nm of PMMA (resist) is deposited on top of PEPI metasurface by spincoating at 3000 rpm in 30 s and followed by an annealing process at 95 o C during 15 min. ARR and ARPL setup: The excitation sources for ARR and ARPL are a halogen light and a picosecond pulsed laser (50 ps, 80 MHz, 405 nm) respectively. The excitation light is focused onto the sample via a microscope objective (x100, NA = 0.8), The excitation spotsize is within 1 µm for the ARPL and 5 µm for the ARR measurements. The emitted light
French
National Research Agency (ANR) under the project POPEYE (ANR-17-CE24-0020) and project EMIPERO (ANR-18-CE24-0016).
C.S., R.M., and N.H.M.D. implemented the fabrication of nano-patterned subtrates. G.T.-A., F.L., and E.D. worked on PEPI perovskite synthesis and its excitonic model. E.D., H.S.N. and N.H.M.D carried out numerical simulation. D.G. developed the quantum model of photonic crystal polaritons. H.S.N. and N.H.M.D worked on perovskite deposition and performed the optical experiments. All the authors contributed to the interpretation of the results.
)Figure 1 :Figure 2 :
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| []
|
[
"THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS",
"THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS"
]
| [
"Thomas Bieske ",
"Erin Martin "
]
| []
| []
| By employing a Carnot parabolic maximum principle, we show existenceuniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity. | 10.1307/mmj/1472066144 | [
"https://arxiv.org/pdf/1501.07181v1.pdf"
]
| 118,869,437 | 1501.07181 | de9e32dcae1282a61b8d88eaf75f0dd92d5c0d25 |
THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS
28 Jan 2015
Thomas Bieske
Erin Martin
THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS
28 Jan 2015
By employing a Carnot parabolic maximum principle, we show existenceuniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.
Motivation
In Carnot groups, the following theorem has been established.
Theorem 1.1. [3,16,5] Let Ω be a bounded domain in a Carnot group and let v : ∂Ω → R be a continuous function. Then the Dirichlet problem
∆ ∞ u = 0 in Ω u = v on ∂Ω
has a unique viscosity solution u ∞ .
Our goal is to prove a parabolic version of Theorem 1.1 for a class of equations (defined in the next section), namely Conjecture 1.2. Let Ω be a bounded domain in a Carnot group and let T > 0. Let ψ ∈ C(Ω) and let g ∈ C(Ω × [0, T )) Then the Cauchy-Dirichlet problem
(1.1) u t − ∆ h ∞ u = 0 in Ω × (0, T ), u(x, 0) = ψ(x)
on Ω u(x, t) = g(x, t) on ∂Ω × (0, T ) has a unique viscosity solution u.
In Sections 2 and 3, we review key properties of Carnot groups and parabolic viscosity solutions. In Section 4, we prove uniqueness and Section 5 covers existence.
Calculus on Carnot Groups
We begin by denoting an arbitrary Carnot group in R N by G and its corresponding Lie Algebra by g. Recall that g is nilpotent and stratified, resulting in the decomposition g = V 1 ⊕ V 2 ⊕ · · · ⊕ V l for appropriate vector spaces that satisfy the Lie bracket relation [V 1 , V j ] = V 1+j . The Lie Algebra g is associated with the group G via the exponential map exp : g → G. Since this map is a diffeomorphism, we can choose a basis for g so that it is the identity map. Denote this basis by X 1 , X 2 , . . . , X n 1 , Y 1 , Y 2 , . . . , Y n 2 , Z 1 , Z 2 , . . . , Z n 3 so that V 1 = span{X 1 , X 2 , . . . , X n 1 } V 2 = span{Y 1 , Y 2 , . . . , Y n 2 } V 3 ⊕ V 4 ⊕ · · · ⊕ V l = span{Z 1 , Z 2 , . . . , Z n 3 }.
We endow g with an inner product ·, · and related norm · so that this basis is orthonormal. Clearly, the Riemannian dimension of g (and so G) is N = n 1 + n 2 + n 3 . However, we will also consider the homogeneous dimension of G, denoted Q, which is given by
Q = l i=1 i · dim V i .
Before proceeding with the calculus, we recall the group and metric space properties. Since the exponential map is the identity, the group law is the Campbell-Hausdorff formula (see, for example, [7]). For our purposes, this formula is given by (2.1) p · q = p + q + 1 2 [p, q] + R(p, q)
where R(p, q) are terms of order 3 or higher. The identity element of G will be denoted by 0 and called the origin. There is also a natural metric on G, which is the Carnot-Carathéodory distance, defined for the points p and q as follows:
d C (p, q) = inf Γ 1 0 γ ′ (t) dt
where the set Γ is the set of all curves γ such that γ(0) = p, γ(1) = q and γ ′ (t) ∈ V 1 . By Chow's theorem (see, for example, [2]) any two points can be connected by such a curve, which means d C (p, q) is an honest metric. Define a Carnot-Carathéodory ball of radius r centered at a point p 0 by B(p 0 , r) = {p ∈ G : d C (p, p 0 ) < r}.
In addition to the Carnot-Carathéodory metric, there is a smooth (off the origin) gauge. This gauge is defined for a point p = (ζ 1 , ζ 2 , . . . , ζ l ) with ζ i ∈ V i by
(2.2) N (p) = l i=1 ζ i 2l! i 1 2l!
and it induces a metric d N that is bi-Lipschitz equivalent to the Carnot-Carathéodory metric and is given by
d N (p, q) = N (p −1 · q)
. We define a gauge ball of radius r centered at a point p 0 by
B N (p 0 , r) = {p ∈ G : d N (p, p 0 ) < r}.
In this environment, a smooth function u : G → R has the horizontal derivative given by ∇ 0 u = (X 1 u, X 2 u, . . . , X n 1 u) and the symmetrized horizontal second derivative matrix, denoted by (D 2 u) ⋆ , with entries
((D 2 u) ⋆ ) ij = 1 2 (X i X j u + X j X i u)
for i, j = 1, 2, . . . , n 1 . We also consider the semi-horizontal derivative given by
∇ 1 u = (X 1 u, X 2 u, . . . , X n 1 u, Y 1 u, Y 2 u, . . . , Y n 2 u).
Using the above derivatives, we define the h-homogeneous infinite Laplace operator for h ≥ 1 by
∆ h ∞ f = ∇ 0 f h−3 n 1 i,j=1 X i f X j f X i X j f = ∇ 0 f h−3 (D 2 f ) ⋆ ∇ 0 f, ∇ 0 f .
Given T > 0 and a function u : G×[0, T ] → R, we may define the analogous subparabolic infinite Laplace operator by u t − ∆ h ∞ u and we consider the corresponding equation
(2.3) u t − ∆ h ∞ u = 0.
We note that when h ≥ 3, this operator is continuous. When h = 3, we have the subparabolic infinite Laplace equation analogous to the infinite Laplace operator in [5]. The Euclidean analog for h = 1 has been explored in [14] and the Euclidean analog for 1 < h < 3 in [15].
We recall that for any open set O ⊂ G, the function f is in the horizontal Sobolev space
W 1,p (O) if f and X i f are in L p (O) for i = 1, 2, . . . , n 1 . Replacing L p (O) by L p loc (O), the space W 1,p loc (O) is defined similarly. The space W 1,p 0 (O) is the closure in W 1,p (O)
of smooth functions with compact support. In addition, we recall a function u : G → R is C 2 sub if ∇ 1 u and X i X j u are continuous for all i, j = 1, 2, . . . n 1 . Note that C 2 sub is not equivalent to (Euclidean) C 2 . For spaces involving time, the space C(t 1 , t 2 ; X) consists of all continuous functions u : [t 1 , t 2 ] → X with max t 1 ≤t≤t 2 u(·, t) X < ∞. A similar definition holds for L p (t 1 , t 2 ; X).
Given an open box
O = (a 1 , b 1 ) × (a 2 , b 2 ) × · · · × (a N , b N ), we define the parabolic space O t 1 ,t 2 to be O × [t 1 , t 2 ]. Its parabolic boundary is given by ∂ par O t 1 ,t 2 = (O × {t 1 }) ∪ (∂O × (t 1 , t 2 ]).
Finally, recall that if G is a Carnot group with homogeneous dimension Q, then G × R is again a Carnot group of homogeneous dimension Q + 1 where we have added an extra vector field ∂ ∂t to the first layer of the grading. This allows us to give meaning to notations such as W 1,2 (O t 1 ,t 2 ) and C 2 sub (O t 1 ,t 2 ) where we consider ∇ 0 u to be X 1 u, X 2 u, . . . , X n 1 u, ∂u ∂t .
Parabolic Jets and Viscosity Solutions
3.1. Parabolic Jets. In this subsection, we recall the definitions of the parabolic jets, as given in [6], but included here for completeness. We define the parabolic superjet of u(p, t) at the point (p 0 , t 0 ) ∈ O t 1 ,t 2 , denoted P 2,+ u(p 0 , t 0 ), by using triples (a, η, X)
∈ R × V 1 ⊕ V 2 × S n 1 so that (a, η, X) ∈ P 2,+ u(p 0 , t 0 ) if u(p, t) ≤ u(p 0 , t 0 ) + a(t − t 0 ) + η, p −1 0 · p + 1 2 Xp −1 0 · p, p −1 0 · p +o(|t − t 0 | + |p −1 0 · p| 2 ) as (p, t) → (p 0 , t 0 )
. We recall that S k is the set of k × k symmetric matrices and n i = dim V i . We define p −1 0 · p as the first n 1 coordinates of p −1 0 · p and p −1 0 · p as the first n 1 + n 2 coordinates of p −1 0 ·p. This definition is an extension of the superjet definition for subparabolic equations in the Heisenberg group [4]. We define the subjet P 2,− u(p 0 , t 0 ) by P 2,− u(p 0 , t 0 ) = −P 2,+ (−u)(p 0 , t 0 ).
We define the set theoretic closure of the superjet, denoted P 2,+ u(p 0 , t 0 ), by requiring (a, η, X) ∈ P 2,+ u(p 0 , t 0 ) exactly when there is a sequence (a n , p n , t n , u(p n , t n ), η n , X n ) → (a, p 0 , t 0 , u(p 0 , t 0 ), η, X) with the triple (a n , η n , X n ) ∈ P 2,+ u(p n , t n ). A similar definition holds for the closure of the subjet. We may also define jets using appropriate test functions. Given a function u : O t 1 ,t 2 → R we consider the set Au(p 0 , t 0 ) given by
Au(p 0 , t 0 ) = {φ ∈ C 2 sub (O t 1 ,t 2 ) : u(p, t) − φ(p, t) ≤ u(p 0 , t 0 ) − φ(p 0 , t 0 ) = 0 ∀(p, t) ∈ O t 1 ,t 2 }.
consisting of all test functions that touch u from above at (p 0 , t 0 ). We define the set of all test functions that touch from below, denoted Bu(p 0 , t 0 ), similarly.
The following lemma relates the test functions to jets. The proof is identical to Lemma 3.1 in [4], but uses the (smooth) gauge N (p) instead of Euclidean distance.
Lemma 3.1.
P 2,+ u(p 0 , t 0 ) = {(φ t (p 0 , t 0 ), ∇φ(p 0 , t 0 ), (D 2 φ(p 0 , t 0 )) ⋆ ) : φ ∈ Au(p 0 , t 0 )}.
3.2. Jet Twisting. We recall that the set V 1 = span{X 1 , X 2 , . . . , X n 1 } and notationally, we will always denote n 1 by n. The vectors X i at the point p ∈ G can be written as
X i (p) = N j=1 a ij (p) ∂ ∂x j
forming the n × N matrix A with smooth entries A ij = a ij (p). By linear independence of the X i , A has rank n. Similarly,
Y i (p) = N j=1 b ij (p) ∂ ∂x j forming the n 2 × N matrix B with smooth entries B ij = b ij .
The matrix B has rank n 2 .
The following lemma differs from [5,Corollary 3.2] only in that there is now a parabolic term. This term however, does not need to be twisted. The proof is then identical, as only the space terms need twisting.
Lemma 3.2. Let (a, η, X) ∈ P 2,+ eucl u(p, t). (Recall that (η, X) ∈ R N × S N .) Then (a, A · η ⊕ B · η, AXA T + M) ∈ P 2,+ u(p, t).
Here the entries of the (symmetric) matrix M are given by
M ij = N k=1 N l=1 a il (p) ∂ ∂x l a jk (p) + a jl (p) ∂a ik ∂x l (p) η k i = j, N k=1 N l=1 a il (p) ∂a ik ∂x l (p)η k i = j.
3.3. Viscosity Solutions. We consider parabolic equations of the form
(3.1) u t + F (t, p, u, ∇ 1 u, (D 2 u) ⋆ ) = 0
for continuous and proper F :
[0, T ] × G × R × g × S n → R. [8]
We recall that S n is the set of n × n symmetric matrices (where dim V 1 = n) and the derivatives ∇ 1 u and (D 2 u) ⋆ are taken in the space variable p. We then use the jets to define subsolutions and supersolutions to Equation (3.1) in the usual way.
Definition 1. Let (p 0 , t 0 ) ∈ O t 1 ,t 2 be as above. The upper semicontinuous function u is a parabolic viscosity subsolution in O t 1 ,t 2 if for all (p 0 , t 0 ) ∈ O t 1 ,t 2 we have (a, η, X) ∈ P 2,+ u(p 0 , t 0 ) produces a + F (t 0 , p 0 , u(p 0 , t 0 ), η, X) ≤ 0. A lower semicontinuous function u is a parabolic viscosity supersolution in O t 1 ,t 2 if for all (p 0 , t 0 ) ∈ O t 1 ,t 2 we have (b, ν, Y ) ∈ P 2,− u(p 0 , t 0 ) produces b + F (t 0 , p 0 , u(p 0 , t 0 ), ν, Y ) ≥ 0.
A continuous function u is a parabolic viscosity solution in O t 1 ,t 2 if it is both a parabolic viscosity subsolution and parabolic viscosity supersolution.
Remark 3.3. In the special case when F (t, p, u, ∇ 1 u, (D 2 u) ⋆ ) = F h ∞ (∇ 0 u, (D 2 u) ⋆ ) = −∆ h ∞ u, for h ≥ 3,
we use the terms "parabolic viscosity h-infinite supersolution", etc. In the case when 1 ≤ h < 3, the definition above is insufficient due to the singularity occurring when the horizontal gradient vanishes. Therefore, following [14] and [15], we define viscosity solutions to Equation (2.3) when 1 ≤ h < 3 as follows:
Definition 2. Let O t 1 ,t 2 be as above. A lower semicontinuous function v : O t 1 ,t 2 → R is a parabolic viscosity h-infinite supersolution of u t − ∆ h ∞ u = 0 if whenever (p 0 , t 0 ) ∈ O t 1 ,t 2 and φ ∈ Bu(p 0 , t 0 ), we have φ t (p 0 , t 0 ) − ∆ h ∞ φ(p 0 , t 0 ) ≥ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 φ t (p 0 , t 0 ) − min η =1 (D 2 φ) ⋆ (p 0 , t 0 ) η, η ≥ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 and h = 1 φ t (p 0 , t 0 ) ≥ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 and 1 < h < 3 An upper semicontinuous function u : O t 1 ,t 2 → R is a parabolic viscosity h-infinite subso- lution of u t − ∆ h ∞ u = 0 if whenever (p 0 , t 0 ) ∈ O t 1 ,t 2 and φ ∈ Au(p 0 , t 0 ), we have φ t (p 0 , t 0 ) − ∆ h ∞ φ(p 0 , t 0 ) ≤ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 φ t (p 0 , t 0 ) − max η =1 (D 2 φ) ⋆ (p 0 , t 0 ) η, η ≤ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 and h = 1 φ t (p 0 , t 0 ) ≤ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 and 1 < h < 3
A continuous function is a parabolic viscosity h-infinite solution if it is both a parabolic viscosity h-infinite subsolution and parabolic viscosity h-infinite subsolution.
Remark 3.4. When 1 < h < 3, we can actually consider the continuous operator
F h ∞ (∇ 0 u, (D 2 u) ⋆ ) = − ∇ 0 u h−3 (D 2 u) ⋆ ∇ 0 u, ∇ 0 u = −∆ h ∞ u ∇ 0 u = 0 0 ∇ 0 u = 0. (3.2)
Definitions 1 and 2 would then agree. (cf. [15])
We also wish to define what [12] refers to as parabolic viscosity solutions. We first need to consider the set
A − u(p 0 , t 0 ) = {φ ∈ C 2 (O t 1 ,t 2 ) : u(p, t)−φ(p, t) ≤ u(p 0 , t 0 )−φ(p 0 , t 0 ) = 0 for p = p 0 , t < t 0 }
consisting of all functions that touch from above only when t < t 0 . Note that this set is larger than Au and corresponds physically to the past alone playing a role in determining the present. We define B − u(p 0 , t 0 ) similarly. We then have the following definition.
Definition 3. An upper semicontinuous function u on O t 1 ,t 2 is a past parabolic viscosity subsolution in O t 1 ,t 2 if φ ∈ A − u(p 0 , t 0 ) produces φ t (p 0 , t 0 ) + F (t 0 , p 0 , u(p 0 , t 0 ), ∇ 1 φ(p 0 , t 0 ), (D 2 φ(p 0 , t 0 )) ⋆ ) ≤ 0. An lower semicontinuous function u on O t 1 ,t 2 is a past parabolic viscosity supersolution in O t 1 ,t 2 if φ ∈ B − u(p 0 , t 0 ) produces φ t (p 0 , t 0 ) + F (t 0 , p 0 , u(p 0 , t 0 ), ∇ 1 φ(p 0 , t 0 ), (D 2 φ(p 0 , t 0 )) ⋆ ) ≥ 0.
A continuous function is a past parabolic viscosity solution if it is both a past parabolic viscosity supersolution and subsolution.
We have the following proposition whose proof is obvious. 3.4. The Carnot Parabolic Maximum Principle. In this subsection, we recall the Carnot Parabolic Maximum Principle and key corollaries, as proved in [6]. where Ω is a (bounded) domain and let τ be a positive real parameter. Let φ(p, q, t) = ϕ(p · q −1 , t) be a C 2 function in the space variables p and q and a C 1 function in t. Suppose the local maximum
(3.3) M τ ≡ max Ω×Ω×[0,T ] {u(p, t) − v(q, t) − τ φ(p, q, t)} occurs at the interior point (p τ , q τ , t τ ) of the parabolic set Ω × Ω × (0, T ). Define the n × n matrix W by W ij = X i (p)X j (q)φ(p τ , q τ , t τ ).
Let the 2n × 2n matrix W be given by
W = 0 1 2 (W − W T ) 1 2 (W T − W ) 0 (3.4)
and let the matrix W ∈ S 2N be given by
W = D 2 pp φ(p τ , q τ , t τ ) D 2 pq φ(p τ , q τ , t τ ) D 2 qp φ(p τ , q τ , t τ ) D 2 qq φ(p τ , q τ , t τ ) . (3.5) Suppose lim τ →∞ τ φ(p τ , q τ , t τ ) = 0.
Then for each τ > 0, there exists real numbers a 1 and a 2 , symmetric matrices X τ and Y τ and vector
Υ τ ∈ V 1 ⊕ V 2 , namely Υ τ = ∇ 1 (p)φ(p τ , q τ , t τ )
, so that the following hold:
A) (a 1 , τ Υ τ , X τ ) ∈ P 2,+ u(p τ , t τ ) and (a 2 , τ Υ τ , Y τ ) ∈ P 2,− v(q τ , t τ ). B) a 1 − a 2 = φ t (p τ , q τ , t τ ). C) For any vectors ξ, ǫ ∈ V 1 , we have X τ ξ, ξ − Y τ ǫ, ǫ ≤ τ (D 2 p φ) ⋆ (p τ , q τ , t τ )(ξ − ǫ), (ξ − ǫ) + τ W(ξ ⊕ ǫ), (ξ ⊕ ǫ) + τ W 2 A(p) T ξ ⊕ A(q) T ǫ 2 . (3.6)
In particular,
(3.7) X τ ξ, ξ − Y τ ξ, ξ τ W 2 ξ 2 . Corollary 3.7. Let φ(p, q, t) = φ(p, q) = ϕ(p·q −1 ) be independent of t and a non-negative function. Suppose φ(p, q) = 0 exactly when p = q. Then lim τ →∞ τ φ(p τ , q τ ) = 0.
In particular, if
(3.8) φ(p, q, t) = 1 m N i=1 (p · q −1 ) i m for some even integer m ≥ 4 where (p · q −1 ) i is the i-th componentA) (a 1 , τ Υ τ , X τ ) ∈ P 2,+ u(p τ , t τ ) and (a 1 , τ Υ τ , Y τ ) ∈ P 2,− v(q τ , t τ ). B) The vector Υ τ satisfies Υ τ ∼ φ(p τ , q τ ) m−1 m . C) For any fixed vector ξ ∈ V 1 , we have (3.9) X τ ξ, ξ − Y τ ξ, ξ τ W 2 ξ 2 τ (φ(p τ , q τ )) 2m−4 m ξ 2 .
Uniqueness of viscosity solutions
We wish to formulate a comparison principle for the following problem.
Problem 4.1. Let h ≥ 1.
Let Ω be a bounded domain and let Ω T = Ω × [0, T ). Let ψ ∈ C(Ω) and g ∈ C(Ω T ). We consider the following boundary and initial value problem:
u t + F h ∞ (∇ 0 u, (D 2 u) ⋆ ) = 0 in Ω × (0, T ) (E) u(p, t) = g(p, t) p ∈ ∂Ω, t ∈ [0, T ) (BC) u(p, 0) = ψ(p) p ∈ Ω (IC) (4.1)
We also adopt the definition that a subsolution u(p, t) to Problem 4.1 is a viscosity subsolution to (E), u(p, t) ≤ g(p, t) on ∂Ω with 0 ≤ t < T and u(p, 0) ≤ ψ(p) on Ω.
Supersolutions and solutions are defined in an analogous matter.
Because our solution u will be continuous, we offer the following remark: Proof. Our proof follows that of [8, Thm. 8.2] and so we discuss only the main parts.
For ε > 0, we substituteũ = u − ε T −t for u and prove the theorem for
u t + F h ∞ (∇ 0 u, (D 2 u) ⋆ ) ≤ − ε T 2 < 0 (4.3) lim t↑T u(p, t) = −∞ uniformly on Ω (4.4)
and take limits to obtain the desired result. Assume the maximum occurs at (p 0 , t 0 ) ∈ Ω × (0, T ) with
u(p 0 , t 0 ) − v(p 0 , t 0 ) = δ > 0. Case 1: h > 1.
Let H ≥ h + 3 be an even number. As in Equation (3.8), we let
φ(p, q) = 1 H N i=1 (p · q −1 ) i H where (p · q −1 ) i is the i-th component of the Carnot group multiplication group law. Let M τ = u(p τ , t τ ) − v(q τ , t τ ) − τ φ(p τ , q τ ) with (p τ , q τ , t τ ) the maximum point in Ω × Ω × [0, T ) of u(p, t) − v(q, t) − τ φ(p, q). If t τ = 0, we have 0 < δ ≤ M τ ≤ sup Ω×Ω (ψ(p) − ψ(q) − τ φ(p, q))
leading to a contradiction for large τ . We therefore conclude t τ > 0 for large τ . Since u ≤ v on ∂Ω × [0, T ) by Equation (BC) of Problem (4.1), we conclude that for large τ , we have (p τ , q τ , t τ ) is an interior point. That is, (p τ , q τ , t τ ) ∈ Ω × Ω × (0, T ). Using Corollary 3.7 Property A, we obtain
(a, τ Υ(p τ , q τ ), X τ ) ∈ P 2,+ u(p τ , t τ ) (a, τ Υ(p τ , q τ ), Y τ ) ∈ P 2,− v(q τ , t τ )
satisfying the equations
a + F h ∞ (τ Υ(p τ , q τ ), X τ ) ≤ − ε T 2 a + F h ∞ (τ Υ(p τ , q τ ), Y τ ) ≥ 0.
If there is a subsequence {p τ , q τ } τ >0 such that p τ = q τ , we subtract, and using Corollary 3.7, we have
0 < ε T 2 ≤ (τ Υ(p τ , q τ )) h−3 τ 2 X τ Υ(p τ , q τ ), Υ(p τ , q τ ) − Y τ Υ(p τ , q τ ), Υ(p τ , q τ ) τ h ϕ(p τ , q τ ) H−1 H h−3 (ϕ(p τ , q τ )) 2H−4 H (ϕ(p τ , q τ )) 2H−2 H (4.5) = τ h (ϕ(p τ , q τ )) Hh+H−h−3 H = (τ ϕ(p τ , q τ )) h ϕ(p τ , q τ ) H−h−3 H . (4.6)
Because H > h + 3, we arrive at a contradiction as τ → ∞.
If we have p τ = q τ , we arrive at a contradiction since
F h ∞ (τ Υ(p τ , q τ ), X τ ) = F h ∞ (τ Υ(p τ , q τ ), Y τ ) = 0. Case 2: h = 1.
We follow the proof of Theorem 3.1 in [14]. We let
ϕ(p, q, t, s) = 1 4 N i=1 (p · q −1 ) i 4 + 1 2 (t − s) 2
and let (p τ , q τ , t τ , s τ ) be the maximum of
u(p, t) − v(q, s) − τ φ(p, q, t, s)
Again, for large τ , this point is an interior point. If we have a sequence where p τ = q τ , then Lemma 3.2 yields
(τ (t τ − s τ ), τ Υ(p τ , q τ ), X τ ) ∈ P 2,+ u(p τ , t τ ) (τ (t τ − s τ ), τ Υ(p τ , q τ ), Y τ ) ∈ P 2,− v(q τ , s τ )
satisfying the equations
τ (t τ − s τ ) + F h ∞ (τ Υ(p τ , q τ ), X τ ) ≤ − ε T 2 τ (t τ − s τ ) + F h ∞ (τ Υ(p τ , q τ ), Y τ ) ≥ 0.
As in the first case, we subtract to obtain
0 < ε T 2 ≤ (τ Υ(p τ , q τ )) −2 τ 2 X τ Υ(p τ , q τ ), Υ(p τ , q τ ) − Y τ Υ(p τ , q τ ), Υ(p τ , q τ ) ϕ(p τ , q τ ) − 3 2 (τ ϕ(p τ , q τ )ϕ(p τ , q τ )3
2 ) = τ ϕ(p τ , q τ ). We arrive at a contradiction as τ → ∞.
If p τ = q τ , then v(q, s) − β v (q, s) has a local minimum at (q τ , s τ ) where
β v (q, s) = − τ 4 N i=1 (p τ · q −1 ) i 4 − τ 2 (t τ − s) 2 .
We then have
0 < ε(T − s τ ) −2 ≤ β v s (q τ , s τ ) − min η =1 (D 2 β v ) ⋆ (q τ , s τ ) η, η .
Similarly, u(p, t) − β u (p, t) has a local maximum at (p τ , t τ ) where
β u (p, t) = τ 4 N i=1 (p · q −1 τ ) i 4 + τ 2 (t − s τ ) 2 .
We then have
0 ≥ β u t (p τ , t τ ) − max η =1 (D 2 β u ) ⋆ (p τ , t τ ) η, η
and subtraction gives us
0 < ε(T − s τ ) −2 ≤ max η =1 (D 2 β u ) ⋆ (p τ , t τ ) η, η − min η =1 (D 2 β v ) ⋆ (q τ , s τ ) η, η + β v s (q τ , s τ ) − β u t (p τ , t τ ) = τ max η =1 (D 2 pp ϕ(p · q −1 τ )) ⋆ (p τ , t τ ) η, η − τ min η =1 (D 2 qq ϕ(p τ · q −1 )) ⋆ (q τ , s τ ) η, η + τ (t τ − s τ ) − τ (t τ − s τ ) = 0.
Here, the last equality comes from the fact that p τ = q τ and the definition of ϕ(p·q −1 ).
The comparison principle has the following consequences concerning properties of solutions: Proof. By Proposition 3.5, past parabolic viscosity h-infinite sub(super-)solutions are parabolic viscosity h-infinite sub(super-)solutions. To prove the converse, we will follow the proof of the subsolution case found in [12], highlighting the main details. Assume that u is not a past parabolic viscosity h-infinite subsolution. Let φ ∈ A − u(p 0 , t 0 ) have the property that φ t (p 0 , t 0 ) − ∆ h ∞ φ(p 0 , t 0 ) ≥ ǫ > 0 for a small parameter ǫ. We may assume p 0 is the origin. Let r > 0 and define S r = B N (r) × (t 0 − r, t 0 ) and let ∂S r be its parabolic boundary. Then the functioñ
φ r (p, t) = φ(p, t) + (t 0 − t) 8l! − r 8l! + (N (p)) 8l!
is a classical supersolution for sufficiently small r. We then observe that u ≤φ r on ∂S r but u(0, t 0 ) >φ(0, t 0 ). Thus, the comparison prinicple, Theorem 4.3, does not hold. Thus, u is not a parabolic viscosity h-infinite subsolution. The supersolution case is identical and omitted.
The following corollary has a proof similar to [14,Lemma 3.2].
Corollary 4.5. Let u : Ω T → R be upper semicontinuous. Let φ ∈ Au(p 0 , t 0 ). If φ t (p 0 , t 0 ) − ∆ 1 ∞ φ(p 0 , t 0 ) ≤ 0 when ∇ 0 φ(p 0 , t 0 ) = 0 φ t (p 0 , t 0 ) ≤ 0 when ∇ 0 φ(p 0 , t 0 ) = 0, (D 2 φ) ⋆ (p 0 , t 0 ) = 0 (4.7)
then u is a viscosity subsolution to (E) of Problem (4.1).
We also have the following function estimates with respect to boundary data. Corollary 4.6. Let h ≥ 1. Let g 1 , g 2 ∈ C(Ω T ) and u 1 , u 2 be parabolic viscosity solutions to Equation 4.1 with boundary data g 1 and g 2 , respectively. Then
sup (p,t)∈Ω T |u 1 (p, t) − u 2 (p, t)| ≤ sup (p,t)∈∂parΩ T |g 1 (p, t) − g 2 (p, t)|.
Proof. The function u + (p, t) = u 2 (p, t) + sup (p,t)∈∂parΩ T |g 1 (p, t) − g 2 (p, t)| is a parabolic viscosity supersolution with boundary data g 1 and the function u − (p, t) = u 2 (p, t) − sup (p,t)∈∂parΩ T |g 1 (p, t) − g 2 (p, t)| is a parabolic viscosity subsolution with boundary data g 1 . Moreover, u − ≤ u 1 ≤ u + on ∂ par Ω T and by Theorem 4
.3 u − ≤ u 1 ≤ u + in Ω T .
Corollary 4.7. Let h ≥ 1. Let g ∈ C(Ω T ). Then every parabolic viscosity solution to Problem 4.1 satisfies sup
(p,t)∈Ω T |u(p, t)| ≤ sup (p,t)∈∂parΩ T |g(p, t)|
Proof. The proof is similar to the previous corollary, but using the functions u ± (p, t) = ± sup (p,t)∈∂parΩ T |g(p, t)| instead. Proof. First note that u is lower semicontinous since every v ∈ L is. Let (p 0 , t 0 ) ∈ Ω T and φ ∈ Au(p 0 , t 0 ). Now let ψ(p, t) = φ(p, t) − (d N (p 0 , p)) 2l! − |t − t 0 | 2 and notice that ψ ∈ Au(p 0 , t 0 ). Then
Existence of Viscosity Solutions
(u − ψ)(p, t) − (d N (p 0 , p)) 2l! − |t − t 0 | 2 = (u − φ)(p, t) ≥ (u − φ)(p 0 , t 0 ) = (u − ψ)(p 0 , t 0 ) = 0 yields (5.1) (u − ψ)(p, t) ≥ (d N (p 0 , p)) 2l! + |t − t 0 | 2 .
Since u is lower semicontinuous, there exists a sequence {(p k , t k )} with t k < t 0 converging to (p 0 , t 0 ) as k → ∞ such that
(u − ψ)(p k , t k ) → (u − ψ)(p 0 , t 0 ) = 0. Since u(p, t) = inf {v(p, t) : v ∈ L}, there exists a sequence {v k } ⊂ L such that v k (p k , t k ) < u(p k , t k ) + 1/k for k = 1,2, . . . . Since v k ≥ u, (5.1) gives us (5.2) (v k − ψ)(p, t) ≥ (u − ψ)(p, t) ≥ (d N (p 0 , p)) 2l! + |t − t 0 | 2 .
Let B ⊂ Ω denote a compact neighborhood of (p 0 , t 0 ). Since v k − ψ is lower semicontinuous, it attains a minimum in B at a point (q k , s k ) ∈ B. Then by (5.1) and (5.2) we have
(u − ψ)(p k , t k ) + 1/k > (v k − ψ)(p k , t k ) ≥ (v k − ψ)(q k , s k ) ≥ (d N (p 0 , q k )) 2l! + |s k − t 0 | 2 ≥ 0
for sufficiently large k such that (p k , t k ) ∈ B. By the squeeze theorem, (q k , s k ) → (p 0 , t 0 ) as k → ∞. Let η = ψ − (d N (q k , p)) 2l! − |s k − t| 2 . Then η ∈ Av k (q k , s k ) and we have that
η t (q k , s k ) + F (s k , q k , v k (q k , s k ), ∇ 1 η(q k , s k ), (D 2 η(q k , s k )) ⋆ ) ≥ 0.
This implies
ψ t (q k , s k ) + F (s k , q k , v k (q k , s k ), ∇ 1 ψ(q k , s k ), (D 2 ψ(s k , s k )) ⋆ ) ≥ 0. Letting k → ∞ yields φ t (p 0 , t 0 ) + F (t 0 , p 0 , u(p 0 , t 0 )∇ 1 φ(p 0 , t 0 ), (D 2 φ(p 0 , t 0 )) ⋆ ) ≥ 0.
and that u is a parabolic viscosity supersolution as desired.
A similar argument yields the following. For the following lemmas, we need to recall the following definition. inf{u(q, s) : |q −1 p| g + |s − t| ≤ r}, respectively.
Lemma 5.3. Let h be a parabolic viscosity supersolution to (3.1) in Ω T . Let S be the collection of all parabolic viscosity subsolutions v of (3.1) satisfying v ≤ h. If forv ∈ S, v * is not a parabolic viscosity supersolution of (3.1) then there is a function w ∈ S and a point (p 0 , t 0 ) such thatv(p 0 , t 0 ) < w(p 0 , t 0 ).
Proof.
Letv ∈ S such thatv * is not a parabolic viscosity supersolution of (3.1). Then there exists (p,t) ∈ Ω T and φ ∈ Av * (p,t) such that
(5.3) φ t (p, t) + F (t, p,v * (p, t), ∇ 1 φ(p, t), (D 2 φ(p, t)) ⋆ ) > 0. Let ψ(p, t) = φ(p, t) − (d N (p, p)) 2l! − |t −t| 2
and notice that ψ ∈ Av * (p,t). As in Lemma 5.1,
(5.4) (v * − ψ)(p, t) ≥ (d N (p, p)) 2l! + |t −t| 2 .
Let B denote a compact neighborhood of (p,t) and let B kǫ = B ∩ (p, t) : (d N (p, p)) 2l! ≤ kǫ and |t −t| 2 ≤ kǫ .
Sincev ∈ S, we have thatv ≤ h and thus ψ(p,t) =v * (p,t) ≤v(p,t) ≤ h(p,t). However, if ψ(p,t) = h(p,t), then ψ ∈ Ah(p,t) and inequality (5.3) would be contradictory. Thus, ψ(p,t) < h(p,t).
Since ψ is continuous and h is lower semicontinuous, there exists ǫ > 0 such that
ψ(p, t) + 4ǫ ≤ h(p, t)
for (p, t) ∈ B 2ǫ . Notice that ψ +4ǫ is a subsolution of (3.1) on the interior of B 2ǫ . Further, by (5.4)
(5.5)v(p, t) ≥v * (p, t) ≥ ψ(p, t) + 4ǫ for (p, t) ∈ B 2ǫ \B ǫ .
We now define ω by
ω = max{ψ(p, t) + 4ǫ,v(p, t)} (p, t) ∈ B ǫ v(p, t) (p, t) ∈ Ω T \B ǫ
But by (5.5) ω(p, t) = max{ψ(p, t) + 4ǫ,v(p, t)} for (p, t) ∈ B 2ǫ , not just for (p, t) ∈ B ǫ . Then by Lemma 5.2, ω is a subsolution in the interior of B 2ǫ and thus a subsolution in Ω T . Therefore, ω ∈ S. Since
0 = (v * − ψ)(p,t) = lim r↓0 inf {(v − ψ)(p, t) : (p, t) ∈ B r } there is a point (p 0 , t 0 ) ∈ B ǫ that satisfieŝ v(p 0 , t 0 ) − ψ(p 0 , t 0 ) < 4ǫ which yieldsv (p 0 , t 0 ) < ψ(p 0 , t 0 ) + 4ǫ = ω(p 0 , t 0 ).
Thus, we have constructed ω ∈ S that satisfiesv(p 0 , t 0 ) < ω(p 0 , t 0 ).
We then have the following existence theorem concerning parabolic viscosity solutions. Proof. Let S = {ν : ν is a parabolic viscosity subsolution to (3.1) in Ω T with ν ≤ g in Ω T } and u(p, t) = sup{ν(p, t) : ν ∈ S}.
Since f ≤ g, the set S is nonempty. Notice that f ≤ u ≤ g by construction. By Lemma (5.2), u is a parabolic viscosity subsolution. Suppose u * is not a parabolic viscosity supersolution. Then by Lemma 5.3, there exists a function w ∈ S and a point (p 0 , t 0 ) ∈ Ω T such that u(p 0 , t 0 ) < w(p 0 , t 0 ). But this contradicts the definition of u at (p 0 , t 0 ). Thus u * is a parabolic viscosity supersolution. By our assumptions on f and g on ∂ par O 0,T ,
u = u * ≤ g * = f * ≤ u * on ∂ par O 0,T .
Then by the (assumed) comparison principle, u ≤ u * on Ω T . Thus we have u is a parabolic viscosity solution such that u ∈ C(O T ).
5.2.
The h = 1 case. We begin by recalling the definition of upper and lower relaxed limit of a function. [8,10].
1 ∞ (∇ 0 u, (D 2 u) ⋆ ) = F 1 ∞ (∇ 0 u, (D 2 u) ⋆ ) = − ∇ 0 u −2 (D 2 u) ⋆ ∇ 0 u, ∇ 0 u ∇ 0 u = 0 0 ∇ 0 u = 0.
We give this operator the label F (∇ 0 u, (D 2 u) ⋆
u t + F (∇ 0 u, (D 2 u) ⋆ ) = 0.
We have the following comparison principle, whose proof is similar to Theorem 4.3 in the case to h = 1 and is omitted.
Lemma 5.5.
Let Ω be a bounded domain in G. If u is a parabolic viscosity subsolution and v a parabolic viscosity supersolution to
u t + F (∇ 0 u, (D 2 u) ⋆ ) = 0. then u ≤ v on Ω T ≡ Ω × [0, T ).
Corollary 5.6. u(p, t) = u(p, t).
Proof. By construction, u(p, t) ≤ u(p, t). By the Lemma, u(p, t) ≥ u(p, t).
Using the corollary, we will call this common relaxed limit u 1 (p, t). By [10, Chapter 2] and [8, Section 6], it is continuous and the sequence {u h (p, t)} converges locally uniformly to u 1 (p, t) as h → 1 + .
We then have the following theorem. Proof. Let {u h (p, t)} and u 1 (p, t) be as above. Let {h j } be a subsequence with h j → 1 + where u h (p, t) → u 1 (p, t) uniformly. We may assume h j < 3.
Let φ ∈ Au 1 (p 0 , t 0 ). Using the uniform convergence, there is a sequence {p j , t j } → (p 0 , t 0 ) so that φ ∈ Au h j (p j , t j ). If ∇ 0 φ(p 0 , t 0 ) = 0, we have ∇ 0 φ(p j , t j ) = 0 for sufficiently large j. We then have
φ t (p j , t j ) − ∆ h j ∞ φ(p j , t j ) ≤ 0 and letting j → ∞ yields φ t (p 0 , t 0 ) − ∆ 1 ∞ φ(p 0 , t 0 ) ≤ 0. Suppose ∇ 0 φ(p 0 , t 0 ) = 0.
By Corollary 4.5, we may assume (D 2 φ) ⋆ (p 0 , t 0 ) = 0. Suppose passing to a subsequence if needed, we have ∇ 0 φ(p j , t j ) = 0. Then
φ t (p j , t j ) − max η =1 (D 2 φ) ⋆ (p j , t j ) η, η ≤ φ t (p j , t j ) − ∆ h j ∞ φ(p j , t j ) ≤ 0. Letting j → ∞ yields φ t (p 0 , t 0 ) = φ t (p j , t j ) − max η =1 (D 2 φ) ⋆ (p 0 , t 0 ) η, η ≤ 0.
In the case ∇ 0 φ(p j , t j ) = 0, since h j < 3, we have φ t (p j , t j ) ≤ 0 and letting j → ∞ yields φ t (p 0 , t 0 ) ≤ 0. We conclude that u 1 is a parabolic viscosity h-infinite subsolution. Similarly, u 1 is a parabolic viscosity h-infinite supersolution.
6. The limit as t → ∞.
We now focus our attention on the asymptotic limits of the parabolic viscosity h-infinite solutions. We wish to show that for 1 ≤ h, we have the (unique) viscosity solution to u t − ∆ h ∞ u = 0 approaches the viscosity solution of −∆ h ∞ u = 0 as t → ∞. Our goal is the following theorem: Theorem 6.1. Let h > 1 and u ∈ C(Ω × [0, ∞)) be a viscosity solution of
(6.1) u t − ∆ h ∞ u = 0 in Ω × (0, ∞), u(p, t) = g(p)
on ∂ par (Ω × (0, ∞)) with g : Ω → R continuous and assuming that ∂Ω satisfies the property of positive geometric density (see [12, pg. 2909]). Then u(p, t) → U(p) uniformly in Ω as t → ∞ where U(p) is the unique viscosity solution of −∆ h ∞ U = 0 with the Dirichlet boundary condition lim q→p U(q) = g(p) for all p ∈ ∂Ω.
We first must establish the uniqueness of viscosity solutions to the limit equation. Note that for future reference, we include the case h = 1. Proof. Let u be a viscosity subsolution to −∆ h ∞ u = 0. Then choose φ ∈ C 2 sub (Ω) such that 0 = φ(p 0 ) − u(p 0 ) < φ(p) − u(p) for p ∈ Ω, p = p 0 . If ∇ 0 φ(p 0 ) = 0, then − (D 2 φ) ⋆ (p 0 )∇ 0 φ(p 0 ), ∇ 0 φ(p 0 ) = 0 ≤ 0. If ∇ 0 φ(p 0 ) = 0, we then have
−∆ h ∞ φ(p 0 ) = − ∇ 0 φ(p 0 ) h−3 (D 2 φ) ⋆ (p 0 )∇ 0 φ(p 0 )
, ∇ 0 φ(p 0 ) ≤ 0. Dividing, we have − (D 2 φ) ⋆ (p 0 )∇ 0 φ(p 0 ), ∇ 0 φ(p 0 ) ≤ 0. In either case, u is a viscosity subsolution to −∆ 3 ∞ u = 0. Similarly, v is a viscosity supersolution to −∆ 3 ∞ u = 0. The theorem follows from the corresponding result for −∆ 3 ∞ u = 0 in [5,3,16]. We state some obvious corollaries: Corollary 6.3. Let 1 ≤ h < ∞ and let g : ∂Ω → R be continuous. Then there exactly one solution to −∆ h ∞ u = 0
in Ω u = g on ∂Ω.
Corollary 6.4. Let 1 ≤ h < ∞ and let g : ∂Ω → R be continuous. The unique viscosity solution to −∆ h ∞ u = 0
in Ω u = g on ∂Ω is the unique viscosity solution to −∆ 3 ∞ u = 0 in Ω u = g on ∂Ω.
Our method of proof for Theorem 6.1 follows that of [12,Theorem 2], the core of which hinges on the construction of a parabolic test function from an elliptic one. In order to construct such a parabolic test function, we need to examine the homogeneity of Equation (6.1). A quick calculation shows that for a fixed h > 1, k 1 h−1 u(x, kt) is a C 2 sub solution to Equation (6.1) if u(x, t) is a C 2 sub solution. A routine calculation then shows parabolic viscosity h-infinite solutions share this homogeneity. We use this property in the following lemma, the proof of which can be found in [9, pg. 170]. (Also, cf. [6, Lemma 6.2] and [12].) Lemma 6.5. Let u be as in Theorem 6.1 and h > 1. Then for every (x, t) ∈ Ω × (0, ∞) and for 0 < T < t, we have
|u(x, t − T ) − u(x, t)| ≤ 2||g|| ∞,Ω h − 1 1 − T t h 1−h T t
Proof. [Theorem 6.1] Fix h > 1. Let u be a viscosity solution of (6.1). The results of [9, Chapter III] imply that the family {u(·, t) : t ∈ (0, ∞)} is equicontinuous. Since it is uniformly bounded due to the boundedness of g, Arzela-Ascoli's theorem yields that there exists a sequence t j → ∞ such that u(·, t j ) converge uniformly in Ω to a function U ∈ C(Ω) for which U(p) = g(p) for all p ∈ ∂Ω. Since it is known from [5,Lemma 5.5] that the Dirichlet problem for the subelliptic p-Laplace equation possesses a unique solution, it is enough to show that U is a viscosity p-subsolution to −∆ p U = 0 on Ω. With
Proposition 3 . 5 .
35Past parabolic viscosity sub(super-)solutions are parabolic viscosity sub(super-)solutions. In particular, past parabolic viscosity h-infinite sub(super-)solutions are parabolic viscosity h-infinite subsub(super-)solutions for h ≥ 1.
Lemma 3 . 6 (
36Carnot Parabolic Maximum Principle). Let u be a viscosity subsolution to Equation (3.1) and v be a viscosity supersolution to Equation (3.1) in the bounded parabolic set Ω × (0, T )
Remark 4. 2 .
2The functions ψ and g may be replaced by one function g ∈ C(Ω T ). This combines conditions (E) and (BC) into one condition(4.2) u(p, t) = g(p, t), (p, t) ∈ ∂ par Ω T (IBC) Theorem 4.3.Let Ω be a bounded domain in G and let h ≥ 1. If u is a parabolic viscosity subsolution and v a parabolic viscosity supersolution to Problem (4.1) then u ≤ v on Ω T ≡ Ω × [0, T ).
Corollary 4 . 4 .
44Let h ≥ 1. The past parabolic viscosity h-infinite solutions are exactly the parabolic viscosity h-infinite solutions.
Lemma 5. 1 .
1Let L be a collection of parabolic viscosity supersolutions to (3.1) and let u(p, t) = inf{v(p, t) : v ∈ L}. If u is finite in a dense subset of Ω T = Ω × [0, T ) then u is a parabolic viscosity supersolution to(3.1).
Lemma 5. 2 .
2Let L be a collection of parabolic viscosity subsolutions to (3.1) and let u(p, t) = sup{v(p, t) : v ∈ L}. If u is finite in a dense subset of Ω T then u is a parabolic viscosity subsolution to(3.1).
Definition 4 .
4The upper and lower semi-continuous envelopes of a function u are given by u * (p, t) := lim r↓0 sup{u(q, s) : |q −1 p| g + |s − t| ≤ r} and u * (p, t) := lim r↓0
Theorem 5. 4 .
4Let f be a parabolic viscosity subsolution to (3.1) and g be a parabolic viscosity supersolution to (3.1) satisfying f ≤ g on Ω T and f * = g * on ∂ par O 0,T . Then there is a parabolic viscosity solution u to (3.1) satisfying u ∈ C(O T ). Explicitly, there exists a unique parabolic viscosity infinite solution to Problem 4.1 when h > 1.
Definition 5 .
5For ε > 0, consider the function h ε : O T ⊂ G → R. The upper relaxed limit h(p, t) and the lower relaxed limit h(p, t) δ (p,t) : O T ∩ B ε (p,t)} Taking the relaxed limits as h → 1 + of the operator F h ∞ (∇ 0 u, (D 2 u) ⋆ ) in Equation 3.2, we have via the continuity of the operator
F
Theorem 5. 7 .
7There exists a unique parabolic viscosity infinite solution to Problem 4.1 when h = 1.
Theorem 6 . 2 .
62Let 1 ≤ h < ∞ and let Ω be a bounded domain. Let u be a viscosity subsolution to ∆ h ∞ u = 0 and let v be a viscosity supersolution to −∆ h ∞ u = 0. Then, sup p∈Ω (u(p) − v(p)) = sup p∈∂Ω (u(p) − v(p)).
of the Carnot group multiplication group law, then for the vector Υ τ and matrices X τ , Y τ , from the Lemma, we have
5.1.Parabolic Viscosity Infinite Solutions: The Continuity Case. As above, we will focus on the equations of the form (3.1) for continuous and proper F : [0, T ]×G×R× g × S n 1 → R that possess a comparison principle such as Theorem 4.3 or [6, Thm. 3.6].
We will use Perron's method combined with the Carnot Parabolic Maximum Principle
to yield the desired existence theorem. In particular, the following proofs are similar to
those found in [10, Chapter 2] except that the Euclidean derivatives have been replaced
with horizontal derivatives and the Euclidean norms have been replaced with the gauge
norm.
). Consider the relaxed limits u(p, t) and u(p, t) of the sequence of unique (continuous) viscosity solutions to Problem 4.1 {u h (p, t)} as h → 1 + . By [10, Thm 2.2.1], we have u(p, t) is a viscosity subsolution and u(p, t) is a viscosity supersolution to
that in mind, let p 0 ∈ Ω and choose φ ∈ C 2 sub (Ω) such that 0 = φ(p 0 )−U(p 0 ) < φ(p)−U(p) for p ∈ Ω, p = p 0 . Using the uniform convergence, we can find a sequence p j → p 0 such that u(·, t j ) − φ has a local maximum at p j . Now definewhere C = 2||g|| ∞,Ω /(h − 1). Note that φ j (p, t) ∈ C 2 sub (Ω × (0, ∞)). Then using Lemma 6.5,for any p ∈ Ω and 0 < t < t j . Thus we have that φ j is an admissible test function at (p j , t j ) on Ω × [0, T ]. Therefore,The theorem follows by letting j → ∞.Combining the results of the previous sections, we have the following theorem:Theorem 6.6. The following diagram commutes:Proof. By Theorem 6.1, Corollary 6.4, and Theorem 5.7, the top, bottom and left limits exist, with the left limit being a uniform limit. By results of iterated limits (see, for example,[1]), we have the fourth limit exists, as does the full limit. In particular, lim h→1 + t→∞ u h,t = lim h→1 + lim t→∞ u h,t = lim t→∞ lim h→1 + u h,t = u 1,∞
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| []
|
[
"Antisymmetric field in string gas cosmology",
"Antisymmetric field in string gas cosmology"
]
| [
"Igmar C Rosas-López \nDepartment of Particle and Nuclear Physics Tsukuba\nThe Graduate University for Advanced Studies (Sokendai)\n305-0801IbarakiJapan\n",
"Yoshihisa Kitazawa \nKEK Theory Center Tsukuba\n305-0801IbarakiJapan\n\nDepartment of Particle and Nuclear Physics Tsukuba\nThe Graduate University for Advanced Studies (Sokendai)\n305-0801IbarakiJapan\n"
]
| [
"Department of Particle and Nuclear Physics Tsukuba\nThe Graduate University for Advanced Studies (Sokendai)\n305-0801IbarakiJapan",
"KEK Theory Center Tsukuba\n305-0801IbarakiJapan",
"Department of Particle and Nuclear Physics Tsukuba\nThe Graduate University for Advanced Studies (Sokendai)\n305-0801IbarakiJapan"
]
| []
| We study how the introduction of a 2-form field flux modify the dynamics of a T-duality invariant string gas cosmology model of Greene, Kabat and Marnerides. It induces a repulsive potential term in the effective action for the scale factor of the spacial dimensions. Without the 2-form field flux, the universe fails to expand when the pressure due to string modes vanishes. With the presence of a homogeneous 2-form field flux, it propels 3 spacial dimensions to grow into a macroscopic 4 dimensional space-time. We find that it triggers an expansion of a universe away from the oscillating phase around the self-dual radius. We also investigate the effects of a constant 2-form field. We can obtain an expanding 4 dimensional space-time by tuning it at the critical value. | 10.1103/physrevd.82.126005 | [
"https://arxiv.org/pdf/1007.1521v1.pdf"
]
| 118,443,215 | 1007.1521 | 835892392d81488c7429d592866258530a215cc1 |
Antisymmetric field in string gas cosmology
June 2010 9 Jul 2010
Igmar C Rosas-López
Department of Particle and Nuclear Physics Tsukuba
The Graduate University for Advanced Studies (Sokendai)
305-0801IbarakiJapan
Yoshihisa Kitazawa
KEK Theory Center Tsukuba
305-0801IbarakiJapan
Department of Particle and Nuclear Physics Tsukuba
The Graduate University for Advanced Studies (Sokendai)
305-0801IbarakiJapan
Antisymmetric field in string gas cosmology
June 2010 9 Jul 2010
We study how the introduction of a 2-form field flux modify the dynamics of a T-duality invariant string gas cosmology model of Greene, Kabat and Marnerides. It induces a repulsive potential term in the effective action for the scale factor of the spacial dimensions. Without the 2-form field flux, the universe fails to expand when the pressure due to string modes vanishes. With the presence of a homogeneous 2-form field flux, it propels 3 spacial dimensions to grow into a macroscopic 4 dimensional space-time. We find that it triggers an expansion of a universe away from the oscillating phase around the self-dual radius. We also investigate the effects of a constant 2-form field. We can obtain an expanding 4 dimensional space-time by tuning it at the critical value.
Introduction
Since it was first proposed by Brandenberger and Vafa [1,2,3,4], the string gas cosmology scenario has generated a significant amount of interest. One of its most appealing characteristics is that it provides a mechanism for dynamically generating a four dimensional space-time. This argument is based on the assumption that strings interact mainly by intersecting each other. If that is the case, the probability of intersection in space-time of two worldsheets has non-zero measure only if the dimension is equal or less than 4. This is a classical argument and it is not obvious that it will remain true if quantum effects are taken into account. There have been several attempts at trying to formulate and prove the Brandenberger-Vafa mechanism with mixed results [5,6,7,8,9]. A recent work [10], for example, succeeds in decompactifying 3 large spacial dimensions for a gas of diluted strings.
Apart from the Brandenberger-Vafa mechanism, there have been other attempts to produce a mechanism for realizing a four dimensional space-time [11,12]. One of these scenarios consists in the inclusion of a two-form field. This field is already present in the supergravity action, hence, it is natural to consider its appearance in the equations of motion. Cosmologies with a two-form field had been studied in the past and several solutions are known [13]. In the context of string gas cosmology, a two-form field flux was introduced for dilaton-gravity in [14,15]. In these solutions the 2-form field flux is restricted to a 4-dimensional submanifold of space-time. The two-form field flux introduces a repulsive potential in the equations of motion for the scale factor of the spacial dimensions. As a result the expansion of three spacial dimensions is enhanced and the corresponding scale factors become large.
In [16], Greene et al. introduced a higher derivative dilaton gravity model. This model replaces the Newtonian-like kinetic terms in the dilaton gravity action by their relativistic counterparts. By doing this, one obtains a model with some nice features: derivatives with respect to the cosmic time become bounded, singularities at finite time are avoided, bounces on the scale factor are produced and loitering phases that solve the horizon problem are realized. Since this model respects T-duality, it has been applied to investigate stringy cosmology such as the Brandenberger-Vafa scenario. The results has shown that this model could lead to a 4 dimensional space-time only with a fine tuning. In general 3 large spacial dimensions are not preferred and any number of dimensions could become large.
Because of the many appealing features of this model, it is interesting to investigate its behavior in the presence of an antisymmetric tensor field. The effects of such kind of field have been studied in several works [13,17] in the context of dilaton gravity. There have been some studies also for the string gas cosmology [14,15] case. For string gas cosmology, it is specially interesting to introduce a 2-form gauge field. Since the initial configuration is supposed to be compactified on a d-dimensional torus, there exist non-trivial effects even for a constant gauge field as strings can wrap the compact dimensions.
In section 2, we briefly recall the low-energy string effective action. In section 3, we introduce a T-duality invariant effective action with 2-form field flux. We obtain analytic solutions of the effective action in several limiting cases. We find they can explain qualitative behaviors of the numerical solutions. One of the main features of the string cosmology model is the introduction of a Hagedorn phase for the early universe. It arguably removes the initial singularity of the universe. We find that a homogeneous 2-form field flux triggers an expansion of a universe away from the oscillating phase around the self-dual radius. An interesting stringy effect, as explained in section 4, is that an electric like two-form field modifies the effective string tension and the Hagedorn temperature [18,19]. In such a case, the energy of the winding modes can vanish for the directions parallel to the electric field. We find that these spacial directions can expand even if the winding modes are present. We conclude in section 5.
Low-energy string effective action
In this section we recall the low energy effective action for string theory.
The low energy effective action in the string frame is given by [20]
S = 1 2κ 2 10 d 10 x √ −ge −φ R + (∇φ) 2 − V − 1 12 H 2 + d 10 x √ −gL matter (2.1)
where κ 2 10 = 8πG 10 and
H 2 = H µνλ H µνλ with H µνλ = ∂ [µ B νλ] .
The sign convention is all + according to the classification in Misner, Thorne and Wheeler [21]. The variation of this action gives the equations of
motion R ν µ − 1 2 g ν µ R = κ 2 10 e φ T ν µ + 1 12 (3H µλκ H νλκ − 1 2 g ν µ H 2 ) − 1 2 g ν µ V − 1 2 g ν µ (∇φ) 2 + (g ν µ g λκ − g λ µ g νκ )∇ λ ∇ κ φ (2.2) ∇ µ (e −φ H µνλ ) = 0 (2.3) 2 φ + R − (∇φ) 2 − V − 1 12 H 2 = 0 (2.4)
where T µν is the energy-momentum tensor derived from the matter Lagrangian. We assume the space-time metric is of the following type in the string frame
ds 2 = −dt 2 + a 2 dx i dx i + b 2 dx I dx I , i = {1, 2, 3} , I = {4, ..., 9} (2.5) with a ≡ e λ(t) , b ≡ e ν(t) (2.6)
By a conformal rescalingg
µν = e − φ 4 g µν (2.7)
we obtain the effective action in the Einstein frame
S = 1 2κ 2 10 d 10 x −g R − 1 8 (∇ µ φ) 2 − V e φ 4 − 1 12 e − φ 2H 2 + d 10 x −g e − 5φ 4 L matter (2.8)
The field equations areR
µν − 1 2g µνR = κ 2 10 (T µν + (H)T µν + (φ)T µν + (V )T µν ) (2.9) ∇ µ (e − φ 2H µνλ ) = 0 (2.10) φ − V e φ 4 + 1 6 e − φ 4H 2 = 0 (2.11)
The homogeneous metric is given by
ds 2 = −dt 2 +ã 2 dx i dx i +b 2 dx I dx I , i = {1, 2, 3} , I = {4, ..., 9} (2.12) withã ≡ e α(t) ,b ≡ e β(t) (2.13)
More detailed relations between the string frame and the Einstein frame are explained in the appendix A.
We need a nontrivial solution for the field strength H µνλ to investigate its effects on the cosmology. The equation of motion for the two-form field (2.10) can be solved using the Freund-Rubin ansatz [22]
H µνα = e φ µναβ ∇ β h , with µνλκ = 4! √ −g δ µ [0 δ ν 1 δ λ 2 δ κ 3]
(2.14)
We assume here that the two-form field flux exists only in three spacial directions. This assumption is consistent with the symmetry of the postulated space-time metric. Because ∇ µ µναβ = 0, the equation of motion is automatically satisfied. It only remains to satisfy the closure condition [13]
∇ [β H µνα] = 0 (2.15)
The equation of motion is then
3 µ,ν=0 g µν ∂ µ (a 3 b −6 e φ h, ν ) = 0 (2.H iλκ H jλκ = 2H 2 o a 6 δ j i for i, j = {1, 2, 3} (2.20)
In particular we have H 2 = 6H 2 o a −6 . As in [1,5,14,16] we consider a very simple setup with 3 types of matter: isotropic winding modes (with all winding numbers W i = W, i = {1, 2, 3}) with energies
E W = 6W e λ (2.21)
isotropic momentum modes (with all momenta K i = K, i = {1, 2, 3}) with energies
E K = 6Ke −λ (2.22)
and string oscillator modes that are modeled as pressureless dust with energy E dust ‡ . The total energy is the sum
E = E W + E K + E dust + V (2.23)
with V = V (ϕ) the potential for the dilaton. In an adiabatic system the pressures are
P ϕ = ∂E ∂ϕ = ∂F ∂ϕ = − ∂L m ∂ϕ = ∂V ∂ϕ (2.24) P λ = − 1 3 ∂F ∂λ = − 1 3 ∂L m ∂λ = − 1 3 ∂E ∂λ = 2Ke −λ − 2W e λ (2.25)
The energy of the string gas is defined as
E s ≡ E W + E K + E dust .
In order to model the behavior of the gas, we consider the following phases as in [16]:
• Hagedorn phase: thermal equilibrium at temperature T H = 1/( √ 8π). The free energy of the gas vanishes (P λ = 0) and E s is conserved
W = √ E s 12 √ π e −λ , K = √ E s 12 √ π e λ (2.26)
• Radiation phase: thermal equilibrium at T < T H with the universe dominated by massless string modes. In d + 1 dimensional space-time, the internal energy is
E s = c d V d T d+1 , c d = 128 2d!ζ(d + 1) (4π) d/2 γ(d/2) (2 − 2 −d ) (2.27) with V d = (2π) d e d|λ| :
the T-duality invariant volume.
F = E s − T S = − 1 d c d V d T d+1 (2.28) P λ = sign(λ)E s /d (2.29) λ > 0 : W = 0 , K = 1 2 P λ e λ (radiation phase) (2.30) λ < 0 : W = − 1 2 P λ e −λ , K = 0 (winding mode dominated phase) (2.31)
Note that the radiation and winding mode dominated phases are T-dual to each other.
• Frozen phase: in this phase the interactions between strings are turned off. The momentum and winding numbers are conserved, so K and W are frozen at the values they have on Hagedorn exit.
• Non-equilibrium phase: In order to model the string gas, we consider a phase in which the the temperature falls below the Hagedorn temperature. Since we also consider the interactions among the strings, the expectation value of the momenta and winding number deviate from their equilibrium values such that the pressure of the string gas does not vanish. ‡ E dust also contains the contributions from strings with momenta and windings along 6 extra dimensions.
T-duality invariant action with two-form field
In order to analyze the effect of the two-form field, we work in the string frame. This allow us to choose a solution where the scale factor ν, defined in (2.6), becomes constant and the analysis can be restricted to a 4-dimensional cosmology in the presence of a two-form field.
In the case of a homogeneous space-time [14,15,17], the action (2.1) can be reduced to
S = dt 4πe −ϕ (dλ 2 −φ 2 − U 0 (λ)) + L m (3.1)
with L m : the matter lagrangian, ϕ: related to the original dilaton φ as ϕ = 2φ − dλ . The potential U (λ) arises due to the nontrivial two-form field strength H µνλ , as shown in [13,14]
H 2 µνα = 12H 2 0 e −2dλ ≡ 24U 0 (λ) (3.2)
The parameter d counts the number of spacial dimensions with the homogeneous scale factor λ. Although our case corresponds to d = 3 such that the space-time is 4-dimensional, we retain d dependence explicitly
in the equations of motion in order to keep track of the algebra.
Now we proceed as [16] and replace the canonical kinetic terms by their higher derivative extensions.
This leads to a phenomenological action with bounded velocitiesφ,λ. It thus rules out singularities at any finite proper time. With this modification we obtain a higher derivative action for the dilaton and scale factor which are coupled to a two-form field strength
S = dt 8πe −ϕ 1 −φ 2 − 1 − dλ 2 − U (λ) + L m (3.3)
L m = −F is the negative of the matter free energy (of the string gas) and U (λ) is a modified potential as explained below.
String gas cosmology model needs to respect T-duality, a fundamental symmetry in string theory originating from the existence of the minimal length (string scale). It is realized as the symmetry between the winding and momentum modes in a toroidal compactification. Since (3.2) is not explicitly invariant under T-duality, it is necessary to modify this potential in order to realize the symmetry. Such a modification allows us to solve the equations of motion near the self-dual radius numerically. An adequate choice is
U (λ) ≡ 1 2 H 2 0 (e 2λ + e −2λ ) −d = 1 2 d+1 H 2 0 (cosh 2λ) −d (3.4)
as (3.4) is not singular at λ = 0 and it reduces to (3.2) for large λ.
Defining the relativistic factors [16] γ ϕ ≡ 1
1 −φ 2 , γ λ ≡ 1 1 − dλ 2 (3.5)
the equations of motion obtained from the action (3.3) arė
γ ϕ =φ(γ ϕ − γ −1 λ − U (λ)) + 1 8π 2φ e ϕ P ϕ (3.6) γ λ =φ(γ λ − γ −1 λ ) −λ ∂U ∂λ + 1 8π 2 dλe ϕ P λ (3.7)
We also need to impose the Hamiltonian constraint
γ ϕ − γ λ − U (λ) = 1 8π 2 Ee ϕ (3.8)
where E is the energy contained in matter. Notice that in the positive energy region γ ϕ − γ λ > U (λ).
The pressure in (3.6), (3.7) for the dilaton and the scale factor are defined in (2.24) and (2.25). Rendering equations (3.6), (3.7) into a more manageable form, we obtain
ϕ = (1 −φ 2 ) 1 − γ −1 ϕ γ −1 λ + U (λ) − 1 8π 2 e ϕ P ϕ (3.9) λ = (1 − dλ 2 ) φλ − γ −1 λ 1 d ∂U ∂λ − 1 8π 2 e ϕ P λ (3.10)
Before trying to find some solutions to the equations of motion, let us examine the equation (3.8) in order to get some idea of the expected behavior. If we put U (λ) on the right side of the equation, we see that λ is subjected to the effective potential figure 1. If we assume, just for the moment, that the dilaton has some fixed value, we can observe the dependence of this potential on λ. The E dependent term in (3.11) grows exponentially as λ increases if the winding modes are present. Consequently, this term tends to confine the scale factor near the self-dual radius. On the other hand U (λ) is a repulsive potential that has its maximun value at the self-dual radius. It decreases exponentially as λ increases. The term containing E, the energy of the string gas, is at the same time modulated by the exponential of the dilaton. Then, as ϕ → −∞, V eff (λ) flattens for large λ. As the confining effect of V eff (λ) diminishes in such a situation, U (λ)
V eff (λ) = U (λ) + 1 8π 2 Ee ϕ (3.11) A schematic plot of V eff (λ) is presented in
becomes dominant and the scale factor is able to continue growing. It is also possible, depending on the initial conditions, for λ to undergo oscillations around one of the minima of the potential or the self dual radius. In general, as the dilaton is going to weak coupling, these oscillations stop and the scale factor is forced to expand by U (λ).
We have explained that the phenomenological action (3.3) with d = 3 is valid for a special class of solutions in superstring theory. In this paper, we investigate these solutions in which only the scale factors for 3 spacial dimensions are time dependent. The dynamics of this kind of cosmology has been studied before in several works [13,17,20] where solutions for the case d = 3 have been found.
We should also be careful to point out that physical interpretation may depend on a chosen conformal frame. Unless we are able to fix the value of the dilaton φ, we can not consistently conclude that the size of a dimension would remain small in the Einstein frame even if it becomes constant in the string frame.
Nevertheless we may argue that the string frame is theoretically preferred to measure the size of the universe as T-duality holds in the string frame. Even with this limitation in mind, we will go ahead to study the cosmology in the string frame [5,10,14,15,23,24,25,26].
Analytic solutions (d = 3)
Now we present some analytic solutions that can be obtained by solving the equations of motion. First, we assume a simple equation of state P λ = wE, with w a constant and P ϕ = 0 (no dilaton potential). Using Λ V Λ Figure 1: Effective potential V (λ) for the scale factor.
(3.8) and the equations of motion we geẗ
ϕ = (1 −φ 2 ) 1 − (1 −φ 2 ) 1/2 (1 − dλ 2 ) 1/2 − H 2 0 2 d+1 (cosh 2λ) −d (3.12) λ = (1 − dλ 2 ) φλ + γ −1 λ H 2 0 2 d (cosh 2λ) −d tanh 2λ + w(γ ϕ − γ λ − H 2 0 2 d+1 (cosh 2λ) −d ) (3.13)
For the equation of state, we have three specific cases of interest: w = 0, w = 1/d and w = −1/d, that correspond to pressureless matter, radiation dominated era and winding mode dominated era respectively.
As the boundary condition for late time asymptotic behavior, we consideṙ
λ → 0 ,φ → 0 , |λ| → ∞ , ϕ → −∞ (3.14)
In this limit, the equations of motion (3.12), (3.13) can be approximated as
λ =φλ + H 2 0 e −2dλ + w 2 (φ 2 − dλ 2 − H 2 0 e −2dλ ) (3.15) ϕ = 1 2φ 2 + 1 2 dλ 2 + H 2 0 2 e −2dλ (3.16) 3.1.1 H 0 = 0, w = 0 case
We start with the standard string gas cosmology without 2-form field flux. For the case when H 0 = 0, we assume the following ansatz
ϕ = A log t + B (3.17) λ = C log t + D (3.18)
After substituting them in (3.15) and (3.16), we find For completeness, we mention that there is an additional solution when d = 1, w = 0. In this special case, assumingφ = −λ, the equations of motion (3.15), (3.16) reduce to a differential equation in one variablë ϕ =φ 2 , (λ = −λ 2 ). Then we get the solution
ϕ = − 2 1 + dw 2 log t + B (3.19) λ = 2w 1 + dw 2 log t + Dϕ = − log(t + A) + B (3.21) λ = log(t + A) + C (3.22)
with A, B, C constants. This solution is not physically relevant since we do not have the correct number of large space dimensions. Nevertheless, it is interesting to observe that a small coordinate can grow large even in the absence of any driving pressure.
H
0 = 0, w = 0 case
Now, we investigate a universe filled with dust (w = 0) and an antisymmetric tensor potential (H 0 = 0).
Under this conditions, we substitute the ansatz (3.17) on equations (3.12), (3.13) to leading order as
ϕ = 1 2φ 2 + 1 2 dλ 2 + H 2 0 2 e −2dλ (3.23) λ =φλ + H 2 0 e −2dλ (3.24)
Using ansatz (3.17), (3.18), we obtain
ϕ = − 2 + 1 d log t + B (3.25) λ = 1 d log t + 1 2d log H 2 0 d 2 d − 1 (3.26)
This analytic solution is plotted as gray colored straight lines in figure 3 for different values of H 0 . Notice that H 0 fixes the initital value of λ(t = 1) in these solutions. In this case we find that H 0 is able, by itself, to induce the growth of a large scale factor, as can be seen in figure 3(a). In this figure we can see how the two-form field flux induces decompactification for different values of H 0 . Since the two-form field flux is along 3 spacial dimensions, in the absence of winding and momentum modes, this field alone is able to induce the growth of 3 large spacial dimensions. We also notice that the moment in which the scale factor is able to "escape" the constant solution depends on the value of H 0 . For larger values of it, the scale factor begins to increase earlier.
This kind of scenario, in which the two-form field flux happens to be the dominant term, can occur if the pressure coming from the winding and momentum modes becomes negligible (P λ ≈ 0). This happens in generic situations, for example, when the scale factor remains near the self-dual radius, the dilaton goes to weak coupling or when the winding and momentum modes have annihilated.
3.1.3 H 0 = 0, w = 0 case
Finally we investigate the generic case when both the flux and the matter pressure are present. In order to find a solution when the antisymmetric tensor potential is present and the pressure fulfills the equation of state P λ = wE, we use (3.12), (3.13) and the ansatz (3.14). Keeping only up to quadratic terms, we find
− A t 2 = A 2 2t 2 + dC 2 2t 2 + H 2 0 2 e −2d(C log t+D) (3.27) − C t 2 = AC t 2 + H 2 0 e −2d(C log t+D) + w 2 A 2 t 2 − dC 2 t 2 − H 2 0 e −2d(C log t+D) (3.28)
By supposing that C = 1 d , we can eliminate the t −2 dependence on the equations. We obtain then
− A = A 2 2 + 1 2d + H 2 0 2 e −2dD (3.29) − 1 d = A d + H 2 0 e −2dD + w 2 (A 2 − 1 d − H 2 0 e −2dD ) (3.30)
Substituting d = 3 explicitly and solving for A and D, we find
A = 5 − 3w 3(w − 1) (3.31) H 2 0 e −6D = 2 − 12w + 6w 2 9(w − 1) 2 (3.32)
In this way we find a solution
ϕ = 5 − 3w 3(w − 1) log t + B (3.33) λ = 1 3 log t − 1 6 log 2 − 12w + 6w 2 9H 2 o (w − 1) 2 (3.34)
We observe, on equation (3.32) that w is constrained by the inequality
w < 1 − 2 3 ≈ 0.1835 (3.35)
It is not consistent with w = 1/3 (radiation). This problem indicates that we cannot smoothly connect this solutions to those with H 0 = 0.
Perturbative solutions
3.2.1 H 0 = 0, w = 0 case
Due to the difficulty we just encountered, we construct perturbative solutions with non-vanishing flux starting from those with no flux. Using the solutions we have obtained for the case when H 0 = 0, we treat the potential term due to H 0 = 0 as a perturbation to the equations of motion. The small expansion parameter is
δ ≡ H 2 0 (3.36)
We expand the solution in terms of the small parameter δ ϕ = ϕ 0 + δϕ 1 + δ 2 ϕ 2 + · · · (3.37) λ = λ 0 + δλ 1 + δ 2 λ 2 + · · · (3.38) and substitute (3.37), (3.38) into the equations of motion. They describe perturbations around the solutions ϕ 0 , λ 0 obtained in (3.19), (3.20).
From the power series expansion of the equation of motion (3.15), we get the differential equation for the first order terms in δφ
1 =φ 0φ1 + dλ 0λ1 + 1 2 e −2dλ0 (3.39) λ 1 =φ 0λ1 +λ 0φ1 + w(φ 0φ1 −λ 0λ1 ) + de −2dλ0 (1 − w 2 ) (3.40)
After substituting ϕ 0 , λ 0 ,φ 0 ,λ 0 into the equation, we obtain
ϕ 1 = − 2 1 + dw 2 t −1φ 1 + 2dw 1 + dw 2 t −1λ 1 + 1 2 t − 4dw 1+dw 2 e −2dD (3.41) λ 1 = − 2 1 + dw 2 t −1λ 1 + 2w 1 + dw 2 t −1φ 1 + w(− 2 1 + dw 2 t −1φ 1 − 2dw 1 + dw 2 t −1λ 1 ) +dt − 4dw 1+dw 2 e −2dD (1 − w 2 ) (3.42)
For λ 1 , we get a second order differential equation in this waÿ
λ 1 + 2t −1λ 1 − dt − 4dw 1+dw 2 e −2dD (1 − w 2 ) = 0 (3.43)
We can integrate this equation easily. Defining x ≡λ 1 ,ẋ ≡λ 1 we geṫ
x + 2t −1 x = t − 4dw 1+dw 2 e −2dD d(1 − w 2 ) (3.44)
This is a differential equation of the formẋ(t) + f (t)x(t) = g(t) and the solution is given by
x(t) = dt g(t)e f (t)dt + c e f (t)dt (3.45)
with a constant c.
After the integration, we find two different class of solutions:
• 4dw 1+dw 2 = 3 case. x = de −2dD (1 − w 2 )(3 − 4dw 1 + dw 2 ) −1 t − 4dw 1+dw 2 +1 + ct −2 (3.46) λ 1 = de −2dD (1 − w 2 )(3 − 4dw 1 + dw 2 ) −1 (2 − 4dw 1 + dw 2 ) −1 t − 4dw 1+dw 2 +2 − ct −1 + c 0 (3.47)
Here, the leading perturbation contains two different time dependent terms. For the perturbation to be small, the exponent on the first term should fulfill the condition − 4dw 1 + dw 2 + 2 < 0 (3.48)
If it is the case, the influence of the two-form flux induced potential is negligible in comparison to the pressure of the string momentum modes. On the other hand, this condition is not satisfied for w = 0 (pressureless dust) case where the perturbation grows as t 2 . In such a situation the solution λ 0 is unstable and the universe is decompactified due to the presence of the two-form field flux.
• 4dw 1+dw 2 = 3. This is the case for d = 3 and a universe filled with radiation (w = 1/d).
x = s log t t 2 + c t 2 (3.49) δλ = − s + c t − s log t t + c 0 , s = − 5 2 H 2 0 e −6D (3.50)
When t → ∞, we find the leading perturbation as δλ ∼ O(t −1 ) . Therefore the correction to the unperturbed solution is negligible at late time.
H 0 = 0 case with both momentum and winding modes
As we observe in equation (3.10), the pressure coming from the winding modes and the momentum modes is multiplied by e ϕ . If |H 0 | 1 and |ϕ| ≈ 1, the scale factor experiences oscillations in the presence of winding and momentum modes. As ϕ goes to weak coupling, oscillations stop and the pressure terms become small with respect to the H 2 0 potential term. Before this terms becomes significant, the solution is characterized asφ
≈ 0,λ ≈ 0, λ ≈ 0, e ϕ 1 (3.51)
We define a small parameter
≡ H 2 0 2 d (3.52)
Under this approximation, keeping terms to the lowest nontrivial order, we geẗ
ϕ = 1 2 (φ 2 + dλ 2 + ) (3.53) λ =φλ + λ (3.54)
We expand the solution in terms of the small parameter
ϕ = ϕ 0 + ϕ 1 + 2 ϕ 2 + · · · (3.55) λ = λ 0 + λ 1 + 2 λ 2 + · · · (3.ϕ 1 =φ 0φ1 + dλ 0λ1 + 1 2 (3.59) λ 1 =φ 0λ1 +λ 0φ1 + λ 0 (3.60)
where the solutions for ϕ 0 , λ 0 is given in equation (3.19), (3.20). Substituting them in (3.59), (3.60), we
obtainφ 1 = − 2 tφ 1 + 1 2 (3.61) λ 1 = − 2 tλ 1 + c 0 (3.62)
with c 0 a constant. These equations are linear differential equations inφ 1 andλ 1 respectively. They can be solved by multiplying them by the integrating factor e ( dt 2 t ) = e (2 log t) = t 2 . The solutions arė
λ 1 (t) = λ 0 3 t + (const.) λ 1 (t) = λ 0 6 t 2 + (const.)t −1 + const. (3.63) ϕ 1 (t) = 1 6 t + (const.) ϕ 1 (t) = 1 12 t 2 + (const.)t −1 + const. (3.64)
We observe the following features: the solution ϕ 0 , λ 0 becomes unstable if we perturbed it with nonvanishing H 0 . To leading order the solution in this regime behaves like ∼ t 2 . This instability initiates an accelerated expansion of a universe away from the oscillating phase around the self-dual radius. However we also observe that the perturbation also affects the dilaton. As the perturbation becomes dominant, the dilaton begins to grow and goes to strong coupling. This indicates that a bounce on the dilaton has been produced. A result like this looks problematic, since a bounce on the dilaton leads to a violation on the positive energy condition as was noted in [16]. We may not be able to trust our solution there as it also takes the dilaton to strong coupling. This behavior can be observed directly in a numerical solution of the equations ( figure 4). We begin with a string gas of equal number of winding modes and momentum modes.
Before the dilaton goes to weak coupling, the scale factor oscillates around the self-dual radius. Once the dilaton reaches weak coupling region, the oscillations stop and the scale factor stabilizes. Then the potential induced by the two-form field flux becomes dominant and the solution begins to grow as predicted by the perturbed solution. In the next sub-section, we investigate the effects of the string interaction on these problems through the Boltzmann equations.
Effects of string interactions
Up to this moment, we have considered situations in which the winding and momentum numbers are frozen at their initial values. When the string gas falls out of equilibrium in an expanding universe, winding strings in the gas can interact and begin to annihilate. In this section we incorporate, together with the two-form field flux induced potential, the Boltzmann equations that take account of the interaction among strings.
These equations, derived by Polchinski [27], are shown beloẇ
W = − e 2λ+ϕ π (W 2 − W 2 ) (3.65) K = − e −2λ+ϕ π (K 2 − K 2 ) (3.66)
We combine these equations with (3.9), (3.10) and evolve the system numerically. The universe we consider is filled with gas of strings that begins at the self dual radius with equal initial winding and momentum numbers (K = W ). The initial conditions areλ ≈ 1,φ ≈ −1 and the dilaton is going from strong coupling to weak coupling. The numerical results including the effects of the Boltzmann equations are presented in figure 5. We have plotted the behavior of the scale factor λ(t), the dilaton ϕ(t), the winding number w(t) and the momentum number k(t). In figure 5 we observe that, as λ(t) grows, the winding modes begin to annihilate.
Then, there is not enough pressure to make the universe contract and experience bounces. Instead, the contribution from the two-form field becomes dominant and the scale factor tends to the solution (3.25), (3.26) with vanishing pressure where the scale factor grows large due to the flux induced potential.
The behavior of the winding and the momentum number is as expected from the following characteristics of the Boltzmann equations (3.65), (3.66). As the dilaton goes to weak coupling, the interaction rate goes to zero and the values of the winding and momentum numbers become constant. When the scale factor grows large, winding modes annihilate more efficiently because their interaction rate goes as the exponential of the scale factor. On the contrary, the rate of annihilation of the momentum modes becomes smaller because the interaction rate between them decays exponentially with the scale factor. In fact this asymmetry between winding and momentum modes can be observed in figure 5(c).
The result obtained by taking account of the Boltzmann equations suggests an interesting scenario when homogeneous H µνλ is present. If the winding modes annihilate rapidly enough, the effect of 2-from field flux becomes important even at early times. The annihilation of the winding modes could take place even in a loitering phase. In that case the two-form field flux becomes dominant and the the expansion of three large spacial dimensions is realized. We emphasize that this mechanism is different from Brandenberger-Vafa mechanism as the presence of homogeneous 2-form field flux is crucial for three spacial dimensions to grow. Without it, the universe remains to be of microscopic size as the blue line in 5(a) indicates.
Effects of constant B µν
So far, the effect of the two-form field has entered only as a modification to the usual dilaton gravity action, as in [14]. The string gas model, as it stands, couples the modified action of dilaton gravity with that of a gas of strings. In this approach the effect of the background field B µν over the string spectrum is usually neglected. The correction on the energy of the string goes as O(B), thus, this approximation is valid for weak fields. In dilaton-gravity, the contribution of B µν to the action enters via U (λ) ∼ |H| 2 = |dB| 2 . In this case, even if B remains small, H is not necessary so as the space-time variation of B µν could be large.
In principle, if we know the two-form field in terms of the scale factors, we can determine |H| 2 as well as their effect on the string spectrum. We can then make use of the adiabatic approximation to study the time dependence of the compactification radii and get the equations of motion. In practice, a homogeneous solution for supergravity is given in terms of H. This presents a problem since we need B µν , not H, in order to get the string spectrum.
With this prospect, we investigate the simplest case, that of a constant B µν . In this case, the H dependent term on the supergravity action vanishes as well as the contribution to the equations of motion. Nevertheless, since strings carry charge under the gauge field, the effect of B field on closed strings wrapping the compact dimensions is non-trivial.
The Polyakov action in the presence of an antisymmetric field B µν
S = − 1 πα d 2 σ[∂ a X µ ∂ a X µ − ab B µν ∂ α X µ ∂ b X µ ] (4.1)
yields the equations of motion
(∂ 2 τ − ∂ 2 σ )X µ (σ, τ ) = − 1 2 H µ λν ab ∂ a X λ ∂ b X ν (4.2)
with H µνλ ≡ ∂ µ B νρ + ∂ ρ B µν + ∂ ν B ρµ . Then, for a constant B µν we obtain the usual two dimensional wave equation
(∂ 2 τ − ∂ 2 σ )X µ = 0 (4.3)
that allows us to give the solution in a Fourier-Laurent expansioṅ
X µ = α 2 (α 0 + α 0 ) + α 2 n =0
(α n e −in(τ +σ) + α µ n e −in(τ −σ) ) (4.4)
X µ = α 2 (α 0 − α 0 ) + α 2 n =0
(α n e −in(τ +σ) − α µ n e −in(τ −σ) ) (4.5)
As it turns out the zero-modes are the only ones that are affected by the B µν field. The components of the energy momentum tensor and their zero modes are given by
T 01 = 1 2πα ∂ 0 X µ ∂ 1 X µ = 1 2πα α p µ − B µ j (wR) j + α 2 n =0 (α µ n e −in(τ +σ) + α µ n e −in(τ −σ) ) × (wR) µ + α 2 n =0
(α nµ e −in(τ +σ) − α nµ e −in(τ −σ) ) (4.6) (T 01 ) zero modes = n i w i + 1 2 n=1 (α n ·α −n +α −n ·α n − α −n · α n − α n · α −n ) (4.7)
T 00 = 1 4πα (Ẋ ·Ẋ + X · X ) = 1 4πα α p µ − B µ j (wR) j + α 2 n =0 (α n e −in(τ +σ) + α µ n e −in(τ −σ) ) 2 + (wR) µ + α 2 n =0 (α µ n e −in(τ +σ) − α µ n e −in(τ −σ) ) 2 (4.8) (T 00 ) zero modes = 1 α − (α p 0 − B 0 j (wR) j ) 2 + (α p i − B i j (wR) j ) 2 + (wR) i (wR i ) +α n=1
(α n ·α −n +α −n ·α n + α n · −n +α −n · α n ) (4.9)
Imposing the physical constraint that the energy momentum tensor must vanish, we get the energy spectrum for the string
p 0 = B 0 j w j R α + 1 α α 2 n R i n R i + (wR) i (wR) i + B i j B ik (wR) j (wR) k + α B ik n i w k + α (Ñ + N ) (4.10)
and the level matching conditionÑ
− N = n i w i (4.11)
Since all the spacial coordinates are compactified with radius R i , the momentum is quantized as
p i = (n/R) i ,
where i denotes the spacial index.
In order to be able to solve the equations of motion, we need to assume some initial winding and momentum distribution of the string gas. The constant B µν field could be either electric or magnetic type.
We find that the effect of electric type field is very interesting as there is a critical value for which the string tension vanishes for winding modes.
Constant electric type field
Let's consider the case of a homogeneous electric type field in 3-spacial dimensions, with B ≡ B 01 = B 02 = B 03 . In order to demonstrate the most dramatic effect, we assume that strings are aligned in the direction of the electric type field. If this is the case, from (4.10), the energy for the winding modes in (2.21) is modified as
E W = 2d(1 − √ 3B)W e λ (4.12)
with W the winding number and d = 3. From this equation, we see immediately that the effect of the field B µν is to reduce the energy of the winding strings. Also, it follows that B is constrained to take values In particular, when the inequality is saturated B = 1/ √ 3, the energy of the winding modes vanishes. As the pressure they exert also vanishes, the spacial dimensions are expected to expand freely because of the presence of the momentum modes.
0 ≤ B ≤ 1 √ 3 (4.13)
In figure 6 we As we get closer to the critical B, the solution bounces and then stabilizes. When we reach the critical value B = 1/ √ 3, the pressure from the winding modes becomes zero and λ expands just like a universe filled with radiation (momentum modes).
Conclusions
In this work we have investigated some effects of the introduction of a two-form field into the model proposed in [16]. This model provides a bouncing and cycling cosmology and also the possibility of long loitering phases.
It avoids singularities at finite times but fails to realize three large spacial dimensions from Brandenberger-Vafa mechanism. Having this in mind, we have included a two-form field into the action, since it may provide an alternative mechanism for the decompactification of 3 spacial dimensions. We have considered two cases:
homogeneous flux H µνρ and constant gauge field B µν .
Homogeneous H µνρ
In order to make the model compatible with T-duality, as the string gas model requires, we have adopted a phenomenological modification on the potential induced by the two-form field entering the gravity action.
The modified potential is non-singular at λ = 0 and reduces to the correct one when the scale factor |λ| is large. In addition, it provides a repulsive potential that can make the universe expand.
In the investigation of the behavior of the scale factor and the dilaton under the influence of the two-form field flux, we find two different cases:
• Matter dominance:
At early times the scale factor can experience bounces as it is governed by the presence of winding and momentum modes. In section 3.2.2 we have observed that the effect of the two form field is not significant at this early stage of the universe. If we assume only the presence of the momentum modes, the late time solutions reduce to those already found in dilaton cosmology. If this solution is perturbed by the introduction of the two-form field flux potential, its influence vanishes as t → ∞. On that account, this kind of solution is stable under the perturbation and the effect of H µνρ is negligible as the universe expands.
• Two-form potential dominance:
We This kind of scenario occurs whenever the dilaton goes to weak coupling and the scale factor settles to a constant value. This behavior is remarkable, since it produces an accelerated expansion analogous to the inflationary universe. However we also need to address the issue that the perturbation to the dilaton also goes as t 2 . Thus the dilaton may eventually bounce and go to strong coupling. The string interaction effect through the Boltzmann equation is observed to resolve these problems as in figure 5.
Constant B µν
We have also investigated the case of a constant B µν in order to test how its presence affects the action of the string gas. For a constant field, the equations of motion for srings reduce to the usual one without B µν .
It is straightforward to include the effects of a constant B field by calculating the spectrum of the string.
The inclusion of a constant B µν has some interesting consequences, one of these is that there is a critical value which makes the energy of the winding modes aligned with B µν field vanish.
We had expected the modification induced by B µν to be significant since its presence makes the energy and the pressure of the winding modes vanish at a critical value. In fact our numerical results indicate that the behavior of the scale factor could be significantly affected. The spacial directions expand like radiation dominated universe even with the presence of the both momentum and winding modes.
In the string frame, the equation of motion for λ contains the dilaton and its time derivative but it does not contain ν terms. In the same way the equation of motion for ν is independent of λ or its time derivatives.
Then, the equations of motion for the scale factors ν and λ decouple and we can proceed to solve them numerically. In comparison, in the Einstein frame, the presence ofH µνλ makes the scale factors couple to each other. In spite of this unfavourable characteristic, the equations of motion in the Einstein frame are also useful, both when trying to solve the equations of motion and also for clarifying the interpretation of the solutions.
In the Einstein frame the fieldH µνλ is included in the equation of motion for both α and β. That is, unless both the dilaton φ and ν are constant in the string frame, there is no solution with β = constant in the Einstein frame.
Figure 2 :
2solutions can be seen in figure 2 for d = 3. We have plotted in the same figure the numerical solutions for the full equations of motion (3.12), (3.13) with H o = 0: momentum mode dominated universe (green line, w = 1/3), dust dominated universe (blue line, w = 0) and winding mode dominated universe (red line, w = −1/3). Of course the green and red lines are T dual to each other. The numerical solutions tend to the late time analytic solutions, which are plotted in figure 2 as gray dotted lines. behavior of λ and ϕ with H 0 = 0 and w = 0, w = 1 3 , w = − 1 3 . The gray dotted lines correspond to the late time analytic solutions.we note that the scale factor goes to a constant value very quickly in the absence of any driving pressure.This behavior can be seen also in figures 2, 3, 4. The blue line in every picture represents the case when the effect of the pressure and the two-form field vanish, leading to the solution(3.19),(3.20) with w = 0. Notice that this solution is valid for arbitrary d.
Figure 3 :
3behavior of λ and ϕ when there is only the two-form field flux present (W = 0, K = 0, H 0 = 0, H 0 = 0.001, H 0 = 0.0001 ).
56 )
56After substituting (3.55), (3.56) into the equation of motion, we have differential equations at each order of
Figure 4 :
4behavior of λ in presence of U (λ), winding and momentum modes (W = 0, K = 0, H 0 = 0, H 0 = 0.01, H 0 = 0.001 ).
Figure 5 :
5(a), (b) behavior of λ and ϕ with initial condition K = W for H 0 = 0, H 0 = 0.001 and H 0 = 0.1. The effect of the Boltzmann equations is included. (c) Evolution of the winding number and momentum number is plotted (the solutions overlap for the three cases considered). .
Figure 6 :
6behavior of the dilaton and scale factor for B = 0, B = 0.5, B = 1/ √ 3 (critical electric field) and W = K.
have plotted the numerical solution for different values of B without including the effect of the Boltzmann equations (3.65). For vanishing B the momentum and winding modes make the scale factor oscillate around the self-dual radius. With small B = 0, the solutions oscillate around positive values of λ.
have obtained the late time analytic solution for vanishing matter pressure and non-vanishing H µνρ . This solution corresponds to an expanding universe, where the initiation time of the expansion is set by the parameter H 0 . This analytical solution matches the leading behavior of the numerical solution for the equations of motion. In generic situations the contribution of the matter pressure becomes negligible and the scale factor becomes constant. This occurs when the dilaton goes to weak coupling, the oscillations on the scale factor stop or the expansion of the universe comes to a halt. Such a possibility is enhanced if we consider the effect of interactions between strings. As momentum and winding modes can annihilate, it drives the pressure to vanish. In all of the above cases, the effect of the matter pressure vanishes and the scale factor becomes approximately constant. Introducing a H µνρ flux, we find that the constant scale factor solution eventually becomes unstable and the scale factor begins to grow as λ ∼ t 2 (5.1)
By looking at the sign of the H o term in (A.10) and (A.11) we can see that the two-form field induces an anisotropic expansion on the scale factors, with φ and α being driven towards positive values while β goes towards negative values. Also, while in the string frame it is possible to find solutions to the equations of motion in which ν becomes constant, this does not imply that the physical scale factor is fixed because it remains to stabilize the value of the dilaton. This can be seen directly from the relations of the Einstein frame to the string frame, where, in the case of ν = constant we have
AcknowledgmentsThis work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan.A Relation between string frame and Einstein frameIn this appendix, we summarize the relation between string frame and Einstein frame in our setup. The parameter d in this appendix which counts the number of the spacial dimensions should be put d = 9 in superstring. From the conformal transformation (2.7) and the corresponding metrics the relation between scale factors isAlso, the shifted dilaton is defined byBecause of the presence of H µνλ for the superstring in ten dimensions, the spacial coordinates factorize as T 3 × T 6 . Defining λ i ≡ λ for i = {1, 2, 3} and λ j ≡ ν for j = {4, · · · , 9}, we have ϕ = φ − 3λ − 6ν. Using the Einstein equations and the solution for the homogeneous two-form field, we obtain the equations of motion for the superstring case (d = 9)φAccordingly, the equations of motion in the Einstein frame are
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| []
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"Deep learning to estimate the physical proportion of infected region of lung for COVID-19 pneumonia with CT image set",
"Deep learning to estimate the physical proportion of infected region of lung for COVID-19 pneumonia with CT image set"
]
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"Wei Wu \nTreatment of Infectious Diseases\nCollaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases\nSchool of Medicine\nState Key Laboratory for Diagnosis\nNational Clinical Research Center for Infectious Diseases\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina\n",
"Yu Shi \nTreatment of Infectious Diseases\nCollaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases\nSchool of Medicine\nState Key Laboratory for Diagnosis\nNational Clinical Research Center for Infectious Diseases\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina\n",
"Xukun Li \nArtificial Intelligence Lab\nHangzhou AiSmartVision Co., Ltd\n310012HangzhouZhejiangPeople's Republic of China\n",
"Yukun Zhou \nArtificial Intelligence Lab\nHangzhou AiSmartVision Co., Ltd\n310012HangzhouZhejiangPeople's Republic of China\n",
"Peng Du \nArtificial Intelligence Lab\nHangzhou AiSmartVision Co., Ltd\n310012HangzhouZhejiangPeople's Republic of China\n",
"Shuangzhi Lv \nDepartment of Radiology The First Affiliated Hospital\nSchool of Medicine\nZhejiang University\n310003HangzhouZhejiangPeople's Republic of China\n",
"Tingbo Liang \nDepartment of Hepatobiliary and Pancreatic Surgery\nInnovation Center for the Study of Pancreatic Diseases\nSchool of Medicine\nZhejiang Provincial Key Laboratory of Pancreatic Disease\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina\n",
"Jifang Sheng \nTreatment of Infectious Diseases\nCollaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases\nSchool of Medicine\nState Key Laboratory for Diagnosis\nNational Clinical Research Center for Infectious Diseases\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina\n"
]
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"Treatment of Infectious Diseases\nCollaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases\nSchool of Medicine\nState Key Laboratory for Diagnosis\nNational Clinical Research Center for Infectious Diseases\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina",
"Treatment of Infectious Diseases\nCollaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases\nSchool of Medicine\nState Key Laboratory for Diagnosis\nNational Clinical Research Center for Infectious Diseases\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina",
"Artificial Intelligence Lab\nHangzhou AiSmartVision Co., Ltd\n310012HangzhouZhejiangPeople's Republic of China",
"Artificial Intelligence Lab\nHangzhou AiSmartVision Co., Ltd\n310012HangzhouZhejiangPeople's Republic of China",
"Artificial Intelligence Lab\nHangzhou AiSmartVision Co., Ltd\n310012HangzhouZhejiangPeople's Republic of China",
"Department of Radiology The First Affiliated Hospital\nSchool of Medicine\nZhejiang University\n310003HangzhouZhejiangPeople's Republic of China",
"Department of Hepatobiliary and Pancreatic Surgery\nInnovation Center for the Study of Pancreatic Diseases\nSchool of Medicine\nZhejiang Provincial Key Laboratory of Pancreatic Disease\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina",
"Treatment of Infectious Diseases\nCollaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases\nSchool of Medicine\nState Key Laboratory for Diagnosis\nNational Clinical Research Center for Infectious Diseases\nthe First Affiliated Hospital\nZhejiang University\n310003HangzhouChina"
]
| []
| Utilizing computed tomography (CT) images to quickly estimate the severity of cases with COVID-19 is one of the most straightforward and efficacious methods. However, due to the three-dimensional structure and ambiguous edge of infected regions, it may be difficult to diagnose quantitatively for clinical physicians and radiologists.Two tasks were studied in this present paper. One was to segment the mask of intact lung in case of pneumonia. Another was to generate the masks of regions infected by COVID-19. The masks of these two parts of images then were converted to corresponding volumes to calculate the physical proportion of infected region of lung.A total of 129 CT image set were herein collected and studied. The intrinsic Hounsfiled value of CT images was firstly utilized to generate the initial dirty version of labeled masks both for intact lung and infected regions. Then, the samples were carefully adjusted and improved by two professional radiologists to generate the final training set and test benchmark. Two deep learning models were evaluated: UNet and 2.5D UNet. For the segment of infected regions, a deep learning based classifier was followed to remove unrelated blur-edged regions that were wrongly segmented out such as air tube and blood vessel tissue etc.For the segmented masks of intact lung and infected regions, the best method could achieve 0.972 and 0.757 measure in mean Dice similarity coefficient on our test benchmark. As the overall proportion of infected region of lung, the final result showed 0.961 (Pearson's correlation coefficient) and 11.7% (mean absolute percent error).The instant proportion of infected regions of lung could be used as a visual evidence to assist clinical physician to determine the severity of the case. Furthermore, a quantified report of infected regions can help predict the prognosis for COVID-19 cases which were scanned periodically within the treatment cycle. | null | [
"https://arxiv.org/pdf/2006.05018v1.pdf"
]
| 219,558,371 | 2006.05018 | 9f619bdbba7e02b4d975190aaa4fd9e05a3ae298 |
Deep learning to estimate the physical proportion of infected region of lung for COVID-19 pneumonia with CT image set
Wei Wu
Treatment of Infectious Diseases
Collaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases
School of Medicine
State Key Laboratory for Diagnosis
National Clinical Research Center for Infectious Diseases
the First Affiliated Hospital
Zhejiang University
310003HangzhouChina
Yu Shi
Treatment of Infectious Diseases
Collaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases
School of Medicine
State Key Laboratory for Diagnosis
National Clinical Research Center for Infectious Diseases
the First Affiliated Hospital
Zhejiang University
310003HangzhouChina
Xukun Li
Artificial Intelligence Lab
Hangzhou AiSmartVision Co., Ltd
310012HangzhouZhejiangPeople's Republic of China
Yukun Zhou
Artificial Intelligence Lab
Hangzhou AiSmartVision Co., Ltd
310012HangzhouZhejiangPeople's Republic of China
Peng Du
Artificial Intelligence Lab
Hangzhou AiSmartVision Co., Ltd
310012HangzhouZhejiangPeople's Republic of China
Shuangzhi Lv
Department of Radiology The First Affiliated Hospital
School of Medicine
Zhejiang University
310003HangzhouZhejiangPeople's Republic of China
Tingbo Liang
Department of Hepatobiliary and Pancreatic Surgery
Innovation Center for the Study of Pancreatic Diseases
School of Medicine
Zhejiang Provincial Key Laboratory of Pancreatic Disease
the First Affiliated Hospital
Zhejiang University
310003HangzhouChina
Jifang Sheng
Treatment of Infectious Diseases
Collaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases
School of Medicine
State Key Laboratory for Diagnosis
National Clinical Research Center for Infectious Diseases
the First Affiliated Hospital
Zhejiang University
310003HangzhouChina
Deep learning to estimate the physical proportion of infected region of lung for COVID-19 pneumonia with CT image set
Correspondence to Wei Wu, PhD, MDCOVID-19deep learningcoronavirus pneumoniaCT image set
Utilizing computed tomography (CT) images to quickly estimate the severity of cases with COVID-19 is one of the most straightforward and efficacious methods. However, due to the three-dimensional structure and ambiguous edge of infected regions, it may be difficult to diagnose quantitatively for clinical physicians and radiologists.Two tasks were studied in this present paper. One was to segment the mask of intact lung in case of pneumonia. Another was to generate the masks of regions infected by COVID-19. The masks of these two parts of images then were converted to corresponding volumes to calculate the physical proportion of infected region of lung.A total of 129 CT image set were herein collected and studied. The intrinsic Hounsfiled value of CT images was firstly utilized to generate the initial dirty version of labeled masks both for intact lung and infected regions. Then, the samples were carefully adjusted and improved by two professional radiologists to generate the final training set and test benchmark. Two deep learning models were evaluated: UNet and 2.5D UNet. For the segment of infected regions, a deep learning based classifier was followed to remove unrelated blur-edged regions that were wrongly segmented out such as air tube and blood vessel tissue etc.For the segmented masks of intact lung and infected regions, the best method could achieve 0.972 and 0.757 measure in mean Dice similarity coefficient on our test benchmark. As the overall proportion of infected region of lung, the final result showed 0.961 (Pearson's correlation coefficient) and 11.7% (mean absolute percent error).The instant proportion of infected regions of lung could be used as a visual evidence to assist clinical physician to determine the severity of the case. Furthermore, a quantified report of infected regions can help predict the prognosis for COVID-19 cases which were scanned periodically within the treatment cycle.
Introduction
Coronavirus disease 2019 or COVID-19 had become a worldwide pandemic and caused great public health problems [1][2][3]. COVID-19 cases can be divided into light, moderate, severe, and extremely severe types from physicians' perspective. Patients of the latter two types exhibited to have a higher intensive care unit (ICU) rates as well as the death rates [4,5] compared with the other two types. It is therefore essential to identify severe and extremely severe patients as early as possible.
In the Diagnosis and Treatment Protocol for COVID-19 (version 7) [6] released by National Health Commission of China, the clinical characteristics of severe cases include: the decreased lymphocyte count, increased level of inflammatory factors, and rapid development of volume for infected regions on CT images. One patient should be classified and treated as severe case if the volume of infected region would be increased more than 50% within 48 hours. Therefore, continuously monitoring the volume of infected regions may provide valid evidence to predict the prognosis for COVID-19 patients.
However, with infected regions in the lung, the Hounsfiled Unit (HU) value of the lesion regions would be difficult to distinguish with healthy tissues. The infected regions may illustrate as a mist of blur-edged cloud or adhere together with normal tissues on CT images. It would be effort costly for a professional radiologist or physician to separate these lesion regions from healthy lung parenchyma. Furthermore, a set of CT images usually consisted of dozens or hundreds of lung images, which made it almost impossible to analyze the lesion regions quantitatively over the images manually. Therefore, it was urgent to find out an automatic method to estimate the proportion of infected region of lung for COVID-19 from chest CT scans.
To date, several researches had concentrated on the deep learning based models for diagnosing COVID-19. Some studies [7][8][9][10] demonstrated that COVID-19 can be distinguished from other types of pneumonia with good accuracy. Compared with the classifications models, the annotation of CT image samples is highly significant and much more time-consuming in the training of segmentation models for the intact lung as well as infected regions. Shan et al. [11] adopted the human-in-the-loop strategy to iteratively update the annotation of their training samples. Liu et al. [12] synthesized part of their training and test dataset with Generative Adversarial Network (GAN). They achieved 0.706 measured in mean Dice similarity coefficient (m-Dice) for the segmentation of infected regions and 0.961 (Pearson correlation coefficient) for the total percent volume of lung parenchyma that was affected by disease. Ma et al. [13] annotated 20 sets of COVID-19 CT images and utilized previous available lung dataset such as lung cancer to assist the segmentation. Yan et al. [14] also investigated the segmentation of infected regions due to COVID-19. They employed a team of six annotators with deep radiology background and proficient annotating skills to label the areas and boundaries of the intact lung and infection regions due to COVID-19. A feature variation block in the segmentation of infected regions was introduced, which could better differentiate the diseased area from the lung. Furthermore, they used the more effective progressive atrous spatial pyramid pooling in the feature extraction stage as well. The optimum m-Dice achieved in their studies for entire lung and infected regions were 0.987 and 0.726 respectively. These studies suffered from the tremendous effort to label the training samples as well as the relatively low accuracy measured in m-Dice.
In this study, we try to establish a fully automatic deep learning system to estimate the physical proportion of infected region of lung for COVID-19 pneumonia with CT image set. The main contribution of this paper can be summarized as:
First, the HU value of CT images was utilized as threshold to generate the initial "dirty" version of labeled masks for both intact lung and infected regions. These first round labeled samples significantly alleviated the labor expenses of annotation compared with start from scratch.
Then these preliminary samples were revised and improved by two professional radiologists to generate the final training set and test benchmark.
Second, it was observed from our experiment that a certain number of blur-edged healthy structures, which had similar appearance as infected regions, were likely to be identified incorrectly. These kind of healthy tissues included air tube, blood vessel, and blur region of lung at the border etc. Therefore, a deep learning based classifier was employed to further clarify candidate regions that could effectively increase the accuracy of the final segmentation results. In addition, the proposed classifier was much easier to be trained compared with pixel-level segmentation models. Figure 2 showed the whole diagnostic process of COVID-19 report generation in this study.
Materials and methods
Study Dataset
Process
As the digital gray scale image had the pixel value ranging [0, 255], the raw data of CT were converted from HU to the aforementioned values interval accordingly. The HU data matrix was clipped within [-1200, 600] (any value beyond this was set to -1200 or 600 accordingly) and then linearly normalized to [0, 255] to fit into the digital image format for further processing. Next, the infected regions and the intact lung were segmented separately to achieve corresponding masks.
For the differentiating of infected regions, a deep learning based classifier was utilized to remove unrelated masks that were wrongly distinguished. Finally, the volumes of two parts were calculated according to the masks achieved in above-mentioned steps, and achieve the proportion of infected regions in lung.
Evaluation criteria
The performance of the proposed method was evaluated using the Dice similarity coefficient (Dice), measuring the similarity between the ground truth and the prediction score maps. It is calculated as follows:
2 | A B | ( , ) 100% | A | | B | Dice A B (1)
where A is the volume of the segmented lesion region and B denotes the ground truth. The mean Dice (m-Dice) of the whole test benchmark was used to evaluate the final outcomes. Two ground truth masks were used in this study: ground truth for intact lung and ground truth for infected regions. The proportion of infected regions of lung (PoIR) was given by:
volume of infected regions 100% volume of intact lung PoIR (2)
Pearson's correlation coefficient was used to evaluate the correlation of two variables:
2 2 2 2 ( ) ( ) i i i i i i i i i i i i i i i N x y x y r N x x N y y (3)
where N is the total number of observations, xi and yi , i=1, ..., N, are observational variables. We used Pearson's correlation coefficient to calculate the correlation between predicted PoIRs and the corresponding value derived from ground truth. Furthermore, mean absolute percent error (m-APE) , which is a assessing of prediction accuracy of a forecasting method, was herein used to measure the relative errors between the mean predicted PoIRs and the ground truth value on the test benchmark.
1 1 | | 100% n predicted ground truth i ground truth PoIR PoIR mAPE n PoIR (3)
Use the value of HU
The most straightforward way to segment desired lung regions was by aid of the value of HU as the threshold, which reflects the degree of X-ray absorption of different tissues. For instance, the HU value for lung parenchyma usually ranges from -800 to -500 and window of other soft tissue is from +100 to +300. This margin usually is solid enough to separate lung with other tissues. However, when there existed infected regions in lung, the HU value of lung parenchyma could extended to from -750 to 150 (based on our statistics on test benchmark in Fig 3). Therefore, the segmentation result with the threshold of HU usually is not typically accurate enough for clinical application. Alternatively, these labeled images could be used as the initial annotated samples in our study.
Deep learning to segment the intact lung and the infectious regions
As mentioned earlier, use of HU threshold cannot properly segmented the intact lung and infected regions. Those infected regions that had a close HU value with other soft tissues cannot be correctly differentiated. Therefore, deep learning techniques were utilized and evaluated in current study.
Training samples with detailed sketch of each infected region and intact lung are highly essential for developing the deep learning models. However, due to ambiguous edge between infected region and normal tissue, it was extremely timing-consuming to annotate thousands of lung CT images. The annotation result achieved by HU threshold was utilized for the preliminary samples. Then, two professional radiologists further manually contoured the intact lung and infected region based on the these "dirty" samples to generate final sample dataset for training and test.
Network structure
Two deep learning models were utilized: 2D UNet [15] (Fig. 5) and 2.5D UNet (Fig. 6). A two-dimensional (2D) deep learning models can well reflect the intra-slice information. However, they may neglect the inter-slice information and cannot fully leverage the spatial architecture of the three-dimensional (3D) slices of CT scans. On the other hand, 3D models [16,17] suffer from tremendous increased parameters and the subsequent of hard to converge and overfitting especially for a limited number of training samples. Furthermore, due to the limitation of GPU memory, the original CT images had to be cropped or resized to small-sized cubes as the input for deep learning models. This crop or resize operation would either restrict the maximum inception regions or attenuate the resolution of original CT images. Therefore, a pseudo-3D segmentation or so called 2.5D UNet [18,19] was used for evaluation purposes, in which the same UNet backbone (with expanded of network parameters) was used. In addition, three neighboring 2D slices were stacked as the inputs during training, so that the 2D network was able to detect a small range of
Using a Classifier to further clarify infected regions
It was observed that a certain number of blue-edged healthy structures, which had similar appearance as infected regions, were likely to be identified incorrectly. These kind of healthy tissues included air tube, blood vessel, and blur region of lung at the border etc as shown in Fig 7. (a) Therefore, a ResNet-18 [20] based binary classifier (Fig. 8), was utilized after the segmentation models to further clarify whether an image patch belonged to infected regions or not.
(b) (c) (d)
The masks corresponding to healthy regions were filtered. respectively. The accuracy of m-Dice significantly decreased as many unrelated regions, e.g.
stomach, were wrongly segmented as lung when the threshold of HU was set to below -150.
Generate mask for intact lung using deep learning models
UNet and 2.5D UNet were utilized in the present study. For regions with light opacity, the threshold of HU along could achieve satisfactory segmentation results. However, for regions with high opacity, deep learning models could achieve obviously superior results. The m-Dice of UNet and 2.5D UNet of intact lung were 0.972 and 0.967, respectively. There was no significant difference between the two deep learning models.
(a) (b) (c) (d) (e) (f) (g) (h)
Generate mask for infected regions
The same UNet and 2.5D UNet deep learning models were evaluated. The m-Dice of UNet and 2.5D UNet for infected regions were 0.684 and 0.693, respectively. The latter model further concentrated on the inter-slice characteristics and demonstrated 1.3% improvement on the results.
It was observed that most of the infected regions could be included in the output of the segmented masks. However, many blur-edged normal tissues were wrongly detected as infected regions, including air tube, blood vessel, stomach, and part of the border of lung etc. Therefore, a classifier was followed to further remove those unrelated regions.
Performance of binary classifier
The receiver operating characteristic curve (ROC) for the ResNet-18 based classifier was depicted in Figure 11. Fig. 12 and Table 1. Table 1. Summarized m-Dice between predicted masks and the measurement derived from ground truth of our test benchmark.
With the aid of the masks of the intact lung and infected regions, the Pearson's correlation coefficient of the PoIRs could be achieved (0.961), which showed a very strong correlation between the predicted masks and those derived from ground truth. Furthermore, the m-APE of the PoIRs on test benchmark also could be obtained (11.7%), which indicated that the average relative errors between predicted PoIRs and the ground truth value was a lightly more than 10%.
Discussion & conclusions
With the rapid development of artificial intelligence technology, experiences of profession radiologists, such as the segmentation of medical images, could be solidified in the deep learning models to accomplish a quantitative analysis report. Several methods were developed to investigate the segmentation of intact lung and infected regions, including the threshold of HU, UNet, and 2.5D UNet. In addition, a fine-tuned classifier was followed to further remove those wrongly segmented healthy regions to improve the accuracy of outputs. We referred to the methodology of the design of nnUNet [21], which had achieved good results in many different medical segmentation tasks. They suggested that if the objected data is very anisotropic then a 2D
UNet may actually be a better choice. For example, in the segmentation of pancreas, which was a blur-edged objective on the images as well, 2D network actually outperformed 3D counterparts.
As a matter of fact, the most challenge work in the calculation of the proportion of infected regions was the annotation of images, especially for the regions that affected by pneumonia. We utilize the intrinsic HU value of CT images to create the initial version of label images. Even though they were dirty samples, it would be much less effort for professional radiologist to further modify and improve on this first round version. Furthermore, the annotation of samples and the training of a fine-tuned binary classifier were much easier than the pixel-level of segmentation.
Compared with direct result of the state-of-the-art segmentation algorithm, the classifier could improve the m-Dice of infected regions around 9%.
For the calculation of proportion of infected regions, the Pearson's correlation coefficient between predicted and the ground truth showed a strong correlation between them, which would be one of an objective indicator for monitoring the progress of one patient at a fixed interval.
Furthermore, the m-APE showed promising outcomes for the reference for the decision of clinical physicians.
In the future, doctors can carry out quantitative analysis of the severity of COVID-19 patients with this model or combined with other clinical data such as blood oxygenation index. At the same time, they can compare the sequential CT scans of the same case to predict the prognosis and provide reliable basis for treatment.
However, this study had several limitations. In some cases, the segmentation models would possible identify healthy tissues together with valid infected regions and the following classifier could not remove this "valid" infected regions. Therefore, the corresponding mask in such scenario would be larger than the ground truth. Moreover, additional COVID-19 CT cases from different subtypes should be included to promote the accuracy of segmentation and classification.
Some atypical infection signs, such as pleural effusions, cannot be distinguished with this model.
Figure 1 .
1A total of 129 transverse-section CT samples were collected, including 105 from 105 patients (mean age 51 years; 58 [55.2%] male patients) with COVID-19 from the First Affiliated Hospital of Zhejiang University, from January 19 to March 31, 2020. Every COVID-19 patient was confirmed with reverse transcription polymerase chain reaction (RT-PCR) kit, and cases with no image manifestations on the chest CT images were excluded. There were 80 (62.0%) COVID-19 from light to moderate types, and the remaining 49 (38.0% ) cases from severe to extremely severe types respectively. All CT imaging was in the format of digital imaging and communications in medicine (DICOM) with 5mm thickness between slices. The study was approved by the ethics committee of the First Affiliated Hospital, School of Medicine, Zhejiang University and all research was performed in accordance with relevant guidelines and regulations. All participants and/or their legal guardians signed the informed consent form prior to commencing the study. A total of 108 CT samples (83.7%) were used for training and validation datasets and the remaining 21 CT sets (16.3%) were used as a test benchmark. Typical labeled CT images: (a) CT image with pneumonia; (b) CT image without pneumonia. The fields within the blue line denote the masks for the intact lung and those within the red line represent the masks for infected regions.
Figure 2 .
2Study flow chart.
Figure 3 .Figure 4 .
34The distribution of HU value base on the ground truth of the proposed test benchmark.(Included some air tube and blood vessel tissue etc as they were hard to separation with lung structures). Most of the pixels were located within[-750, 150].First, arithmetic progression of HU value (from -800 to 0 with increment of 50) was used as the threshold to segment the intact lung to achieve their corresponding masks. The segmented masks with maximum m-Dice (compared with the ground truth of intact lung) were used as the mask for intact lung. In the next step, use this mask for intact lung to minus the masks obtained with different HU value to obtain the mask of their difference values. The masks with difference in the maximum m-Dice (compared with the ground truth of infected regions) (a) Ground truth of intact lung (green line) and infected regions (yellow line); (b) mask of lung obtained with HU = -200 (red line); (c) mask of lung obtained with HU = -750 (red line); (d) mask of infected regions was obtained by (b) minus (c) (red line).
3D contexts each time. Three masks of image would be generated each time and the average value of segmentation maps would be used as the final masks for overlaps. The proposed two networks would be used both for the segmentation of intact lung and infected regions.
Figure 5 .Figure 6 .
56UNet model 2.5D UNet model.
Figure 7 .
7Regions in the red line are healthy regions (a) air tube and blood vessel; (b) (c) (d) healthy blur-edged regions at the border.
Compared with time-consuming pixel-level annotation on the blur-edged infected region, the training samples of this binary classification model could be relatively easily to be labeled and trained. The candidate images from the output of segmentation model were firstly enclosed in a minimum circumscribed square bounding box. Then these image patches were used as the input data for the binary classifier to determine the valid existence of infected regions. Classical ResNet-18 network backbone was employed for image feature extraction part of the classifier. At the same time, generic data expansion mechanisms such as random clipping and left-right flipping were performed on specimens to increase the number of training samples and prevent data overfitting and improve the problem of generalization. The output of the convolution layer was flattened to a 256-dimensional feature vector, followed by three full-connection layers to export the final binary classification result.
Figure 8 .
8The network structure of ResNet18-based binary classification modelExperiment resultsSegment the intact lung and infected region with the threshold of HU as the initial annotated samples. Arithmetic progression value of HU (from -800 to 0 with the increment of 50) was used as the threshold to segment the intact lung directly to generate corresponding masks. Each outcome masks were evaluated on the test benchmark cases, as shown inFig 9.The maximum m-Dice for intact lung and infected region was 0.921 (HU = -200) and 0.530 (HU = -750),
Figure 9 .
9The maximum m-Dice for intact lung (HU = -200) and infected regions (HU = -750) could be achieved.
Figure 10 .
10The first row of CT images were belongs to a case with light opacity, and the second row belongs to a case with high opacity. (a) Ground truth mask (green line); (b) Segmentation with HU (-200); (c) Segmentation with UNet; (d) Segmentation with 2.5D UNet; (e) Ground truth mask (green line); (f) Segmentation with HU (-200); (g) Segmentation with UNet; (h) Segmentation with 2.5D UNet.
The value of area under curve (AUC) was 0.913, and when that valve equals to 0.45, the classification exhibited the best performance with accuracy of 93.8%. The m-Dice of UNet and 2.5D UNet of infected regions were 0.743 and 0.758 after the filtering this classification model, respectively. Compared with the results directly from the segmentation, the m-Dice improved 8.6% and 9.2%, respectively. The illustration of the segmentation result of infected regions and the summarized m-Dice for different methods were showed in
Figure 11 .Figure 12 .
1112The ROC for the binary classifier. The first row of CT images belongs to a case with light opacity and the second row belongs to a case with high opacity. Ground truth mask marked with green line and predictedinfected region marked in red line. (a) The prediction of UNet; (b) Further clarified by classifier; (c) The predictions of 2.5D UNet; (d) Further clarified by classifier; (e) The prediction of UNet; (f) Further clarified by classifier; (g) The prediction of 2.5D UNet; (h) Further clarified by classifier.
AcknowledgementsThis study was supported by the Zhejiang province natural science fund for emergency research (LED20H190003) This study was supported by the China national science and technology major project fund (20182X10101-001)Compliance with ethics guidelinesAll authors declare that they have no conflict of interest or financial conflicts to disclose.
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"Richard F Lebed [email protected] \nJefferson Lab\n12000 Jefferson Avenue Newport News23606VAUSA\n"
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"Jefferson Lab\n12000 Jefferson Avenue Newport News23606VAUSA"
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| Since QCD is believed to be the underlying theory of the strong interaction, it is appropriate to study techniques that take into account more features of its rich and complex structure. We begin by discussing aspects of physics that are ill-reproduced by the usual one-or two-meson exchange approaches and identify the source of the deficiencies of these models. We then reveal promising methods for curing some of these ills, such as new quark potential models, baryon chiral perturbation theory, soluble strongly-interacting field theories, and large Nc QCD. | null | [
"https://arxiv.org/pdf/nucl-th/9809093v1.pdf"
]
| 14,466,136 | nucl-th/9809093 | c98dc8b429fd69bbe310feddac7ac9d2d69aa1c2 |
Sep 1998
Richard F Lebed [email protected]
Jefferson Lab
12000 Jefferson Avenue Newport News23606VAUSA
Sep 1998arXiv:nucl-th/9809093v1 30 NN INTERACTIONS IN QCD: OLD AND NEW TECHNIQUES a
Since QCD is believed to be the underlying theory of the strong interaction, it is appropriate to study techniques that take into account more features of its rich and complex structure. We begin by discussing aspects of physics that are ill-reproduced by the usual one-or two-meson exchange approaches and identify the source of the deficiencies of these models. We then reveal promising methods for curing some of these ills, such as new quark potential models, baryon chiral perturbation theory, soluble strongly-interacting field theories, and large Nc QCD.
Introduction
As a particle theorist, I was invited to speak at this conference about what can be said regarding the nucleon-nucleon interaction in the context of quantum chromodynamics. Hadronic physics is probably the most difficult problem in the entire panoply of particle theory, and the primary quest of its practitioners (myself included) is to uncover some particularly simple physical picture that not only respects the physics of observed hadrons, but arises as a natural consequence of the substructure of quarks and gluons. To this day we remain hindered in our ability to provide an eloquent and definitive solution to the problem.
However, even on that day of QCD's eventual solution, the meson exchange picture of nuclear physics will remain the natural picture for the NN interaction in almost all situations, in precisely the same way that NASA quite successfully employs Newtonian gravitation for calculating trajectories of spacecraft, the existence of general relativity as a more "fundamental" theory of gravitation notwithstanding. Quarks and gluons are certainly present in all NN interactions, but it is not always necessary to take them explicitly into account.
Nevertheless, we have learned quite a bit about QCD in its first quarter century. We know the Lagrangian and its symmetries, some properties of the quarks themselves, and a bit about the nature of color confinement. Moreover, one can study versions of QCD-like theories that have been simplified so that a Invited talk presented at Mesons and Light Nuclei, Aug. 31-Sept. 4, 1998, Pru some of the difficult physics becomes tractable. These sorts of advances can be applied to the NN problem, to constrain types of possible dynamics or to reduce allowed parameter spaces. The significance of such physics becomes most apparent when one considers circumstances in which the standard meson exchange picture begins to falter, so we begin with a discussion of the successes and problems of this picture. We then consider exactly what it means to obtain QCD improvements, and exhibit a number of techniques designed to accomplish this goal.
The Age of the Meson Exchange Picture
The NN interaction is perhaps the most studied of all problems in nuclear physics, and the decades of careful scrutiny and hard work are now coming to fruition in the convincing numerical success of various famous potential models. With only a score or two of parameters, the models of Paris, 1 Bonn, 2 Nijmegen, 3 and Argonne 4 have begun to achieve a global fit to the numerous experimentally observed partial waves and hundreds of extracted data points approaching the all-important statistical criterion of χ 2 /d.o.f. ≈ 1. The fundamental physics incorporated in these models is obtained from the current understanding of the dynamics of meson exchange: The unknown fit parameters appear in the context of Yukawa potentials, form factors, and so forth.
Are we then to conclude that the NN interaction is essentially a solved problem, with the few remaining discrepancies requiring only minor adjustments of functional forms or numerical values of the parameters? From a strictly reductionist point of view, as the fits of models to data become ever more precise, questions about the origin of these numerical values become more pressing. Moreover, one has in QCD the apparent underlying fundamental theory of strong interactions. Even if one looks askance at future prospects of describing a nuclear problem such as the NN interaction with its complicated phenomenology in the language of quarks and gluons, one cannot believe that all of the masses and couplings of the common mesons are independent quantities. The proper question is, how do nuclear phenomena arise as a limit of QCD?
In the traditional picture of the NN potential, at large internuclear separations ( > ∼ 3 fm) one observes exponential saturation of nuclear forces with a residual attraction, which is explained by one-pion exchange. At intermediate ranges, between roughly 0.6 and 3 fm, there is an attractive region typically ascribed to exchange of a scalar σ meson or scalar combination of two π's, while at smaller distances one finds an effective short-range repulsion, which is identified with ω exchange. Including the ρ, whose primary role is to can-N N π cel most of the tensor interaction of the π at short distance, completes this picture. One is immediately faced with questions about the identity of these mesons, π, σ, ρ, ω. Are they the fundamental mesons appearing in the Particle Data Book, or do they merely serve as placeholders to parametrize more complicated physics? For example, the troublesome "σ" might actually indicate the exchange of scalar current between nucleons arising from many different sources at different momentum scales, including but not limited to ππ pairs.
The fundamental problem with one-meson exchanges as a universal explanation for NN phenomena can be described in terms of the following simple "parable" (Fig. 1): Think of hadrons as classical hard spheres. The charge radius of the proton is r 2 ≈ 0.6 fm, while the typical size of light mesons is set by the QCD scale Λ QCD , r ≈ 1/Λ QCD ≈ 1/(300 MeV) = O(0.5 fm). Then a meson cannot mediate an NN interaction at distances below (0.6 fm + 2 · 0.5 fm + 0.6 fm) = 2.2 fm: It simply won't fit between the nuclei! Of course, real hadrons are quantum mechanical, but the point of the parable remains: Onemeson exchange only makes sense inasmuch as the meson (and, to a smaller extent, the nucleon) can be treated as a point particle. At distances smaller than a couple of fermi, one probes the inner structure of the hadrons, which is to say their short distance or high momentum components. In meson models this is typically taken into account with form factors. But which type of form factor is correct? Obviously, this is a situation in which more input from QCD would provide valuable clarification.
Perhaps you are unconvinced by this parable, and would prefer a more direct demonstration of some situation where one-meson exchange fails. To be suitably rigorous, one needs a theory in which quark and hadron degrees of freedom can be handled equally well. There in fact exists a completely soluble, strongly interacting theory, which is called the 't Hooft model; 5 by definition, it is QCD in one space and one time dimension with a large number of color charges N c . "Completely soluble" here means that one can obtain meson masses, wavefunctions, and transition amplitudes precisely in terms of quark masses. Within this framework, consider a typical hadronic quantity, the meson electromagnetic form factor: 6
F (q 2 ) = ∞ n=0 Λ n µ 2 n q 2 − µ 2 n + iǫ ,(1)
where µ n and Λ n are the mass and pole residue of the nth meson. That the form factor may be written as a sum of pole terms is a consequence of large N c . Single-meson exchange models tell us to expect Λ 0 ≫ Λ 1 , Λ 2 , . . ., i.e., that the contribution of the lightest meson is the most important. Carrying out the calculation in the 't Hooft model, one finds that this is true only for light quarks (Fig. 2). As the quark mass increases, one sees not only that a larger number of poles become significant, but that they alternate in the signs of their residues, meaning that a one-meson exchange picture becomes progressively more inadequate. Indeed, as the quark mass becomes very large, one can show that the residues arrange themselves to give the same predictions as the nonrelativistic quark model.
Including More QCD
Since we have argued that there are circumstances in which input from QCD is essential for physical understanding, one must define exactly what is meant by QCD improvement. Let us adopt the broadest definition possible, in order to represent a complete spectrum of the many features of the strong interaction. Then QCD improvements fall into two basic categories. First, since QCD is a quantum field theory, it must satisfy the properties of causality, relativistic covariance, crossing symmetry, unitarity, and unrestricted production of virtual particles. Second, a number of field-theoretic properties are special to QCD, namely, the presence of quarks and gluons (of course), as well as gauge invariance and color conservation, color confinement, and the discrete symmetries of parity, charge conjugation, and time reversal invariance. In addition, QCD provides for many kinds of hadrons with numerous types of nonlinear interactions, and -very importantly -approximate chiral symmetry.
Inverse Scattering
First consider an improvement that takes into account only field theoretic QCD properties. The premise of inverse scattering is that one uses data directly from the S-matrix and phase shifts as functions of momentum transfer k via which, by construction, automatically satisfies field theoretic aspects of QCD, since the S-matrix obeys unitarity, crossing symmetry, and so forth. One then inverts (2) using standard mathematical techniques such as Gel'fand-Levitan or Marchenko inversions, 7 to obtain an equivalent local potential V (r) that, by construction, agrees with the data. If one then compares to usual potential models, the agreement is quite good for most partial waves, 8 since these potentials were designed expressly to fit the phase shifts. In particular, V (r) obtained in this method exhibits a repulsive core. Moreover, the inversion may be continued off-shell to produce interesting results relevant to processes like nucleon bremsstrahlung. 9 However, a local potential V (r) depends on only one coordinate r, which is the separation of the two nucleon centers. This is a natural picture when the nucleons may be considered (nonrelativistic) point particles, but may be inadequate when nucleonic substructure is taken into account, in which case there is more than just one relevant separation coordinate.
S ℓ (k) = exp (2iδ ℓ (k)) ,(2)
Local and Nonlocal Potentials
The nucleonic substructure of quarks and gluons can create nonlocality in the NN potential, which may be expressed as an energy dependence V (E, r). It is true that relativistic effects also produce an energy-dependent potential, but one can study the effects of substructure separately by considering a nonrelativistic toy model. An interesting example of this approach appears in Ref. 10 , where the p-Σ + potential, known to be have a highly nonlocal potential in the quark model, is considered. The nonlocality is introduced through a potential term
∆V (r N , r Σ ) ∼ exp − (r N + r Σ ) 2 4a 2 c exp − (r N − r Σ ) 2 4a 2 d ,(3)
which depends on not only the separation (r N − r Σ ) but also the average position (r N + r Σ )/2. The phase shifts obtained from this nonlocal potential can then be used to generate an equivalent local potential. The result of this calculation shows that the height of the repulsive core in the nonlocal potential is greatly reduced compared to that from the local potential (see esp. their Fig. 3). Such a conclusion suggests that the repulsive core of potential models is actually due to nucleon substructure; it would certainly agree with our earlier comments that single-meson exchange at short distance should not be a good description of the NN interaction.
Quark Models
One form of quark model phenomenology uses nothing more than valence quarks either interacting in some phenomenological potential, or with some chosen wavefunctions within the nucleon. Such quark model studies of features of the NN interaction have a very old history, dating back to the dawn of the quark model itself in the mid 1960s. The unique feature of quarks, however, is that they possess the color degree of freedom. Once one determines that color exists, situations become inevitable in which the color degree of freedom must be considered explicitly. As an example, consider a second parable in the form of Fig. 3. Starting with one-meson exchanges in the NN interaction (Fig. 3a), once one decides that N is a 3-quark state while mesons areqq states, the interpretation of the meson exchange in terms of colored quark lines becomes clear (Fig. 3b). However, just as likely are diagrams in which the quark lines are tangled (Fig. 3c). In such a case, the intermediate state is clearly not a single meson, nor is it even a color singlet. It is still possible to describe it in the meson language, but to do so requires a large number of carefully correlated meson exchanges; this is the same phenomenon that we saw in the 't Hooft model (Fig. 2) for large quark masses. The promotion of this argument from parable to rigor involves the inclusion as well of all possible gluon exchanges, but it seems reasonable that such a modification cannot completely screen all color from our notice. To date, however, the best quark model studies still only include the gluon degrees of freedom through field-theoretic reductions of one-gluon exchanges. Nevertheless, this is enough to capture quite a bit of physics, such as spinorbit couplings and hyperfine terms. Consider, for example, the results of one particular study, 11 in which six-quark states are placed in an interaction derived from single-gluon exchange plus an explicit confining potential. It is then found that the repulsive core originates as anti-binding from the spinspin coupling of the hyperfine interaction, while the intermediate attraction is a result of the excitation of color nonsinglet P -wave clusters of quarks. Clearly, these are phenomena that have no simple interpretation in terms of one-meson exchange.
Ν Ν π (a) (b) (c)
Another interesting idea for the suppression of the repulsive core arises in the context of Moscow potentials. 12 One begins with the physical observation that it is rare to observe baryons with very small separation. The standard explanation, of course, is that one is seeing a potential with a repulsive core. However, many researchers suggest that the same physics may be obtained through an NN wavefunction with a node at small separation, effectively suppressing such observations. In the case of the Moscow potential, the same sort of wavefunction suppression is achieved through a "two-phase" model: Starting with six quarks, at large separation there is a high probability for segregation into two three-quark nucleon clusters, with combined wavefunction Φ. At small separation there is a high probability to form a "bag-like" six-quark state Ψ. The wavefunctions Φ and Ψ are then taken to be orthogonal. Thus, it is not terribly difficult to push six quarks into a very small volume, but then the dominant part of the wavefunction no longer resembles two distinct nucleons. Such models allow for good fits to many of the phase shifts, and the residual meson interaction potentials (to account for long-distance physics not incorporated into the quark interactions) may then be taken as local. Moreover, the ωNN coupling falls to values consistent with that predicted from SU(3) symmetry, since the ω is no longer required to serve the special role of providing the repulsive core interaction.
Quark models typically explain the short-and intermediate-distance features of the NN interaction; both nuclear and particle physicists can agree that the long-distance tail is due to single-pion exchange. Nevertheless, it is fruitful to compare the tortuous discovery of this fact with its current explanation in many textbooks. Historically, Yukawa proposed in 1935 the exchange of mesons to explain nuclear binding; the π was the first true meson discovered (in 1947), and was subsequently found to explain the long-distance behavior of NN interactions well. Since the π is still the lightest observed meson, by the Heisenberg principle it has the longest range. This bottom-up process of discovery is to be contrasted with our current top-down understanding of the same phenomenon: The QCD Lagrangian possesses chiral symmetry, which is spontaneously broken to an approximate flavor symmetry. The breaking produces a multiplet of pseudoscalar Nambu-Goldstone bosons, of which the π is the lightest, since it contains no heavy strange quarks. Therefore, again by the Heisenberg principle, it should have the largest range of any strongly interacting particle.
Effective Theories
The discovery of the approximate chiral symmetry of strong interactions has been one of the primary achievements of the extensive efforts placed in understanding the NN interaction over the years. It is exploited to great effect in chiral Lagrangians and chiral perturbation theory (χPT), and yet such theories are only one specific type of what are now collectively called effective theories. Let us explain how such theories are constructed in general, with reference to the familiar χPT case.
1. Choose a set of fields as dynamical degrees of freedom. In χPT these are pions, as well as K's and η's in the 3-flavor case, and nucleons can also be incorporated into this scheme.
2. Identify the symmetries obeyed by interactions of these fields. In χPT these are Lorentz covariance, (approximate) chiral symmetry, and the discrete symmetries P, C, and T.
3. Express fields in forms that transform appropriately under the given symmetries. For example, one convenient representation containing the pion field π in χPT is Σ ≡ exp(2iπ ·τ /f π ), for then under SU(2) L × SU(2) R chiral rotations L and R one has Σ → LΣR † . Here τ are the isospin generators and f π is the pion decay constant.
4. Construct the Lagrangian that explicitly obeys all symmetries. In general, this procedure produces an infinite list of terms, which gives the initial naive impression that the theory has no predictive power at all. In the case of χPT, however, more complicated terms with more fields or derivatives enter, by virtue of simple dimensional analysis, with more inverse powers of some characteristic mass scale Λ. In practice, Λ for χPT is typically taken at the scale of m ρ or 1 GeV, where describing physics solely in terms of pion interactions is no longer adequate. Therefore, all but a small finite number of the possible infinite set are insignificant for a given physical process. In the general case, an effective theory is useful if the more complicated terms are suppressed numerically in physical quantities, which means that the characteristic momenta of the process must be below some scale Λ. In this sense, Λ acts as a radius of convergence for the perturbative organization of the series.
5.
Each term in the Lagrangian has an unknown coefficient, expected to be of order unity, which must be fit to data. Once the effective Lagrangian has been truncated by the process described above, a (hopefully small) number of such coefficients remain. Of course, the usefulness of the theory depends on few enough coefficients remaining that the Lagrangian may then be used to predict other observables. The expectation that the coefficients are of order unity once the known physics is taken into account is called the naturalness assumption; if a coefficient turns out to be too small, one suspects a hidden symmetry, while if it is too large, one suspects that important physics has not been taken properly into account.
The relevance of this construction in the current context is that a great deal of effort has recently been invested in developing effective chiral theories to compute nucleon properties. A nice talk on the importance of chiral symmetry in nuclear interactions appears in 13 , while 14 provides a very thorough review through 1995. The subtlety in the nucleon case is that the development of the effective theory runs into complications because of the presence of several mass scales. For suppose, in the construction described above, one finds not one but two scales of physics, Λ 1 ≪ Λ 2 , relevant to a given process. Then it is not enough to merely choose processes with characteristic momenta p satisfying p ≪ Λ 1 and p ≪ Λ 2 , for the combination Λ 2 /Λ 1 ≫ 1 might appear in the dimensionless unknown coefficients, making them unnaturally large and thus defeating the predictivity of the theory.
In the single nucleon case, in addition to the scale of the onset of nonpionic interactions Λ, one must also deal with the appearance of the nucleon mass, as discussed first in 15 . One particularly successful treatment 16 is to use a Foldy-Wouthuysen transformation b to remove nucleon mass terms from the Lagrangian, a method that effectively replaces nuclear momenta with velocities.
However, in the case of two or more nucleons, one typically has a three-scale problem: momentum p, nucleon mass M , and nuclear binding energy p 2 /2M . In this case, a typical approach is to remove the scale M as described above, and then to sum up diagrams with the small scale p 2 /2M in nucleon propagators -a chain of loop diagrams -using nonperturbative quantum mechanics in the form of the Lippmann-Schwinger or Schrödinger equation, into an effective potential. 18 A very new approach 19 eliminates the small binding scale by regularizing loop integrals using minimal subtraction near D = 3 dimensions rather than D = 4, as is usually done in field-theoretic calculations. Then the loop diagrams are summed by means of renormalization group equations, thus avoiding the necessity of picking a kernel for a particular wave equation. That such an approach might work is perhaps not so surprising: Binding energy scales are very small compared to the nucleon masses, so the fundamental dynamics of the problem involves perturbations about an essentially static nucleon, and therefore is three-dimensional.
Before leaving this topic, it should be pointed out that many theories and models can be promoted to an effective theory. All that is needed is a set of symmetry principles for deciding what interactions are allowed, and an organizing principle (e.g., a perturbation series) for deciding which of these interactions are important. For example, meson potential models have neglected corrections in the form of nontrivial form factors or meson-meson couplings, b Actually, this is also how the Heavy Quark Effective Theory is developed. See, e.g., Ref. 17 . while quark potential models are typically organized in a series in 1/m quark . All in all, the concept of the effective theory is not unlike the famous Wigner-Eckart theorem. Both divide physics into a symmetry part and a dynamical part. In the case of the W-E theorem, the symmetry part is represented by spin SU(2) Clebsch-Gordan coefficients, while the dynamical part is the so-called reduced matrix element. In effective theories, the symmetry part consists of Lorentz, chiral, and parity invariances, and other conditions we impose upon the interactions, while the dynamical part is represented by the unknown coefficients that must be fit to data. In this sense, effective theories are very minimal in their dynamical content, but provide a very useful starting point for deeper inquiries into the dynamics.
Large N c QCD
It is a remarkable fact that considering the limit in which the number N c of QCD color charges, which is 3 in our universe, becomes infinite, 20 actually simplifies strong interaction physics. How can increasing the number of degrees of freedom actually lead to a simplification? Think of statistical mechanics as an analogy, where Avogadro's number of particles can be described by just a few collective quantities, such as pressure, temperature, etc. In large N c QCD, baryons are treated similarly, in a Hartree-Fock picture: 21 To first approximation, each of the N c quarks feels only the collective effect of the other N c − 1.
However, taking the large N c limit seriously means that one expands physical quantities in a series in 1/N c . If we apply this to our universe, the expansion parameter is 1/3, which certainly does not seem small! However, for many quantities, the first correction to the large N c limiting value appears not at relative order 1/N c but 1/N 2 c = 1/9, which is arguably a small parameter. Even if this does not occur, one may simply adopt the expansion anyway, fit to the data using the 1/N c expansion and set N c = 3 at the end of the calculation. Then one can see a posteriori whether the factors of 1/3 truly are supported by experiment. A simple example was first pointed out in 22 where it is observed that the relative mass splitting between nucleons and ∆ resonances is suppressed by 1/N 2 c . Writing this relation in a scale-independent way,
m ∆ − m N 1 2 (m ∆ + m N ) = O J 2 N 2 c .(4)
Experimentally, the l.h.s. is 0.27, whereas the r.h.s. is 3/N 2 c , which is 3 if we dismiss the factors of N c as irrelevant, but 0.33 if they are retained. In fact, one can study the entire spectrum of the ground state baryons this way, 23 and indeed the explicit factors of N c are essential to account properly for all masses.
In fact, studies of the large N c expansion for nuclear (as opposed to nucleon) systems are in their infancy; only a handful of papers studying this problem have yet appeared, but the prospects look quite promising. The basic lesson is that large N c provides a kind of effective theory for nuclear systems, in that the old spin-flavor SU(6) is known 24 to hold in the large N c limit (the symmetry), while interaction operators suppressed in this limit are accompanied by powers of 1/N c (the organizing principle).
One direction that such a theory may be used is to note that, if the leading interactions in 1/N c obey some symmetry, then so do the corresponding physical observables. For example, nature obeys an approximate symmetry under interchanges of the states (p ↑, p ↓, n ↑, n ↓), the famous Wigner supermultiplet. In fact, this phenomenon has a large N c explanation 25 in that the operators that would lift this degeneracy are suppressed by powers of 1/N c .
Another direction is to find exactly which operators appear at leading order in 1/N c for a given process, and study their symmetry properties. This is what is done in the first large N c analysis of the NN interaction, 26 where it is shown that one of the leading operators acting on nucleons 1 and 2 is the combined spin-isospin operator (σ 1 · σ 2 ) (τ 1 · τ 2 ) .
It is important to realize that this sort of analysis is independent of the particular dynamical origin of the given operator. If one assumes that the dynamics arises from one-meson exchanges, for example, then one concludes that mesons with strong couplings to the given operator will be important for the NN interaction. In the given example, we know that the π and ρ tensor couplings contain pieces like (5), and so large N c explains why we might have expected that meson potential models require large tensor couplings to these mesons. But any model, meson exchange or not, that successfully describes the NN interaction must recognize the importance of nucleon operators such as (5).
Conclusions
In the final analysis, we return to a variant of our original question: Why does the meson exchange picture work so well for the NN interaction, when the underlying theory of QCD is so much more complicated? The ultimate contribution of QCD to the understanding of the NN interaction will almost certainly not be in the form of a solution to some as yet unknown field equation, but rather the realization of how a complicated collection of quarks and gluons possesses some limiting case in which the system achieves a collective degree of simplicity, which we observe as a pair of interacting nucleons. We have already begun to see such simplifications take place in effective theories, and especially in large N c QCD. Even though we cannot yet solve the strong interaction problem, we have begun to nibble at the edges. For example, we have argued that the famous "repulsive core" of the NN potential appears to be due to quark effects. It would be exciting to find more evidence for a six-quark "bag" or colored particle exchanges as QCD suggests, phenomena that are quite exotic from the onemeson exchange perspective. Obviously these are topics of interest to both the nuclear and particle communities.
This last observation lies at the crux of my optimism on the future of NN studies: After following divergent paths for some decades, nuclear and particle physics are again making great strides together. We will see much more of the fruits of this combined effort in the future.
), Czech Republic; to appear in Proceedings.
Figure 1 :
1A parable for the short-distance failure of one-meson exchange in the NN interaction.
Figure 2 :
2Size of form factor pole residue contributions Λn vs. µ 2 n /µ 2 0 , shown for quark masses decreasing in the order (a)-(f ) (after Ref.6 ).
Figure 3 :
3NN interactions in (a) meson and (b),(c) quark exchange pictures. In (b) the intermediate quark exchange is easily described as a single-meson exchange, while in (c) it is not.
AcknowledgmentsI would like to extend a special dȇkuji vám to the conference organizers for their kind invitation and hospitality, and to Franz Gross and Wally van Orden for valuable comments on the content of the talk. This work was supported by the U.S. Department of Energy under contract No. DE-AC05-84ER40150.
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| []
|
[
"A direct test of the integral Yang-Mills equations through SU (2) monopoles",
"A direct test of the integral Yang-Mills equations through SU (2) monopoles"
]
| [
"C P Constantinidis ",
"L A Ferreira ",
"G Luchini ",
"\nDepartamento de Física\nUniversidade Federal do Espírito Santo (UFES)\nCEP 29075-900, Vitória-ESBrazil\n",
"\nInstituto de Física de São Carlos\nIFSC/USP\nUniversidade de São Paulo Caixa Postal 369\n13560-970São Carlos-SPCEPBrazil\n"
]
| [
"Departamento de Física\nUniversidade Federal do Espírito Santo (UFES)\nCEP 29075-900, Vitória-ESBrazil",
"Instituto de Física de São Carlos\nIFSC/USP\nUniversidade de São Paulo Caixa Postal 369\n13560-970São Carlos-SPCEPBrazil"
]
| []
| We use the SU (2) 't Hooft-Polyakov monopole configuration, and its BPS version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary it then implies that non-abelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Polyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results points to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the Yang-Mills theory. arXiv:1710.03359v1 [hep-th] | 10.1103/physrevd.96.105024 | [
"https://arxiv.org/pdf/1710.03359v1.pdf"
]
| 119,666,219 | 1710.03359 | 708010747856abfc18d981913bf5dffaab64bc93 |
A direct test of the integral Yang-Mills equations through SU (2) monopoles
10 Oct 2017
C P Constantinidis
L A Ferreira
G Luchini
Departamento de Física
Universidade Federal do Espírito Santo (UFES)
CEP 29075-900, Vitória-ESBrazil
Instituto de Física de São Carlos
IFSC/USP
Universidade de São Paulo Caixa Postal 369
13560-970São Carlos-SPCEPBrazil
A direct test of the integral Yang-Mills equations through SU (2) monopoles
10 Oct 2017
We use the SU (2) 't Hooft-Polyakov monopole configuration, and its BPS version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary it then implies that non-abelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Polyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results points to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the Yang-Mills theory. arXiv:1710.03359v1 [hep-th]
Introduction
The purpose of this paper is to perform a test of the integral equations of Yang-Mills theories, recently proposed in [1,2], using the SU (2) 't Hooft-Polyakov monopole solution [3,4] as well as its exact analytical BPS version [5,6]. The main motivation for such a test is that these integral equations involve two complex parameters that are not present in the Yang-Mills partial differential equations. If those parameters are arbitrary, it means that contrary to abelian electromagnetism, Yang-Mills theories possess in fact an infinity of integral equations. Indeed, by expanding the Yang-Mills integral equations in power series of those parameters, we check that the SU (2) 't Hooft-Polyakov monopole, and its BPS version, do satisfy the integral equations appearing in that expansion, up to second order in one of the parameters. The cancelations involved in such a check are highly non-trivial and give strong evidence on the arbitrariness of those parameters.
As shown in [1,2] the integral Yang-Mills equations lead in a quite natural way to gauge invariant conserved charges. Such charges involve those two parameters in a way that if they are indeed arbitrary it would imply that in principle, the number of charges is infinite. However, due to some special properties of BPS multi-dyon solutions [7,8], shown in [9], the higher charges are not really independent for such solutions, being in fact powers of the first ones (the electric and magnetic charges). The same is true for the SU (2) 't Hooft-Polyakov monopole. There remains to be investigated if other non-BPS solutions also present such special properties or not, and so possess or not an infinity of charges.
In order to discuss the role of such parameters in a more concrete way let us start by the theory of electromagnetism described by the Maxwell equations
∂ µ f µν = j ν ∂ µf µν = 0 (1.1)
where f µν = ∂ µ a ν − ∂ ν a µ ,f µν = 1 2 ε µνρλ f ρλ , j µ being the external four current, and a µ the electromagnetic four vector potential. The integral version of those equations is obtained through the abelian Stokes theorem for a rank-two antisymmetric tensor b µν on a space-time 3-volume Ω, aś
∂Ω b =´Ω d ∧ b,
where ∂Ω is the border of Ω. Taking b µν as a linear combination of f µν and its Hodge dual, and using (1.1), one getŝ ∂Ω α f µν + βf µν dΣ µν =ˆΩ βj µνρ dV µνρ (1.2) wherej µνρ = ε µνρλ j λ is the Hodge dual of the external current and α and β are arbitrary parameters used in the liner combination. By considering α and β to be arbitrary, the integral equations (1.2) correspond to the four usual integral equations of electromagnetic theory, which in fact preceded Maxwell differential equations. Indeed, taking α = 0 and Ω to be a purely spatial 3-volume one gets the Gauss law. On the hand, taking β = 0 and Ω to be a solid cylinder with its height in the time direction, and its base on a spatial plane, one gets the Faraday law, and so on. The role of the parameters α and β are not really important here because (1.2) is linear in them. The situation becomes more complex in a non-abelian gauge theory.
The Yang-Mills theories were formulatedà la Maxwell in terms of partial differential equations, the so-called Yang-Mills equations [10]
D µ F µν = J ν D µF µν = 0 (1.3) where F µν = ∂ µ A ν −∂ ν A µ +i e [ A µ , A ν ]
, with e being the gauge coupling constant,F µν = 1 2 ε µνρλ F ρλ , J µ being the external matter current, and D µ = ∂ µ + i e [ A µ , ], and A µ being the non-abelian gauge field taking value on the Lie algebra of the gauge group G.
In order to construct the integral form of Yang-Mills equations (1.3) one needs the non-abelian version of the Stokes theorem for a (non-abelian) rank-two antisymmetric tensor B µν on a space-time 3-volume Ω. Even though the non-abelian Stokes theorem for a one-form connection on a 2-surface was known for some time, the same theorem for a two-form connection was constructed only more recently in [11,12] using concepts on generalized loop spaces. Conceptually everything becomes more clear if one uses the two-form B µν , defined on space-time, to construct a one-form connection on the generalized loop space. Using such generalized non-abelian Stokes theorem, the integral form of Yang-Mills equations were constructed in [1,2]. The formulas involve path, surface and volume ordered integrals as follows.
Consider a space-time 3-volume Ω, and choose a reference point x R on its border ∂Ω. Scan Ω with closed 2-surfaces based on x R , labelled by a variable ζ, such that ζ = 0 corresponds to the infinitesimal surface around x R , and ζ = ζ 0 to the border ∂Ω. Then scan each closed 2-surface with loops, starting and ending at x R , labelled by a variable τ . Each loop is parameterized by a variable σ. The integral form of Yang-Mills equations (1.3) is given by [1,2]
V (∂Ω) ≡ P 2 e ie´∂ Ω dτ dσW −1 (αFµν+β Fµν )W ∂x µ ∂σ ∂x ν ∂τ = P 3 e´Ω dζdτ V J V −1 ≡ U (Ω) ,(1.4)
where P 2 and P 3 mean surface and volume ordered integration respectively, as explained above, and
J =ˆσ f σ i dσ ieβ J W µνλ dx µ dσ dx ν dτ dx λ dζ + e 2ˆσ σ i dσ (α − 1) F W κρ + β F W κρ σ , αF W µν + β F W µν (σ) × d x κ d σ d x µ d σ d x ρ (σ ) d τ d x ν (σ) d ζ − d x ρ (σ ) d ζ d x ν (σ) d τ (1.5)
with J µνλ = ε µνλρ J ρ , being the Hodge dual of the external matter current. In order to simplify the formulas we have used the notation
X W ≡ W −1 X W (1.6)
with X standing for the field tensor, its Hodge dual, or the dual of the matter currents. The quantity W appearing above stands for the Wilson line, defined on a path parameterized by σ through the equation
dW dσ + ieA µ dx µ dσ W = 0 (1.7)
and so
W = 1 − i eˆσ σ i dσ A µ σ dx µ dσ + (i e) 2ˆσ σ i dσ A µ σ dx µ dσ ˆσ σ i dσ A ν σ dx ν dσ − . . . (1.8)
The quantity V , called the Wilson surface, is defined on a surface parameterized by σ and τ , through the equation
dV dτ − V T (τ ) = 0, (1.9) with T (τ ) = ieˆσ f σ i dσ W −1 αF µν + β F µν W ∂x µ ∂σ ∂x ν ∂τ .
(1.10) and the integration being on the closed loops used in the scanning of Ω, as explained above. The initial and final values of σ, denoted σ i and σ f respectively, correspond to the initial and final points of the loop, which in fact are the same point since the loop is always closed. Therefore the solution of (1.9) is the surface-ordered series
V (τ ) = 1 +ˆτ τ i dτ T (τ ) +ˆτ τ i dτ ˆτ τ i dτ T (τ )T (τ ) + . . . (1.11)
The l.h.s. of (1.4) is obtained by integrating (1.9) on the 2-surface ∂Ω, i.e. the border of Ω. On the other hand the r.h.s. of (1.4) is obtained by integrating the equation
dU dζ − K U = 0 (1.12)
on the 3-volume Ω, and where
K =ˆΣ dτ V J V −1 (1.13)
with Σ being the closed 2-surfaces scanning Ω, labelled by ζ, and J given by (1.5). The solution of (1.12) is given by the volume-ordered series
U (ζ) = 1 +ˆζ 0 dζ K(ζ ) +ˆζ 0 dζ ˆζ 0 dζ K(ζ ) K(ζ ) + . . .
(1.14)
Note that (1.4) does reduce to (1.2) in the case that the gauge group G is U (1). However, for non-abelian gauge groups the dependence of both sides of (1.4) on the parameters α and β are highly non-linear. Indeed, if such parameters are arbitrary one can expand both sides of (1.4) in a power series on them. The coefficient of each term of such series on the l.h.s. of (1.4) will have to equal the corresponding coefficient of the series on the r.h.s., leading to an infinity of integral equations. Consequently any solution of the Yang-Mills equations (1.3) will have to satisfy such an infinity of integral equations. It is this test that we want to perform with the 't Hooft-Polyakov monopole, and its exact analytical BPS version [8]. We shall consider the 3-volume Ω to be purely spatial, and consequently only the spatial components of the field tensor and its dual, i.e. F ij and F ij , i, j = 1, 2, 3, will be present on both sides of (1.4). However, F ij is proportional to the electric field and so it vanishes for those static monopole solutions. In addition, only the componentJ 123 ∼ J 0 appears on the r.h.s. of (1.4), and that vanishes because the solution is static and we shall work in the gauge where A 0 = 0. Remember that the only contribution for the matter current for such a solution comes from the triplet Higgs field φ, and that is of the form J µ ∼ [ φ , D µ φ ]. Therefore, all terms involving the parameter β are not present on both sides of (1.4), for static monopoles when Ω is purely spatial, and so it reduces to
P 2 e i e α´∂ Ω dτ dσW −1 F ij W ∂x i ∂σ ∂x j ∂τ = P 3 e α(α−1)´Ω dζdτ VĴ V −1 , (1.15) withĴ = e 2 2 ˆσ f σ i dσ F W k,l (σ ),ˆσ f σ i dσF W ij (σ) ∂x k ∂σ ∂x i ∂σ ∂x l ∂τ (σ ) ∂x j ∂ζ (σ) − ∂x l ∂ζ (σ ) ∂x j ∂τ (σ) (1.16)
where i, j, k, l = 1, 2, 3, repeated indices are summed, and where we have denoted F W ij ≡ W −1 F ij W . Note that we have explored the symmetry ofĴ in σ and σ to replace´σ
f σ i dσ´σ σ i dσ → 1 2´σ f σ i dσ´σ f σ i dσ . Equation (1.15
) is what we call generalized integral Bianchi identity. Note that one would expect the integral Bianchi identity to be (1.15) for α = 1, i.e.
P 2 e i e´∂ Ω dτ dσW −1 F ij W ∂x i ∂σ ∂x j ∂τ = 1l (1.17)
Indeed, that is what leads to the quantization of the magnetic charge. From a physical point of view it is intriguing that by rescaling the field tensor (magnetic field) as F ij → α F ij , leads to the appearance of a term like the r.h.s of (1.15), making the magnetic flux through ∂Ω to change drastically. However, the validity of (1.4), and so of (1.15), is guaranteed by the generalized nonabelian Stokes theorem for a two-form B µν and the partial differential Yang-Mills equations (1.3) as proved in [11,12,1,2]. The intriguing non-linear phenomenon that we want to directly check in this paper, is if one can expand both sides of (1.15) in powers of α, and if the SU (2) 't Hoot-Polyakov monopole and its exact analytical BPS version, satisfy each one of the integral equations obtained through such an expansion.
The paper is organized as follows. In section 2 we perform the expansion of the generalized integral Bianchi identity (1.15) in powers of the parameter α, and we show that each term of the expansion can be expressed solely in terms of the Wilson line operator. In section 3 we calculate explicitly the Wilson line operator for the SU (2) 't Hooft-Polyakov and BPS monopoles using a suitable scanning of surfaces and volumes. The result is quite simple and it is given in (3.14). In section 4 we check the validity of the integral equation in first order of the α-expansion, and in section 5 we do the same for the integral equation in second order of that same expansion. We present our conclusions in section 6 and in the appendix A we give the results of the numerical calculations of the integrals needed to perform the check of the integral equations.
The expansion of the Yang-Mills integral equations
Assuming that α and β are indeed arbitrary we expand both sides of (1.4) in power series in those parameters. As we have said the l.h.s. of (1.4) is obtained by integrating (1.9), and its r.h.s. by integrating (1.12). By writing the expressions on the l.h.s and on the r.h.s of the integral equation (1.4) in terms of (1.11) and (1.14), and collecting the coefficients at first order in α and zeroth order in β, we get the integral equation at first order in α
τ f τ i dτˆσ f σ i dσ F W µν ∂x µ ∂σ ∂x ν ∂τ | ζ=ζ 0 = ieˆζ 0 0 dζˆτ f τ i dτˆσ f σ i dσˆσ σ i dσ F W κρ σ , F W µν (σ) × d x κ d σ d x µ d σ d x ρ (σ ) d τ d x ν (σ) d ζ − d x ρ (σ ) d ζ d x ν (σ) d τ (2.1)
where ζ 0 is the value of ζ corresponding to the closed surface ∂Ω, in the scanning of the 3-volume Ω, which is the border of Ω (see explanation of the scanning in the paragraph above (1.4)). On the other hand, the integral equation appearing in order β and zeroth order in α, in the expansion of (1.4) is given bŷ
τ f τ i dτˆσ f σ i dσ F W µν ∂x µ ∂σ ∂x ν ∂τ | ζ=ζ 0 =ˆζ 0 0 dζˆτ f τ i dτˆσ f σ i dσ J W µνλ dx µ dσ dx ν dτ dx λ dζ + i eˆσ σ i dσ F W κρ σ , F W µν (σ) (2.2) × d x κ d σ d x µ d σ d x ρ (σ ) d τ d x ν (σ) d ζ − d x ρ (σ ) d ζ d x ν (σ) d τ
Note that in the case where the gauge group G is the abelian group U (1), the equation (2.1) corresponds to (1.2) for α = 1 and β = 0. Equation (2.2) corresponds to (1.2) for α = 0 and β = 1. Note in addition that in the case where the 3-volume Ω is purely spatial, the commutator term in (2.1) involving the field tensors can be interpreted as a density of non-abelian magnetic charge associated to the gauge field configuration inside Ω. The commutator term in (2.2) involving the field tensor and its Hodge dual can be interpreted as a density of non-abelian electric charge associated to the gauge field configuration inside Ω. In the case where Ω has time components, those commutators will be associated to flows of non-abelian electric and magnetic charges. We have explored further those facts to obtain the integral form of the non-abelian Gauss, Faraday, etc., laws, and the physical implications of these new terms (the commutator terms) should be further explored in some other opportunity.
As one goes higher in the expansion, the integral equations become more and more complex. However, for the case we are considering in this paper, namely the static 't Hooft-Polyakov monopole and its BPS version, and where the 3-volume Ω is purely spatial, there is an important simplification. As we have argued in the paragraph above (1.15), only the spatial components of the field tensor (magnetic field) appear in the formulas, since the spatial components of its Hodge dual (electric field) vanish. As explained in section 2 of [11], or in the appendix of [2], if one performs an infinitesimal variation, x µ (σ) → x µ (σ) + δx µ (σ), of the curve where the Wilson line W (1.7) is calculated, but keeping its end points fixed, the infinitesimal variation of the Wilson line operator is given by
W −1 (σ f ) δW (σ f ) = ieˆσ f σ i dσ W −1 F µν W d x µ d σ δx ν (2.3)
The Wilson line operators W appearing in the Yang-Mills integral equations (1.4) are evaluated on the paths that scan the closed surfaces which in their turn scan the 3-volume Ω. Thefore, as we vary the parameter τ which labels the loops, we vary the loop along a given surface, and so δx µ = dx µ dτ δτ . When we vary the parameter ζ which labels the surfaces, the loops vary perpendicular to that surface and so δx µ = dx µ dζ δζ. Consequently, from (2.3) we get the following two useful formulas
ieˆσ f σ i dσ W −1 F µν W dx µ dσ dx ν dτ = W −1 dW dτ ieˆσ f σ i dσ W −1 F µν W dx µ dσ dx ν dζ = W −1 dW dζ . (2.4)
As we have shown, for the static 't Hooft-Polyakov monopole and its BPS version, and a purely spatial 3-volume Ω, the integral Yang-Mills equation (1.4) becomes the generalized integral Bianchi identity (1.15). Therefore, from (1.11), (1.10) and (2.4), one gets that the l.h.s. of (1.15) is given by
V (∂Ω) ≡ P 2 e i e α´∂ Ω dτ dσW −1 F ij W ∂x i ∂σ ∂x j ∂τ = 1 + αˆτ f τ i dτ W −1 dW dτ (τ ) | ζ=ζ 0 (2.5) + α 2ˆτ f τ i dτˆτ τ i dτ W −1 dW dτ (τ )W −1 dW dτ (τ ) | ζ=ζ 0 + . . . = 1 + α V (1) + α 2 V (2) + . . .
From (2.4) one gets that (1.16) becomeŝ
J = − W −1 dW dτ , W −1 dW dζ . (2.6)
Therefore, from (1.14) and (2.6), the r.h.s. of (1.15) becomes
U (Ω) ≡ P 3 e α(α−1)´Ω dζdτ VĴ V −1 = 1 − α (α − 1)ˆζ 0 0 dζˆτ f τ i dτ V W −1 dW dτ , W −1 dW dζ V −1 + [α (α − 1)] 2ˆζ 0 0 dζˆζ 0 dζ ˆτ f τ i dτˆτ f τ i dτ V W −1 dW dτ , W −1 dW dζ V −1 (τ , ζ) × V W −1 dW dτ , W −1 dW dζ V −1 τ , ζ + . . . (2.7) = 1 + α U (1) + α 2 U (2) + . . .
where V in (2.7) is evaluated with the same expansion as in (1.11) with β = 0, and so an expansion similar to (2.5).
Therefore, by equating (2.5) to (2.7) one gets that the term in first order in α leads to the integral equation
V (1) =ˆτ f τ i dτ W −1 dW dτ | ζ=ζ 0 =ˆζ 0 0 dζˆτ f τ i dτ W −1 dW dτ , W −1 dW dζ = U (1) . (2.8)
Similarly, the term in order α 2 gives the following integral equation
V (2) =ˆτ f τ i dτˆτ τ i dτ W −1 dW dτ W −1 dW dτ | ζ=ζ 0 = −ˆζ 0 0 dζˆτ f τ i dτ W −1 dW dτ , W −1 dW dζ +ˆζ 0 0 dζˆτ f τ i dτˆτ τ i dτ W −1 dW dτ , W −1 dW dτ , W −1 dW dζ (2.9) +ˆζ 0 0 dζˆζ 0 dζ ˆτ f τ i dτˆτ f τ i dτ W −1 dW dτ , W −1 dW dζ W −1 dW dτ , W −1 dW dζ = U (2) .
We are going to verify if the SU (2) 't Hooft-Polyakov monopole and its BPS version [3,4,5,6,8], satisfy the integral equations (2.8) and (2.9). Note that the only quantity appearing in (2.8) and (2.9) is the Wilson loop W . In the next section we evaluate it for those monopole solutions.
The Wilson loop for 't Hooft-Polyakov and BPS monopoles
The spherically symmetric 't Hooft-Polyakov ansatz [3,4] for a SU (2) static magnetic monopole reads
φ = 1 e r H (ζ)r · T A 0 = 0 (3.1) A i = − 1 e ijk x j r 2 (1 − K(ζ)) T k with r = x 2 1 + x 2 2 + x 2 3 ,r i = x i /r, ζ = ear,
a being the vacuum expectation value of the Higgs field in the triplet representation, and T i being the generators of the SU (2) Lie algebra
[ T i , T j ] = i ε ijk T k (3.2)
The exact analytical BPS monopole solution corresponds to the functions [5,6]
K(ζ) = ζ sinh ζ H(ζ) = ζ coth ζ − 1 (3.3)
For the 't Hooft-Polyakov monopole the functions K(ζ) and H(ζ) are obtained numerically, but they have qualitatively the same behaviour as (3.3), i.e. we have that K(0) = 1, and then it decays monotonically (exponentially) to zero as r → ∞, and H(0) = 0, and then it grows monotonically with r and for r → ∞, such grow is linear in r. The function H(ζ) will not be important in our calculations because the Higgs field does not appear in our integral equations for the case of static solutions and for Ω being purely spatial (see (2.8) and (2.9)). The important simplification we obtain in our calculations is due to the fact that K(ζ) is a monotonic function of ζ, and so it admits an inverse function. We will then trade the parameter ζ by the function K, and our calculations will not depend upon the explicit form of the function K(ζ).
We have chosen to evaluate both sides of the integral equations (2.8) and (2.9) on a purely spatial 3-volume Ω which is a ball centered at the origin of the Cartesian coordinate system x i , i = 1, 2, 3, used in the ansatz (3.1). We then scan that volume Ω with closed surfaces which are spheres also centered at the origin of the Cartesian coordinate system, with radii varying from zero to the radius of Ω. However, since the reference point x R have to lie on the border ∂Ω of Ω, and since the surfaces scanning it have to be based at x R , we shall attach to the ball Ω a infinitesimally thin cylinder lying on the negative x 1 -axis, and locate the reference point 0 , 0). The cylinder has a radius ε, which will be taken to zero at the end of the calculations. The surfaces scanning Ω will have the form depicted in Figure 2, i.e. an infinitesimally thin cylinder on the negative x 1 -axis and a sphere centered at the origin of the Cartesian coordinate system. With the attachment of the thin cylinder we can keep the surfaces based at x R , and centered at the origin. In addition, x R being at infinity allows us to have the volume Ω with any radius. We shall label the surfaces scanning Ω with the parameter ζ, which is the same as the one appearing in the ansatz (3.1). Then ζ = 0 corresponds to the surface made of the thin cylinder and a sphere of radius zero attached to it, and ζ = ζ 0 corresponds to the border ∂Ω, made of the thin cylinder attached to a sphere of radius ζ 0 , the same as the radius of Ω, centered at the origin. The loops will be labelled by a parameter τ , they start and end at the reference point x R , and there will be three types of loops, as follows:
x R at (x 1 , x 2 , x 3 ) = (−∞ ,
1. Loops of type (I), scanning the thin cylinder, as depicted in Figure 1. For such loops the parameter τ varies from −∞ to − π 2 , with τ = −∞ corresponding to the infinitesimal loop around x R , and τ = − π 2 corresponding to a straight line from x R to the border of the sphere, then encircling the joint of the cylinder with the sphere, and coming back to x R through the same straight line. The three parts of such loops will be denoted (I.1), the first straight line, (I.2), the circle and (I.3) the second straight line. We parameterize the loops with σ, such that the points on the loops have the following coordinates:
(I.1) x 1 = τ + σ − ζ + π 2 x 2 = 0 x 3 = −ε (−∞ ≤ σ ≤ 0) (I.2) x 1 = τ − ζ + π 2 x 2 = ε sin σ x 3 = −ε cos σ (0 ≤ σ ≤ 2π) (I.3) x 1 = τ + 2π − σ − ζ + π 2 x 2 = 0 x 3 = −ε (2π ≤ σ ≤ ∞) with fixed ζ and −∞ ≤ τ ≤ − π 2 . "
x 1
x R Figure 1: Scanning of type (I). The gap between the straight lines is only a visual resource.
2. Loops of type (II), scanning the thin sphere, as depicted in Figure 2. For such loops the parameter τ varies from − π 2 to π 2 . A loop of this type is made of a straight line from x R to the border of the sphere, then making a circle on the surface of the sphere, starting and ending at the junction of the cylinder with the sphere, and lying on a plane perpendicular to the plane x 1 x 3 , that makes an angle τ with the plane x 1 x 2 . Finally, it returns to x R through the same straight line. Again, the three parts of such loops will be denoted as (II.1) for the first straight line, (II.2) for the circle and (II.3) labels the second straight line. We parameterize the loops with σ, such that the points on the loops have the following coordinates:
(II.1) x 1 = σ − ζ x 2 = 0 x 3 = −ε (−∞ ≤ σ ≤ 0) (II.2) x 1 = ζ cos 2 τ (1 − cos σ) − 1 x 2 = ζ cos τ sin σ (0 ≤ σ ≤ 2π) x 3 = ζ cos τ sin τ (1 − cos σ) (II.3) x 1 = −σ + 2π − ζ x 2 = 0 x 3 = −ε (2π ≤ σ ≤ ∞).
with fixed ζ and where in (II.2) the parameter τ varies from − π 2 to π 2 .
x R . . . 3. Loops of type (III), scanning the thin cylinder backwards, as depicted in Figure 3. For such loops the parameter τ varies from π 2 to ∞, and they are made of two straight lines. The first one starting at x R and ending on some point on the side of the cylinder with coordinates (x 1 , x 2 , x 3 ) = (x 1 , 0 , −ε). The second part of the loop is the same straight line (reversed) going back to x R . We shall denote the first straight line (III.1) and the second (III.2). We parameterize the loops with σ, such that the points on the loops have the following coordinates:
(III.1) x 1 = π 2 − τ − ζ + σ x 2 = 0 x 3 = −ε (−∞ ≤ σ ≤ 0) (III.2) x 1 = π 2 − τ − ζ − σ x 2 = 0 x 3 = −ε (0 ≤ σ ≤ ∞)
with fixed ζ, and where π 2 ≤ τ ≤ ∞.
An important simplification is made by observing that the Wilson line is constant along loops scanning the thin cylinder. Indeed, we observe that on the segments (I.1), (I.3), (II.1), (II.3), (III.1) and (III.2), the coordinate x 1 is linear in σ, and x 2 and x 3 are independent of it. Therefore, using
x R x 1 " Figure 3: Scanning of type (III).
(3.1), we have that
A i dx i dσ | straight lines = ±A 1 = ∓ 1 e ε r 2 (1 − K) T 2 → 0 as ε → 0 (3.4)A i dx i dσ | (I.2) = ε [cos σ A 2 + sin σ A 3 ] (3.5) = − ε e (1 − K) r 2 −ε T 1 + τ − ζ + π 2 (− cos σ T 3 + sin σ T 2 ) → 0 as ε → 0
The only non-vanishing contribution comes from the segment (II.2) which gives
A i dx i dσ | (II.2) = 1 e (1 − K) cos τ cos τ sin τ (1 − cos σ)T 1 + sin τ sin σ T 2 + sin 2 τ (1 − cos σ) − 1 T 3 = − 1 e (1 − K) cos τ e i τ T 2 e i σ T 3 e −i τ T 2 T 3 e i τ T 2 e −i σ T 3 e −i τ T 2(→Ā i = g A i g −1 + i e ∂ i g g −1 , with g = e i τ T 2 e −i σ T 3 e −i τ T 2 , one gets that W 2 →W 2 = g f W 2 g −1 i = e i τ T 2 e −i 2 π T 3 e −i τ T 2 W 2 (3.7)
where g i and g f are the values of g at the initial and final points of the loop (II.2), and so g i = 1l, and g f = e i τ T 2 e −i 2 π T 3 e −i τ T 2 . Therefore, one gets that
A i dx i dσ | (II.2) →Ā i dx i dσ | (II.2) = 1 e [K cos τ T 3 − sin τ T 1 ](3.dW 2 d σ + i [K cos τ T 3 − sin τ T 1 ]W 2 = 0. (3.9)
Since the connection term [K (ζ) cos τ T 3 − sin τ T 1 ] is independent of σ it follows that the path ordering is unimportant and the integration on the loops (II.2) gives
W 2 = e −i 2 π [K cos τ T 3 −sin τ T 1 ](3.10)
Using the fact that e i 2 π T 3 = ±1l, depending if the representation used is of integer (+) or half-integer (−) spin, we get that g f = ± 1l, and so
W = W (II) = W 2 = ± e −i 2 π [K(ζ) cos τ T 3 −sin τ T 1 ] (3.11)
where we have equated W to W (II), because, as shown above W (I) = W (III) = 1l. Therefore, in (3.11) we have τ varying from − π 2 to π 2 . The calculations concerning the Wilson line can be simplified defining γ as
cos γ = K cos τ F ; sin γ = sin τ F (3.12) with F (ζ , τ ) = K 2 (ζ) cos 2 τ + sin 2 τ . (3.13)
Then (3.11) can be written as
W = ± e i γ T 2 e −i 2 π F T 3 e −i γ T 2 ,(3.14)
from which we get
W −1 ∂ W = i e i γ T 2 {−2 π ∂F T 3 + ∂γ [(cos (2 π F ) − 1) T 2 + sin (2 π F ) T 1 ]} e −i γ T 2 = i {[2 π ∂F sin γ + ∂γ sin (2 π F ) cos γ] T 1 + ∂γ [cos (2 π F ) − 1] T 2 + [−2 π ∂F cos γ + ∂γ sin (2 π F ) sin γ] T 3 } . (3.15)
We then have
W −1 d W d τ = i cos τ N j (K , τ ) T j (3.16) with N 1 (K , τ ) = 1 F 2 2 π 1 − K 2 sin 2 τ + K 2 sin (2 π F ) F N 2 (K , τ ) = − K F 2 cos τ [1 − cos (2 π F )] (3.17) N 3 (K , τ ) = K sin τ F 2 cos τ −2 π 1 − K 2 cos 2 τ + sin (2 π F ) F .
In addition
W −1 d W d τ , W −1 d W d ζ = 2 π i d F d τ d γ d ζ − d γ d τ d F d ζ × e i γ T 2 [(1 − cos (2 π F )) T 1 + sin (2 π F ) T 2 ] e −i γ T 2 = −i 2 π K cos 2 τ M j (K , τ ) T j (3.18) with M 1 (K , τ ) = K cos τ F 2 [1 − cos (2 π F )] M 2 (K , τ ) = sin (2 π F ) F (3.19) M 3 (K , τ ) = sin τ F 2 [1 − cos (2 π F )]
where K stands for dK dζ , and where we have used the formulas
d F d τ = sin τ cos τ F 1 − K 2 , d F d ζ = K cos 2 τ F K , d γ d τ = K F 2 , d γ d ζ = − sin τ cos τ F 2 K . (3.20)
With these expressions we are ready to perform the calculations of section 2 for the SU (2) monopoles.
Check of first order integral equations for SU (2) monopoles
The integral equation for a purely spatial volume Ω, in first order in α, for the SU (2) monopoles ('t Hooft-Polyakov or BPS) is given by expression (2.8). However, since the Wilson line is unit for the loops of type I and III (see section 3) we get that (2.8) is only non-trivial for loops of type II, where τ varies from − π 2 to π 2 , and so (2.8) becomes
V (1) =ˆπ 2 − π 2 dτ W −1 dW dτ ζ=ζ 0 =ˆζ 0 0 dζˆπ 2 − π 2 dτ W −1 dW dτ , W −1 dW dζ = U (1) (4.1)
where the l.h.s is an integration on a closed surface of radius ζ 0 and the r.h.s is an integration in the volume contained inside that surface. Our goal is to evaluate both sides of this equation using the results obtained in the expressions (3.16) and (3.18). In order to perform the integration of the l.h.s term, a better choice of variables is the following:
y = sin τ ; −1 ≤ y ≤ 1 ; z = K(ζ) cos τ ; 0 ≤ z ≤ 1 (4.2) with 0 ≤ ζ ≤ ∞, 0 ≤ K(ζ) ≤ 1, and − π 2 ≤ τ ≤ π 2 .
Note that we are using the fact that K(ζ) is monotonically decreasing function of ζ for both, the 't Hooft-Polyakov and BPS monopole solutions. The explicit form of the function K(ζ) is not important here. In these variables we get
F 2 = y 2 + z 2 = K 2 + (1 − K 2 )y 2 (4.3)
and so using (3.16) and (3.17) we get that the l.h.s. of (4.1) becomes
V (1) = iˆ1 −1 dy N j (K 0 , y) T j (4.4)
with K 0 ≡ K(ζ 0 ), and
N 1 (K , y) = 2π F 2 y 2 (1 − K 2 ) + K 2 sin(2πF ) 2πF N 2 (K , y) = − K 1 − y 2 F 2 {1 − cos (2πF } (4.5) N 3 (K , y) = 2π K y 1 − y 2 F 2 F 2 − 1 + sin (2πF ) 2πF .
Note that N 3 is an odd function of y and so integrating we get
V (1) (K 0 ) = i J 1 (K 0 )T 1 + i J 2 (K 0 )T 2 ,(4.6)
with
J 1 (K 0 ) = 2πˆ1 −1 dy 1 K 2 0 + (1 − K 2 0 )y 2 (1 − K 2 0 )y 2 + K 2 0 sin 2π K 2 0 + (1 − K 2 0 )y 2 2π K 2 0 + (1 − K 2 0 )y 2 J 2 (K 0 ) = −ˆ1 −1 dy K 0 1 − y 2 K 2 0 + (1 − K 2 0 )y 2 1 − cos 2π K 2 0 + (1 − K 2 0 )y 2 .
Note that as ζ varies from 0 to ζ 0 , one has that K varies from 1 to K 0 ≡ K (ζ 0 ) < 1. Therefore, the integration domain on the r.h.s. of (4.1) is a truncated semi-disc shown in Figure 4. The absolute value of the Jacobian of the variable transformation (ζ , τ ) → (y , z), given in (4.2), is | K | cos 2 τ = −K cos 2 τ , since K is strictly negative. In addition, it is more appropriate to perform a further change of variables to evaluate the integration on the r.h.s. of (4.1). We define the polar type coordinates (s , θ) as
y = s cos θ ; z = s sin θ ; S (K 0 , θ) ≤ s ≤ 1 ; 0 ≤ θ ≤ π (4.7) with S(K , θ) ≡ K 1 − cos 2 θ(1 − K 2 ) (4.8)
Therefore one has that
ζ 0 0 dζˆπ 2 − π 2 dτ K cos 2 τ = −ˆt runcated semi-disc dzdy = −ˆπ 0 dθˆ1 S(K 0 ,θ) ds s (4.9) with J 1 (K 0 ) = 2⇡ˆ1 1 dy 1 K 2 0 + (1 K 2 0 )y 2 8 < : (1 K 2 0 )y 2 + K 2 0 sin ⇣ 2⇡ p K 2 0 + (1 K 2 0 )y 2 ⌘ 2⇡ p K 2 0 + (1 K 2 0 )y 2 9 = ; J 2 (K 0 ) = ˆ1 1 dy K 0 p 1 y 2 K 2 0 + (1 K 2 0 )y 2 ⇢ 1 cos ✓ 2⇡ q K 2 0 + (1 K 2 0 )y 2 ◆ .
Note that as ⇣ varies from 0 to ⇣ 0 , one has that K varies from 1 to K 0 ⌘ K (⇣ 0 ) < 1. Therefore, the integration domain on the r.h.s. of (4.1) is a truncated semi-disc shown in Figure 4. The absolute value of the Jacobian of the variable transformation (⇣ , ⌧ ) ! (y , z), given in (4.2), is | K 0 | cos 2 ⌧ = K 0 cos 2 ⌧ , since K 0 is strictly negative. In addition, it is more appropriate to perform a further change of variables to evaluate the integration on the r.h.s. of (4.1). We define the polar type coordinates (s , ✓) as
y = s cos ✓ ; z = s sin ✓ ; S (K 0 , ✓) s 1 ; 0 ✓ ⇡ (4.7) with S(K , ✓) ⌘ K p 1 cos 2 ✓(1 K 2 ) (4.8)
Therefore one has that We then have that the M i 's, defined in (3.19), become
⇣ 0 0 d⇣ˆ⇡ 2 ⇡ 2 d⌧ K 0 cos 2 ⌧ = ˆt runcated semi-disc dzdy = ˆ⇡ 0 d✓ˆ1 S(K 0 ,✓)M 1 = z F 2 [1 − cos (2 π F )] = sin θ s [1 − cos (2 π s)] M 2 = sin (2 π F ) F = sin (2 π s) s (4.10) M 3 = y F 2 [1 − cos (2 π F )] = cos θ s [1 − cos (2 π s)]
Using (3.18) we get that in these coordinates the r.h.s. of (4.1), denoted by U (1) , reads
U (1) = i2πˆπ 0 dθˆ1 S(K 0 ,θ)
ds {sin θ(1 − cos(2πs))T 1 + sin(2πs)T 2 + cos θ(1 − cos(2πs))T 3 } (4.11) from where we can easily perform the integration in s, obtaining
U (1) (K 0 ) = iI 1 (K 0 )T 1 + iI 2 (K 0 )T 2 (4.12)
where
I 1 (K 0 ) =ˆπ 0 dθ sin θ 2π − 2πK 0 1 − cos 2 θ(1 − K 2 0 ) + sin 2πK 0 1 − cos 2 θ(1 − K 2 0 ) I 2 (K 0 ) = −ˆπ 0 dθ 1 − cos 2πK 0 1 − cos 2 θ(1 − K 2 0 ) (4.13)
The integral along the T 3 -direction in (4.11) vanishes since the integrand is odd, under reflection around θ = π 2 , in the interval 0 ≤ θ ≤ π (note that S 0 (θ) is even in that interval). Therefore, in order to check the validity of the integral equation at first order in α, given in (4.1), we have to verify the equalities of the coefficients of T i in (4.6) and in (4.12). We have performed the numerical integration of the quantities I i (K 0 ) and J i (K 0 ) for several values of K 0 , covering the range 1 ≥ K 0 ≥ 0, corresponding to 0 ≤ ζ 0 ≤ ∞. Note that the actual value of K 0 for a given value of ζ 0 is different for the 't Hooft-Polyakov and BPS monopoles. However, the fact that K (ζ) is a monotonically decreasing function of ζ, for both solutions, allowed us to trade the coordinate ζ by K, and perform one check that is valid for the two monopole solutions. In section A.1 we give the results of the numerical integrations of the quantities I i (K 0 ) and J i (K 0 ), i = 1, 2. As one observes in those tables the values of I i (K 0 ) and J i (K 0 ) are remarkably identical, differing in the worst case around the eighth decimal place, due to the numerical approximation. This indicates that the 't Hooft-Polyakov and BPS SU (2) monopoles are indeed solutions of the first order integral equation (2.1), or equivalently (4.1), appearing in the expansion in α of the integral non-abelian Gauss law in (2.5) and (2.7).
Check of second order integral equations for SU (2) monopoles
The integral equation for a purely spatial volume Ω, in second order in α, for the SU (2) monopoles ('t Hooft-Polyakov or BPS) is given by expression (2.9). However, since the Wilson line is unit for the loops of type I and III (see section 3) we get that (2.9) is only non-trivial for loops of type II, where τ varies from − π 2 to π 2 , and so (2.9) becomes
V (2) =ˆπ 2 − π 2 dτˆτ − π 2 dτ W −1 dW dτ W −1 dW dτ ζ=ζ 0 = −ˆζ 0 0 dζˆπ 2 − π 2 dτ W −1 dW dτ , W −1 dW dζ +ˆζ 0 0 dζˆπ 2 − π 2 dτˆτ − π 2 dτ W −1 dW dτ , W −1 dW dτ , W −1 dW dζ (5.1) +ˆζ 0 0 dζˆζ 0 dζ ˆπ 2 − π 2 dτˆπ 2 − π 2 dτ W −1 dW dτ , W −1 dW dζ W −1 dW dτ , W −1 dW dζ ≡ −U (1) + G 2 + G 3 = U (2)
where we have denoted G 2 and G 3 the terms appearing on the second and third lines respectively of (5.1). In addition, we have used the fact that the first term on r.h.s. of the first line of (5.1) is the same (up to a minus sign) as U (1) given on the r.h.s. of (4.1).
We start by evaluating the l.h.s. of (5.1), using (3.16), and (4.5) to get
π 2 − π 2 dτˆτ − π 2 dτ W −1 dW dτ W −1 dW dτ ζ=ζ 0 = −ˆ1 −1 dyˆy −1 dy 3 i,j=1 N i K 0 , y N j (K 0 , y) T i T j = − 1 2 3 i=1 ˆ1 −1 dy N i (K 0 , y) 2 T 2 i −ˆ1 −1 dyˆy −1 dy 3 i =j=1 N i K 0 , y N j (K 0 , y) T i T j (5.2)
where in the first term on the r.h.s. of (5.2) we have used the symmetry of the integrand in y and y to transform the integral on the triangle −1 ≤ y ≤ 1 and y ≤ y, to the integral on the square −1 ≤ y , y ≤ 1. We now use the fact that N i (K 0 , −y) = ε i N i (K 0 , y), with ε i = 1 for i = 1, 2 and ε 3 = −1 (see (4.5)). Then we can writê
1 −1 dyˆy −1 dy N i K 0 , y N j (K 0 , y) = 1 2ˆ1 −1 dyˆy −1 dy N i K 0 , y N j (K 0 , y) + ε i ε j 2ˆ1 −1 dyˆ1 y dy N i K 0 , y N j (K 0 , y) (5.3)
Therefore for the case where ε i ε j = 1, one can write further that
1 −1 dyˆy −1 dy N i K 0 , y N j (K 0 , y) = 1 2ˆ1 −1 dyˆ1 −1 dy N i K 0 , y N j (K 0 , y) ; ε i ε j = 1
For the case ε i ε j = −1, we do not use (5.3), but instead write
T i T j = 1 2 {T i , T j } + 1 2 [ T i , T j ] = 1 2 {T i , T j } + i ε ijk T k (5.4)
Note that we are dealing here with products, and not only commutators, of the SU (2) Lie algebra generators. We have therefore to work with a basis in the enveloping algebra of SU (2), which in the case of quadratic terms we shall take to be the 9 quantities T i , and the anti-commutators {T i , T j }, i, j = 1, 2, 3. If one works with the spinor representation given by the Pauli matrices σ i , i = 1, 2, 3, then one has σ i σ j = i ε ijk σ k + δ ij 1l, and non-diagonal terms vanish, i.e. {σ i , σ j } = 0, for i = j. However, if one works with the triplet or higher representations one has {T i , T j } = 0 even for i = j. So, we have to consider the coefficients of all the 9 elements of the basis of the enveloping algebra to be independent. Therefore, using (5.4) one gets that
V (2) =ˆπ 2 − π 2 dτˆτ − π 2 dτ W −1 dW dτ W −1 dW dτ ζ=ζ 0 = − N 1 (K 0 )T 2 1 + N 2 (K 0 )T 2 2 + N 12 (K 0 ) {T 1 , T 2 } + N + 13 (K 0 ) {T 1 , T 3 } + N + 23 (K 0 ) {T 2 , T 3 } − i N − 13 (K 0 ) T 2 + i N − 23 (K 0 ) T 1 (5.5) where N i (K 0 ) = 1 2 ˆ1 −1 dy N i (K 0 , y) 2 i = 1, 2 N 12 (K 0 ) = 1 2 ˆ1 −1 dy N 1 (K 0 , y) ˆ1 −1 dy N 2 (K 0 , y ) (5.6) N ± 13 (K 0 ) = 1 2 ˆ1 −1 dyˆy −1 N 1 (K 0 , y )N 3 (K 0 , y) ±ˆ1 −1 dyˆy −1 dy N 3 (K 0 , y )N 1 (K 0 , y) N ± 23 (K 0 ) = 1 2 ˆ1 −1 dyˆy −1 dy N 2 (K 0 , y )N 3 (K 0 , y) ±ˆ1 −1 dyˆy −1 dy N 3 (K 0 , y )N 2 (K 0 , y)
with the N i 's defined in (4.5), and where we have dropped the term proportional to T 2 3 because N 3 is an odd function of y, and so its integral on the interval −1 ≤ y ≤ 1, vanishes.
Using (3.16), (3.18) and (4.2), the term on the second line of (5.1), denoted G 2 , becomes
G 2 = −i 2 π ε ijk T kˆK 0 1 dKˆ1 −1 dy 1 − y 2 M j (K , y)ˆy −1 dy N i K , y ≡ −i 4 π 2 R k (K 0 ) T k (5.7)
Using (3.18) and (4.10) the term on the third line of (5.1), denoted G 3 , becomes
G 3 = −4 π 2ˆπ 0 dθˆ1 S(K 0 ,θ) ds sˆπ 0 dθ ˆ1 S(K,θ ) ds s 3 i,j=1 M i (s , θ) M j s , θ T i T j (5.8)
with K ≥ K 0 , and so ζ ≤ ζ 0 . Note that in the (θ , s )-integration, K has to be taken as a function of θ and s. From (4.2) and (4.7) one gets that K = s sin θ √ 1−s 2 cos 2 θ . Note that the (θ , s )-integration is the same as the one performed in (4.11), with K 0 replaced by K. Therefore, similar to what happened, there will be no terms in the direction of T j for j = 3, since M 3 (s , θ ) is odd under reflection of θ around θ = π 2 (see (4.10)). Since K and S (K 0 , θ) are even under the reflection of θ around θ = π 2 , there will be no terms in the direction of T i for i = 3, since M 3 (s , θ) is odd under that reflection. Using (5.4) one gets that
G 3 = −4π S 1 (K 0 )T 2 1 + S 2 (K 0 )T 2 2 + S 12 {T 1 , T 2 } + iS 3 (K 0 )T 3 with S a (K 0 ) = πˆπ 0 dθˆ1 S(K 0 ,θ) ds sˆπ 0 dθ ˆ1 S(K,θ )
ds s M a (s , θ) M a s , θ ; a = 1, 2
S 12 (K 0 ) = π 2ˆπ 0 dθˆ1 S(K 0 ,θ) ds sˆπ 0 dθ ˆ1 S(K,θ ) ds s × M 1 (s , θ) M 2 s , θ + M 2 (s , θ) M 1 s , θ (5.9) S 3 (K 0 ) = π 2ˆπ 0 dθˆ1 S(K 0 ,θ) ds sˆπ 0 dθ ˆ1 S(K,θ ) ds s × M 1 (s , θ) M 2 s , θ − M 2 (s , θ) M 1 s , θ
The s -integration can be performed analytically and so, using (4.10) and the fact that K = s sin θ √ 1−s 2 cos 2 θ , we get Note that the above integrals are symmetric under the reflection of θ and θ around π 2 . The quantities M 1 (s , θ), M 2 (s , θ), S (K , θ) and K (s , θ) are also symmetric under the reflection of θ around π 2 . Therefore, the integration in θ and θ can be performed in the interval from zero to π 2 , by multiplying the result by two. So, we then get that where we have introduced
S 1 (K 0 ) = 2ˆπ 2 0 dθˆ1 S(K 0 ,θ)S 12 = 1 2 (P 12 (K 0 ) + P 21 (K 0 )) S 3 = 1 2 (P 12 (K 0 ) − P 21 (K 0 )) ,
Summarizing, we have obtained both sides of the integral equation in second order in α for a given K 0 , given in (5.1). From (5.5) we have that
V (2) = −iN − 23 T 1 + iN − 13 T 2 − N 1 T 2 1 − N 2 T 2 2 − N 12 {T 1 , T 2 } − N + 13 {T 1 , T 3 } − N + 23 {T 2 , T 3 } (5.16)
and, from (5.1), (4.12), (5.7) and (5.9) we have that
U (2) = −i(I 1 +4π 2 R 1 )T 1 +−i(I 2 +4π 2 R 2 )T 2 −i(4π 2 R 3 +4πS 3 )T 3 −4πS 1 T 2 1 −4πS 2 T 2 2 −4πS 12 {T 1 , T 2 } .
(5.17) We have to check the equality between the coefficients of each element of the basis of the SU (2) enveloping algebra on the expansion of V (2) and U (2) . Those coefficients involve integrals which are calculated numerically for a set of values of K 0 . The results are presented in the tables of section A.2 in the Appendix. The consistency is remarkable and with that check we can state clearly that the the 't Hooft-Polyakov monopole and its BPS version satisfy the integral Yang-Mills equations up to second order in α.
Conclusions
The integral Yang-Mills equations appeared from an attempt to understand integrability in higher dimensional space-times [1,2]. Through a loop space formulation [11,12] one can construct a suitable generalization of the non-abelian Stokes theorem for two-form fields that can be used naturally to define conservation laws, thus mimicking the so-called zero curvature representation of integrable field theories in (1 + 1)-dimensions. That has led us to consider the applications of such non-abelian Stokes theorem to construct the integral equations for non-abelian gauge theories, generalizing the well known abelian version of such integral equations used to describe the laws of electrodynamics. That was indeed possible, as we have shown in [1,2], and the usual differential Yang-Mills equations are obtained from these integral equations when the appropriate limit is taken.
The present paper shows that there is more to be explored. The integral Yang-Mills equations allow the introduction of two c-numbers as parameters which arise naturally in the construction of the equations, and as being non-linear, produce a quite non-trivial dependence on those parameters of the surface and volume ordered integrals appearing on both sides of the equation.
We have tested the assumption that the integral Yang-Mills equations are in fact a collection of an infinite number of equations, each one corresponding to the coefficients of the above mentioned expansion in powers of those parameters. This was done by considering the fact that, by construction, a solution of the differential Yang-Mills equation is also a solution of the integral Yang-Mills equation. Thus, using the 't Hooft-Polyakov and BPS monopoles as such configurations, we tested the validity of the equations arising at first and second order in the parameter expansion of the integral Yang-Mills equation. Despite the quite different structures of the terms resulting from the surface and volume ordered integrals, we have checked their equalities with a high numerical precision of at least one part in 10 7 . In addition, much of the check has been done analytically, and we have obtained an exact expression for the Wilson line operator, on each loop scanning the surfaces and volumes, for the SU (2) 't Hooft-Polyakov monopole solution and its BPS version (see (3.14)). That result can certainly be useful in many other applications.
The fact that those configurations are solutions of both of the highly non-trivial equations at each order of the expansion, indicates that the parameters could indeed be arbitrary. The arbitrariness of the parameters leads to a variety of important consequences which can now be considered, such as their role in the conserved charges that arise dynamically from the integral equations and the significance of having an infinite number of integral equations.
Aknowkedgments: The authors are grateful to partial financial support by FAPES under contract number 0447/2015. LAF is partially supported by CNPq-Brazil.
A Numerical Results
In this section we show the results of the numerical integrations related to the terms on the l.h.s and r.h.s of the expansion of the integral equation performed at first and second order in α. The coefficients of the generators of the algebra (eventually, up to a common factor of i ≡ √ −1) are compared for different values of K 0 . For each integral estimative there is an associated upper bound on the error, which we represent by using the following notation: 1.372 ± 0.008 ≡ 1.37(2 ± 8).
A.1 Equation
V (1) = U (1) Coefficients of T 1 K 0 I 1 (K 0 ) J 1 (K 0 ) 0.
A.2 Equation V
(2) = U (2) The tables below show the values of the coefficients of the algebra elements of (5.16) and (5.17). The fact that they agree implies on the validity of the equation obtained after expanding the Yang-Mills integral equation to second order in α and therefore, on the validity of the integral equation itself for any value of α, at least up to that order. Coefficients of T 3 , {T 2 , T 3 } and {T 1 , T 3 } K 0 4π 2 R 3 (K 0 ) + 4πS 3 (K 0 ) N + 23 (K 0 ) N + 13 (K 0 ) 0.01 ± 2 × 10 −8 ± 7 × 10 −11 ± 9 × 10 −10 0.1 ± 7 × 10 −8 ± 2 × 10 −9 ± 6 × 10 −9 0.2 ± 1 × 10 −7 ± 7 × 10 −9 ± 1 × 10 −8 0.3 ± 2 × 10 −11 ± 1 × 10 −8 ± 4 × 10 −9 0.4 ± 7 × 10 −8 ± 1 × 10 −8 ± 6 × 10 −9 0.5 ± 7 × 10 −10 ± 9 × 10 −9 ± 7 × 10 −9 0.6 ± 5 × 10 −9 ± 4 × 10 −9 ± 7 × 10 −9 0.7 -4.44(5± 2) × 10 −11 ± 6 × 10 −11 ± 5 × 10 −9 0.8 -1± 7× 10 −11 ± 9 × 10 −10 ± 3 × 10 −9 0.9 -3.(7± 7) × 10 −13 ± 3 × 10 −10 ± 5 × 10 −9 0.99 9.599(0± 4) × 10 −16 -1.(6± 2) × 10 −11 ± 8× 10 −9
Coefficients of T 1 K 0 I 1 (K 0 ) + 4π 2 R 1 (K 0 ) N − 23 (K 0 ) 0.
Figure 2 :
2Scanning of type (II).
with the upper signs valid for the segments (I.1), (II.1) and (III.1), and the lower signs valid for (I.3), (II.3) and (III.2). On the segment (I.2), on the other hand, we get that
Figure 4 :
4The integration domain in the new "polar" coordinates. Each value of K 0 fixes a new domain by shortening the area of the disk from below.
Figure 4 :
4The integration domain in the new "polar" coordinates. Each value of K 0 fixes a new domain by shortening the area of the disk from below.
sin 2 θ + (1 − s 2 ) sin 2 θ (5.11)
Table 1 :
1Numerical verification of the validity of equation (4.1): the coefficients of T 1 and T 2 in (4.6) and in (4.12) agree up to the eighth order.
Table 2 :
2Comparison between the coefficients of T 1 and T 2 of equations (5.16) and (5.17).Coefficients of T 2
1
K 0
4πS 1 (K 0 )
N 1 (K 0 )
0.01
78.894398(6± 2)
78.8943986254
0.1
72.92922798(7± 8)
72.9292279882
0.2
57.2526284(6± 2)
57.2526284639
0.3
37.7395933527
37.7395933527
0.4
20.4073345(1± 3)
20.4073345139
0.5
8.690611129(8± 2)
8.69061112984
0.6
2.7109923(7± 3)
2.71099237533
0.7
0.5387264(6± 1)
0.538726461126
0.8
0.0496472951(1± 2)
0.0496472951165
0.9
0.0007667310202(6± 1)
0.000766731020264
0.99
7.21318539905×10 −10
7.213185398(9± 2)×10 −10
Coefficients of T 2
2
K 0
4πS 2 (K 0 )
N 2 (K 0 )
0.01
0.018377674(1± 2)
0.0183776(7± 1)
0.1
1.7266117(8± 1)
1.7266117(8± 2)
0.2
5.7018878(2± 4)
5.7018878(2± 7)
0.3
9.2357861093(6± 1)
9.235786(1± 1)
0.4
10.1438262(5± 4)
10.1438262(5± 5)
0.5
8.172899770(8± 4)
8.1728997(7± 4)
0.6
4.82243658089
4.8224365(8± 3)
0.7
1.9452173(4± 2)
1.9452173(4± 1)
0.8
0.4400963774(2± 5)
0.4400963(7± 1)
0.9
0.0285636848378
0.0285636(8± 1)
0.99
2.72985908954e-06
2.729(8± 8)e-06
Table 3 :
3Comparison between the coefficients of T 2 1 and T 2 2 of equations (5.16) and (5.17).Coefficients of {T 1 , T 2 }
K 0
4πS 12 (K 0 )
N 12 (K 0 )
0.01
-1.2041160882(4± 2)
-1.20411608813
0.1
-11.2214288(0± 5)
-11.2214288084
0.2
-18.0678738(4± 7)
-18.0678738427
0.3
-18.669622708
-18.6696227077
0.4
-14.3877884(2± 1)
-14.3877884224
0.5
-8.4277810668(8± 6)
-8.42778106684
0.6
-3.6157418(0± 4)
-3.6157418051
0.7
-1.02368943(3± 4)
-1.02368943394
0.8
-0.1478160841(3± 1)
-0.147816084139
0.9
-0.0046798144427(1± 3)
-0.00467981444135
0.99
-4.43745194072×10 −8
-4.437470(7± 7)×10 −8
Table 4 :
4Comparison between the coefficients of {T 1 , T 2 } of equations (5.16) and (5.17).
Table 5 :
5The coefficients above are the ones that should vanish in the equation obtained at second order in α; indeed, within a numerical precision, they are zero.
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